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**A**: This research was supported by the grant ”Agreement on the provision of a grant in the form of subsidies from the federal budget for implementation of state support for the creation and development of world-class scientific centers, including international world-class centers and scientific centers, carrying out research and development on the priorities of scientific technological development No. 075-15-2019-1619” dated 8th November 2019**B**: We are grateful to Olga Postnova, Nicolai Reshetikhin, Pavel Nikitin, Fedor Petrov for fruitful discussions**C**: The author expresses gratitude for the support by the Xing Hua Scholarship program of Tsinghua University.
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**A**:
In particular, several recent research works present the benefit of utilizing the polarization domain in recently proposed communication schemes including, but not limited to, MIMO spatial multiplexing [1]; spatial modulation (SM) [2, 3, 4]; non-orthogonal multiple access (NOMA) [5]; and beamforming [6, 7]**B**: Hybrid beamforming can adopt dual polarization and the associated codebook design to improve the system performance [7, 25]. Although polarization multiplexing without spatial diversity is promising [18], polarization diversity can be combined with spatial diversity to further improve the performance of wireless communication systems [15, 16, 17, 1].**C**: It is validated that the deployment of dual-polarized antennas can not only increase channel capacity [19, 14, 21, 24]; but also improve SER [6, 15, 17, 16, 18]
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**A**: They generalize MZV and MSZV in a natural way and also build a bridge to the theory of partitions**B**: Related works can be found in [Ba18], [BY18], [BC20], and [BKSYY23]. One of the referees of our paper pointed out that it would be interesting to see if the RHS of (5.18) can be expressible in terms of SMZV. The author hopes to consider it in the near future.
**C**: Recently Schur multiple zeta values (SMZV for short) were introduced and investigated in [NPY18] by utilizing the semi-standard Young tableaux
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**A**: As shown by Example 8.5, the non-toric generalization has to involve all valuations instead of just one.Lastly, let us mention that our generalization of Boucksom–Chen theorem has important consequences in Archimedean pluripotential theory as well**B**: It is of interest to generalize D to the non-toric setting as well**C**: When applied to generalized deformation to the normal cone in the sense of (biblatex) Package biblatex Error: Command ’\cite’ undefinedSee the biblatex package documentation for explanation.The citation command ’\cite’
has not been definedby the
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**A**: Notably, our result may extend to non-monotone profiles**B**: Additionally, the result depends only on the front solution, not on the initial condition.
**C**: Therefore, for a broad class of nonlinear diffusive-dispersive equations of Burgers type, a traveling front solution is asymptotically, nonlinearly, and orbitally stable, provided that the profile lies within some L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT distance from an appropriate translate of the corresponding ideal shock
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**A**: Later, Takahashi and Nunokawa [15], studied the class 𝒮α,βsubscript𝒮𝛼𝛽\mathcal{S}_{\alpha,\beta}caligraphic_S start_POSTSUBSCRIPT italic_α , italic_β end_POSTSUBSCRIPT for the range α,β∈(0,1]𝛼𝛽01\alpha,\beta\in(0,1]italic_α , italic_β ∈ ( 0 , 1 ] and denoted it as 𝒮𝒮∗(α,β).𝒮superscript𝒮𝛼𝛽\mathcal{SS}^{*}(\alpha,\beta).caligraphic_S caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_α , italic_β ) . Note that 𝒮𝒮∗(min{α,β})⊂𝒮𝒮∗(α,β)⊂𝒮𝒮∗(max{α,β})𝒮superscript𝒮𝛼𝛽𝒮superscript𝒮𝛼𝛽𝒮superscript𝒮𝛼𝛽\mathcal{SS}^{*}(\min\{\alpha,\beta\})\subset\mathcal{SS}^{*}(\alpha,\beta)%**B**:
where α,β∈(0,2)𝛼𝛽02\alpha,\beta\in(0,2)italic_α , italic_β ∈ ( 0 , 2 ) with α+β<2𝛼𝛽2\alpha+\beta<2italic_α + italic_β < 2. It holds significant connections with other important classes in Univalent function theory, prominent among them are 𝒮∗=𝒮1,1superscript𝒮subscript𝒮11\mathcal{S}^{*}=\mathcal{S}_{1,1}caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = caligraphic_S start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT and 𝒮𝒮∗(β)=𝒮β,β𝒮superscript𝒮𝛽subscript𝒮𝛽𝛽\mathcal{SS}^{*}(\beta)=\mathcal{S}_{\beta,\beta}caligraphic_S caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_β ) = caligraphic_S start_POSTSUBSCRIPT italic_β , italic_β end_POSTSUBSCRIPT**C**: Authors in [2] gave the extremal function and the structural formula for the function f∈𝒮α,β𝑓subscript𝒮𝛼𝛽f\in\mathcal{S}_{\alpha,\beta}italic_f ∈ caligraphic_S start_POSTSUBSCRIPT italic_α , italic_β end_POSTSUBSCRIPT. Further, they derived the sharp bounds of |zf′(z)/f(z)|𝑧superscript𝑓′𝑧𝑓𝑧|zf^{\prime}(z)/f(z)|| italic_z italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) / italic_f ( italic_z ) |
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**A**: Our contribution consists in providing a unified version of Biswas-Gómez-Muñoz proof, which holds for any almost-simple group G𝐺Gitalic_G**B**: Instead of using the theory of spectral covers, we will use the theory of cameral covers.
