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<|MaskedSetence|> <|MaskedSetence|> Dunfield and Bell found monodromies for many of the fibered, orientable 1-cusped manifolds that can be triangulated with at most 9 tetrahedra [BN]. Using Bell’s program flipper, they were able to find invariant laminations for about 25,700 of them [Bel13]. <|MaskedSetence|> The first few such examples with genus ≥2absent2\geq 2≥ 2 are listed in Table 1.
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Building on work of Dunfield and Bell, we were able to find 2598 manifolds in the Hodgson-Weeks census which can be constructed by a surgery satisfying the hypotheses of 1.1.
**B**: This represents about 44.7% of the 5801 non-L-spaces in the Hodgson-Weeks census [HW94, Dun19].
**C**: About 800 of these have orientable invariant laminations and monodromy preserving these orientations.
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In dynamics, invariant measures and periodic orbits are both indispensable tools and basic objects for investigations themselves. The locations of periodic points in terms of certain combinatorial structures were studied, and an exact formula for the number of periodic points for each period was found in [Li16] for expanding Thurston maps. Equidistribution and large deviation results for periodic points were studied in [Li18, Li15]. An asymptotic formula for the number of periodic orbits with a certain weight induced by a Hölder continuous potential similar to the prime number theorem in number theory was established in [LZ24a, LZ24b, LZ24c] (see also [LZ18]). <|MaskedSetence|> The first-named author has been informed that similar results on equilibrium states have also been obtained independently by P. Haïssinsky.
In this paper, we investigate some other basic properties of periodic points of expanding Thurston maps and initiate the studies of another two classes of invariant measures, namely, (potential-energy-)maximizing measures and ground states. More precisely, we establish in our context the Livšic Theorem and various closing lemmas, including a Bressaud–Quas closing lemma and a local version of Anosov closing lemma. The Livšic Theorem, dating back to the work of Livšic [Liv72], has played important roles in the study of rigidity problems in dynamics (see for example, [Sp04]). It basically states that a real-valued function (called a potential) is determined uniquely (up to a coboundary of the same regularity) by the sum of its values along periodic orbits (see Theorem 1.1). <|MaskedSetence|> Our current approach is to establish a stronger result, called the bilateral Mañé lemma, using tools from ergodic optimization. <|MaskedSetence|> A version of the (global) Anosov closing lemma was established by the first-named author in [Li18, Lemma 8.6], albeit coarse in the temporal direction, i.e., it holds only for sufficiently high iterates of the map and sufficiently long orbits. In this paper, we establish a local version (away from critical points) of the Anosov closing lemma (Lemma 8.6) without the assumptions on the iterate of the map or lengths of the orbits. On the other hand, we formulate fairly general rules to deduce the Bressaud–Quas shadowing property between related systems, such as factors and iterations, before verifying this property for our maps, proving the Bressaud–Quas closing lemma. Combining these two kinds of closing lemmas, we are able to formulate and establish another local closing lemma in Lemma 8.1, which is crucial in our investigations on the new classes of invariant measures, i.e., the maximizing measures and ground states. We prove the existence, uniqueness, and periodicity of the maximizing measures for generic Hölder continuous potentials for expanding Thurston maps, establishing the Typically Periodic Optimization (TPO) Conjecture ([YH99, Conjecture 1.1]) in our setting. We say that an invariant measure is periodic if it is supported on a periodic orbit. Outside of complex dynamics, the TPO Conjecture has previously been fully verified mostly in uniformly expanding and uniformly hyperbolic systems (see [Co16] for distance expanding maps and [HLMXZ19] for Axiom A attractors and Anosov diffeomorphisms). The TPO Conjecture has connections to various fields such as the Finiteness Conjecture in control theory [Boc18] and the random minimum (or maximum) mean cycle problems in probability theory [BZ16, DLZ24]. Our theorem may be the first to verify the TPO Conjecture in a non-uniformly expanding setting for Hölder continuous potentials to the best of our knowledge. Finally, we prove that for an expanding Thurston map and a generic Hölder potential, there exists a unique ground state which the equilibrium state converges to at zero temperature.
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**A**: Regarding important classes of invariant measures of these maps, the measures of maximal entropy were studied in [BM10, BM17, HP09, Li16], the absolutely continuous invariant measures in [BM17], and the equilibrium measures by the first-named author [Li18] and later by Das et al. [DPTUZ21] in broader settings with a different approach building upon prior works of P. Haïssinsky and K. M. Pilgrim [HP09].
**B**: The usual approach to Livšic Theorems only yielded a partial result [Li18, Proposition 8.8] for expanding Thurston maps.
**C**: Due to the presence of critical points, the lack of Markov partitions, and the non-uniform expansion nature in our setting, the full version of the classical Anosov closing lemma is beyond reach (c.f. [CKY88] in a one-dimensional real dynamics setting).
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The goal of this paper is to study the question of existence of optimal sets, that is, minimizers of (1.4). <|MaskedSetence|> Indeed, as said above, relaxation arguments allow to deduce that a minimizing set exists if and only if there is a function minimizing (1.5) which is a characteristic function, and this is, of course, a peculiar situation. As a matter of fact, in all the results where existence of optimal sets is established, as in the ones described above, the optimal sets are actually balls, and there is not really an argument which provides existence, but rather the existence is simply obtained as a consequence of the optimality of the balls. <|MaskedSetence|> <|MaskedSetence|>
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**A**: First of all, we underline that existence should not be expected in general.
**B**: However, they are not of the form (1.1), but of the form
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**C**: It is worth noting that there are, in fact, non-local energies for which existence of optimal sets (different from balls) is known.
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This paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> In section 3, we prove the inequality in (1.8). <|MaskedSetence|> The author would also like to thank Hao Fang, Lihe Wang and Biao Ma for helpful discussions, and Hao Fang for all the guidance on writing this paper..
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**A**: In section 4, we study when equality in (1.8) holds.
The author would like to thank Biao Ma for introducing the work of Shen and Wang [MR4308060].
**B**: In section 2 we prove a technical lemma.
**C**: The proof of Theorem 1.3 is split into two parts.
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The model (1.1), with i=1𝑖1i=1italic_i = 1 or 2222, is a kinetic description of the probability distribution of a certain system of interacting particles, submitted to an external force derived from the potential ϕitalic-ϕ\phiitalic_ϕ, at time t𝑡titalic_t located at the position x𝑥xitalic_x in the physical space Ω⊂ℝdΩsuperscriptℝ𝑑\Omega\subset\mathbb{R}^{d}roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT with the velocity v∈ℝd𝑣superscriptℝ𝑑v\in\mathbb{R}^{d}italic_v ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. It is complemented with the proper boundary conditions corresponding to the behaviour of particles at the boundary. The boundary condition in (1.1) is prescribed as zero influx when α=β=0𝛼𝛽0\alpha=\beta=0italic_α = italic_β = 0, and it is generally governed by the balance relations between the distribution of particles at the incoming and outgoing boundaries. The operator ℒ1subscriptℒ1\mathcal{L}_{1}caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT serves as a prototype for describing collision processes of linear relaxation type, including the case of neutron transport. The operator ℒ2subscriptℒ2\mathcal{L}_{2}caligraphic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT derived from the Ornstein-Uhlenbeck velocity process captures the impact of collisions with particles of a surrounding bath. The collision operators act only on the velocity variable and conserve local mass. <|MaskedSetence|> The hypocoercivity theory seeks to understand how the interaction between transport and collisions leads to time-decay convergence for inhomogeneous collisional kinetic equations such as (1.1). <|MaskedSetence|> <|MaskedSetence|>
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**A**: The (small) parameter ε∈(0,1]𝜀01\varepsilon\in(0,1]italic_ε ∈ ( 0 , 1 ] representing the ratio of the collisional mean free path and the observation length is introduced to measure the balance between the transport part v⋅∇x⋅𝑣subscript∇𝑥v\cdot\nabla_{x}italic_v ⋅ ∇ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and the collision part ℒisubscriptℒ𝑖\mathcal{L}_{i}caligraphic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, while diffusion phenomena can be observed on long time scales of order ε−1superscript𝜀1\varepsilon^{-1}italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT.
**B**: They are both dissipative in the velocity variable and contribute to the relaxation process towards local equilibrium.
**C**: This regime with ε→0→𝜀0\varepsilon\rightarrow 0italic_ε → 0 is often referred to as the diffusion limit.
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<|MaskedSetence|> When ω𝜔\omegaitalic_ω is restricted to linear sums of ΔgjsubscriptΔsubscript𝑔𝑗\Delta_{g_{j}}roman_Δ start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT for gj∈ℤ/njℤsubscript𝑔𝑗ℤsubscript𝑛𝑗ℤg_{j}\in\mathbb{Z}/n_{j}\mathbb{Z}italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_Z / italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT blackboard_Z, which we will denote by ωjsubscript𝜔𝑗\omega_{j}italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, then ωjsubscript𝜔𝑗\omega_{j}italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is a multiplicative linear functional of ℓ01(ℤ/njℤ)superscriptsubscriptℓ01ℤsubscript𝑛𝑗ℤ\ell_{0}^{1}(\mathbb{Z}/n_{j}\mathbb{Z})roman_ℓ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_Z / italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT blackboard_Z ). Thus, there must exist some kj∈{1,2,…,nj−1}subscript𝑘𝑗12…subscript𝑛𝑗1k_{j}\in\{1,2,\ldots,n_{j}-1\}italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ { 1 , 2 , … , italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - 1 } such that ωj=ωj(kj)subscript𝜔𝑗superscriptsubscript𝜔𝑗subscript𝑘𝑗\omega_{j}=\omega_{j}^{(k_{j})}italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT. Equation (5.7) then follows by previous comments and Lemma 5.3. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Thus, we see that we have n1n2⋯nm=card(G)subscript𝑛1subscript𝑛2⋯subscript𝑛𝑚card𝐺n_{1}n_{2}\cdots n_{m}=\mathrm{card}(G)italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_n start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = roman_card ( italic_G ) possible choices for ω𝜔\omegaitalic_ω, including the zero map.
This leads to the following theorem and corollary..
**B**: Note that ω𝜔\omegaitalic_ω is the zero map exactly when all the ωj(kj)superscriptsubscript𝜔𝑗subscript𝑘𝑗\omega_{j}^{(k_{j})}italic_ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT are zero maps.
**C**:
Suppose ω:ℓ01(G)→ℂ:𝜔→superscriptsubscriptℓ01𝐺ℂ\omega\colon\ell_{0}^{1}(G)\rightarrow\mathbb{C}italic_ω : roman_ℓ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_G ) → blackboard_C is a multiplicative linear functional.
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2.2. Automorphic representations
Let E/F𝐸𝐹E/Fitalic_E / italic_F be a quadratic extension of number fields with η=ηE/F𝜂subscript𝜂𝐸𝐹\eta=\eta_{E/F}italic_η = italic_η start_POSTSUBSCRIPT italic_E / italic_F end_POSTSUBSCRIPT the quadratic character associated to the extension E/F𝐸𝐹E/Fitalic_E / italic_F via global class field theory. The global Jacquet-Langlands transfer is an injective map from the set of irreducible discrete series representations of Gr(𝔸F)subscript𝐺𝑟subscript𝔸𝐹G_{r}(\mathbb{A}_{F})italic_G start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( blackboard_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) to that of GLrd(𝔸F)subscriptGL𝑟𝑑subscript𝔸𝐹\operatorname{GL}_{rd}(\mathbb{A}_{F})roman_GL start_POSTSUBSCRIPT italic_r italic_d end_POSTSUBSCRIPT ( blackboard_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) [Bad08, Theorem 5.1]. <|MaskedSetence|> Let π𝜋\piitalic_π be an irreducible cuspidal automorphic representation of GLn(𝔸F)subscriptGL𝑛subscript𝔸𝐹\operatorname{GL}_{n}(\mathbb{A}_{F})roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) such that π≇π×η𝜋𝜋𝜂\pi\not\cong\pi\times\etaitalic_π ≇ italic_π × italic_η. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Moreover, by [AC89, §3, Theorem 5.1], BC(π)v=BC(πv)BCsubscript𝜋𝑣BCsubscript𝜋𝑣\mathrm{BC}(\pi)_{v}=\mathrm{BC}(\pi_{v})roman_BC ( italic_π ) start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = roman_BC ( italic_π start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) for all places v𝑣vitalic_v of F𝐹Fitalic_F.
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**B**: Then its base change to GLn(𝔸E)subscriptGL𝑛subscript𝔸𝐸\operatorname{GL}_{n}(\mathbb{A}_{E})roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_A start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ), denoted by BC(π)BC𝜋\mathrm{BC}(\pi)roman_BC ( italic_π ), exists and is unique, which is an irreducible cuspidal automorphic representation of GLn(𝔸E)subscriptGL𝑛subscript𝔸𝐸\operatorname{GL}_{n}(\mathbb{A}_{E})roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_A start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ).
**C**: We denote this map still by JLJL\mathrm{JL}roman_JL, since there is no chance of confusion.
For the following fact about global base change lift, we refer the reader to [AC89, §3, Theorem 4.2].
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<|MaskedSetence|> In particular, we focus on the notion of gait controllability, an intermediate one (in terms of strength), which we argue to be the most apt for the task. With gait controllability (see Definition 2.4 below) we mean that we prescribe the initial and final position and the same initial and final shape. <|MaskedSetence|> <|MaskedSetence|> The possibility for the swimmer to change the size of some body parts has been considered, for example, for a volume change in a model of two-sphere swimmer [5, 13, 42]; here we study instead the elongation and shortening of a slender filament..
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**A**: Moreover, on a more technical perspective, gait controllability is usually the gateway through which the stronger total controllability is proven, making an independent notion even more useful.
Classical models of slender microswimmers are based on the capability to control curvature or angles along the swimmer body: in addition to the above-mentioned N𝑁Nitalic_N-link swimmers, we recall travelling waves [40] or the rotation of a corkscrew flagellum [22, 33, 37].
**B**: In this way, the feature of a periodic shape change becomes embedded in the structure of the problem, resulting in a more fitting treatment of some situations, see Remarks 2.14 and 2.17.
**C**: In this paper, we propose four models for an enhanced two-link swimmer to achieve controllability, thus overcoming the Scallop Theorem in the context of microswimming.
To present our results, we discuss different concepts of controllability, such as total controllability (to prescribe the initial and final positions and shapes) and fiber controllability (to prescribe the initial position and shape and the final position).
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<|MaskedSetence|> The embedding tree is a full binary tree where each leaf vertex is associated with a dangling edge/mode of the subnetwork made from composing the two partitions that are being contracted. <|MaskedSetence|> All tensors within this network have an order of three, except for the tensor located at the root vertex.
An example of such an embedding tree is illustrated in the second left diagram of Fig. 4b.
The selection of the embedding tree is guided by an analysis of the structure of the input tensor network graph G𝐺Gitalic_G, its partitioning, and the contraction path. This analysis aims to identify a tree structure that optimizes the efficiency of both the current contraction and any subsequent contractions involving the contracted output. <|MaskedSetence|>
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**A**: When two tensor network partitions are contracted, an embedding tree is first constructed which specifies the structure of the network that will result from the contraction.
**B**: The determination of each embedding tree structure occurs in lines 5-9..
**C**: Furthermore, each non-leaf vertex in the embedding tree corresponds to a tensor within the resulting binary tree tensor network.
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Selection 1
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Let U⊆[n]𝑈delimited-[]𝑛U\subseteq[n]italic_U ⊆ [ italic_n ] and denote by 𝒢n,Usubscript𝒢𝑛𝑈\mathcal{G}_{n,U}caligraphic_G start_POSTSUBSCRIPT italic_n , italic_U end_POSTSUBSCRIPT the set of frames from ℱn∩FrLsubscriptℱ𝑛Fr𝐿\mathcal{F}_{n}\cap\operatorname{Fr}Lcaligraphic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∩ roman_Fr italic_L where U𝑈Uitalic_U is the maximal connected component. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Moreover, G⊧Lmodels𝐺𝐿G\models Litalic_G ⊧ italic_L implies that G↑([n]∖U)⊧L,↑𝐺delimited-[]𝑛𝑈models𝐿G{\uparrow}([n]\setminus U)\models L,italic_G ↑ ( [ italic_n ] ∖ italic_U ) ⊧ italic_L , so G′⊧Lmodelssuperscript𝐺′𝐿G^{\prime}\models Litalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊧ italic_L since G′≅G↑([n]∖U)⊎F2superscript𝐺′𝐺↑⊎delimited-[]𝑛𝑈subscript𝐹2G^{\prime}\cong G{\uparrow}([n]\setminus U)\,\uplus\,F_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≅ italic_G ↑ ( [ italic_n ] ∖ italic_U ) ⊎ italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and F2⊧L.modelssubscript𝐹2𝐿F_{2}\models L.italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊧ italic_L . Thus G′∈ℱn∩FrL.superscript𝐺′subscriptℱ𝑛Fr𝐿G^{\prime}\in\mathcal{F}_{n}\cap\operatorname{Fr}L.italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∩ roman_Fr italic_L ..
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**A**: Let us consider any F1,F2∈ℱ|U|∩ConFrLsubscript𝐹1subscript𝐹2subscriptℱ𝑈ConFr𝐿F_{1},\,F_{2}\in\mathcal{F}_{|U|}\cap\operatorname{Con}\operatorname{Fr}Litalic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_F start_POSTSUBSCRIPT | italic_U | end_POSTSUBSCRIPT ∩ roman_Con roman_Fr italic_L.
**B**: For any G=([n],R)∈𝒢n,U𝐺delimited-[]𝑛𝑅subscript𝒢𝑛𝑈G=([n],R)\in\mathcal{G}_{n,U}italic_G = ( [ italic_n ] , italic_R ) ∈ caligraphic_G start_POSTSUBSCRIPT italic_n , italic_U end_POSTSUBSCRIPT such that G↑U↑𝐺𝑈G{\uparrow}Uitalic_G ↑ italic_U equals F1subscript𝐹1F_{1}italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (up to monotone relabeling, which we assume hereinafter), we construct a frame G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT by changing the relation of G𝐺Gitalic_G on U𝑈Uitalic_U in such a way that G′↑([n]∖U)=G↑([n]∖U)↑superscript𝐺′delimited-[]𝑛𝑈𝐺↑delimited-[]𝑛𝑈G^{\prime}{\uparrow}([n]\setminus U)=G{\uparrow}([n]\setminus U)italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ↑ ( [ italic_n ] ∖ italic_U ) = italic_G ↑ ( [ italic_n ] ∖ italic_U ) and G′↑U=F2.↑superscript𝐺′𝑈subscript𝐹2G^{\prime}{\uparrow}U=F_{2}.italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ↑ italic_U = italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .
