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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 fi... | **A**:
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 o... | CBA | ABC | ABC | ABC | Selection 2 |
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 exp... | **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 wi... | BAC | ABC | ABC | ABC | Selection 2 |
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, o... | **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
.
**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.. | **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 ... | **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_POSTSUBSCRI... | ABC | BAC | BAC | BAC | Selection 4 |
<|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 ... | **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 ) pos... | CBA | CBA | BAC | CBA | Selection 4 |
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 glo... | **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.
.
*... | CAB | CBA | CBA | CBA | Selection 3 |
<|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... | **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 swimme... | CBA | CBA | CBA | ACB | Selection 3 |
<|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 r... | **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... | ACB | ACB | ACB | CAB | Selection 1 |
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_POST... | **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_POSTS... | ABC | BAC | ABC | ABC | Selection 1 |
<|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... | **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).
2.
**B**: The rest of the paper is organized as follows.
**C**: In the next section, we discuss the functional setting of the probl... | BCA | ACB | BCA | BCA | Selection 1 |
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|>... | **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 ... | BCA | BCA | BCA | BAC | Selection 3 |
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... | **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 star... | BAC | BAC | BAC | BAC | Selection 3 |
<|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 incur... | **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 b... | CAB | ACB | CAB | CAB | Selection 1 |
<|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|> | **A**: (2014) and Qiao and Polonik (2016).
.
**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_POST... | **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 c... | **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 composit... | BCA | BCA | BCA | BCA | Selection 4 |
<|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 o... | **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... | ACB | ACB | ACB | ABC | Selection 2 |
<|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. <|Mas... | **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 ... | ACB | CBA | CBA | CBA | Selection 4 |
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 prot... | **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... | BAC | BCA | BAC | BAC | Selection 3 |
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.
. | **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, th... | ACB | CAB | CAB | CAB | Selection 4 |
<|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. ... | **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 tw... | **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 o... | ACB | ABC | ABC | ABC | Selection 4 |
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 althou... | **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 ... | ACB | ABC | ABC | ABC | Selection 4 |
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 equi... | **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 a... | ACB | BCA | ACB | ACB | Selection 3 |
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|... | **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 th... | CAB | CAB | CAB | ACB | Selection 2 |
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 i... | **A**: Final discussions and future work directions are then reported in section 5.
.
**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 fun... | CBA | CBA | ABC | CBA | Selection 2 |
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 ... | **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 practi... | BAC | ABC | BAC | BAC | Selection 4 |
<|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... | **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.
.
**B**: Automating statistical inference for these stochastic systems requires knowledge of their distribution; that i... | ABC | BCA | BCA | BCA | Selection 2 |
Conventions. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We are thankful to K. Arun Kumar for helpful discussions during the preparation of this manuscript.. | **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... | BCA | ABC | BCA | BCA | Selection 4 |
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→subscript𝒟subscript𝑏1s... | **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_POSTSUBSCRI... | CBA | CAB | CAB | CAB | Selection 4 |
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 cu... | **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 effect... | CAB | CAB | BCA | CAB | Selection 4 |
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 ... | **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 Sc... | CAB | CAB | BAC | CAB | Selection 1 |
<|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 it... | **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 ... | BCA | BCA | BCA | BAC | Selection 3 |
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 ex... | **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 actio... | CAB | BAC | BAC | BAC | Selection 2 |
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|> <|Mask... | **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 dynamic... | CBA | CBA | ACB | CBA | Selection 2 |
<|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 gradi... | **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**: O... | ACB | ABC | ACB | ACB | Selection 4 |
<|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 , ∞ ... | **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_P... | ACB | ACB | ACB | ACB | Selection 3 |
{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 + ... | **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\pri... | CBA | CBA | CBA | ABC | Selection 2 |
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 ... | **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 di... | CBA | ACB | ACB | ACB | Selection 3 |
<|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 subcarri... | **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 late... | BAC | BAC | BAC | BAC | Selection 4 |
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, ... | **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 no... | CBA | CBA | CBA | BAC | Selection 2 |
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 3... | **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 ... | CAB | CAB | CAB | CBA | Selection 2 |
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... | **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 (... | BCA | CBA | BCA | BCA | Selection 4 |
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... | **A**: Consequently,
.
**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_... | CAB | BCA | BCA | BCA | Selection 4 |
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 ... | **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 follo... | ABC | ABC | ABC | ABC | Selection 4 |
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_{\m... | **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\ran... | ABC | ABC | BCA | ABC | Selection 4 |
<|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... | **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, ... | CAB | CAB | CAB | ABC | Selection 2 |
<|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... | **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 int... | ABC | CAB | ABC | ABC | Selection 1 |
<|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] consid... | **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 c... | ACB | CAB | CAB | CAB | Selection 4 |
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 ... | **A**: By Fact 15, the discrete group ΓΓ\Gammaroman_Γ is non-amenable.