**C**: The main difference between our paper and the other ones consists in the study of the Hitchin fibration
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**A**: In Section 3 we define strongly Lech-independence and expansion property and prove some equivalent conditions. There are also some examples showing the relation between strongly Lech-independence and other notions. In Section 4 we use strongly Lech-independence to analyze the colength of powers of ideals and derive inequalities on multiplicities.**B**:
The paper is organized in the following way**C**: In Section 2 we start with the definition of a standard set, along with some basic definitions and properties on the set of monomials in a polynomial ring
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**A**: However, this requires additional caveats when discussing the factorization**B**: Therefore, we prefer not to consider them as snakes.
**C**: If we consider meanders of total order one as snakes, the generating function and the corresponding numerical sequences take a somewhat more natural form
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**A**: In addition, the isomorphism (46) is provided in [DHSSM22, Thm.14.10]**B**: At this stage, all that remains, amongst the assumptions of Theorem B is to prove that this “Betti χχ\upchiroman_χ-independence” isomorphism preserves mixed Hodge structures. Moreover, the original P=W conjecture has been proven [MS22, HMMS22, MSQ23], so that this Betti χχ\upchiroman_χ-independence conjecture is now the only missing component to finish proving all of the conjectures mentioned in this paper.**C**:
Since this paper first appeared, proofs of Conjectures 4.6 and 5.6, along with the construction of the isomorphism (16) required for Conjectures B0superscriptB0\textrm{B}^{0}B start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and B, have all been provided in [DHSSM22]
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**A**: However, there are numerous approaches to community detection [12, 33], and it is worthwhile to adapt other approaches, such as modularity maximization [29] and statistical influence using stochastic block models [31], to account for node-absorption rates**B**: Absorption-scaled graphs provide a useful way to adapt community-detection methods (including ones that are not based on random walks) to account for heterogeneous node-absorption rates. Community structure depends not only on network structure but also on network dynamics (see, e.g., [17]), and it is important to use a variety of perspectives to examine the “effective community structure” that is associated with different dynamical processes.**C**:
The community-detection algorithm InfoMap is based on random walks, so it is natural to adapt it to absorbing random walks
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**A**: However, their methods can not cover the regime when the expected degree is Ω(1)Ω1\Omega(1)roman_Ω ( 1 ) due to the lack of concentration**B**: In subsequent works [25, 71] we proposed algorithms to achieve weak consistency**C**: Additionally, [72] proposed Projected Tensor Power Method as the refinement stage to achieve strong consistency, as long as the first stage partition is partially correct, as ours.
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**A**: Instead of passing a critical state of high energy, the process has to cross a valley of negative fitness through a sequence of deleterious mutations**B**: Similarly to the fast dynamics after passing a high energy state, the adaptive dynamics system quickly attains a new metastable equilibrium once a fit mutant is reached due to fast exponential growth. The results of [8] and this paper even confirm classical definitions of the mean time for a metastable transition (e.g. [9]), by proving that the waiting times for jumps between equilibrium states are exponentially distributed when considering the correct time scale.