Let us show that G′∈𝒢n,U.superscript𝐺′subscript𝒢𝑛𝑈G^{\prime}\in\mathcal{G}_{n,U}.italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_G start_POSTSUBSCRIPT italic_n , italic_U end_POSTSUBSCRIPT .
**C**: By the construction, U𝑈Uitalic_U is the maximal connected component of G′.superscript𝐺′G^{\prime}.italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .
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<|MaskedSetence|> <|MaskedSetence|> After defining necessary function spaces, we define the linear and nonlinear operators and show that these operators satisfy a monotonicity property for r≥3𝑟3r\geq 3italic_r ≥ 3 (see Theorems 2.5 and 2.6). Furthermore, we show that the sum of linear and nonlinear operators satisfy demicontinuous property also (Lemma 2.8). After providing an abstract formulation of the system (1.1), the existence and uniqueness of a global weak solution in the Leray-Hopf sense is examined in Section 3. The monotonicity and hemicontinuity properties of the linear and nonlinear operators as well as the Minty-Browder techniques are exploited in the proofs (Theorem 3.5). In the final section, we discuss the global strong solutions to the system (1.1). Due to technical difficulties described above, we are able to prove the regularity of the weak solutions in the class (1.3) in periodic domains only (Theorem 4.2). <|MaskedSetence|> Mathematical Formulation.
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**A**: Moreover, using the abstract theory developed in [3, 4], we prove the existence and uniqueness of global strong solutions for the system (1.1) in the class (1.3) (Theorem 4.4).
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**B**: The rest of the paper is organized as follows.
**C**: In the next section, we discuss the functional setting of the problem described in (1.1).
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In POD the modes are ranked by energy level through the POD singular values. There is no such criteria for ranking the contributions of the different DMD modes. <|MaskedSetence|> The DMD modes can then be selected based on their amplitude or based on their frequency/growth rate. <|MaskedSetence|> <|MaskedSetence|>
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**A**: The selection based on frequency/growth rate has also disadvantages because it relies on a priori physical knowledge.
Additionally, spatial non-orthogonality of the DMD modes
may introduce a poor quality of approximation.
**B**: Different criteria are developed depending on what can be considered important for the models used.
**C**: The amplitude criterion is also not perfect because there exist modes with very high amplitudes but which are very fast damped.
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It remains for us to prove that we can find a family {mx∈𝒮x},x∈Lsubscript𝑚𝑥subscript𝒮𝑥𝑥𝐿\{m_{x}\in\mathcal{S}_{x}\},x\in L{ italic_m start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∈ caligraphic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT } , italic_x ∈ italic_L so that the resulting embedding F~~𝐹\tilde{F}over~ start_ARG italic_F end_ARG will not have any new complex tangents off of L𝐿Litalic_L. Note that it will suffice to prove this is true for a tubular neighborhood of N⊂M𝑁𝑀N\subset Mitalic_N ⊂ italic_M of L𝐿Litalic_L and then proceed using Gromov’s h-principle for extensions (see [5]) to extend our solution beyond the tubular neighborhood above to all of M𝑀Mitalic_M.
Define the set: 𝒮≡∪𝒮x⊂GL6(ℝ)𝒮subscript𝒮𝑥𝐺subscript𝐿6ℝ\mathcal{S}\equiv\cup\mathcal{S}_{x}\subset GL_{6}(\mathbb{R})caligraphic_S ≡ ∪ caligraphic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⊂ italic_G italic_L start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ( blackboard_R ) over all x∈L𝑥𝐿x\in Litalic_x ∈ italic_L. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Also, let N𝑁Nitalic_N be a tubular neighborhood of L⊂M𝐿𝑀L\subset Mitalic_L ⊂ italic_M..
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**A**: Consider further the set of sections to this bundle Γ={s:L→𝒮}double-struck-Γconditional-set𝑠→𝐿𝒮\mathbb{\Gamma}=\{s:L\rightarrow\mathcal{S}\}blackboard_Γ = { italic_s : italic_L → caligraphic_S }.
**B**: This forms a fiber bundle over L𝐿Litalic_L with fibers 𝒮xsubscript𝒮𝑥\mathcal{S}_{x}caligraphic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT over each x∈L𝑥𝐿x\in Litalic_x ∈ italic_L.
**C**: By our work above, for each section s𝑠sitalic_s there exists an automorphism E:ℂ3→ℂ3:𝐸→superscriptℂ3superscriptℂ3E:\mathbb{C}^{3}\rightarrow\mathbb{C}^{3}italic_E : blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT → blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT so that the function f~s=Es∘fsubscript~𝑓𝑠subscript𝐸𝑠𝑓\tilde{f}_{s}=E_{s}\circ fover~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ∘ italic_f has complex tangents along L𝐿Litalic_L with holomorphic tangent space exactly the desired complex line field η𝜂\etaitalic_η.
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<|MaskedSetence|> <|MaskedSetence|> If the series terms and the error bound are rational, the algorithm can be implemented using only rational arithmetic.
Standard simulation methods, as implemented in computer software, typically define the parameter τ𝜏\tauitalic_τ as a floating-point value. This inevitably incurs a loss of precision. <|MaskedSetence|> The method presented in this paper avoids that problem, and generates a random variable with the exact parameter τ𝜏\tauitalic_τ..
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**A**: The algorithm requires a positive series representation of τ𝜏\tauitalic_τ, and a bound for the truncation error that converges to 00.
**B**: More specifically, it is not possible to represent irrational values (or even certain rational values) exactly using floating-point variables.
**C**:
An algorithm has been proposed that generates a Bernoulli random variable of arbitrary parameter τ𝜏\tauitalic_τ, using a sequence of independent Bernoulli variables of parameter 1/2121/21 / 2 as input.
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<|MaskedSetence|> These ODEs provide the mathematical (population level) model for our algorithm. <|MaskedSetence|> (2016) and Arias-Castro and Qiao (2022). For the Subspace Constraint Mean Shift algorithm, see Genovese et al. <|MaskedSetence|>
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**A**: (2014) and Qiao and Polonik (2016).
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**B**: For the original mean shift algorithm, this analogy has been used in Arias-Castro et al.
**C**:
The population level results are using theory for Ordinary Differential Equations (ODE).
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1.3. What is done in this paper?
In this paper we deal with the case M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 for generic q𝑞qitalic_q. The advantage of the M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 assumption is that in this case the group UM,Nsubscript𝑈𝑀𝑁U_{M,N}italic_U start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT is
trivial, so GL(M,𝐎)⋉UM,N(𝐅)GLleft-normal-factor-semidirect-product𝑀𝐎subscript𝑈𝑀𝑁𝐅{\mathop{\operatorname{\rm GL}}}(M,{\mathbf{O}})\ltimes U_{M,N}({\mathbf{F}})roman_GL ( italic_M , bold_O ) ⋉ italic_U start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT ( bold_F ) is just equal to GL(N−1,𝐎)GL𝑁1𝐎{\mathop{\operatorname{\rm GL}}}(N-1,{\mathbf{O}})roman_GL ( italic_N - 1 , bold_O ) (and the character χ𝜒\chiitalic_χ is trivial as well).
The current paper should be thought of as a sequel to [BFGT]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: There we consider (among other things) the case q=1𝑞1q=1italic_q = 1.
**B**: As was noted above, one has to be careful about specializing to non-generic q𝑞qitalic_q.
**C**: It turns out that for q=1𝑞1q=1italic_q = 1 the correct statement is as follows..
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<|MaskedSetence|> However, we get around this issue by doubling. <|MaskedSetence|> <|MaskedSetence|> Since we can also consider singular composition as doing nonsingular composition and then reducing along a reducing annulus and capping off (see Section 2), this approach also works to prove tg-hyperbolicity in the case of singular composition when both factor knots have genus two or greater. This approach, however, fails to work when generating volume bounds for singular composition of a genus one knot with a genus two (or more) knot. This is why Theorem 4.2 is stated to exclude the composition of a genus one knot with a genus at least two knot. The case of singular composition when both factor knots have genus one is simple because the result is a genus one knot, so the boundary is not included.
If instead of knots in thickened surfaces, we consider knots in handlebodies, results on compositions appear in [6].
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**A**: Then we take half of this double to compose, and therefore everything being composed is boundary compatible in that both pieces include their boundary.
**B**: At first, this seems to present an issue with composing genus one and genus two (or more) knots.
**C**: For any genus one knot in a nonsingular composition, we first remove its cork, and then double it over what remains of the cork boundary to obtain a genus two surface that includes its boundary.
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<|MaskedSetence|> The work [42] studies the asynchronous voter rule and the asynchronous majority rule dynamics with Poisson clocks when the opinion set is binary.
The authors use mean-field techniques, and focus on two different scenarios: In the first, some agents have a probability (which depends on their current opinion) not to update when the clock ticks. In the second, there are stubborn agents. <|MaskedSetence|> If the two sizes are close to each other and not too large, then agreement on both opinions is possible. Otherwise, either no agreement is possible, or the process converges to an agreement towards a single opinion, which is that of the largest stubborn community. <|MaskedSetence|>
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**A**: In [49], the authors show that, in the voter model, the presence of stubborn agents with opposite opinions precludes the convergence to consensus.
**B**: The case in which the two stubborn communities have equal size corresponds to the uniform communication noise model.
We remark, however, that in our work we consider a different setting: First, we consider the synchronous version of the 3-Majority dynamics, which cannot be analyzed with the same tools of the asynchronous version..
**C**: In the second case, which directly relates to our work, they show that for the 3-Majority dynamics, there are either one or two possible stable equilibria, depending on the sizes of the stubborn communities, which are reached in logarithmic time.
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<|MaskedSetence|> Top-ψ𝜓\psiitalic_ψ for strata with residue conditions
A central ingredient in our study will be spaces of meromorphic differentials on curves satisfying residue conditions at some of their poles and the intersection numbers of their fundamental classes with powers of ψ𝜓\psiitalic_ψ-classes. <|MaskedSetence|> [BCG+19b]). Moreover, their fundamental classes have many nice properties and explicit descriptions. <|MaskedSetence|>
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**A**: For instance, as shown in [BRZ21] these classes form a partial cohomological field theory when requiring the residues to vanish at all of the poles, and for the case of differentials with precisely two zeroes there exists a connection to the KP hierarchy..
**B**: These spaces are very natural since they appear in the description of boundary strata of the usual spaces of abelian differentials (see e.g.
**C**:
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We compare an oracle (red) with our proposals for the surrogate model (light blue) and the missing-at-random model (dark blue).
We also visualize benchmarks from previous work, either with (black, dashed) or without (black, solid) covariates. In the background, we indicate ranges of class sizes in the Project STAR protocol (gray).
The oracle, visualized in red, is estimated from long term experimental data, i.e. joint observations of the randomized action D𝐷Ditalic_D and long term reward Y𝑌Yitalic_Y in Project STAR. <|MaskedSetence|> Figure 4 shows that the oracle curve is typically decreasing: larger class sizes appear to cause lower test scores, across horizons. In particular, the oracle estimates are nonlinearly decreasing, from positive counterfactual test scores (above average) to negative counterfactual test scores (below average). <|MaskedSetence|> <|MaskedSetence|>
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**A**: As the long term horizon increases, i.e.
**B**: Our goal is to recover similar estimates without access to long term experimental data.
**C**: as the definition of Y𝑌Yitalic_Y corresponds to later grades, the oracle curves flatten: the effect of kindergarten class size on test scores appears to attenuate over time..
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1.1. Related Work
Mandel in [Man21] has previously proven the full Frobenius structure conjecture for all cluster varieties. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Note however that the non-degeneracy of the trace form is not known in general without the existence of a dense torus.
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**A**: This allows for tropical curve counting techniques to be utilized in the study of the enumerative geometry of cluster varieties.
**B**: Instead of assuming a toric model, we use the maximal degeneracy assumption on the boundary to facilitate the connection with tropical geometry.
**C**: In these situations, the target geometries can be tamed by considering toric models associated with various seeds of the cluster variety, corresponding to different choices of dense torus.
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<|MaskedSetence|> For a nonnegative integer m𝑚mitalic_m, we write 12[m]12delimited-[]𝑚12\left[m\right]12 [ italic_m ]
(resp. <|MaskedSetence|> 212…212…212\ldots212 …) of length m𝑚mitalic_m. A quadratic normalisation (A,N)𝐴𝑁\left(A,N\right)( italic_A , italic_N )
is said to be of left-class m𝑚mitalic_m (resp. <|MaskedSetence|>
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**A**: the notion of class, which measures the complexity of normalising
length-three words.
**B**: 21[m]21delimited-[]𝑚21\left[m\right]21 [ italic_m ]) for the alternating sequence 121…121…121\ldots121 …
(resp.
**C**: right-class.
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Analyzing policies beyond SAA to achieve rate-optimality.
In Section 5.1 we complete the picture for the pricing problem under Wasserstein heterogeneity. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We combine this observation with a critical relation between the Wasserstein distance (in 1 dimension) of two probability measures and their associated cumulative distribution functions to obtain the desired result. We believe that this problem-specific analysis may be of independent interest.
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**A**: We propose a policy which appropriately deflates the price selected by SAA, and show that this policy achieves a worst-case regret which has a ϵitalic-ϵ\sqrt{\epsilon}square-root start_ARG italic_ϵ end_ARG dependence in the radius of heterogeneity.
**B**: We also show that this performance is rate-optimal.
To our understanding, analyzing these non-SAA policies (which are required for good performance in pricing) deviates from standard analyses used in learning theory and critically requires the reduction we derive in Theorem 1, as we elaborate on in Section 5.1.
**C**: To derive our result, we leverage the structure of the objective function in pricing: while it is not continuous in general, it is ensured to be one-sided Lipschitz-continuous (when deflating the price).
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3.6 Sensitivity analysis
In this subsection, we discuss the sensitivity of the model to the ‘maximum trolley limit’ constraint for each PCB on dataset A. <|MaskedSetence|> <|MaskedSetence|> From the experiments, we find that there is no change in the use of the number of trolleys used to load the components although their specific loading does change. However, there is a large change in the time to solve the problem as the model takes 30 minutes (m), 2 m, 0.35 m, 0.18 and 0.18 m for the case study setting, 16, 18, 20 and 22 trolleys, respectively. <|MaskedSetence|> Since 20 and 22 trolleys take almost the same time so the problem has no effect after 20 trolleys. Thus, we find that the problem is sensitive to the maximum trolley limit constraint and is one of the reasons for the better performance of the model on dataset B..
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**A**: We solve the model for three more settings, in addition to the case study.
**B**: For the simplicity of the experiments, first, we set the same limits for all the PCBs as 16, 20 and 22 trolleys.
**C**: This large difference in time to solve the problem is because an increase in the maximum trolley limit makes the problem less and less restrictive until the maximum trolley limit has no effect.
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In order to solve the NEP (1), some studies deal with a reformulation of the problem, such as a variational inequality problem or as a complementarity problem; see for instance [33] and [17]. One problem with this approach is that several solutions of the reformulated problem may not be solutions of the associated equilibrium problem. Another possibility is to reformulate the NEP as an optimization problem via the Nikaido-Isoda function (see [36]), which transforms the NEP into a minimax problem. A penalty update scheme was also proposed in the works of [22] and [19]. Those indirect approaches often require solving a nontrivial optimization problem at each step of the algorithm, an exception being [42] which approximates the Nikaido-Isoda function by replacing the minimization problem by a Cauchy step. In general Nikaido-Isoda based formulations deal with convex objective functions since otherwise the subproblem that arises could be hard to deal with. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> These methods came up from the analysis of equilibrium problems with discrete strategy sets, where the actions of each player are taken considering an aggregate of the other player’s previous strategies instead of only their last action. However, expanding on this idea considering non-discrete strategy sets often leads to control problems such as [8], which may be hard to solve.
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**A**: There are some exceptions, such as [46], which requires pseudoconvexity of a regularized Nikaido-Isoda operator, or [43], which presents a branch and bound algorithm on the case of discrete constrained non-convex games.
**B**: In order to avoid this kind of computations, the method proposed in this paper aims at solving the system (1) without rewriting it as an indirect problem, and as we shall see, this leads to solving a linear system of equations at each iteration, while still being able to deal with the non-convex case.
Another recent trend in the literature is considering reinforced learning, see for instance [38], [37] or [14].
**C**: Recent advances deal with the subproblems via similar branch and bound schemes (such as [29]) or genetic algorithms (see [25]), thus being able to treat a broader class of problems, such as general non-convex or non-differentiable games; even so, the subproblems that arise can still be hard to solve.
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For many important applications entanglement has been proven to be a powerful resource. An example of resourceful states are the absolutely maximally entangled (AME) states which maximized the entanglement in the bipartitions, but are notoriously difficult to characterize Scott (2004); Facchi et al. <|MaskedSetence|> <|MaskedSetence|> (2022); Rather et al. <|MaskedSetence|>
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**A**: (2017, 2018); Contreras and Goyeneche (2022).
Still, the analysis of AME states is important for understanding quantum error correction and regarded as one of the central problems in the field Horodecki et al.
**B**: (2022).
However, multiparticle entanglement offers a complex and rich structure resulting in the impossibility of quantification by means of a single number..
**C**: (2008); Reuvers (2018); Gour and Wallach (2010); Huber et al.
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The paper is organised as follows: RCDT, POD and interpolation methodology are introduced in section 2, and their numerical implementation with simple test cases is reported in section 3. To observe and test the errors described above and assess the capabilities of RCDT in the context of MOR, we consider a number of images and flow case studies in section 3.1 and section 4. Test cases to quantify specific errors can be found in section 3.1. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: Final discussions and future work directions are then reported in section 5.
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**B**: We then test our workflow for a multi-phase fluid wave and the flow around an airfoil using high-resolution CFD data.
**C**: These include discontinuous images of a unit circle and a circular ring and continuous Gaussian functions to observe the intrinsic error in the RCDT workflow, and the flow field given by a twin jet at different separation widths to test the interpolation in RCDT space compared to the physical.
In section 4, instead, we focus on the complete MOR procedure starting with a simple moving Gaussian distribution, transformed into RCDT space and order-reduced using POD, compared alongside ’standard’ POD in physical space.