∎
.
**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 schem... | CBA | CAB | CBA | CBA | Selection 3 |
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 expectatio... | **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 iss... | ABC | ABC | ABC | BCA | Selection 3 |
<|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... | **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... | CAB | BCA | BCA | BCA | Selection 3 |
<|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 sim... | **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 se... | CBA | CBA | CBA | CBA | Selection 4 |
<|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... | **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 ... | BCA | BCA | BCA | BCA | Selection 3 |
<|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:�... | **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\ma... | ACB | CBA | CBA | CBA | Selection 2 |
(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 ∗ ... | **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... | ACB | ACB | ACB | ACB | Selection 3 |
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... | **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)00... | CAB | BAC | BAC | BAC | Selection 4 |
<|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... | **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 w... | BAC | BAC | ABC | BAC | Selection 1 |
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. . | **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.
| CAB | ACB | ACB | ACB | Selection 3 |
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... | **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... | BAC | BAC | BAC | BCA | Selection 2 |
<|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_P... | **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... | CAB | CAB | ABC | CAB | Selection 1 |
<|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 i... | **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 s... | CAB | BCA | CAB | CAB | Selection 4 |
<|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 deri... | **A**:
Remark.
**B**: Additionally, the eigenvalue condition does not hold at singularity, so Lemma 1 does not hold.
.
**C**: This is notable as this definition is not used for all definitions of singularity.
| ACB | ACB | ACB | ABC | Selection 2 |
<|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, ... | **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 thei... | ACB | ACB | ACB | ACB | Selection 3 |
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]. <|MaskedS... | **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 [K... | ABC | ABC | ABC | BAC | Selection 1 |
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 auth... | **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, ... | ACB | ACB | ACB | BCA | Selection 2 |
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 dynami... | **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 form... | BCA | BCA | BAC | BCA | Selection 2 |
<|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 interestin... | **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 limi... | ABC | CBA | ABC | ABC | Selection 3 |
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 l... | **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... | BAC | CBA | CBA | CBA | Selection 3 |
<|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.... | **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.
| ACB | ACB | ABC | ACB | Selection 4 |
<|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 ... | **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 ... | ACB | BAC | BAC | BAC | Selection 4 |
<|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. <|Mask... | **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 objec... | ABC | ABC | ABC | ABC | Selection 4 |
<|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.... | **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
.
**C**: Unique ergodicity of the dynamics follows from results in the literature... | CAB | CAB | CAB | CAB | Selection 3 |
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... | **A**: However, in [BGK+23], A.
**B**: Burgin, S.
**C**: Keleti, C.
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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... | **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)... | BAC | BAC | BAC | CBA | Selection 2 |
<|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... | **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→𝑉𝑀Ult𝑉𝑈... | BCA | BCA | BCA | ACB | Selection 3 |
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 contributi... | **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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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 stocha... | **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 prob... | BAC | CAB | BAC | BAC | Selection 1 |
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 s... | **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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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∈Γγ𝜆subscr... | **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 star... | ABC | ABC | CBA | ABC | Selection 4 |
<|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... | **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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<|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.... | **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 th... | ABC | ABC | ABC | CAB | Selection 3 |
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. T... | **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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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 nonlin... | **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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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 reliabi... | **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 a... | BCA | BCA | BCA | ABC | Selection 2 |
<|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 parti... | **A**: Then
.
**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 ... | CBA | CAB | CBA | CBA | Selection 4 |
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 s... | **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... | CBA | BAC | CBA | CBA | Selection 4 |
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 propose... | **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 specifi... | CBA | CBA | CBA | ACB | Selection 3 |
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... | **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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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 Widg... | **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 Ma... | BCA | CAB | CAB | CAB | Selection 4 |
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 S... | **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 fo... | ACB | ACB | CBA | ACB | Selection 1 |
<|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}... | **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... | CAB | CAB | CAB | ABC | Selection 3 |
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 main... | **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 tractab... | ABC | ABC | ABC | ABC | Selection 3 |
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 th... | **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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<|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\... | **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}_... | ACB | BCA | ACB | ACB | Selection 1 |
*
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|> | **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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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 paramete... | **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 w... | ABC | ABC | BCA | ABC | Selection 2 |
<|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... | **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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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 dynami... | **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 d... | BCA | BCA | BCA | BCA | Selection 3 |
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