**C**: In the former case, as well as in this paper, the role of the traditional physical energy (landscape) is taken over by the fitness (landscape)
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**A**: Though we have stated the modularity score of a partition for binary edge weights it is simple to take the weight of edges inside each part (instead of the number of edges) and to take the degree of a vertex v𝑣vitalic_v to be the sum of the weights of the edges incident to v𝑣vitalic_v (instead of the number of edges)**B**: This weighted modularity is often used, and indeed the popular community detection algorithm Louvain can take weighted networks as input [4]. Our Theorems 1.1 and 1.2 have analogs for weighted networks - see Section 10.**C**:
Network data which is of interest to cluster often has weights associated with each edge
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**A**: Then α(x)𝛼𝑥\alpha(x)italic_α ( italic_x )
is a subshift of X𝑋Xitalic_X and α(x)=α(x′)𝛼𝑥𝛼superscript𝑥′\alpha(x)=\alpha(x^{\prime})italic_α ( italic_x ) = italic_α ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) implies**B**: Conversely, for every x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X, let α(x)={y∈Aℤ∣ℒ(y)⊂ℒ(x)}𝛼𝑥conditional-set𝑦superscript𝐴ℤℒ𝑦ℒ𝑥\alpha(x)=\{y\in A^{\mathbb{Z}}\mid\mathcal{L}(y)\subset\mathcal{L}(x)\}italic_α ( italic_x ) = { italic_y ∈ italic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∣ caligraphic_L ( italic_y ) ⊂ caligraphic_L ( italic_x ) }**C**: and thus (ii) implies (i)
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**A**: Further, sj−si∈Δupsubscript𝑠𝑗subscript𝑠𝑖superscriptΔups_{j}-s_{i}\in\Delta^{\mbox{\tiny up}}italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Δ start_POSTSUPERSCRIPT up end_POSTSUPERSCRIPT**B**: On the other hand,
the induction hypothesis implies sj−(0,1)−si∈ℐsubscript𝑠𝑗01subscript𝑠𝑖ℐs_{j}-(0,1)-s_{i}\in\operatorname{{\mathcal{I}}}italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - ( 0 , 1 ) - italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_I, and since bj−1−bi−ℓ2≥0subscript𝑏𝑗1subscript𝑏𝑖subscriptℓ20b_{j}-1-b_{i}-\ell_{2}\geq 0italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - 1 - italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ 0 we even have sj−(0,1)−si∈Δupsubscript𝑠𝑗01subscript𝑠𝑖superscriptΔups_{j}-(0,1)-s_{i}\in\Delta^{\mbox{\tiny up}}italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - ( 0 , 1 ) - italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Δ start_POSTSUPERSCRIPT up end_POSTSUPERSCRIPT**C**: sj−(0,1)∈s↑subscript𝑠𝑗01superscript𝑠↑s_{j}-(0,1)\in s^{\uparrow}italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - ( 0 , 1 ) ∈ italic_s start_POSTSUPERSCRIPT ↑ end_POSTSUPERSCRIPT
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**A**: More precisely,
these authors established the third term on the right-hand side in**B**: (2017)**C**: The result in Theorem 4 for s≥1/2𝑠12s\geq 1/2italic_s ≥ 1 / 2 (that is, 2k+2≥d2𝑘2𝑑2k+2\geq d2 italic_k + 2 ≥ italic_d) was already derived in Sadhanala et al
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**A**: In this study, we explore a nonlinear stochastic COVID-19 system, incorporating the influence of non-Gaussian noise**B**: This consideration is crucial for a comprehensive understanding of the system’s behavior and its response to unpredictable environmental factors.
**C**: The presence of non-Gaussian noise adds a layer of complexity to the modeling framework, allowing for a more realistic representation of the uncertainties and random fluctuations inherent in the dynamics of the COVID-19 epidemic
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**A**:
Recent studies have illustrated the versatility of persistent homology in analyzing complex networks, including brain networks**B**: (2019); Xing et al. (2022) highlighted the application of persistent homology in evaluating temporal changes in topological network features.**C**: Sizemore et al
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**A**: Thanks are also due to Colin Guillarmou for pointing out to us that we could use Fried’s result for the proof of Theorem 3**B**: Finally, we would like to thank the anonymous referee for his comments and suggestions that helped to improve the manuscript. The first author is supported from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme with agreement No. 725967.**C**:
We would like to thank Frédéric Faure, Benjamin Delarue, Stéphane Nonnenmacher, Tobias Weich and Luchezar Stoyanov for very interesting discussions
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**A**: Next, in Section 4, we show that the pISSf conditions on control gains additionally guarantee ISSt for the system in the sense of (10).
**B**: First, in Section 3, we find the conditions on control gains that satisfy the pISSf criterion in (9)**C**: In the subsequent sections, our approach of finding the control gains are as follows
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**A**: Financial support was provided by the POSCO Science Fellowship, as well as Korea NRF grants (nos. 2020R1F1A1A01048645 and RS-2023-00237811).
**B**: The author expresses gratitude to both institutions for providing conducive environments for focused research and generous support**C**: This work was undertaken during the author’s research fellowship at Yonsei University and Catholic Kwandong University
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**A**: The development of the following lemma is motivated by the convenience and simplicity of considering points on interior edges of a given origin face and destination face of an arbitrary landscape in any convex unit polyhedron**B**: By doing so, we eliminate the need for four variables when constructing arguments relating to trails incident to some landscape**C**: This becomes extremely useful in the proof of Theorem 5.6.