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A deterministic or random selection of the columns and rows is possible, with the option of obtaining additive or relative approximation errors. <|MaskedSetence|> The DEIM is another deterministic way to select a matrix’s columns or rows and relies on the top singular vectors [6]. The three probability distributions that are most typically used to choose columns and rows are uniform, length-squared, and leverage-score distributions. <|MaskedSetence|> <|MaskedSetence|>
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**A**: For additional details on these sampling techniques, see [21].
**B**: It is generally known that the columns or rows with the highest volume might provide virtually optimal solutions in a deterministic scenario [29].
**C**: It has been shown that approximations with relative error accuracy that are more practical in practice can be produced by sampling columns with the leverage-score probability distribution [30].
The MCUR approximation of matrix pairs (matrices with the same number of columns) as the extension of the classical MCUR approximations were presented in [2]..
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<|MaskedSetence|> <|MaskedSetence|> This approach was first implemented in the Mora (Bartocci et al., 2020) tool and later further improved in the Polar tool (Moosbrugger et al., 2022) to also support multi-path probabilistic loops with if-statements, symbolic constants, circular linear dependency among program state variables and drawing from distributions that depend on program state variables. More recently, (Amrollahi et al., 2022) proposed a method to handle more complex state variable dependencies that make both probabilistic and deterministic loops in general unsolvable. Furthermore, the work in (Karimi et al., 2022) shows how to use a finite set of high-order moment-based invariants to estimate the probability distribution of the program’s random variables. <|MaskedSetence|>
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**A**: The core theory underlying all these approaches combines techniques from computer algebra such as symbolic summation and recurrence equations (Kauers and Paule, 2011) with statistical methods.
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**B**: Automating statistical inference for these stochastic systems requires knowledge of their distribution; that is, the distribution(s) of the random variable(s) generated by executing the probabilistic program that encodes them.
Statistical moments are essential quantitative measures that characterize many probability distributions.
**C**: In (Bartocci et al., 2019) the authors introduced the notion of Prob-solvable loops, a class of probabilistic programs with a non-nested loop with polynomial updates and acyclic state variable dependencies for which it is possible to automatically compute moment-based invariants of any order over the program state variables as closed-form expressions in the loop iteration.
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Conventions. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We are thankful to K. Arun Kumar for helpful discussions during the preparation of this manuscript..
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**A**: All the schemes will be assumed to be noetherian and of finite Krull dimension.
Acknowledgements.
**B**: In this paper the coefficient ring ΛΛ\Lambdaroman_Λ will be assumed to be of finite characteristic not equal to 2222 and all the schemes considered will have the property that their residue characteristics are prime to the characteristic of ΛΛ\Lambdaroman_Λ.
**C**: Whenever we consider complex algebraic varieties, we may further restrict ΛΛ\Lambdaroman_Λ to be a finite ring.
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in ℝℝ\mathbb{R}blackboard_R.
Let ε1>0subscript𝜀10\varepsilon_{1}>0italic_ε start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0, b1>b0>0subscript𝑏1subscript𝑏00b_{1}>b_{0}>0italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 and the map Φ:𝒟b1→𝒟b1:Φ→subscript𝒟subscript𝑏1subscript𝒟subscript𝑏1\Phi:{\mathcal{D}}_{b_{1}}\to{\mathcal{D}}_{b_{1}}roman_Φ : caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT → caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT be as in the proof of Lemma 2.1.
Then by (2.40), (2.41) and an argument similar to the proof of Lemma 2.1, we get (2.42), (2.43) and
(2.44). <|MaskedSetence|> Thus (w~3,z3)subscript~𝑤3subscript𝑧3(\widetilde{w}_{3},z_{3})( over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) is a fixed point of the mapping Φ:𝒟b1→𝒟b1:Φ→subscript𝒟subscript𝑏1subscript𝒟subscript𝑏1\Phi:{\mathcal{D}}_{b_{1}}\to{\mathcal{D}}_{b_{1}}roman_Φ : caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT → caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. <|MaskedSetence|> We now choose b2<−b1subscript𝑏2subscript𝑏1b_{2}<-b_{1}italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < - italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Then both z𝑧zitalic_z and z3subscript𝑧3z_{3}italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT satisfies (2.4) in (b2,∞)subscript𝑏2(b_{2},\infty)( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ∞ ) with w~=w~,w~3~𝑤~𝑤subscript~𝑤3\widetilde{w}=\widetilde{w},\widetilde{w}_{3}over~ start_ARG italic_w end_ARG = over~ start_ARG italic_w end_ARG , over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, respectively and z(b2)=z3(b2)𝑧subscript𝑏2subscript𝑧3subscript𝑏2z(b_{2})=z_{3}(b_{2})italic_z ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). <|MaskedSetence|> Thus z=z3𝑧subscript𝑧3z=z_{3}italic_z = italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT in ℝℝ\mathbb{R}blackboard_R and the theorem follows.
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**A**: Hence by the uniqueness of the fixed point for the mapping Φ:𝒟b1→𝒟b1:Φ→subscript𝒟subscript𝑏1subscript𝒟subscript𝑏1\Phi:{\mathcal{D}}_{b_{1}}\to{\mathcal{D}}_{b_{1}}roman_Φ : caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT → caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, w~3=w~subscript~𝑤3~𝑤\widetilde{w}_{3}=\widetilde{w}over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = over~ start_ARG italic_w end_ARG, z3=zsubscript𝑧3𝑧z_{3}=zitalic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_z, in (−∞,−b1)subscript𝑏1(-\infty,-b_{1})( - ∞ , - italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ).
**B**: Hence by the standard ODE theory, z=z3𝑧subscript𝑧3z=z_{3}italic_z = italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT in [b2,∞)subscript𝑏2[b_{2},\infty)[ italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ∞ ).
**C**: These together with (2.48) implies that (w~3,z3)∈𝒟b1subscript~𝑤3subscript𝑧3subscript𝒟subscript𝑏1(\widetilde{w}_{3},z_{3})\in{\mathcal{D}}_{b_{1}}( over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∈ caligraphic_D start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and (w~3,z3)=Φ(w~3,z3)subscript~𝑤3subscript𝑧3Φsubscript~𝑤3subscript𝑧3(\widetilde{w}_{3},z_{3})=\Phi(\widetilde{w}_{3},z_{3})( over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = roman_Φ ( over~ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ).
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Although the concept of the fractional derivatives and dispersion has a long history, their realization in physical systems, such as nonlinear fiber optics, is a relatively new and emerging topic. This inspires several promising perspectives: 1) The immediate goal is to explore additional solutions of the FNLSE. The current results focus primarily on positive fractional dispersion lengths, corresponding to the “repulsion” case in the force model, which explains the observed spectral valleys. However, equally intriguing results are expected in the “attraction” case, which would illustrate spectral squeezing. Recently two works have demonstrated an “attraction” effect (narrowed spectrum) in similar contexts [65, 66]. <|MaskedSetence|> <|MaskedSetence|> This approach has significant potential for applications to optical encoding and spatiotemporal mode-locked laser architectures [67, 68]. <|MaskedSetence|>
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**A**: 2) The second perspective is the realization of the spatial-temporal light synthesis by incorporating both fractional dispersion and fractional diffraction.
**B**: 3) Finally, due to the mathematical similarity between the FNLSE in nonlinear optics and its quantum counterpart, this regime may serve as an effective model for emulating fractional quantum mechanics [1, 69].
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**C**: By involving additional parameters, such as higher-order dispersion or stronger nonlinearity, even greater diversity in the pulse dynamics could be uncovered.
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CAB
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CAB
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BCA
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CAB
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Selection 4
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Part of the work has been done when CD was a visiting reseacher at the Institut des Hautes Études and she would like to thank Laure Saint-Raymond for her support and hospitality. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> 426365943. CD also acknowledges the support from the Jean-Paul Gimon Fund and from the Erasmus+ programme.
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**A**: Frank for helpful suggestions, leading to our consideration of Lorentz norms in Theorem 2.
**B**: We also thank Hajer Bahouri, Morris Brooks, Albert Cohen, Patrick Gérard, Kihyun Kim, Mathieu Lewin, Hoai-Minh Nguyen, Julien Sabin, Nikita Simonov, Thomas Sørensen, Jakob Stern, Hanne van den Bosch and Jean van Schaftingen for interesting and inspiring discussions.
We acknowledge the support from the Deutsche Forschungsgemeinschaft through the DFG project Nr.
**C**: We are grateful to Rupert L.
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CAB
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CAB
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BAC
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CAB
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Selection 1
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<|MaskedSetence|> From (3.4) it follows that θ𝜃\thetaitalic_θ is convex, and strictly convex if D<n−22c.𝐷𝑛22𝑐D<\frac{n-2}{2}c.italic_D < divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c . As in part (i) we obtain a contradiction if D<n−22c,𝐷𝑛22𝑐D<\frac{n-2}{2}c,italic_D < divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c , or if D=n−22c𝐷𝑛22𝑐D=\frac{n-2}{2}citalic_D = divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c and one of the limits lims→±∞θ(s)subscript→𝑠plus-or-minus𝜃𝑠\lim_{s\rightarrow\pm\infty}\theta(s)roman_lim start_POSTSUBSCRIPT italic_s → ± ∞ end_POSTSUBSCRIPT italic_θ ( italic_s ) is infinite. Otherwise θ𝜃\thetaitalic_θ must be a constant function and ψ,𝜓\psi,italic_ψ , and hence ϕ,italic-ϕ\phi,italic_ϕ , must be constant along γ.𝛾\gamma.italic_γ . Since ∂jϕ=0, 2≤j≤n,formulae-sequencesubscript𝑗italic-ϕ02𝑗𝑛\partial_{j}\phi=0,\ 2\leq j\leq n,∂ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ϕ = 0 , 2 ≤ italic_j ≤ italic_n , it follows that ϕitalic-ϕ\phiitalic_ϕ must be a constant function on Mn.superscript𝑀𝑛M^{n}.italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . From (3.3) we obtain ∂tvj=0subscript𝑡subscript𝑣𝑗0\partial_{t}v_{j}=0∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0 for each j, 2≤j≤n.𝑗2𝑗𝑛j,\ 2\leq j\leq n.italic_j , 2 ≤ italic_j ≤ italic_n . Hence we obtain ϕ=K(∂t,∂j)=0italic-ϕ𝐾subscript𝑡subscript𝑗0\phi=K(\partial_{t},\partial_{j})=0italic_ϕ = italic_K ( ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , ∂ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = 0 using (3.2). <|MaskedSetence|> <|MaskedSetence|>
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**A**: This gives 0=ϕ=c+⟨η1,η2⟩=c,0italic-ϕ𝑐subscript𝜂1subscript𝜂2𝑐0=\phi=c+\langle\eta_{1},\eta_{2}\rangle=c,0 = italic_ϕ = italic_c + ⟨ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ = italic_c , which is a contradiction..
**B**: -2}{2}c>0.italic_ψ = divide start_ARG 2 italic_D end_ARG start_ARG italic_n end_ARG - italic_ϕ = divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c + divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG | | italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c > 0 .
Moreover, (2.6) implies that ϕ+n−22c≤D,italic-ϕ𝑛22𝑐𝐷\phi+\frac{n-2}{2}c\leq D,italic_ϕ + divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c ≤ italic_D , or ϕ≤D−(n−22c)≤0.italic-ϕ𝐷𝑛22𝑐0\phi\leq D-(\frac{n-2}{2}c)\leq 0.italic_ϕ ≤ italic_D - ( divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c ) ≤ 0 .
**C**: Since D=n−22c,𝐷𝑛22𝑐D=\frac{n-2}{2}c,italic_D = divide start_ARG italic_n - 2 end_ARG start_ARG 2 end_ARG italic_c , using (2.6) we obtain η2=0.subscript𝜂20\eta_{2}=0.italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 .
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BCA
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BCA
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BCA
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BAC
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Selection 3
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The globalization problem has been investigated in several other contexts. <|MaskedSetence|> For categories acting partially on sets, a globalization is not unique, but a globalization with an additional universal property, called universal globalization, is unique up to equivalence, and such a globalization always exists [29, Theorem 4]. For semigroups acting partially on sets, there exist two non-isomorphic universal globalizations [25]. For Hopf algebras acting partially on a unital algebra, a globalization is not unique, but a globalization with an additional minimal property, called minimal globalization, is [1, Theorem 3].
The problem of determining when a partial action can be realized as a restriction of a global one is very relevant, because it allows us to undestand how the partial theory behaves compared to the global one. The issue of uniqueness is particularly crucial, as it establishes a well-defined framework for transitioning to global actions. <|MaskedSetence|> In Section 2, we provide a background on the partial ordered actions of ordered groupoids and establish some notation. In Section 3, we prove that a partial ordered action of an ordered groupoid on a ring has a globalization if and only if it is unital. <|MaskedSetence|> Finally, in Section 5, we present two applications: we construct a Morita context between the crossed products of an ordered groupoid acting partially on a ring and its globalization, and we apply our results to the case of inverse semigroups acting on rings via the Ehresmann-Schein-Nambooripad (ESN) Theorem.
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**A**: Thus, the primary aim of this work is to analyze the existence and uniqueness of ordered globalizations of partial ordered actions of ordered groupoids on rings.
**B**: For instance, in [14] it was proved that a partial action of a group on a unital algebra is globalizable if and only if each ideal of the action is unital, and it is unique up to equivalence.
**C**: Section 4 introduces strong partial ordered actions, pseudoassociative groupoids and minimal globalizations, showing that a unital strong partial ordered action of a pseudoassociative groupoid on a ring always has a unique minimal globalization.
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CAB
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BAC
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BAC
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BAC
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Selection 2
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Understanding the long-term behaviour of dynamical systems is a fundamental research challenge. <|MaskedSetence|> The fundamental results presented in the previous section form the cornerstone of our study of the long-term behaviour of the equation (7) using the theory of global attractors. <|MaskedSetence|> <|MaskedSetence|> For a more comprehensive understanding, readers are encouraged to refer to [11]..
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**A**: We thus establish the existence of a global attractor for our model.
Let us first revisit some key properties that are derived from the theory of global attractors and will help us in establishing our results in this section.
**B**: In this section, we carry out a complete examination of the long-term dynamics of the equation (7).
**C**: A crucial concept in the study of the behaviour of such systems is that of global attractors [11].
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CBA
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CBA
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ACB
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CBA
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Selection 2
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<|MaskedSetence|> Such difficulties have been tackled by extending the work of Bakry-Émery to hypoelliptic settings, see [8] in the context of obtaining Villani’s [51] hypocoercivity estimates, [18, 17] for gradient estimates of Kolmogorov diffusions, [16] for Langevin dynamics with singular potentials, [10] for gradient estimates of sub-elliptic heat kernels on SU(2)SU2\operatorname{SU}(2)roman_SU ( 2 ), and using coupling for gradient estimates on the Heisenberg group [6]. <|MaskedSetence|> <|MaskedSetence|> In general, the results obtained through these methods may lead to dimension-dependent functional inequalities. Besides geometric and probabilistic techniques such as coupling, functional inequalities in such degenerate settings can be approached by using the structure of the underlying spaces as in [21, 26, 32, 27, 47, 48, 35, 36].
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**A**:
However, for hypoelliptic diffusions several fundamental issues arise due to lack of such geometric methods in general, and in particular, for not having a Dirichlet form corresponding to the hypoelliptic differential generator.
**B**: For a detailed account on such techniques, we refer to [11, 15].
**C**: One of the key tools in such a context is the generalized curvature-dimension condition, which implies reverse Poincaré and reverse logarithmic Sobolev inequalities, Li-Yau type gradient estimates for the associated Markov semigroups.
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ACB
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ABC
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ACB
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ACB
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Selection 4
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<|MaskedSetence|> Assume by contradiction that (λn,un)→(0,0)→subscript𝜆𝑛subscript𝑢𝑛00(\lambda_{n},u_{n})\rightarrow(0,0)( italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) → ( 0 , 0 ) in [0,∞)×C(Ω¯)0𝐶¯Ω[0,\infty)\times C(\overline{\Omega})[ 0 , ∞ ) × italic_C ( over¯ start_ARG roman_Ω end_ARG ) for some positive solution (λn,un)subscript𝜆𝑛subscript𝑢𝑛(\lambda_{n},u_{n})( italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) of (1.1) with λn>0subscript𝜆𝑛0\lambda_{n}>0italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > 0. Then, ‖un‖→0→normsubscript𝑢𝑛0\|u_{n}\|\rightarrow 0∥ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ → 0. Problem (4.3) admits the positive solution (λn,Un)subscript𝜆𝑛subscript𝑈𝑛(\lambda_{n},U_{n})( italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) with
Un=λn−1/(1−q)unsubscript𝑈𝑛superscriptsubscript𝜆𝑛11𝑞subscript𝑢𝑛U_{n}=\lambda_{n}^{-1/(1-q)}u_{n}italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / ( 1 - italic_q ) end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, and then, Unsubscript𝑈𝑛U_{n}italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is bounded in H1(Ω)superscript𝐻1ΩH^{1}(\Omega)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) by Lemma 4.3. Up to a subsequence, Un⇀U∞≥0⇀subscript𝑈𝑛subscript𝑈0U_{n}\rightharpoonup U_{\infty}\geq 0italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⇀ italic_U start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≥ 0, and Un→U∞→subscript𝑈𝑛subscript𝑈U_{n}\rightarrow U_{\infty}italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_U start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT in Lp+1(Ω)superscript𝐿𝑝1ΩL^{p+1}(\Omega)italic_L start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT ( roman_Ω ) and L2(∂Ω)superscript𝐿2ΩL^{2}(\partial\Omega)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ∂ roman_Ω ) for some U∞∈H1(Ω)subscript𝑈superscript𝐻1ΩU_{\infty}\in H^{1}(\Omega)italic_U start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ). <|MaskedSetence|> <|MaskedSetence|>
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**A**:
(ii) For (4.11), we employ Lemmas 4.3 through 4.5.
**B**: As a matter of fact,.
**C**: Owing to Lemma 4.5, Lemma 2.2 provides that U∞≠0subscript𝑈0U_{\infty}\neq 0italic_U start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≠ 0.
On the other hand, we infer U∞=0subscript𝑈0U_{\infty}=0italic_U start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = 0.