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**A**: More recently, Ding, Fukushima, Sun and Xu [17, 18, 19] gave detailed results in the case of Bernoulli hard obstacles in dimension d≥2𝑑2d\geq 2italic_d ≥ 2. We will give a more complete description of the results for Bernoulli (soft or hard) obstacles, as well as a comparison with our own work and techniques, in Comment 5 (Section 1.3).**B**:
Random motion among random obstacles has been intensively studied in the past decades**C**: We refer to Sznitman’s monograph [44] for a thorough analysis of Brownian motion among Poissonian obstacles in dimension d≥1𝑑1d\geq 1italic_d ≥ 1
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**A**:
This paper is organized as follows**B**: In Section 3 we discuss the Gaussian modified torsional rigidity. In Section 4 we define the Gaussian version of the Kohler-Jobin rearrangement, study its properties, and prove Theorem 1.1.**C**: In Section 2 we discuss some preliminaries
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**A**:
The transpose network in (5) is defined as the conceptually dual dynamical model of (1), and it has the same set of vertices with every edge reversed compared to the orientation of the corresponding edge in the original network (1)**B**: The relation between the identifiability properties in the original and transpose networks is presented in the following lemma, see its proof in Appendix A.**C**: Furthermore, all the excited (measured) vertices in the original network become measured (excited) ones in the transpose network
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**A**:
We have established the asymptotic theory in a class of directed random graph model parameterized by the differentially private bi-sequence and illustrated application to the Probit model. The result shows that statistical inference can be made using the noisy bi-sequence**B**: However, the asymptotic normality of the estimator is not clear. To avoid this problem, we need appropriately select a probability distribution for directed random graphs when using the existing method. In the further, we may relax our theoretical conditions to ignore the independence of edges.**C**: We assume that the edges are mutually independent in this work. We should be able to obtain consistent conclusion if the edges are dependent, provided that the conditions stated in Theorem 1 are met
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**A**: At the time when the project was performed, Jose Agudelo was an undergraduate student at North Dakota State University, Brooke Dippold was an undergraduate student at Longwood University, Ian Klein was an undergraduate student at Carleton College, Alex Kokot was an undergraduate student at the University of Notre Dame, and Eric Geiger was a graduate student at NCSU**B**: Irina Kogan is a Professor of Mathematics at NCSU. The project was mentored by Eric Geiger and Irina Kogan. A poster based on this project received a honorable mention at JMM 2021.**C**:
This work was performed during the REU 2020 program at the North Carolina State University (NCSU) and was supported by the Department of Mathematics at NCSU and the NSA grant H98230-20-1-0259
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**A**: Obviously, this is not guaranteed for each instance of the algorithm. The Gauss-Southwell method leads to faster convergence at the cost of extra computations and evaluations of gradients during the selection of coordinates which can be an issue in large-scale problems [25].**B**: On the other hand, the use of an irregular order is then considered by researchers to accelerate convergence. Particularly, it is shown in [31] that randomization leads to faster convergence in terms of expectation**C**:
A substantial review of variants of coordinate descent algorithms can be found in [4, Section 6.5.1]. The cyclic selection of coordinates is normally assumed to ensure convergence of the algorithm
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**A**: Gicquaud and T. Marsh, we give in this paper a geometric characterization of complete non-compact Kähler manifolds admitting a compactification by a strictly pseudoconvex CR structure.
The study is more intricate in the complex hyperbolic setting than in the real hyperbolic one, due to the anisotropic nature of complex hyperbolic geometry.**B**: Inspired by the work of E**C**: Bahuaud, R
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**A**:
In this section we discuss the general geometric structure with which ℋℋ{\mathscr{H}}script_H and ℒℒ{\mathscr{L}}script_L are eventually endowed**B**: For example, in [LS-2014] the introduction of a ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operator to define the holomorphic structure of a smooth (quasi)-Hilbert field is not abstracted, but introduced in the ‘direct image’ constructions of [LS-2014, Part II] for the bundle of holomorphic sections. And the discussion of subfields is not as explicit, but many of the elements are hinted at in both [LS-2014] and [B-2009]. In fact, [B-2009] considers subfields in the case of trivial families of complex manifolds.**C**: We start with a slight reorganization and minor refinement of ideas introduced by Lempert and Szőke in [LS-2014], and then we introduce definitions that are already suggested or implied in [LS-2014]
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**A**: We would like to express our sincere gratitude to our scientific adviser Nikolai Vavilov for formulating the problem and for a constant support, without which this paper would never have been written**B**:
Acknowledgment**C**: The authors are grateful to Alexei Stepanov for carefully reading our original manuscript and for numerous remarks and corrections. Also, we would like to thank an anonymous referee for bringing our attention to the paper [32].