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ACB
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ACB
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ACB
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ACB
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Selection 3
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{sign}(q){\bf a})_{+}]_{i}[ bold_y start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG italic_q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_d start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG [ ( sign ( italic_q ) bold_a ) start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < [ ( sign ( italic_q ) bold_a ) start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for some i∈{m1+1,m1+2}𝑖subscript𝑚11subscript𝑚12i\in\{m_{1}+1,m_{1}+2\}italic_i ∈ { italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 , italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 }. <|MaskedSetence|> <|MaskedSetence|> Suppose f𝐛+|f𝐲+conditionalsuperscriptsubscript𝑓𝐛superscriptsubscript𝑓𝐲f_{\bf b}^{+}|f_{\bf y}^{+}italic_f start_POSTSUBSCRIPT bold_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT. Since either m1+1∈supp(𝐛+)subscript𝑚11suppsubscript𝐛m_{1}+1\in\mbox{supp}({\bf b}_{+})italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 ∈ supp ( bold_b start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) or m1+2∈supp(𝐛+)subscript𝑚12suppsubscript𝐛m_{1}+2\in\mbox{supp}({\bf b}_{+})italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 ∈ supp ( bold_b start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ), then comparing signs of 𝐛𝐛{\bf b}bold_b and 𝐲𝐲{\bf y}bold_y at the ithsuperscript𝑖thi^{\mbox{th}}italic_i start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT component for some i∈{m1+1,m1+2}𝑖subscript𝑚11subscript𝑚12i\in\{m_{1}+1,m_{1}+2\}italic_i ∈ { italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 , italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 }, we get (−1)i+1sign(r)=(−1)isign(qp)superscript1𝑖1sign𝑟superscript1𝑖sign𝑞𝑝(-1)^{i+1}\mbox{sign}(r)=(-1)^{i}\mbox{sign}(\frac{q}{p})( - 1 ) start_POSTSUPERSCRIPT italic_i + 1 end_POSTSUPERSCRIPT sign ( italic_r ) = ( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT sign ( divide start_ARG italic_q end_ARG start_ARG italic_p end_ARG ). This implies that sign(pqr)sign𝑝𝑞𝑟\mbox{sign}(pqr)sign ( italic_p italic_q italic_r ) is negative. Again, either m1+m2+1∈supp(𝐛+)subscript𝑚1subscript𝑚21suppsubscript𝐛m_{1}+m_{2}+1\in\mbox{supp}({\bf b}_{+})italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 1 ∈ supp ( bold_b start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) or m1+m2+2∈supp(𝐛+)subscript𝑚1subscript𝑚22suppsubscript𝐛m_{1}+m_{2}+2\in\mbox{supp}({\bf b}_{+})italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 ∈ supp ( bold_b start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ). Then comparing signs of 𝐛𝐛{\bf b}bold_b and 𝐲𝐲{\bf y}bold_y at the ithsuperscript𝑖thi^{\mbox{th}}italic_i start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT component for some i∈{m1+m2+1,m1+m2+2}𝑖subscript𝑚1subscript𝑚21subscript𝑚1subscript𝑚22i\in\{m_{1}+m_{2}+1,m_{1}+m_{2}+2\}italic_i ∈ { italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 1 , italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 }, we get (−1)isign(q)=(−1)isign(rp)superscript1𝑖sign𝑞superscript1𝑖sign𝑟𝑝(-1)^{i}\mbox{sign}(q)=(-1)^{i}\mbox{sign}(\frac{r}{p})( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT sign ( italic_q ) = ( - 1 ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT sign ( divide start_ARG italic_r end_ARG start_ARG italic_p end_ARG ). This implies that sign(pqr)sign𝑝𝑞𝑟\mbox{sign}(pqr)sign ( italic_p italic_q italic_r ) is positive which is contradiction. If f𝐛−|f𝐲+conditionalsuperscriptsubscript𝑓𝐛superscriptsubscript𝑓𝐲f_{\bf b}^{-}|f_{\bf y}^{+}italic_f start_POSTSUBSCRIPT bold_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, then f(−𝐛)+|f𝐲+conditionalsuperscriptsubscript𝑓𝐛superscriptsubscript𝑓𝐲f_{(-{\bf b})}^{+}|f_{\bf y}^{+}italic_f start_POSTSUBSCRIPT ( - bold_b ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and proceeding similarly as above, we get a contradiction.
Therefore supp(f𝐧)=E(D)subscript𝑓𝐧𝐸𝐷(f_{\bf n})=E(D)( italic_f start_POSTSUBSCRIPT bold_n end_POSTSUBSCRIPT ) = italic_E ( italic_D ) and 𝐧𝐧{\bf n}bold_n be in the form (7). Since 𝐧+≤𝐲+subscript𝐧subscript𝐲{\bf n_{+}}\leq{\bf y_{+}}bold_n start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ≤ bold_y start_POSTSUBSCRIPT + end_POSTSUBSCRIPT and 𝐧−≤𝐲−subscript𝐧subscript𝐲{\bf n_{-}}\leq{\bf y_{-}}bold_n start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ≤ bold_y start_POSTSUBSCRIPT - end_POSTSUBSCRIPT, then by comparing the expression of 𝐲𝐲{\bf y}bold_y above with that of 𝐧𝐧{\bf n}bold_n in (11), we get that pi′′′di+ri≤qidi<1superscriptsubscript𝑝𝑖′′′subscript𝑑𝑖subscript𝑟𝑖subscript𝑞𝑖subscript𝑑𝑖1\frac{p_{i}^{\prime\prime\prime}}{d_{i}}+r_{i}\leq\frac{q_{i}}{d_{i}}<1divide start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG + italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ divide start_ARG italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG < 1 for i=3,4𝑖34i=3,4italic_i = 3 , 4. <|MaskedSetence|> By the minimality of 𝐲𝐲{\bf y}bold_y in E2subscript𝐸2E_{2}italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, this gives that (p3′′′,p4′′′)=(q3,q4)superscriptsubscript𝑝3′′′superscriptsubscript𝑝4′′′subscript𝑞3subscript𝑞4(p_{3}^{\prime\prime\prime},p_{4}^{\prime\prime\prime})=(q_{3},q_{4})( italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT ) = ( italic_q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ). Thus, 𝐲=𝐧𝐲𝐧{\bf y}={\bf n}bold_y = bold_n and hence f𝐧=f𝐲subscript𝑓𝐧subscript𝑓𝐲f_{\bf n}=f_{\bf y}italic_f start_POSTSUBSCRIPT bold_n end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT, which is a contradiction. Therefore f𝐲∈GrDsubscript𝑓𝐲𝐺subscript𝑟𝐷f_{\bf y}\in Gr_{D}italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ∈ italic_G italic_r start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT.
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**A**: This implies that r3=0=r4subscript𝑟30subscript𝑟4r_{3}=0=r_{4}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 = italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT and (p3′′′,p4′′′)≤(q3,q4)superscriptsubscript𝑝3′′′superscriptsubscript𝑝4′′′subscript𝑞3subscript𝑞4(p_{3}^{\prime\prime\prime},p_{4}^{\prime\prime\prime})\leq(q_{3},q_{4})( italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT ) ≤ ( italic_q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ).
**B**: Proceeding similarly as above, we get a contradiction if f𝐭+|f𝐲+conditionalsuperscriptsubscript𝑓𝐭superscriptsubscript𝑓𝐲f_{\bf t}^{+}|f_{\bf y}^{+}italic_f start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT or f𝐭−|f𝐲+conditionalsuperscriptsubscript𝑓𝐭superscriptsubscript𝑓𝐲f_{\bf t}^{-}|f_{\bf y}^{+}italic_f start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, 𝐭=sign(r)𝐜𝐭sign𝑟𝐜{\bf t}=\mbox{sign}(r){\bf c}bold_t = sign ( italic_r ) bold_c.
**C**: If fsign(q)𝐚−|f𝐲+conditionalsuperscriptsubscript𝑓sign𝑞𝐚superscriptsubscript𝑓𝐲f_{\mbox{sign}(q){\bf a}}^{-}|f_{\bf y}^{+}italic_f start_POSTSUBSCRIPT sign ( italic_q ) bold_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | italic_f start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , then by comparing the second components of sign(q)𝐚sign𝑞𝐚\mbox{sign}(q){\bf a}sign ( italic_q ) bold_a and 𝐲𝐲{\bf y}bold_y, we get a contradiction since 2∈supp((sign(q)𝐚)−)2suppsubscriptsign𝑞𝐚2\in\mbox{supp}((\mbox{sign}(q){\bf a})_{-})2 ∈ supp ( ( sign ( italic_q ) bold_a ) start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ), 2∉supp(𝐲+)2suppsubscript𝐲2\notin\mbox{supp}({\bf y}_{+})2 ∉ supp ( bold_y start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ).
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The study of non-monotonicity and fluctuation of convergent stochastic processes is a topic of intrinsic interest in probability theory. For a symmetric random walk, non-monotonicity was first investigated by Erdős and Rényi [Erdos3]. They proved that the length of the longest run of heads in n𝑛nitalic_n tosses of a fair coin will approximately be equal to logn𝑛\log nroman_log italic_n. <|MaskedSetence|> <|MaskedSetence|> The former one refers to strict inequality, whereas the later one does not require the inequality to be strict. <|MaskedSetence|>
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**A**: The question of fluctuation for such stochastic processes was first studied by Chung and Hunt [Chung-Hunt], and later by Chung and Erdős [Erdos1], and Csáki, Erdős and Révész [Erdos2].
**B**: in a symmetric random variable.
While our primary focus is on various ergodic averages, we will also study Lebesgue differentiation,.
**C**: They found bounds on the length of the longest excursion 111There is a subtle difference between fluctuation and excursion.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The radio resource management scheme with max-marginal-rate subcarrier assignment and equal power allocation, proposed and shown to be near-optimal in[50], is adopted. Then the communication latency is calculated using Shannon capacity given the assigned subcarrier and power. After receiving data from all agents, the server decodes the bits stream to reconstruct features.
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**A**: The scheme corresponds to the conventional digital broadband orthogonal-access approach, where each agent is assigned a subset of subcarriers for feature uploading.
**B**:
Digital air interface.
**C**: On the agent side, each feature coefficient is encoded into 2222 to 5555 bits, depending on the desired latency-precision tradeoff, via uniform quantization.
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The D-Wave QPU solver could only compute local optimum solutions for instances with less than 150150150150 vertices. <|MaskedSetence|> To address these, we developed a post-processing procedure that enables us to detect samples that could lead to improved solutions and to extract solutions that are stable sets. Also, we provide theoretical guarantees about the quality of the extracted solutions. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Table 2 contains detailed numerical results of the solution provided by the QPU solver as part of Algorithm 1.
3..
**B**: The procedure is outlined in Algorithm 1.
**C**: Our results show that very often, these solutions are quite far away from the global solutions in terms of the objective value and may not even be stable sets.
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However, there seems to be a significant gap between the practical performance of the DDPM sampler and the existing theory. For example, for two widely used image datasets, CIFAR-10 (dimension d=32×32×3𝑑32323d=32\times 32\times 3italic_d = 32 × 32 × 3) and ImageNet (dimension d≥64×64×3𝑑64643d\geq 64\times 64\times 3italic_d ≥ 64 × 64 × 3), it is known that 50 and 250 steps (also known as NFE, the number of function evaluations) are sufficient to generate good samples (Nichol and Dhariwal,, 2021; Dhariwal and Nichol,, 2021). This is in stark contrast with the existing theoretical guarantees discussed above, which suggest that the number of steps T𝑇Titalic_T should exceed the order of the dimension d𝑑ditalic_d to achieve good performance.
Empirical evidence suggests that the distributions of natural images are concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside (Simoncelli and Olshausen,, 2001; Pope et al.,, 2021). In view of this, a reasonable conjecture is that the convergence rate of the DDPM sampler actually depends on the intrinsic dimension rather than the ambient dimension. However, the theoretical understanding of diffusion models when the support of the target data distribution has a low-dimensional structure remains vastly under-explored. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: However, their error bound has linear dependence on the ambient dimension d𝑑ditalic_d and exponential dependence on the diameter of the low-dimensional manifold.
**B**: Another line of works (Chen et al., 2023b, ; Tang and Yang,, 2024; Oko et al.,, 2023) focused mainly on score estimation with properly chosen neural networks that exploit the low-dimensional structure, which is also different from our main focus..
**C**: As some recent attempts, De Bortoli, (2022) established the first convergence guarantee under the Wasserstein-1 metric.
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5. Infinite and finite continued fractions
A continued fraction is determined by a sequence (ai)i=0∞superscriptsubscriptsubscript𝑎𝑖𝑖0(a_{i})_{i=0}^{\infty}( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT with ai∈ℤsubscript𝑎𝑖ℤa_{i}\in\mathbb{Z}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_Z and ai≥0subscript𝑎𝑖0a_{i}\geq 0italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0 for i>0𝑖0i>0italic_i > 0. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For each n𝑛nitalic_n we let.
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**A**: in ℝ∪{∞}ℝ\mathbb{R}\cup\{\infty\}blackboard_R ∪ { ∞ }) can (often) be associated to a continued fraction as follows.
**B**: The continued fraction is called simple if ai>0subscript𝑎𝑖0a_{i}>0italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 for i>0𝑖0i>0italic_i > 0.
**C**: An extended real number (i.e.
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Firstly, we prove that Φ:Cp(X,Y)→Cp(Z):Φ→subscript𝐶𝑝𝑋𝑌subscript𝐶𝑝𝑍\Phi:C_{p}(X,Y)\to C_{p}(Z)roman_Φ : italic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_X , italic_Y ) → italic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_Z ). Fix u∈Cp(X,Y)𝑢subscript𝐶𝑝𝑋𝑌u\in C_{p}(X,Y)italic_u ∈ italic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_X , italic_Y ) and c=(a,α)∈Z𝑐𝑎𝛼𝑍c=(a,\alpha)\in Zitalic_c = ( italic_a , italic_α ) ∈ italic_Z. Let us prove that w=Φ(u)𝑤Φ𝑢w=\Phi(u)italic_w = roman_Φ ( italic_u ) is continuous at c𝑐citalic_c. Consider, ε>0𝜀0\varepsilon>0italic_ε > 0. Set b=u(a)𝑏𝑢𝑎b=u(a)italic_b = italic_u ( italic_a ) and U=u−1(B(b,ε2))𝑈superscript𝑢1𝐵𝑏𝜀2U=u^{-1}\big{(}B(b,\frac{\varepsilon}{2})\big{)}italic_U = italic_u start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ( italic_b , divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG ) ). Since u𝑢uitalic_u is continuous, U𝑈Uitalic_U is a neighborhood of a𝑎aitalic_a. <|MaskedSetence|> <|MaskedSetence|> Therefore, W=U×O𝑊𝑈𝑂W=U\times Oitalic_W = italic_U × italic_O is a neighborhood of c𝑐citalic_c in Z𝑍Zitalic_Z. Consider a point z=(x,λ)∈W𝑧𝑥𝜆𝑊z=(x,\lambda)\in Witalic_z = ( italic_x , italic_λ ) ∈ italic_W. Then y=u(x)∈B(b,ε2)𝑦𝑢𝑥𝐵𝑏𝜀2y=u(x)\in B(b,\frac{\varepsilon}{2})italic_y = italic_u ( italic_x ) ∈ italic_B ( italic_b , divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG ) and |λ(b)−α(b)|<ε2𝜆𝑏𝛼𝑏𝜀2\big{|}\lambda(b)-\alpha(b)\big{|}<\frac{\varepsilon}{2}| italic_λ ( italic_b ) - italic_α ( italic_b ) | < divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG. <|MaskedSetence|>
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**A**: Consequently,
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**B**: Put O={λ∈Λ:|λ(b)−α(b)|<ε2}𝑂conditional-set𝜆Λ𝜆𝑏𝛼𝑏𝜀2O=\Big{\{}\lambda\in\Lambda:\big{|}\lambda(b)-\alpha(b)\big{|}<\frac{%
\varepsilon}{2}\Big{\}}italic_O = { italic_λ ∈ roman_Λ : | italic_λ ( italic_b ) - italic_α ( italic_b ) | < divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG }.
**C**: Then O𝑂Oitalic_O is a neighborhood of α𝛼\alphaitalic_α in ΛΛ\Lambdaroman_Λ.
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remain magic squares for 1⩽k⩽K1𝑘𝐾1\leqslant k\leqslant K1 ⩽ italic_k ⩽ italic_K. Before we can state our problem of interest we must first discuss trivial multimagic squares.
It is clear any multiple of the N×N𝑁𝑁N\times Nitalic_N × italic_N matrix of all ones is trivially a MMS(K,N)𝐾𝑁(K,N)( italic_K , italic_N ) for every K⩾2𝐾2K\geqslant 2italic_K ⩾ 2. However, this is not the only family of “trivial” multimagic squares one must consider. Suppose 𝐙𝐙{\mathbf{Z}}bold_Z is an N×N𝑁𝑁N\times Nitalic_N × italic_N matrix in which every row, column, and both main diagonals contain precisely N𝑁Nitalic_N distinct symbols. <|MaskedSetence|> <|MaskedSetence|> Then for N⩾4𝑁4N\geqslant 4italic_N ⩾ 4 any mapping of these N𝑁Nitalic_N symbols to the integers yields a MMS(K,N)𝐾𝑁(K,N)( italic_K , italic_N ) for every K⩾2𝐾2K\geqslant 2italic_K ⩾ 2. <|MaskedSetence|>
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**A**: Such matrices are known as doubly diagonalized Latin squares of order N𝑁Nitalic_N, or DDLS(N𝑁Nitalic_N) for short.
**B**: DDLS(N𝑁Nitalic_N) are known to exist for all N⩾4𝑁4N\geqslant 4italic_N ⩾ 4, see [7].
**C**: Consideration of these “trivial” MMS(K,N)𝐾𝑁(K,N)( italic_K , italic_N ) motivates the following definition.
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Let a𝑎aitalic_a be an ad-nilpotent element of 𝒰𝒰\mathcal{U}caligraphic_U. <|MaskedSetence|> For a 𝒰𝒰\mathcal{U}caligraphic_U-module M𝑀Mitalic_M by D⟨a⟩M=D⟨a⟩𝒰⊗𝒰Msubscript𝐷delimited-⟨⟩𝑎𝑀subscripttensor-product𝒰subscript𝐷delimited-⟨⟩𝑎𝒰𝑀D_{\langle a\rangle}M=D_{\langle a\rangle}{\mathcal{U}}\otimes_{\mathcal{U}}Mitalic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT italic_M = italic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT caligraphic_U ⊗ start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT italic_M we denote the ⟨a⟩delimited-⟨⟩𝑎\langle a\rangle⟨ italic_a ⟩-localization of M𝑀Mitalic_M. Note that if a𝑎aitalic_a is injective on M𝑀Mitalic_M, then M𝑀Mitalic_M is isomorphic to a submodule of D⟨a⟩Msubscript𝐷delimited-⟨⟩𝑎𝑀D_{\langle a\rangle}Mitalic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT italic_M. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Then the set ⟨a⟩={an|n≥0}delimited-⟨⟩𝑎conditional-setsuperscript𝑎𝑛𝑛0\langle a\rangle=\{a^{n}\;|\;n\geq 0\}⟨ italic_a ⟩ = { italic_a start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT | italic_n ≥ 0 } is an Ore subset of 𝒰𝒰\mathcal{U}caligraphic_U which allows us to define the ⟨a⟩delimited-⟨⟩𝑎\langle a\rangle⟨ italic_a ⟩-localization D⟨a⟩𝒰subscript𝐷delimited-⟨⟩𝑎𝒰D_{\langle a\rangle}\mathcal{U}italic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT caligraphic_U of 𝒰𝒰\mathcal{U}caligraphic_U.