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**A**: For this we mainly follow [UN]. Then we recall the classification of T-invariant Cartier divisors in terms of Cartier divisorial support functions, following the work in [AIPSV] and [PS].**B**: In Section 3 we restrict to the complexity one case and we describe the invariant subvarieties induced by the polyhedra in the slices of the divisorial fan. It turns out that these invariant subvarieties generate the pseudoeffective cone of the T-variety and we give a list of its generators (Theorem 3.3)**C**:
In section 2 we recall some basic facts of T-varieties and the combinatorial framework to describe them. In particular, we recall the definitions of p-divisors and of divisorial fans
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**A**: Then there exists two prime numbers ℓ1,ℓ2≠psubscriptnormal-ℓ1subscriptnormal-ℓ2𝑝\ell_{1},\ell_{2}\neq proman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ italic_p, ℓ1≠ℓ2subscriptnormal-ℓ1subscriptnormal-ℓ2\ell_{1}\neq\ell_{2}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and two non-cuspidal unipotent representations π1,π2∈Irr(𝐆𝖥)subscript𝜋1subscript𝜋2normal-Irrsuperscript𝐆𝖥\pi_{1},\pi_{2}\in\operatorname{Irr}(\operatorname{\mathbf{G}}^{\mathsf{F}})italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ roman_Irr ( bold_G start_POSTSUPERSCRIPT sansserif_F end_POSTSUPERSCRIPT ) such that π∼ℓ1π1subscriptsimilar-tosubscriptnormal-ℓ1𝜋subscript𝜋1\pi\sim_{\ell_{1}}\pi_{1}italic_π ∼ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and π∼ℓ2π2subscriptsimilar-tosubscriptnormal-ℓ2𝜋subscript𝜋2\pi\sim_{\ell_{2}}\pi_{2}italic_π ∼ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.**B**: Let 𝐆𝐆\operatorname{\mathbf{G}}bold_G be a finite reductive group with positive
semisimple rank, and π∈Irr(𝐆𝖥)𝜋normal-Irrsuperscript𝐆𝖥\pi\in\operatorname{Irr}(\operatorname{\mathbf{G}}^{\mathsf{F}})italic_π ∈ roman_Irr ( bold_G start_POSTSUPERSCRIPT sansserif_F end_POSTSUPERSCRIPT ) a unipotent cuspidal representation of 𝐆𝐆\operatorname{\mathbf{G}}bold_G**C**: Let us assume that q≠2𝑞2q\neq 2italic_q ≠ 2
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**A**:
On the contrary, we show that the Bayes optimal algorithm performs sub-optimally with some of the worst model parameters, which implies that maximizing the Bayesian objective differs substantially from maximizing the frequentist objective**B**: The fact that the Bayes optimal algorithm is suboptimal means that even if we enhance KG and EI to plan more than one step ahead, their performance in a frequentist measure might not improve.**C**: A Bayesian measure requires it to optimize the performance averaged over the prior, whereas a frequentist measure requires an algorithm to optimize the performance against any case
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**A**: For many of our results, we will be operating in a regime in which k𝑘kitalic_k is fixed, m𝑚mitalic_m is large (as a function of k𝑘kitalic_k) and λ𝜆\lambdaitalic_λ is arbitrary**B**: We will employ asymptotic notation with respect to this regime.**C**:
Over the course of this section and the next one, we will prove Theorem 1.3
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**A**:
Acknowledgement: The authors would like to thank the anonymous referees for their many useful comments and suggestions**B**: The second author is supported by CSIR (India). The third author is supported by the Prime Minister’s Research Fellows (PMRF) Scheme.**C**: The first author is supported by the Science and Engineering Research Board (CRG/2021/000859)
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**A**: Also, it can be extended to give individual non-trivial eigenvalues with non-trivial coboundaries to general S𝑆Sitalic_S-adic sequences, even when they are neither finitary, nor constant-length, nor otherwise, recognizable**B**:
Finally we remark that this kind of construction is essentially the only kind that yields height in the constant length S𝑆Sitalic_S-adic case, see [BGMnY23]**C**: We thank the referee for pointing this out.
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**A**: quickly found four other infinite families by looking at the eigenmatrices of various primitive rank 3333 graphs but did not publish the results.
In Winter 2021, C.-Y.L**B**: and W.-H.Y. joined the project of finding all nontrivial real ETFs obtained as spherical embeddings of primitive rank 3333 graphs.**C**: Then, Ei.B., Et.B., and H.T
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**A**: Lastly, dash-dotted curves take into account both the corralated interference and the correlated MRC.**B**: The dashed lines depict the scenario with finite pool of Q=24𝑄24Q=24italic_Q = 24 pilot sequences and the solid horizontal line is the corresponding lower bound as given by (29)**C**: The markers represent the performance of the probability of the outage for the idealized version of the Random selection scheme and serve as a reference.
The dotted lines show the actual performance of MRC in the presence of correlated interference
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**A**: In fact, in our algorithm, the yielded path has length at
most (300)9/2log300superscript30092300(300)^{9/2}\log{300}( 300 ) start_POSTSUPERSCRIPT 9 / 2 end_POSTSUPERSCRIPT roman_log 300 times the length of the smallest spanning tree**B**: Moreover, the exponent 3333 in our theorem is sharp and can not be lowered; see Section 5.4.**C**: We remark that although time-wise this algorithm is fast, the ratio constant of the yielded path over the optimal path has not been computed and is much larger that Christofides’ 3/2323/23 / 2 ratio
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**A**: In the case of a minimizing T𝑇Titalic_T around a spherical inclusion, T𝑇Titalic_T will approach the particle surface since the nematic and magnetic exchange length become small w.r.t**B**: 6]).
In the case of a peanut-shaped particle aligned with the magnetic field we expect one of three different minimizing configurations, depending on β𝛽\betaitalic_β, see Figure 3.**C**: the particle radius and thus T𝑇Titalic_T forms a half-sphere (compare with [5, Ch
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**A**: is an ∞\infty∞-stack**B**:
In order to be able to compute this, it would be convenient to know that W¯A¯𝑊𝐴\overline{W}Aover¯ start_ARG italic_W end_ARG italic_A is fibrant in the Čech model structure, i.e**C**: This follows thanks to the following wonderful theorem.
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**A**: The polymatroids of this form are now called Chow polymatroids,333This naming is unfortunate for us because we will later associate a polymatroid to the formats of non-degenerate Chow forms of a variety, which are not themselves Chow polymatroids**B**: an interesting (and proper) subclass of polymatroids.
**C**: For the remainder of the section, the chief example of a polymatroid to us is the support of a multiprojective variety (and its downward closure)
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**A**: This suggests that geometric semilattices should be an “affine” counterpart to geometric lattices (among whose main examples we find posets of intersections of arrangements of linear hyperplanes). The following structure theorem is an abstract counterpart to this linear-affine relationship.
**B**: The poset of intersections of any arrangement of hyperplanes in a vector space is a geometric semilattice, as is the poset of all affine subspaces of a finite-dimensional vector space**C**: A main source of intuition on geometric semilattices comes from finite-dimensional vector spaces
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**A**: We first show a suitable mapping class exists, and then relate the parity to a certain determinant of the mapping class group element acting on a symmetrically self-dual representation**B**: Because the determinant, which is always ±1plus-or-minus1\pm 1± 1, is preserved by reduction mod p𝑝pitalic_p for odd p𝑝pitalic_p, we will use this to deduce the odd characteristic case, i.e. Theorem 1.1 (3).**C**:
We now reinterpret Lemma 2.3 as a result about the mapping class group, using Heegaard splittings
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**A**: I would also like to thank the anonymous referee for carefully reading the paper and making a number of helpful suggestions.
**B**: I am also very grateful to Alessio Martini for bringing an error in Lemma 2.1 of an earlier version of this article to my attention**C**: I am deeply grateful to my advisor Detlef Müller for constant support and numberless helpful suggestions
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**A**: The second named author was supported by National Key R&D Program of China 2021YFA1003100, NSFC-11825101, NSFC-11522101 and NSFC-11431013**B**: The third named author was supported by China Postdoctoral Science Foundation BX20230402 and 2023M743719.
**C**: Acknowledgements
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**A**:
Section 2 contains all necessary preliminaries on integral, superintegral and Kleinian sphere packings**B**: Sections 3.2, 3.3 and 3.4 contain the proofs of our main theorems. Finally, in Section 3.5 we provide an example of a properly integral packing and study its properties.**C**: There we also recall Vinberg’s arithmeticity criterion and algorithm
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**A**: 2022YFA1004900, the National Natural Science Foundation of China under Grant No**B**:
The work is supported by the National Key Research and Development Program of China under Grant No**C**: 12201637 and No. 62202475, the Natural Science Foundation of Hunan Province of China under Grant No.2021JJ40701, and the Innovation Program for Quantum Science and Technology under Grant No. 2021ZD0302902.
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**A**: Almost surely either a player was chosen on Step 1 or 2 or the sum of just the odd-numbered terms (given by B’s moves) of expression (2) diverges to ∞\infty∞, by Lemma 3.6**B**: Thus player B is chosen on this step with probability 00, finishing the proof.
∎**C**: In the latter case, the sum of the even-numbered terms must diverge to −∞-\infty- ∞ (as the sum of all terms is convergent), and therefore cannot diverge to ∞\infty∞
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**A**: We refer to Section S13 of the Supplementary Materials [Balocchi et al., 2024+] for more details.**B**: For all these SSP, the prediction is somewhat accurate, and most often the credible intervals contain the “true” value of the functional of interest.
When the data was generated from a power-law Zipf distributios, the predictive performance gets worse, with median errors less then 40%percent4040\%40 % for the number of unseen and the unseen prevalence and 60%percent6060\%60 % for the coverage of prevalences**C**: Figure 2 displays the predicted and actual value for a synthetic dataset that can be thought as “representative”, as it achieved the median error across the several generated datasets
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**A**: It also contains a Red join gadget in [1.5,13.5]1.513.5[1.5,13.5][ 1.5 , 13.5 ]. Its long intervals terminate at 1.5,7.51.57.51.5,7.51.5 , 7.5 and start from 10.5,13.510.513.510.5,13.510.5 , 13.5.