**B**: In the latter case we will identify M𝑀Mitalic_M with that submodule.
We next recall the definition of the generalized conjugation of D⟨a⟩𝒰subscript𝐷delimited-⟨⟩𝑎𝒰D_{\langle a\rangle}\mathcal{U}italic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT caligraphic_U relative to x∈ℂ𝑥ℂx\in{\mathbb{C}}italic_x ∈ blackboard_C, see Lemma 4.3 in [Mat00].
**C**: This is the automorphism ϕx:D⟨a⟩𝒰→D⟨a⟩𝒰:subscriptitalic-ϕ𝑥→subscript𝐷delimited-⟨⟩𝑎𝒰subscript𝐷delimited-⟨⟩𝑎𝒰\phi_{x}:D_{\langle a\rangle}\mathcal{U}\to D_{\langle a\rangle}\mathcal{U}italic_ϕ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT caligraphic_U → italic_D start_POSTSUBSCRIPT ⟨ italic_a ⟩ end_POSTSUBSCRIPT caligraphic_U given by.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We show that, for appropriate parameter choices, such models are sparse, i.e., lead to sparse exponential random graphs. In Section 4, we show how our main results can be used to consistently estimate the exponential random graph parameters. We close in Section 5 with a discussion and a list of open problems.
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**A**: In Section 2, we estimate the second-order of the large-deviation probabilities of the rare event that a sparse Erdős–Rényi random graph has a linear number of vertices in triangles, study the structure of the graph conditionally on this rare event, and provide proofs for our main results.
**B**: In Section 3, we use these results, as well as the key insights developed in their proofs, to study exponential random graphs based on the number of vertices in triangles.
**C**: 1.3 Organisation
This paper is organised as follows.
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<|MaskedSetence|> The library incorporates robust parallel data structures, making it highly effective for tackling large-scale scientific computing problems across diverse domains [30, 32]. <|MaskedSetence|> <|MaskedSetence|> Readers interested in a detailed explanation of this partitioning approach are referred to Sundar et al. [34].
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**A**:
3.2.1 Hanging nodes and 2:1 balancing
This paper utilizes the DENDRO-KT library, an octree-based meshing and finite element solver framework designed for parallel scalability and efficiency.
**B**: To distribute the computational domain across multiple processors while ensuring load balance and minimizing inter-processor communication, we employ a space-filling curve (SFC) for partitioning.
**C**: The SFC algorithm organizes the octants to preserve spatial locality, which is crucial for reducing communication overhead.
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<|MaskedSetence|> <|MaskedSetence|> Existence and regularity of solutions for (1.5), as well as maximum principles are among the results obtained in [3], [4], [5], and [11], where the advection q𝑞\displaystyle qitalic_q is absent and the boundary condition is of Dirichlet type. <|MaskedSetence|> The work [8] considers a purely nonlocal diffusion and provides existence results for the problem with nonlocal Neumann conditions. It is important to note that [8] does not consider a PDE with a mixed diffusion and it does not account for advection.
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**A**: Tat is, u≡0𝑢0\displaystyle u\equiv 0italic_u ≡ 0 in ℝN∖Ω.superscriptℝ𝑁Ω\displaystyle\mathbb{R}^{N}\setminus\Omega.blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∖ roman_Ω .
**B**: The authors of this paper studied (1.5) in the recent work [6], where an advection term is present and (1.5) is coupled with the Dirichlet condition u≡0𝑢0\displaystyle u\equiv 0italic_u ≡ 0 on ℝ∖Ω¯.ℝ¯Ω\displaystyle\mathbb{R}\setminus\overline{\Omega}.blackboard_R ∖ over¯ start_ARG roman_Ω end_ARG .
The recent work [7] considers (1.5) with q≡0𝑞0\displaystyle q\equiv 0italic_q ≡ 0 and a≡0𝑎0\displaystyle a\equiv 0italic_a ≡ 0 to provide spectral properties of the mixed diffusion operator.
**C**: has been extensively studied when q≡0𝑞0\displaystyle q\equiv 0italic_q ≡ 0 and the boundary condition is of Dirichlet type.
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The right-hand side is independent of x𝑥xitalic_x and finite since the ΓΓ\Gammaroman_Γ-action is proper. <|MaskedSetence|> Therefore, ΛΛ\Lambdaroman_Λ is a quasi lattice. <|MaskedSetence|> Since it is coarsely equivalent to (Γ,ℰRΓ)ΓsubscriptsuperscriptℰΓ𝑅(\Gamma,\mathcal{E}^{\Gamma}_{R})( roman_Γ , caligraphic_E start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ), (Γ,ℰRΓ)ΓsubscriptsuperscriptℰΓ𝑅(\Gamma,\mathcal{E}^{\Gamma}_{R})( roman_Γ , caligraphic_E start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ) also has a Ponzi scheme. <|MaskedSetence|>
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**A**: By Fact 15, the discrete group ΓΓ\Gammaroman_Γ is non-amenable.
∎
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**B**: By Theorem 6.1, the coarse space (X,ℰΓX)𝑋subscriptsuperscriptℰ𝑋Γ(X,\mathcal{E}^{X}_{\Gamma})( italic_X , caligraphic_E start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT ) has a Ponzi scheme.
**C**: Thus, ΛΛ\Lambdaroman_Λ is uniform locally finite.
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To be more precise, once a noncommutative Minkowski spacetime is constructed, this noncommutativity is commonly interpreted as an algebraic means of introducing a certain fuzziness generated by quantum gravity effects (for a recent review, see [24]). The spacetime position of a given particle is given by the expectation value of appropriately defined operators, and their noncommutativity leads to non-vanishing uncertainty relations among these operators. <|MaskedSetence|> Similarly, for massive particles, their spacetime fuzziness should remain within their respective timelike or spacelike regions. However, this causality issue cannot be easily solved within the conventional approach to noncommutative Minkowski spacetimes as ambient spaces aiming to describe quantum gravity effects for all kinds of particles simultaneously, since the fuzziness around a point in the lightcone could, in principle, invade both the timelike and the spacelike regions.
In this paper, we present a novel approach to this problem by explicitly constructing noncommutative lightcones in three spacetime dimensions, seen as quantum homogeneous spaces of a quantum SO(2,1) group. <|MaskedSetence|> <|MaskedSetence|>
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**A**: If, for instance, massless particles are considered, causality preservation requires that the induced fuzziness remains strictly confined to the lightcone.
**B**: In this framework, the noncommutative lightcone is constructed with no reference to any ambient noncommutative space, thus avoiding any causality issues arising from the fuzziness of the embedding.
**C**: In this way, five new noncommutative lightcones with quantum conformal group invariance will be presented and analyzed.
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<|MaskedSetence|> <|MaskedSetence|> Notwithstanding, in this paper, we successfully analyzed and reconstructed the surface using the proposed AP-HSOH approach. Figure 9A shows the reconstruction results using the AP-HSOH approach. <|MaskedSetence|> However, the texture of the reconstructed surface on the back of the bunny and the ears’ connectivity to the head is noisy.
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**A**: The AP-HSOH approach successfully reconstructed the global features of the mesh.
**B**:
4.2.2 Preservation of sharp features in the harmonic reconstruction
The Stanford bunny, with an open base, is a challenging surface that usually causes reconstruction issues with harmonic approaches.
**C**: In our DHA [27] work, we failed to correctly reconstruct the ears of the bunny that comprised a relatively small patch on the unit disk after parameterization.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Once we have a well defined trace map, we can translate the Galois extensions equivalence theorem from partial orthogonal groupoid actions [2, Theorem 5.3] to the case of global 00-E𝐸Eitalic_E-unitary, categorical at zero inverse semigroup actions. Following similar steps from Sections 5 and 6, with minimal adjustments, we achieve a Galois correspondence for finite inverse semigroups categorical at zero. This paper concludes by presenting this theorem. Thus, all the theory developed so far remains applicable in this setting.
Theorem 7.15..
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**A**: This relationship allows us to define an invariant trace map for global actions of inverse semigroups with zero, constructed similarly to Section 3.
**B**: Hence, by Theorem 7.12, also with partial orthogonal groupoid actions.
**C**: With arguments very similar to those used in the previous sections, we can see that there is a global-partial relation between inverse semigroup with zero actions and partial primitive inverse semigroup actions.
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<|MaskedSetence|> Scope of the paper and outline
In this paper, we would like to provide a simple, rigorous, and self-contained introduction to the Liouville–von Neumann equation generated by unbounded Hamiltonians on infinite-dimensional Hilbert spaces. <|MaskedSetence|> <|MaskedSetence|> By this, we mean that all theorems will be directly presented (and, wherever possible, proven) in the Hilbert space setting, instead of deriving them—as mostly done in the aforementioned references—as particular cases deducted from the general scenario of parametric semigroups of linear transformations on a Banach space. Apart from a concise and self-contained introduction to the Liouville–von Neumann equation in infinite dimensions, this paper also reports new results on the domains of the Liouville superoperator and its powers.
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**A**: Rather, it should be regarded as a self-contained guide to the mathematically-oriented working physicist who would like (and possibly need111Admittedly, this was the very situation in which the authors of the present manuscript found themselves (cf. [18]), and which eventually inspired them to write the present note.) to face the aforementioned mathematical subtleties rather than sweeping them under the carpet—and, at the same time, would like to embark in this journey, at least in a first moment, with the exact degree of mathematical generality that is strictly needed to describe the Liouville–von Neumann equation rigorously.
**B**: 1.2.
**C**: This paper should not be interpreted as a comprehensive review of the subject, neither at the mathematical nor physical level.
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<|MaskedSetence|> Two W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-categories 𝒞,𝒟𝒞𝒟\mathscr{C},\mathscr{D}script_C , script_D are called equivalent (as W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-categories) if there is a fully faithful ∗*∗-functor F:𝒞→𝒟:𝐹→𝒞𝒟F:\mathscr{C}\to\mathscr{D}italic_F : script_C → script_D that is essentially surjective. Such a functor is automatically normal. Equivalently, there exist normal ∗*∗-functors F:𝒞→𝒟:𝐹→𝒞𝒟F:\mathscr{C}\to\mathscr{D}italic_F : script_C → script_D and G:𝒟→𝒞:𝐺→𝒟𝒞G:\mathscr{D}\to\mathscr{C}italic_G : script_D → script_C such that F∘G≅id𝒟𝐹𝐺subscriptid𝒟F\circ G\cong\operatorname{\mathrm{id}}_{\mathscr{D}}italic_F ∘ italic_G ≅ roman_id start_POSTSUBSCRIPT script_D end_POSTSUBSCRIPT and G∘F≅id𝒞.𝐺𝐹subscriptid𝒞G\circ F\cong\operatorname{\mathrm{id}}_{\mathscr{C}}.italic_G ∘ italic_F ≅ roman_id start_POSTSUBSCRIPT script_C end_POSTSUBSCRIPT . <|MaskedSetence|> <|MaskedSetence|> If 𝒞𝒞\mathscr{C}script_C also obtains the structure of a (right) 𝒟𝒟\mathscr{D}script_D-module C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-category [DCY13], then we call 𝒞𝒞\mathscr{C}script_C a (right) 𝒟𝒟\mathscr{D}script_D-module W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-category..
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**A**: A leading role will be played by the W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-category CorrG(M,N)Corr𝐺𝑀𝑁\mathrm{Corr}{G}(M,N)roman_Corr italic_G ( italic_M , italic_N ) where G𝐺\mathbb{G}italic_G is a locally compact quantum group and M,N𝑀𝑁M,Nitalic_M , italic_N are G𝐺\mathbb{G}italic_G-W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras.
Let 𝒞𝒞\mathscr{C}script_C be a W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-category and let 𝒟𝒟\mathscr{D}script_D be a C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-tensor category [NT14].
**B**: All W∗superscript𝑊W^{*}italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-categories in this paper will be very concrete.
**C**:
is normal.
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(3.27)
which is easily seen to be injective when restricted to ℂ∗superscriptℂ\mathbb{C}^{*}blackboard_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT; only the pair of points f=0𝑓0f=0italic_f = 0 and f=∞𝑓f=\inftyitalic_f = ∞ have the same image, namely the singularity N∗subscript𝑁N_{*}italic_N start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT. These are also the only two points mapped to N∗subscript𝑁N_{*}italic_N start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT.
The image of the mapping (3.27) is an irreducible subset of the curve at infinity. <|MaskedSetence|> It follows that the curve at infinity is irreducible and that the mapping (3.27) is surjective. <|MaskedSetence|> <|MaskedSetence|> So, the curve at infinity is a singular, rational, irreducible, quartic curve with a unique singularity..
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**A**: If the curve at infinity were reducible, then any of its irreducible components would have degree less than four, which contradicts the fact that one of them necessarily contains the image of the mapping (3.27).
**B**: It further follows that N∗subscript𝑁N_{*}italic_N start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is the only singularity on the curve at infinity.
**C**: In particular, (3.27) defines a rational parametrisation of the curve at infinity of the Segre surface.
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For six collinear triples, we get a=4𝑎4a=4italic_a = 4. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> they are on the lines of the sides and diagonals of the square. The unique positioning of the remaining three points are (1/2,1/2,1),(1,0,0),(0,1,0)12121100010(1/2,1/2,1),(1,0,0),(0,1,0)( 1 / 2 , 1 / 2 , 1 ) , ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ), which gives a rational realization.
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**A**: Then, the remaining points have degree two.
**B**: The four degree-three points span no collinear triple since that would require seven collinear triples.
If the arrangement has a real realization, then let’s send the four degree-three points to the vertices of the unit square, (0,0,1),(0,1,1),(1,0,1),(1,1,1)001011101111(0,0,1),(0,1,1),(1,0,1),(1,1,1)( 0 , 0 , 1 ) , ( 0 , 1 , 1 ) , ( 1 , 0 , 1 ) , ( 1 , 1 , 1 ).
**C**: By Assumption 2, every collinear triple has two points out of the four, i.e.
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<|MaskedSetence|> This reduction depends on a scaling function, which plays a role similar to equivariant momentum maps in symplectic reduction. <|MaskedSetence|> In systems on cotangent bundles with a Riemannian metric on the configuration space, there is often a natural choice for a scaling function, something that is implicitly apparent in [3]. For instance, in the two-body problem, a power of the configuration variable is a natural choice, while in the n𝑛nitalic_n-body system, a power of the moment of inertia serves this purpose. <|MaskedSetence|>
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**A**: A key feature of their approach is its flexibility, allowing for various choices of scaling functions to obtain the reduced equations.
**B**: Scaling symmetries are ubiquitous in physical problems, appearing in fields such as fluid dynamics, field theory, and classical mechanics.
Despite the extensive body of work on this topic, the treatment of scaling symmetries and their corresponding reduction in classical mechanics has historically been approached on a case-by-case basis, focusing on specific systems (see [3] for some examples).
Bravetti, Jackman, and Sloan [3] have recently shown that a symplectic Hamiltonian system with scaling symmetry can be reduced to a contact Hamiltonian system with one fewer degree of freedom.
**C**: In this paper, we introduce a new distinguished scaling function, which we refer to as the conformal momentum map, extending the notion of the momentum map in symplectic reduction..
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The considered model has many applications. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> From our results, it follows that the complex cone provides a bridge from von Neumann Algebras to Monge–Ampère manifolds and to Frobenius manifolds.
0.0.3. .
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**A**: For instance by taking the real cone, we provide a Monge–Ampère domain and this space parametrises complex tori, forming the simplest example of Calabi–Yau manifolds.
**B**: 8.3].
**C**: This is reminiscent to the construction à la Strominger–Yau–Zaslow (SYZ) in [KoS01, Sec.
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Forcing is a technique invented by Cohen in [cohen1] to prove that the continuum hypothesis is independent of 𝖹𝖥𝖢𝖹𝖥𝖢\mathsf{ZFC}sansserif_ZFC. <|MaskedSetence|> The modern treatment of forcing is largely due to Scott, Solovay, Silver, and Rowbottom, as communicated by Shoenfield in [shoenfield]. <|MaskedSetence|> The usual forcing argument can be rewritten to occur entirely in the ground model with respect to a forcing notion that lives therein. Exactly because of this, we often forget the fact that our ground model is a CTM, or at least we eschew mentioning it. <|MaskedSetence|>
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**A**: We shall give a very brief and high-level introduction to forcing, following the layout found in Section 2.4 of [myself].
In a typical application of forcing, we start with a CTM, called the ground model.
**B**: It has since taken on a life of its own, becoming an indispensable tool in set theory, and even in other branches of logic.
**C**: This is also why our ground model is conventionally taken to be V𝑉Vitalic_V itself.
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<|MaskedSetence|> <|MaskedSetence|> θ1⋅θ2⋅θ3⋅η4⋅subscript𝜃1subscript𝜃2subscript𝜃3subscript𝜂4\theta_{1}\cdot\theta_{2}\cdot\theta_{3}\cdot\eta_{4}italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is nontrivial). There is no (ȷ,ι)∈2𝒥(S),F×I(S)Fitalic-ȷ𝜄superscript2𝒥𝑆𝐹𝐼superscript𝑆𝐹(\jmath,\iota)\in 2^{\mathcal{J}(S),F}\times I(S)^{F}( italic_ȷ , italic_ι ) ∈ 2 start_POSTSUPERSCRIPT caligraphic_J ( italic_S ) , italic_F end_POSTSUPERSCRIPT × italic_I ( italic_S ) start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT corresponding to (2), (3), (4), (5), (8), (9) in Remark 5.4. For pairs (ȷ,ι)∈2𝒥(T),F×I(T)Fitalic-ȷ𝜄superscript2𝒥𝑇𝐹𝐼superscript𝑇𝐹(\jmath,\iota)\in 2^{\mathcal{J}(T),F}\times I(T)^{F}( italic_ȷ , italic_ι ) ∈ 2 start_POSTSUPERSCRIPT caligraphic_J ( italic_T ) , italic_F end_POSTSUPERSCRIPT × italic_I ( italic_T ) start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT corresponding to (6) or (7) in Remark 5.4, the corresponding tγ¯,θ,ȷ,ι=0subscript𝑡¯𝛾𝜃italic-ȷ𝜄0t_{\bar{\gamma},\theta,\jmath,\iota}=0italic_t start_POSTSUBSCRIPT over¯ start_ARG italic_γ end_ARG , italic_θ , italic_ȷ , italic_ι end_POSTSUBSCRIPT = 0 (resp. tγ¯,θ,ȷ,ι=1subscript𝑡¯𝛾𝜃italic-ȷ𝜄1t_{\bar{\gamma},\theta,\jmath,\iota}=1italic_t start_POSTSUBSCRIPT over¯ start_ARG italic_γ end_ARG , italic_θ , italic_ȷ , italic_ι end_POSTSUBSCRIPT = 1) if either θ1subscript𝜃1\theta_{1}italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or θ2⋅θ3⋅η4⋅subscript𝜃2subscript𝜃3subscript𝜂4\theta_{2}\cdot\theta_{3}\cdot\eta_{4}italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is nontrivial (resp. <|MaskedSetence|> And all terms in Theorem 3.24 corresponding to (6) and (7) in Remark 5.4 sum up to 00 (resp. 1111) if either θ1subscript𝜃1\theta_{1}italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or θ2⋅θ3⋅η4⋅subscript𝜃2subscript𝜃3subscript𝜂4\theta_{2}\cdot\theta_{3}\cdot\eta_{4}italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is nontrivial (resp. both θ1subscript𝜃1\theta_{1}italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and θ2⋅θ3⋅η4⋅subscript𝜃2subscript𝜃3subscript𝜂4\theta_{2}\cdot\theta_{3}\cdot\eta_{4}italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT are trivial)..
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**A**: 00) in Theorem 3.24 by Lemma 5.1 if θ1⋅θ2⋅θ3⋅η4=1⋅subscript𝜃1subscript𝜃2subscript𝜃3subscript𝜂41\theta_{1}\cdot\theta_{2}\cdot\theta_{3}\cdot\eta_{4}=1italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 1 (resp.
**B**: both θ1subscript𝜃1\theta_{1}italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and θ2⋅θ3⋅η4⋅subscript𝜃2subscript𝜃3subscript𝜂4\theta_{2}\cdot\theta_{3}\cdot\eta_{4}italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT are trivial) by Lemma 5.1.
**C**:
Proof.
The pair (ȷ,ι)∈2𝒥(S),F×I(S)Fitalic-ȷ𝜄superscript2𝒥𝑆𝐹𝐼superscript𝑆𝐹(\jmath,\iota)\in 2^{\mathcal{J}(S),F}\times I(S)^{F}( italic_ȷ , italic_ι ) ∈ 2 start_POSTSUPERSCRIPT caligraphic_J ( italic_S ) , italic_F end_POSTSUPERSCRIPT × italic_I ( italic_S ) start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT corresponding to (1) in Remark 5.4 contributes to the sum −11-1- 1 (resp.
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<|MaskedSetence|> This procedure for renormalizing Einstein-AdS gravity by embedding it into CG was dubbed Conformal Renormalization.
In the computation of holographic EE, the Conformal Renormalization prescription provides a natural way to isolate the finite term in odd-dimensional CFTs dual to Einstein-AdS. This is because the EE can be computed directly from the gravitational on-shell action, using the generalized gravitational entropy formula Lewkowycz:2013nqa . <|MaskedSetence|> Following this idea, in Ref. Anastasiou:2022ljq the holographic EE functional for Einstein-AdS gravity in four bulk dimensions was derived starting from CG. This is achieved by applying the generalized gravitational entropy formula to the CG action, which is evaluated on the conically singular orbifold obtained via the replica trick Calabrese:2004eu and using the relations given in Ref. Fursaev:2013fta . <|MaskedSetence|> This functional was explicitly used to derive not only the renormalized holographic EE of Einstein-AdS, but also the so-called “reduced Hawking mass” and Willmore energy functionals which, in other contexts, are related to interesting quantities such as the entanglement susceptibility and to global bounds on information Fonda:2015nma .
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**A**: The finiteness of the latter gets inherited by the former.
**B**: Then, the resulting functional was identified with the integrand of the Graham-Witten anomaly Graham:1999pm , which corresponds to a pointwise conformally invariant functional defined on the codimension-two hypersurface localized at the conical singularity.
**C**: Furthermore, when the LPP CG action is evaluated in Einstein-AdS spacetimes, it becomes finite as it reduces to the renormalized Einstein-AdS action Anastasiou:2020mik .
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<|MaskedSetence|> We define a singularity as the case that g𝑔gitalic_g is the zero matrix. <|MaskedSetence|> Our definition is consistent with the notion that as the manifold becomes arbitrarily small, its metric becomes arbitrarily small in all elements. Observe the functional is 0 at singularity, but the time derivative is nonzero. Also, the condition defined in equation 26 holds at singularity. <|MaskedSetence|>
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**A**:
Remark.
**B**: Additionally, the eigenvalue condition does not hold at singularity, so Lemma 1 does not hold.
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**C**: This is notable as this definition is not used for all definitions of singularity.
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<|MaskedSetence|> Much of existing multi-agent BO literature studies batch BO, in which a central coordinator has access to each agent’s acquired information [21], [22]. It then computes the sampling decisions for all agents, and communicates these decisions to each agent. These decisions are disseminated in batches, allowing multiple agents to simultaneously sample points, parallelizing the optimization process [23], [24].
Centralized approaches are inapplicable in distributed cases, in which there is no centralized coordinator and each agent must possess a local instance of the optimization algorithm [25]. <|MaskedSetence|> Distributed networks are prevalent in real-world applications, such as in multi-robot source seeking and sensor networks [21], [26]. It may not be the case that all agents have access to all prior sampled points as in the batch setting - communication may be constrained, where some agents are only able to communicate with specific other agents [27]. These constraints may be due to limited communication capacity or computational capacity of the agents, or due to physical proximity constraints. Prior literature providing theoretical guarantees for distributed Bayesian optimization require fully connected communication graphs, even in asynchronous cases [22], [28], and thus are inapplicable in settings with constrained communication. <|MaskedSetence|>
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**A**:
We are interested in multi-agent BO, where multiple agents can sample the objective function at a single timestep.
**B**: In this work, we study the distributed setting with constrained communication, in which at each round, agents send their sampled points to their neighbors and receive points sampled by their neighbors..
**C**: Additionally, they often do not scale well, as they require a central coordinator to manage the processing of all agents’ data.
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For K𝐾Kitalic_K equal to a ball, the inequality (1.4) was stated, without a proof, by V. Klimov in his paper [Kl1]. Over the years, Klimov has authored several papers around this inequality and its applications, but,
as far as we know, he never published a full proof, although some hints are given in [Kl1]. <|MaskedSetence|> Thanks to the latter result, a proof of (1.4), when K𝐾Kitalic_K is a ball, was accomplished in [Ci4] via an approximation process by sequences of Steiner symmetrizations. <|MaskedSetence|> The argument of [VSch] again rests upon approximations, which exploit sequences of polarizations.
The inequality (1.4) is the point of departure in the approach to Sobolev type inequalities in anisotropic Orlicz-Sobolev spaces. <|MaskedSetence|> These embeddings have a role in the existence and regularity theory of solutions to anisotropic elliptic equations and variational problems – see e.g. [Al, ADF, ACCZ, Ba, Ci2, Ci4]..
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**A**: In [Kl3] he established a variant of (1.4), where symmetrization with respect to a convex set is replaced with Steiner symmetrization about a hyperplane.
**B**: In the paper [VSch], the inequality (1.4) is proved for a general convex body K𝐾Kitalic_K.
**C**: Early results in this direction are contained in [Kl1]; sharp embeddings are the subject of [Ci1, Ci3].
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A-primitive222Agievich in [6] introduced the term ”primitive” for this concept (in more general case of partitioning into affine subspaces), but the use of this term is controversial, since it rather characterizes a certain non-degeneracy of the partition. <|MaskedSetence|> <|MaskedSetence|> Note that in [2] the authors used the term ”tight” for the same concept, which also does not seem to us to be ”tight”., if each component is fixed in at least one of the subcubes of the partition.
The most well-known problem is the partitioning problem into small-dimensional subcubes. <|MaskedSetence|> In [1], the problems of partitioning a Boolean cube into subcubes are considered, mainly of small dimensions, which can also be different within a single partition.
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**A**: In addition, such non-degeneracy of the partition can be defined in different ways, and the word ”primitive” is generally overloaded in mathematics.
**B**: Thus, if all subcubes of a partition of a Boolean cube have dimension 1111, then these subcubes are edges, and the partitions are called perfect matchings, and the problem of their number is well known.
**C**: At the same time, giving another term also seems incorrect, therefore in [5] it was proposed to call such a partition Agievich-primitive or A-primitive.
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1.3. <|MaskedSetence|> <|MaskedSetence|> However, a generalization of statements such as “stochastically dominated by a sub-critical branching process” for complex measures appears very challenging. <|MaskedSetence|> Our key observation is that a factorization property, which arises in decomposing the Glauber dynamics, can be translated to the complex plane.
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**A**: In particular, the monotonicity of probability measures crucially relies on the non-negativity axiom.
**B**: Technical overview
A few challenges arise when trying to locate complex zeros through a percolation-type argument.
**C**: To extend the notion of probability measures to the complex plane, one can formally define complex normalized measures as ratios between partition functions.
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<|MaskedSetence|> From the perspective of state realization on a quantum processor, this is a natural starting point since these machines have a built-in choice of sites and local observables. <|MaskedSetence|> <|MaskedSetence|> For example, coarse spaces have (co)-homology and K-theories, and it would be interesting to work out if states have invariants valued in these groups, as in the case of disordered band insulators [EM19]. Another interesting issue is the question of actually computing aspects of the emergent coarse structure in practice, either theoretically or from a large but finite data set (See section 4.1). This problem has many similarities to the problems addressed in topological data anlysis [CM21] and in particular, persistance homology which we hope to explore in the future.
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**A**: Another aspect of our analysis we wish to emphasize is that the choice of the “sites” is taken as given for us.
**B**: In the abstract setting, one could wonder if we could derive a site structure itself from just a state on an algebra.
**C**: This seems unlikely, since any two states in the thermodynamic limit of a spin system are related by an automorphism, but it would be very interesting to see if some natural additional structure, such as a Hamiltonian, could give rise to a decomposition into sites
A natural follow-up question is whether we can find interesting order parameters that are valued in algebraic structure related to coarse spaces.
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Figure 7: RFDVC data savings, for both H.264 and H.265-based RFDVC variants, utilizing masks obtained using DS method and GT masks in noon, evening and wet conditions. <|MaskedSetence|> While static RF models are applied, the conditions within these models do not align precisely with actual conditions, resulting in lighting discrepancies, particularly under wet conditions scenarios. <|MaskedSetence|> <|MaskedSetence|> With GT masks, data savings vary from approximately 65% to 90% using the H.264 codec and from 38% to 63% using the H.265 codec..
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**A**: Higher resolutions provide greater data savings as larger black regions are more effectively encoded.
**B**: For RFDVC with the DS algorithm, the interquartile range of data savings spans 48% to 71% with the H.264 codec and 24% to 44% with the H.265 codec.
**C**: H.264-based RFDVC savings are measured relative to plain H.264 maskless frame compression, while H.265-based RFDVC savings are measured relative to plain H.265 codec.
Fig. 7 displays a box plot comparing the data savings achieved by RFDVC and VC when using either the DS algorithm or GT masks for transmitting data from camera sensors under noon, evening, and wet conditions.
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<|MaskedSetence|> We prove the Theorem 4.1.0.1 which states that a complex compact Kähler manifold with vanishing curvature is a Hermitian Frobenius manifold and we discuss some properties around this statement. In particular, by Kähler-Frobenius manifold we mean a Kähler manifold being a Hermitian Frobenius manifold. <|MaskedSetence|> <|MaskedSetence|> We consider some properties in Section 5.2 related to coverings by a torus and pre-Frobeniusity.
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**A**: In section 4, we consider Kähler-Frobenius manifolds.
**B**: 4.2 we prove that Chern’s conjecture holds for pre-Frobenius Kähler manifolds (Theorem 4.2.0.1).
In section 5, we investigate properties of Kähler-Frobenius manifolds, we particularly study the Kähler–Einstein case (section 5.1).
**C**: In Sec.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The Brown measure of the natural limit operator is the candidate limit law of empirical spectral distributions of the random matrices. There are counterexamples to this (see [[]Chapter 11, Exercise 15]SpeicherBook for a simple counterexample), but typically this is true (see [SniadyPaper] for a precise statement of this). In the case of the Circular Law and the Single Ring Theorem, the limit law is indeed the Brown measure of the limit operator.
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**A**: Of the random matrix models that are considered, typically there is already a natural limit operator of the random matrix model that comes from free probability.
**B**:
Before proving the proving the convergence, we must first determine what the limit of the random matrix model should even be.
**C**: To any operator in a tracial von Neumann algebra, there is an associated complex Borel probability measure, the Brown measure.
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<|MaskedSetence|> In this experiment, the data tensors are large. So, computing the stable solutions of the continuous neurodynamic is prohibitive because we usually need a large time-step for finding the stable solutions. To resolve this problem, we have used discrete neurodynamic with a population size of 5. <|MaskedSetence|> <|MaskedSetence|>
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**A**:
We used tensor rank R=10𝑅10R=10italic_R = 10 for all data tensors to represent them in the CPD format.
**B**: The relative errors of the algorithms for the COIL20, the ORL and the YALE datasets are reported in Figures 9, respectively.
From Figure 9, we see that the CNO-based algorithm can achieve lower objective values.
**C**: This clearly demonstrates the superiority of the CNO-based algorithms over the baseline methods.
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Selection 4
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<|MaskedSetence|> Notably, a uniform and geometric controlled filter stability is not required even though this would be sufficient. Therefore, due to the weak Feller property of controlled non-linear filters, we can apply the Q-learning algorithm to also belief-based models to arrive at near optimal control policies. <|MaskedSetence|> If the invariant measure under the exploration policy is the initial state, [67, Prop 2.1] implies that the time averages will converge as imposed in Assumption V.2. <|MaskedSetence|>
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**A**: Nonetheless, since positive Harris recurrence cannot typically be assumed for the filter process, the initial state may not be arbitrary.
**B**: A sufficient condition for unique ergodicity then is the following.
Assumption V.5
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**C**: Unique ergodicity of the dynamics follows from results in the literature, such as, [46, Theorem 2] and [67, Prop 2.1], which holds when the randomized control is memoryless under mild conditions on the process notably, the hidden variable is a uniquely ergodic Markov chain and the measurement structure satisfies filter stability in total variation in expectation (one can show that weak merging in expectation also suffices); we refer the reader to [49, Figure 1] for mild conditions leading to filter stability in this sense, which is related to stochastic observability in Definition II.7 (see also [49, Definition II.1]).
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CAB
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CAB
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CAB
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CAB
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Selection 3
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It is currently not known if the non-universality of sequences in ℛ(p)ℛ𝑝\mathscr{R}(p)script_R ( italic_p ) implies the non-universality of sequences in sets of positive measure, or vice versa. <|MaskedSetence|> <|MaskedSetence|> Goldberg, T. <|MaskedSetence|> MacMahon, and X. Wang showed that some of the results of subsection 5.1 can be used to construct sets of positive measure not containing any decreasing geometric progressions.
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**A**: However, in [BGK+23], A.
**B**: Burgin, S.
**C**: Keleti, C.
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ABC
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CAB
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ABC
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ABC
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Selection 3
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(1)
where c1,c2subscript𝑐1subscript𝑐2c_{1},c_{2}italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are positive constants (independent of m𝑚mitalic_m and k𝑘kitalic_k). While the upper bound holds for every m≥k+1𝑚𝑘1m\geq k+1italic_m ≥ italic_k + 1, the lower bound was only proved for m𝑚mitalic_m sufficiently large w.r.t. k𝑘kitalic_k.
The exponential gap between lower and upper bounds (1) is a well-known difficult problem in the area. <|MaskedSetence|> Starting with the probabilistic lower bound on R3(m,m)subscript𝑅3𝑚𝑚R_{3}(m,m)italic_R start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_m , italic_m ), the induction argument is based on Erdős-Hajnal stepping-up lemma that allows one to bound Rk+1(m,m)subscript𝑅𝑘1𝑚𝑚R_{k+1}(m,m)italic_R start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ( italic_m , italic_m ) in terms of Rk(m′,m′)subscript𝑅𝑘superscript𝑚′superscript𝑚′R_{k}(m^{\prime},m^{\prime})italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for some m′superscript𝑚′m^{\prime}italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT depending on m𝑚mitalic_m [5, 8]. <|MaskedSetence|> <|MaskedSetence|>
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**A**: The stepping-up lemma has been improved by Conlon, Fox, and Sudakov in [2].
**B**: The lower bound in (1) is proved by induction.
**C**: While the original version required the restriction m≥m0(k)𝑚subscript𝑚0𝑘m\geq m_{0}(k)italic_m ≥ italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_k ) with m0(k)subscript𝑚0𝑘m_{0}(k)italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_k ) exponential in k𝑘kitalic_k, the improved version from [2] can be applied as long as m≥52k+4𝑚52𝑘4m\geq\frac{5}{2}k+4italic_m ≥ divide start_ARG 5 end_ARG start_ARG 2 end_ARG italic_k + 4.111This bound is not explicitly stated in [2], but is a direct consequence of their Theorem 4 combined with the base case proved in [12]..