**B**: It also contains the second part of the third switch gadget in [14.5,17.5]14.517.5[14.5,17.5][ 14.5 , 17.5 ]. Its long intervals terminate at 14.514.514.514.5 and start from 16.516.516.516.5**C**: The sixth buffer contains a Blue join gadget in [18,21]1821[18,21][ 18 , 21 ]. Its long intervals terminate at 18181818 and start from 21212121
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**A**: ∎
**B**: Next consider the case where G𝐺Gitalic_G is abelian-by-compact**C**: Then exactly by the same argument as above, G𝐺Gitalic_G has the same asymptotic dimension and Hirsch length as its cocompact abelian subgroup (which could clearly assumed to be closed and so locally compact) and the result follow from the previous case
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**A**: We give a further simplification of this latter property into a condition of bifilteredness thanks to our key observation, which harmonizes this result with our definitions of bi-accessibility. We then prove the 2-categories of flat pseudofunctors to be themselves bi-accessible and bipresentable whenever their domain 2-category have finite weighted bilimits.
**B**: They were defined in [DDS18], from which we give the following definitions and elementary property. As in the 1-dimensional case, those are pseudofunctors that virtually preserves finitely weighted bilimits whenever they exist (which amounts to testing real preservation at the level of the left biKan extension); it was also remarked that this amounted to requiring their category of elements to be σ𝜎\sigmaitalic_σ-cofiltered relatively to their opcartesian morphisms**C**: We deduced several properties of the bi-accessible and bipresentable 2-categories from analysing their binerve pseudofunctors, which allowed to see them as 2-categories of 𝐂𝐚𝐭𝐂𝐚𝐭{\bf Cat}bold_Cat-valued pseudofunctors. Here we describe the precise class of pseudofunctors obtained through this process, the analog of the ordinary flat functors
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**A**: The third author was supported by EPSRC grant EP/W522338/1 and London Mathematical Society grant ECF-2021-27**B**: The authors have no relevant financial or non-financial interests to disclose. There are no datasets associated with the article. The authors thank the two reviewers for their helpful comments and Grahame Erskine for useful discussion.**C**:
First of all, we thank Sudev Naduvath for suggesting this topic
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**A**:
The main goal of this paper is to study finite and infinite horizon risk averse MDPs in a similar framework as above, but with latent costs and randomized actions, under a weakly continuous transition mechanism, subject to state-dependent DRMs. Below we provide a high level discussion on some of the open problems we address in this paper. We would like to note that in some of the earlier risk-averse MDP frameworks, latent costs were not considered. We believe that in many circumstances, latent costs can be used to account for the risk related to factors such as time-discretization and processing delays**B**: We note that randomized actions are allowed in [23], but are evaluated in an arguably risk-neutral manner. Lastly, an adequate discussion on weakly continuous transition kernels is missing from the existing literature on risk averse MDPs. Weakly continuous transition kernels are often preferred in practice, as they provide the flexibility to consider transition dynamics that are partly random and partly deterministic. Moreover, they facilitate data-based modeling by removing the need of working with the density of the kernel or the underlying probability spaces, which typically requires additional model assumptions.**C**: To the best of our knowledge, [6] is the first to consider latent costs within risk-averse MDP frameworks. Their DRMs, however, are constructed from compositions of static risk measures. A framework that allows for state-dependent risk measures allows for greater flexibility in adjusting the level of risk aversion depending on where one is in state space (e.g., in the context of portfolio allocation, an investor may become more risk averse as their wealth increase). Apart from the points mentioned above, it is worth questioning whether it is possible to modify the existing risk-averse MDP framework to account for the randomness in randomized actions in a risk-averse manner
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**A**: Motivated by applications in fluid dynamics, we suppose that u𝑢uitalic_u is divergence-free,
**B**: In spite of its mathematical simplicity, advection equations are of fundamental importance in a variety of models in physics**C**: By considering this linear model it is supposed that the velocity field has no or negligible feedback on the transported quantities, and θ𝜃\thetaitalic_θ is accordingly commonly referred to as a passive scalar or simply tracer
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**A**:
All the symmetry calculation results of this paper can immediately be obtained from our general result for the given coefficients of PDE (3.37)**B**: All that is needed is the calculation of the semi-invariant K(x)𝐾𝑥K(x)italic_K ( italic_x ) defined by (2.5)**C**: Then, the condition for the equation to have a six-dimensional symmetry algebra is simply obtained by forming the condition (see formula (2.19))
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**A**: The proofs of these results are given in the subsequent sections.**B**: The main results obtained in the present paper are Theorems 3.1 and 4.1 from sections 3 and 4
respectively**C**: The next section briefly recalls only the essential part of [8] used in the sequel
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**A**: The second is to explore more complicated Turan-type properties, in the spirit of the paper by Keller and Lifshitz [33]**B**: Below an example of such a statement for several families.**C**:
The first direction is to broaden the class of potential ‘ambient’ families
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**A**: Two curves are stably homeomorphic if there is a homeomorphism of their regular neighborhoods in the ambient surfaces mapping the first curve onto the second one preserving the orientations of the curve and the surface.**B**: A curve is the image of a generic immersion of oriented circles into an oriented (closed) surface 111The condition of “closed” is not essential, but is supposed here to avoid a detailed argument of boundaries..