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BAC
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BAC
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BAC
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CBA
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Selection 2
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<|MaskedSetence|> <|MaskedSetence|> We have that |iU(κ)|≤2κsubscript𝑖𝑈𝜅superscript2𝜅|i_{U}(\kappa)|\leq 2^{\kappa}| italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) | ≤ 2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT since any ordinal α<iU(κ)𝛼subscript𝑖𝑈𝜅\alpha<i_{U}(\kappa)italic_α < italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) is represented by a function f:κ→κ:𝑓→𝜅𝜅f:\kappa\to\kappaitalic_f : italic_κ → italic_κ, of which there are 2κsuperscript2𝜅2^{\kappa}2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT. To see that iU(κ)≥2κsubscript𝑖𝑈𝜅superscript2𝜅i_{U}(\kappa)\geq 2^{\kappa}italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) ≥ 2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT, notice that κ𝜅\kappaitalic_κ is measurable in V𝑉Vitalic_V, so iU(κ)subscript𝑖𝑈𝜅i_{U}(\kappa)italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) is measurable in M𝑀Mitalic_M. In particular, iU(κ)subscript𝑖𝑈𝜅i_{U}(\kappa)italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) is a strong limit cardinal in M𝑀Mitalic_M. <|MaskedSetence|> Then μ𝜇\muitalic_μ is also less than (2κ)Msuperscriptsuperscript2𝜅𝑀(2^{\kappa})^{M}( 2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT. Hence any V𝑉Vitalic_V-cardinal less than 2κsuperscript2𝜅2^{\kappa}2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT is not strong limit in M𝑀Mitalic_M, so iU(κ)≥2κsubscript𝑖𝑈𝜅superscript2𝜅i_{U}(\kappa)\geq 2^{\kappa}italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) ≥ 2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT.
Now, let i=iU↾K:K→KM:𝑖subscript𝑖𝑈↾𝐾→𝐾superscript𝐾𝑀i=i_{U}\upharpoonright K:K\to K^{M}italic_i = italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ↾ italic_K : italic_K → italic_K start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT be the restriction of the embedding iUsubscript𝑖𝑈i_{U}italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT to K𝐾Kitalic_K. By Lemma 2.2(i), i𝑖iitalic_i is an iterated ultrapower of K𝐾Kitalic_K, so let ⟨Nν:ν≤θ⟩delimited-⟨⟩:subscript𝑁𝜈𝜈𝜃\langle N_{\nu}:\nu\leq\theta\rangle⟨ italic_N start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT : italic_ν ≤ italic_θ ⟩ be the iterates, so that N0=Ksubscript𝑁0𝐾N_{0}=Kitalic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_K and Nθ=KMsubscript𝑁𝜃superscript𝐾𝑀N_{\theta}=K^{M}italic_N start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = italic_K start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT. If ν<θ𝜈𝜃\nu<\thetaitalic_ν < italic_θ is a limit ordinal then there are ξν<νsubscript𝜉𝜈𝜈\xi_{\nu}<\nuitalic_ξ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT < italic_ν and Uν∈Nξνsubscript𝑈𝜈subscript𝑁subscript𝜉𝜈U_{\nu}\in N_{\xi_{\nu}}italic_U start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∈ italic_N start_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_POSTSUBSCRIPT such that Nν+1=Ult(Nν,iξν,ν(Uν))subscript𝑁𝜈1Ultsubscript𝑁𝜈subscript𝑖subscript𝜉𝜈𝜈subscript𝑈𝜈N_{\nu+1}=\mathrm{Ult}(N_{\nu},i_{\xi_{\nu},\nu}(U_{\nu}))italic_N start_POSTSUBSCRIPT italic_ν + 1 end_POSTSUBSCRIPT = roman_Ult ( italic_N start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , italic_ν end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ) ). (The reader should be careful and notice that Uν∉Nνsubscript𝑈𝜈subscript𝑁𝜈U_{\nu}\notin N_{\nu}italic_U start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∉ italic_N start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT, but iξν,ν(Uν)∈Nνsubscript𝑖subscript𝜉𝜈𝜈subscript𝑈𝜈subscript𝑁𝜈i_{\xi_{\nu},\nu}(U_{\nu})\in N_{\nu}italic_i start_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , italic_ν end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ) ∈ italic_N start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT, so that Nν+1subscript𝑁𝜈1N_{\nu+1}italic_N start_POSTSUBSCRIPT italic_ν + 1 end_POSTSUBSCRIPT is actually the ultrapower of Nνsubscript𝑁𝜈N_{\nu}italic_N start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT by a measure in Nνsubscript𝑁𝜈N_{\nu}italic_N start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT.) Moreover, for ν<θ𝜈𝜃\nu<\thetaitalic_ν < italic_θ we write κνsubscript𝜅𝜈\kappa_{\nu}italic_κ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT for the ordinal i0,ν(κ)subscript𝑖0𝜈𝜅i_{0,\nu}(\kappa)italic_i start_POSTSUBSCRIPT 0 , italic_ν end_POSTSUBSCRIPT ( italic_κ ). We first claim that this iteration is of length at least κ++superscript𝜅absent\kappa^{++}italic_κ start_POSTSUPERSCRIPT + + end_POSTSUPERSCRIPT.
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**A**: Let μ𝜇\muitalic_μ be any cardinal less than (2κ)Vsuperscriptsuperscript2𝜅𝑉(2^{\kappa})^{V}( 2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT.
**B**: Let U𝑈Uitalic_U be any measure on κ𝜅\kappaitalic_κ and let iU:V→M=Ult(V,U):subscript𝑖𝑈→𝑉𝑀Ult𝑉𝑈i_{U}:V\to M=\mathrm{Ult}(V,U)italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT : italic_V → italic_M = roman_Ult ( italic_V , italic_U ) be the usual ultrapower embedding.
**C**: First, we claim that |iU(κ)|=2κsubscript𝑖𝑈𝜅superscript2𝜅|i_{U}(\kappa)|=2^{\kappa}| italic_i start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_κ ) | = 2 start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT.
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BCA
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BCA
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BCA
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ACB
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Selection 3
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The case of extreme statistics over sequences of i.i.d. RVs are very well studied, and we have rules to identify the limiting distribution and the corresponding normalization constants for most of the practical cases. The case of i.n.i.d. RVs is relatively new in the communication literature, and only a few contributions discuss the limiting distribution of the extremes in such scenarios [26, 11]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: The following sub-sections discuss the key results for the cases of i.i.d.
**B**: and i.n.i.d.
**C**: RVs.
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ABC
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ABC
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BAC
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ABC
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Selection 1
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The literature on EV charging station location embraces various approaches to tackle the congestion challenge, from stochastic programming to economic assessments, exhibiting the multi-criteria nature of the problem. Hung and Michailidis [2022] addresses the location-routing problem for EV charging systems under stochastic conditions. <|MaskedSetence|> This methodology focuses on minimizing travel time through efficient routing and optimal location of charging stations. This approach is shown to be effective in urban settings adapting to various distances, vehicle speeds, and request distribution scenarios. To handle congestion, the mean waiting time at each charging station is assumed from the beginning.
In contrast, Tran et al. [2021] uses a bi-level nonlinear optimization framework to address the deployment of fast-charging stations. The upper level minimizes total system costs, including capital, travel, and environmental costs, while the lower level models traveler routing behaviors under stochastic demand. The integration of the Cross-Entropy Method and the Method of Successive Average provides a meta-heuristic solution, showcasing how increased EV range and strategic infrastructure can mitigate congestion and improve system performance. To analyze charging congestion at stations, the arrival rate at each charging station is defined as a function of the electric vehicles’ path flow.
Economic considerations are crucial in the deployment of charging infrastructure. Xiang et al. <|MaskedSetence|> To address the challenges of uncertain operational states and economic planning for charging stations from both power and transportation perspectives, Xiang et al. <|MaskedSetence|> It includes origin-destination analysis to represent EV behaviors, utilizes an optimization assignment model to determine equilibrium traffic flow, and employs an M/M/s Queuing System to determine the capacity of the charging stations..
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**A**: [2016] presents a novel planning framework for determining the location and size of charging stations by integrating the interactions between distribution and transportation networks.
**B**: They propose a data-driven method to solve the charging station location by formulating a partition-based clustering problem with size constraints.
**C**: [2016] introduces a novel planning framework.
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CAB
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BAC
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BAC
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Selection 1
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As shown in Fig. 4(a) for the (4,4) scenario, both the classical AE and the proposed QAE schemes outperform BPSK across all block fading channels. <|MaskedSetence|> <|MaskedSetence|> 4(b) for the (7,4) scenario, both the classical AE and QAE schemes exhibit nearly identical BLER performance to the Hamming code with soft decoding. <|MaskedSetence|> Finally, in Fig. 4(c) for the (8,8) scenario, the proposed QAE system performs similarly to the AE scheme, with both surpassing the conventional BPSK.
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**A**: In Fig.
**B**: In particular, the proposed QAE system shows a slight BLER improvement over the AE, with a more noticeable performance gap in Rayleigh and 3GPP channels.
**C**: This suggests that the proposed QAE scheme achieves comparable performance to the existing near-optimal channel coding baseline.
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CBA
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BAC
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Selection 3
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The upper bounds (i) follow from our analysis of λ↦φλmaps-to𝜆subscript𝜑𝜆\lambda\mapsto\varphi_{\lambda}italic_λ ↦ italic_φ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT and λ↦Πλmaps-to𝜆subscriptΠ𝜆\lambda\mapsto\Pi_{\lambda}italic_λ ↦ roman_Π start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT for complex λ∈Γγ𝜆subscriptΓ𝛾\lambda\in\Gamma_{\gamma}italic_λ ∈ roman_Γ start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT, promised above. To explain our derivation of the lower bound (ii), we now write Δλ≡Δ(λ,b)subscriptΔ𝜆Δ𝜆𝑏\Delta_{\lambda}\equiv\Delta(\lambda,b)roman_Δ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ≡ roman_Δ ( italic_λ , italic_b ) to denote the spectral gap arising from the Hamiltonian Hλsubscript𝐻𝜆H_{\lambda}italic_H start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT for a given λ𝜆\lambdaitalic_λ and b𝑏bitalic_b. <|MaskedSetence|> <|MaskedSetence|> Consequently, for |b|𝑏|b|| italic_b | small enough – say, |b|<b0(λ0)𝑏subscript𝑏0subscript𝜆0|b|<b_{0}(\lambda_{0})| italic_b | < italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) – we have Δ(λ0,b)>12δ1Δsubscript𝜆0𝑏12subscript𝛿1\Delta(\lambda_{0},b)>\frac{1}{2}\delta_{1}roman_Δ ( italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_b ) > divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. <|MaskedSetence|>
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**A**: First set b=0𝑏0b=0italic_b = 0.
Then, Δ(λ,0)Δ𝜆0\Delta(\lambda,0)roman_Δ ( italic_λ , 0 ) is the spectral gap for the non-magnetic Hamiltonian.
**B**: By classical results for the non-magnetic Hamiltonian (see, for example, (3, 2, 9)), for fixed real λ0>C1subscript𝜆0subscript𝐶1\lambda_{0}>C_{1}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (see [9]), we have Δ(λ0,0)>δ1Δsubscript𝜆00subscript𝛿1\Delta(\lambda_{0},0)>\delta_{1}roman_Δ ( italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 0 ) > italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT for some positive constant δ1subscript𝛿1\delta_{1}italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
**C**: This is the required lower bound (ii) for |Δ(λ,b)|2superscriptΔ𝜆𝑏2|\Delta(\lambda,b)|^{2}| roman_Δ ( italic_λ , italic_b ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for a single value of λ𝜆\lambdaitalic_λ and small b𝑏bitalic_b.
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ABC
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ABC
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CBA
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ABC
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Selection 4
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This means that barring the aforementioned case, there is always a pivot resolution that is shorter than the Taylor resolution. It is natural to ask which pivot resolution is the “smallest” for a fixed monomial ideal I𝐼Iitalic_I. For this purpose, we introduce a new notion.
Definition 3.6..
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**A**: In general, this may be the only pivot resolution.
**B**: In fact, by the preceding theorem, it is clear that this happens exactly when the Taylor resolution is minimal.
**C**: By definition, the Taylor resolution is a pivot resolution.
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CAB
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CAB
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ACB
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Selection 2
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<|MaskedSetence|> <|MaskedSetence|> The idea of the argument there is as follows. The induced metric from the spectral norm of the Gaussian random matrix is shown to be bounded by the induced metric of another Gaussian process plus a second term. <|MaskedSetence|> Thus the Slepian-Fernique inequality can be applied.
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**A**: 3.
**B**: Proof of the main result
The proof of Theorem 1.1 below is partly based on the argument in [14, Proof of Theorem 4.1] that uses the Slepian-Fernique inequality to bound the expected spectral norm of a Gaussian random matrix.
**C**: Using a spectral decomposition, one can bound this second term by the induced metric of some Gaussian process.
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ABC
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ABC
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ABC
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CAB
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Selection 3
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2 Experimental setup
An experimental investigation was carried out using a specially constructed setup, as illustrated in Fig. 1a.
The setup consists of a magnetic pendulum (1) with a neodymium magnet (2) attached to one end of a axis (3). An electric coil (4) is positioned on a fixed platform beneath the pendulum. The platform’s vertical position can be adjusted by a linear lift (5). The other end of the axis connects to a fixed base via an elastic rubber element (6). <|MaskedSetence|> The coil current signal follows a voltage signal from an NI USB-6341 card, controlled by LabView software. <|MaskedSetence|> The angular position of the pendulum is recorded by an optical incremental sensor (7). <|MaskedSetence|>
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**A**: The materials used for the setup, including the frame (9), are non-magnetic..
**B**: The electric coil is powered by a laboratory power supply.
**C**: During experiments, a positive current repels the magnet from the coil, while a negative current causes attraction.
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CBA
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BCA
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BCA
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BCA
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Selection 2
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The inhomogeneous nonlinear equation of Schrödinger type (INLS) models propagation of the beam in nonlinear optics and plasma physics. Indeed, stable high-power propagation in a plasma could be achieved by sending a precursor laser beam to create a channel with reduced electron density. This, in turn, lowers the nonlinearity within the channel [39, 25]. The inhomogeneous nonlinear equation of Hartree type (INLH) describes various physical phenomena. Specific instances of this model emerge in the mean-field limit of large systems of non-relativistic atoms and molecules, as well as in the propagation of electromagnetic waves in plasmas, among other applications [42, 6, 23].
The existence of energy subcritical solutions to (INLS) was first established in [24]. This result was later revisited in [29], where solutions in Strichartz spaces were studied under additional conditions for N=2,3𝑁23N=2,3italic_N = 2 , 3. The distinction between global existence and scattering versus finite-time blow-up below the ground state threshold was explored in [20, 21, 22] through the concentration-compactness argument introduced by Kenig and Merle [32]. This work was further extended in [7, 8] using the Dodson-Murphy method [19], and the assumption of spherical symmetry was relaxed in [9]. <|MaskedSetence|> See also [30, 37, 33, 12, 13, 4, 2] for the critical regime. The local/global well-posedness of (INLH) was first established in [1] using an adapted Gagliardo-Nirenberg type identity. Then, in [44], the scattering theory was studied using the approach from [19] under the spherically symmetric assumption. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Additional discussions on more general inhomogeneous terms can be found in [3, 15].
**B**: See also [34, 31] for the critical regime..
**C**: More recently, [48] proved the scattering theory for the non-radial case, inspired by [40].
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BCA
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ACB
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ACB
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ACB
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Selection 2
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This paper is organized as follows. Section 3 introduces a few uncertainty model types for the perturbed scenarios. <|MaskedSetence|> <|MaskedSetence|> The reliability analysis of the resulting designs is studied in Section 7. This analysis enables the designer to determine if the resulting design meets the reliability specifications imposed upon the system. <|MaskedSetence|> Finally, we state a few conclusions and outline some directions of future work.
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**A**: If this is not the case, he/she can either expand the training set 𝒟𝒟{\mathcal{D}}caligraphic_D or choose a better design architecture before redesigning.
**B**: This is followed by Section 4, where the strategies proposed are qualitatively explained using an engineering example.
**C**: Several risk-averse and risk-agnostic formulations to robust design are presented and exemplified in Sections 5 and 6.
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BCA
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BCA
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BCA
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ABC
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Selection 2
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<|MaskedSetence|> <|MaskedSetence|> Next, we know that the sum of the 𝐮isubscript𝐮𝑖\mathbf{u}_{i}bold_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPTs is 0; observe, then, since the α~isubscript~𝛼𝑖\widetilde{\alpha}_{i}over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPTs form a partition of n𝑛nitalic_n, that 𝐮α~k=−∑i=1k−1𝐮α~isubscript𝐮subscript~𝛼𝑘superscriptsubscript𝑖1𝑘1subscript𝐮subscript~𝛼𝑖\mathbf{u}_{\widetilde{\alpha}_{k}}=-\sum_{i=1}^{k-1}\mathbf{u}_{\widetilde{%
\alpha}_{i}}bold_u start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT bold_u start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT. It is clear that p=∑i=1k−1pα~i𝐮α~i𝑝superscriptsubscript𝑖1𝑘1subscript𝑝subscript~𝛼𝑖subscript𝐮subscript~𝛼𝑖p=\sum_{i=1}^{k-1}p_{\widetilde{\alpha}_{i}}\mathbf{u}_{\widetilde{\alpha}_{i}}italic_p = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT bold_u start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT. If each pα~i≥0subscript𝑝subscript~𝛼𝑖0p_{\widetilde{\alpha}_{i}}\geq 0italic_p start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ 0, we are done; suppose, then, that this is not the case. Let λk=maxi≤k−1{−pα~i}subscript𝜆𝑘subscript𝑖𝑘1subscript𝑝subscript~𝛼𝑖\lambda_{k}=\max_{i\leq k-1}\{-p_{\widetilde{\alpha}_{i}}\}italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_max start_POSTSUBSCRIPT italic_i ≤ italic_k - 1 end_POSTSUBSCRIPT { - italic_p start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT } and for i<k𝑖𝑘i<kitalic_i < italic_k, let λi=pα~i+λksubscript𝜆𝑖subscript𝑝subscript~𝛼𝑖subscript𝜆𝑘\lambda_{i}=p_{\widetilde{\alpha}_{i}}+\lambda_{k}italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Since −λk≤pα~isubscript𝜆𝑘subscript𝑝subscript~𝛼𝑖-\lambda_{k}\leq p_{\widetilde{\alpha}_{i}}- italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ italic_p start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT for each i𝑖iitalic_i, λi≥0subscript𝜆𝑖0\lambda_{i}\geq 0italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0. Moreover, points in 𝒫n−1subscript𝒫𝑛1\mathcal{P}_{n-1}caligraphic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT satisfy xj−xi≤1subscript𝑥𝑗subscript𝑥𝑖1x_{j}-x_{i}\leq 1italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 1 and xj−xi≥−1subscript𝑥𝑗subscript𝑥𝑖1x_{j}-x_{i}\geq-1italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ - 1, for i<j𝑖𝑗i<jitalic_i < italic_j, so λi≤1subscript𝜆𝑖1\lambda_{i}\leq 1italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 1. <|MaskedSetence|>
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**A**: Then
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**B**: Since p∈𝒫n−1𝑝subscript𝒫𝑛1p\in\mathcal{P}_{n-1}italic_p ∈ caligraphic_P start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT, we know that 0≤|pj|≤10subscript𝑝𝑗10\leq|p_{j}|\leq 10 ≤ | italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | ≤ 1 for each j𝑗jitalic_j.
**C**:
Now, we need to show that p=∑i=1kλi𝐮α~i𝑝superscriptsubscript𝑖1𝑘subscript𝜆𝑖subscript𝐮subscript~𝛼𝑖p=\sum_{i=1}^{k}\lambda_{i}\mathbf{u}_{\widetilde{\alpha}_{i}}italic_p = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_u start_POSTSUBSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT for 0≤λi≤10subscript𝜆𝑖10\leq\lambda_{i}\leq 10 ≤ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 1.
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CBA
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CAB
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CBA
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CBA
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Selection 4
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This section serves two purposes. <|MaskedSetence|> Second, we collect facts about toric varieties and stacks that are needed to establish our results. <|MaskedSetence|> <|MaskedSetence|> The subsections of this section are largely modular and can be read independently on an as-needed basis.
Section 2.1 recalls some fundamentals of the birational geometry of toric varieties and the secondary fan, following [CLSToricVarieties]*Chapters 14–15. In Section 2.2, we summarize how to compute cohomology on the toric stacks of interest. Section 2.3 defines the Bondal–Thomsen collection and collects some of its basic properties..
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**A**: For instance, standard references like [CLSToricVarieties] do not cover toric stacks, and so we provide independent proofs of the necessary cohomological vanishing theorems.
**B**: In many cases, we need a version of a statement that is not clearly stated elsewhere in the literature.
**C**: First, we establish our notation.
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BAC
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Selection 4
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Important subtleties of the BP model have been discussed in the late reference Ji:2019phv where, besides being directly related to Pauli-Villars regularization, a thorough analysis of static density charge configurations has been worked out, and important natural higher-order gauge-fixing functions have been proposed for both covariant and light-front approaches. <|MaskedSetence|> BP model’s main feature amounts to describing a higher-order derivative generalized electrodynamics, still linear in the fields, providing mass for the gauge field in a gauge invariant way in terms of a nontrivial mass parameter a𝑎aitalic_a. A couple of years ago, in Bonin:2022tmg , Bonin and Pimentel showed how BP’s generalized electrodynamics might arise from an Abelian Higgs model containing a nonminimally coupled scalar field for a certain regime, thus suggesting that the Podolsky mass parameter could emerge from a specific Higgs mechanism. <|MaskedSetence|> <|MaskedSetence|> The role played by BP electrodynamics in a possibly Lorentz symmetry violating scenario has been investigated in reference Ferreira:2024qkd , where the authors have also considered contributions from a Carroll-Field-Jackiw term.
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**A**: The authors of deMelo:2024dxu have found differences between Maxwell and BP electrodynamics regarding phase shift predictions, proposing experimental bounds to a possible physical mass for the photon.
**B**: Also, the Aharonov-Bohm effect has been discussed in the context of the BP model.
**C**: Those specific gauge-fixing functions have been shown to lead to simpler, neater expressions for the field propagators, a key fact for perturbative calculations.
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CBA
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CBA
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CBA
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ACB
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Selection 3
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Parts (ii) and (iii) are relatively simple to prove, for which we refer to [book:sato, p. 40] and [book:sato, p. 41] (or [book:applebaum, p. <|MaskedSetence|> <|MaskedSetence|> In short, there is a 1-1 correspondence between infinitely divisible distributions and Lévy processes; see [book:sato, Thm. <|MaskedSetence|> I.31]; the natural filtration of any Lévy process combined with all the ℙℙ\mathbb{P}blackboard_P-null sets is known to be right continuous.
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**A**: 7.10].
The “usual conditions” part of (iii) follows from [book:protter, Thm.
**B**: 30] for instance) respectively.
**C**: Observe the theorem above is often stated in terms of infinitely divisible distributions.
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BCA
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BCA
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BCA
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BCA
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Selection 1
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A particular characteristic of our approach is that our result has an intermediate epistemic status between a theorem and a heuristic, albeit in our opinion much closer to the former. The situation, more in general, is closely related to the question of how much power randomness gives to computation, for which Avi Widgerson recently received a Turing Award [9]. <|MaskedSetence|> However, in order to practically obtain knowledge of such deterministic facts, we leverage the computational benefit of randomness, which allows us (at least in current practice), to determine facts that otherwise would be out of reach. The cost, however, is the possibility of error in the associated randomized algorithms, which forbids us from claiming to have definite proofs of the facts of interest. As usual, we can get such probability of error to be as small as we deem necessary for convincing ourselves, at a modest computational price, while keeping the curse of never reaching 100%percent100100\%100 % confidence.
An interesting counterpoint is to discuss whether traditional proofs equal certainty, as it is not evident that when reading traditional proofs we can reliably reach 100%percent100100\%100 % of confidence either. We might claim that for simple proofs like the irrationality of 22\sqrt{2}square-root start_ARG 2 end_ARG, the elementary proof of Theorem 2 in this paper, or even the Central Limit Theorem. <|MaskedSetence|> <|MaskedSetence|> For a more general discussion of the impact of computation in modern mathematics, and how our understanding of “proofs” can be affected by computation, we refer the interested reader to the work of Avigad [1, 2].
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**A**: However, proofs that span dozens or even hundreds of pages, covering a multitude of cases, and including non-trivial calculations, are much more delicate from a trust perspective.
**B**: For instance, the proof of Kepler’s conjecture by Thomas Hales took years before reviewers, from the prestigious Annals of Mathematics, accepted the paper while saying they were only “99% sure of its correctness” [10, 16].
**C**: In the context of mathematical results, such a question may be phrased as follows:
Are there properties of finite mathematical objects that can only be certified efficiently to a high degree of confidence by probabilistic algorithms, but that we never can be certain of through a short proof?
For instance, consider primality testing; whether a given number is prime or not is a fully deterministic fact, in the same way as the average distance of the Rubik’s cube graph is.
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BCA
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CAB
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CAB
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CAB
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Selection 4
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The bound on both numbers can be tight. <|MaskedSetence|> In particular, the list chromatic number of the Fano plane is also three.
A bound for the paintability number of a hypergraph can be obtained from the Alon–Tarsi number as well. In the case of graphs the respective relation was proved for the first time by Schauz [19], whose proof was presented in a simplified form by Zhu and Balakrishnan [27, Section 3.9]. Recently, a different approach was used by Grytczuk, Jendroľ, Zając in [8]. <|MaskedSetence|> <|MaskedSetence|>
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**A**: For instance, the Fano plane has edge density one and chromatic number 3333, while the Alon–Tarsi number of the hypergraph polynomial defined above is upper bounded by 3333.
**B**: The exposition below follows very closely that of [8], with the necessary adjustments..
**C**: To describe analogous connection for hypergraphs we shall now describe in some detail the concept of hypergraph paintability.
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ACB
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ACB
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CBA
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ACB
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Selection 1
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<|MaskedSetence|> The function f(x)𝑓𝑥f(x)italic_f ( italic_x ) of the system (1.1) is linear and locally independent of the interval number in each interval. We assume that the function f(x)=Λx𝑓𝑥Λ𝑥f(x)=\Lambda xitalic_f ( italic_x ) = roman_Λ italic_x in the interval
I0=[−12,12)subscript𝐼01212I_{0}=[-\frac{1}{2},\frac{1}{2})italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = [ - divide start_ARG 1 end_ARG start_ARG 2 end_ARG , divide start_ARG 1 end_ARG start_ARG 2 end_ARG ), where Λ=2m+1Λ2𝑚1\Lambda=2m+1roman_Λ = 2 italic_m + 1 is an odd number, for simplicity. <|MaskedSetence|> <|MaskedSetence|> Such a dynamical system leads to the evolution of the measure as follows. If the measure has a constant density ρksubscript𝜌𝑘\rho_{k}italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT on each interval Iksubscript𝐼𝑘I_{k}italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT at the initial moment, and the measure of the entire axis ∑k=−∞+∞ρk=1superscriptsubscript𝑘subscript𝜌𝑘1\displaystyle\sum_{k=-\infty}^{+\infty}\,\rho_{k}=1∑ start_POSTSUBSCRIPT italic_k = - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + ∞ end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1, then the Perron–Frobenius operator ℱℱ\mathcal{F}caligraphic_F preserves the structure of the constancy of the densities in all intervals Iksubscript𝐼𝑘I_{k}italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. However, their density changes in such a way that the measure μk=ρksubscript𝜇𝑘subscript𝜌𝑘\mu_{k}=\rho_{k}italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of each interval Iksubscript𝐼𝑘I_{k}italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT transforms into measures of 2m+12𝑚12m+12 italic_m + 1 intervals Ik−m,…,I0,I1,…,Ik+msubscript𝐼𝑘𝑚…subscript𝐼0subscript𝐼1…subscript𝐼𝑘𝑚I_{k-m},...,I_{0},I_{1},...,I_{k+m}italic_I start_POSTSUBSCRIPT italic_k - italic_m end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_k + italic_m end_POSTSUBSCRIPT
with densities 1Λρk1Λsubscript𝜌𝑘\displaystyle\frac{1}{\Lambda}\rho_{k}divide start_ARG 1 end_ARG start_ARG roman_Λ end_ARG italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, and the initial measure transforms into the sum of specified changes of the measures from each interval.
Let ρ0subscript𝜌0\rho_{0}italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be the piecewise-constant density of the initial measure in intervals, and ρ1=ℱρ0subscript𝜌1ℱsubscript𝜌0\rho_{1}=\mathcal{F}\rho_{0}italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = caligraphic_F italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be the density of the measure under the action of the Perron–Frobenius operator ℱℱ\mathcal{F}caligraphic_F, then using convolution, the explicit form of the density is
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**A**: In each interval Iksubscript𝐼𝑘I_{k}italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT the function has the form f(k+x)=k+Λx𝑓𝑘𝑥𝑘Λ𝑥f(k+x)=k+\Lambda xitalic_f ( italic_k + italic_x ) = italic_k + roman_Λ italic_x, where |x|<12𝑥12|x|<\frac{1}{2}| italic_x | < divide start_ARG 1 end_ARG start_ARG 2 end_ARG.
**B**: That is, the function f(x)𝑓𝑥f(x)italic_f ( italic_x ) is 1111–periodic with lift.
**C**: Let the entire axis ℝ1superscriptℝ1\mathbb{R}^{1}blackboard_R start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT be splitted into disjoint, semi–closed intervals Ik=[k−12,k+12)subscript𝐼𝑘𝑘12𝑘12I_{k}=[k-\frac{1}{2},k+\frac{1}{2})italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ italic_k - divide start_ARG 1 end_ARG start_ARG 2 end_ARG , italic_k + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ), where k∈Z𝑘𝑍k\in Zitalic_k ∈ italic_Z are integers.
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CAB
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CAB
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CAB
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ABC
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Selection 3
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The proposed model is broadly applicable to various domains, including social interactions, biological systems (e.g., neural or protein interactions), and technological networks (e.g., the spread of computer viruses or resilience of infrastructure systems). <|MaskedSetence|> <|MaskedSetence|> The model not only maintains a strong fit to empirical data but also reveals hidden structural features of the contact network underlying the disease’s spread. <|MaskedSetence|> This stochastic SIR framework thus provides a versatile tool for modeling infectious diseases and other dynamic processes beyond the scope of traditional SIR models..
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**A**: By transforming the SIR model using dynamical survival analysis within the edge-based configuration network framework, the resulting system of equations captures the intricate dynamics of network-based interactions.
**B**: Despite the complexity of these interactions, the equations remain mathematically tractable, often enabling precise predictions of disease trends (see, for instance, the discussions of related DSA-based approaches given in [8, 24]).
The utility of the Poisson SIR network model is demonstrated through secondary analysis of the data from 2018–2020 Ebola outbreak in the Democratic Republic of the Congo.
**C**: Identifying such networks is crucial for effectively targeting at-risk populations—such as through vaccination campaigns—to prevent further transmission.
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ABC
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ABC
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ABC
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ABC
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Selection 3
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Figure 11: Zonal prices in Italy on October 24th, 2024, at 8pm. <|MaskedSetence|> <|MaskedSetence|>
Figure 12: Zonal prices in Central-northern Italy on October 24th, 2024, at 8pm. This graph can be obtained through the GME website by selecting the button MAP and the zone CNORD. In red, we show for comparison the total quantity purchased in CNORD and the flows at Nash equilibrium. <|MaskedSetence|>
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**A**: This map can be obtained through the GME website by selecting the button MAP.
**B**: .
**C**: In red, we show for comparison the predicted prices at Nash equilibrium.
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ACB
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ACB
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ACB
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ACB
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Selection 1
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<|MaskedSetence|> for any i∈𝐒N𝑖subscript𝐒𝑁i\in\mathbf{S}_{N}italic_i ∈ bold_S start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and any t∈ℝ𝑡ℝt\in\mathbb{R}italic_t ∈ blackboard_R (i.e. <|MaskedSetence|> in A(x¯)𝐴¯𝑥A\left(\overline{x}\right)italic_A ( over¯ start_ARG italic_x end_ARG )). <|MaskedSetence|> in ℝN\ℝ+N\superscriptℝ𝑁superscriptsubscriptℝ𝑁\mathbb{R}^{N}\backslash\mathbb{R}_{+}^{N}blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT \ blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT.
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**A**:
that is u(x¯±tei)=0𝑢plus-or-minus¯𝑥𝑡subscript𝑒𝑖0u\left(\overline{x}\pm te_{i}\right)=0italic_u ( over¯ start_ARG italic_x end_ARG ± italic_t italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 0 holds a.e.
**B**: Since u∈C2(ℝ+N)𝑢superscript𝐶2superscriptsubscriptℝ𝑁u\in C^{2}\left(\mathbb{R}_{+}^{N}\right)italic_u ∈ italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ), the arbitrariness of x¯¯𝑥\overline{x}over¯ start_ARG italic_x end_ARG leads to the conclusion that u≡0𝑢0u\equiv 0italic_u ≡ 0 in ℝ+Nsubscriptsuperscriptℝ𝑁\mathbb{R}^{N}_{+}blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT and u=0𝑢0u=0italic_u = 0 a.e.
**C**: u=0𝑢0u=0italic_u = 0 a.e.
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ACB
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BCA
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ACB
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ACB
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Selection 1
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*
MN-BaB with 600s timeout threshold for all models. “-” indicates that we could not run a model due to unsupported model structure or other errors. <|MaskedSetence|> <|MaskedSetence|> However, we can still achieve better verified accuracy than all other baselines. <|MaskedSetence|>
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**A**: We run β𝛽\betaitalic_β-CROWN, GCP-CROWN with MIP cuts and BICCOS with a shorter 200s timeout for all models.
**B**: Other results are reported from [61]..
**C**: The increased timeout for MN-BaB may increase the percentage of verified instances.
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BAC
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ACB
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ACB
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ACB
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Selection 3
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Moreover, angular positions of targets may not be known precisely due to factors such as motion. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> To address this, we propose establishing strict hierarchies through careful weight design. Our approach consistently prioritizes communications regardless of parameter settings, ensuring its full optimization before addressing the sensing requirements, thus leading to a strictly tiered resource allocation framework.
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**A**: Thus, accounting for this aspect in the resource allocation design can help mitigate potential performance degradation in sensing, a crucial aspect explored in only a few studies, such as [11].
In ISAC systems, one functionality may be more critical than the other [12].
**B**: Particularly, this view aligns with industry’s pragmatic stance of preserving communication performance, while enabling sensing opportunistically when feasible.
**C**: While tradeoff functions can balance the importance of functionalities by using weights [13], changes in parameter settings (e.g., number of users, transmit power) can skew objective function values, rendering preset weights ineffective and shifting the intended operating point.
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ABC
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ABC
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BCA
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ABC
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Selection 2
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<|MaskedSetence|> <|MaskedSetence|> In Section 4, we detail the routine to verify that a flow does not have perfect fits (Algorithm 4.1). This includes a review of the Agol-Guéritaud construction (4.2) and a characterisation of the absence of perfect fits by veering triangulations (4.4). The main algorithm (Algorithm 5.1) of Figure 1 and that of Figure 1 are presented in Section 5. <|MaskedSetence|>
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**A**: We conclude with a list of questions in Section 6..
**B**:
Outline.
In Section 2, we give background on flows (Section 2.1), perfect fits (Sections 2.2 and 2.3), and veering triangulations (Section 2.4).
**C**: The routine for verifying that a flow has perfect fits (Algorithm 3.1) is given in Section 3.
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BCA
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BCA
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BCA
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BCA
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Selection 4
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I-B Restless Multi-armed Bandit Problem
In this paper, we will formulate our channel allocation problem as a restless multi-armed bandit process (RMAB). RMAB is a generalization of the classical multi-armed bandit problem (MAB). MAB is a well-known mathematical model that serves as a foundational framework for dynamic resource allocation problems and has been widely used in multiple fields [13, 14, 15]. <|MaskedSetence|> The chosen arm undergoes a state transition according to a Markovian rule while other arms do not change their states. <|MaskedSetence|> <|MaskedSetence|> The inception of the general MAB concept can be traced back to its original exposition in 1933 [16], and despite subsequent research endeavors, it remains partially unresolved. Gittins made noteworthy advancements by addressing the curse of dimensionality of the classical MAB problem, effectively reducing the complexity from an N𝑁Nitalic_N-dimensional problem to N𝑁Nitalic_N individual 1111-dimensional problems [17, 18].
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**A**: The player’s goal is to maximize their cumulative discounted reward over an infinite time interval according to a specific arm selection policy.
**B**: In the classical formulation, a player is challenged by the task of selecting a single arm out of N𝑁Nitalic_N options, subsequently receiving a random reward dependent on the state of the arm.
**C**: At all time, the states of all arms are perfectly observable.
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BCA
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BCA
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BCA
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BCA
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Selection 3
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