Each self-intersection is called a crossing 222Only in Section 4, since both curves and link diagrams appear, we call them separately.**C**: A one-component curve with a base point, which is not an intersection, is called pointed
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**A**: This does not correspond to a valid move, since the support of ν𝜈\nuitalic_ν would be {−10,−9,9,10}109910\{-10,-9,9,10\}{ - 10 , - 9 , 9 , 10 }. Intuitively, such moves should not ultimately help push mass far away, and indeed in Lemma 4.2 we show for a different relaxation of highest Elo that they don’t help.**B**:
There is, however, one big discrepancy: the case of z<−1𝑧1z<-1italic_z < - 1**C**: In particular, in the highest Elo problem it is possible for a player rated −1010-10- 10 to beat a player rated 10101010, garnering a full point of Elo
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**A**: We will show in this subsection that such kind of bijection does not exist; see Corollary 5.5.**B**: One may think that it might work with a different choice of bijection in Theorem 3.8**C**:
We shall show that the Prasad conjecture does not hold for ℓℓ\ellroman_ℓ-modular representations with the V𝑉Vitalic_V correspondence
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**A**: Finally, we prove structure results for primal and dual solutions.
**B**: Then, we obtain a Strassen-type result for a variant of the convex order involving positively 1111-homogenous functions**C**: We first obtain an improved duality result showing that under mild conditions there is dual attainment
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**A**: [duc, 1.3.1]), so it makes sense to speak of branches issued from a point of C𝐶Citalic_C**B**: See [duc, 1.7] for some elements from the branch language which we use here. See also [duc, pg. 210] for the notion of a virtual disc.
**C**: A Berkovich curve has the structure of a real graph (cf
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**A**: We will need that the following constructions yield algebraic representations, all clear from the definitions:
**B**: Equivalently, the adjoint maps G×V→V→𝐺𝑉𝑉G\times V\to Vitalic_G × italic_V → italic_V satisfy the usual equations for an action and are morphisms of varieties**C**: We will also refer to this as an algebraic action
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**A**: We also flag up the very recent [spiliopolus], which treats the case of SDEs which are non-linear in the sense of McKean.**B**:
The literature on multiscale methods is extremely vast so in the above we have mentioned only the works which are most relevant to our discussion**C**: Other related works (without any claim to completeness of references) are [XU2018116, veretennikovAvging, liumsstochastic, stochAvgingXu, hairer2004periodic], some of which do cover the case of linearly growing coefficients
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**A**: L𝐿Litalic_L) is finitely generated**B**: In most of the applications, we assume F𝐹Fitalic_F (resp**C**: Since the ring is local,
any finitely generated flat module is free.***for the case F𝐹Fitalic_F (resp. L𝐿Litalic_L) is not-necessarily finitely generated, see Remark 3.32 (resp. Proposition 3.20).
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**A**: Choose the singular cycle with the smallest
initial point (in the standard representation)**B**: Choose a complex linear subdigraph**C**: Find its smallest point of singularity. Either it is an enclosed point or a corner point of an IDI path, as described earlier.
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**A**: Despite many works, complete classifications of real and strongly real elements are known only for a very few families of Lie groups**B**: Mostly, the equivalence between real and strongly real classes has been understood for certain groups, e.g., [Wo, ST].
A complete classification of such elements is known only for a very few families. We refer to the monograph [OS] for a survey.**C**: The latter viewpoint is of importance in representation theory
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**A**: After this paper was written, we realized that the analogue of γ𝛾\gammaitalic_γ-coisotropic- with γ𝛾\gammaitalic_γ replaced by the Hofer distance (and under the name of “locally rigid”) had already been defined in [Ush19]**B**: Using such a notion, the property that for a submanifold, the equivalence between local rigidity and being coisotropic was stated by Usher and he proved that the Humilière-Leclercq-Seyfaddini theorem ([HLS15]) follows from the definition and its invariance by symplectic homeomorphism.
**C**: We start with a new definition that will play a central role in this paper
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**A**:
Section 2 details the mathematical set-up of translationally invariant stabilizer codes in terms of commutative algebra. Rudiments of symplectic geometry over group rings of ΛΛ\Lambdaroman_Λ are laid out here. Section 3 makes the connection between topological excitations and the functor ExtExt\mathrm{Ext}roman_Ext. Section 4 discusses operations on stabilizer codes, e.g. coarse-graining and stacking**B**: In particular we prove that charge modules are invariant to coarse-graining and that they provide obstructions to obtaining a system from a lower dimensional one by stacking. Section 5 ventures a definition of mobility for excitations in any dimension. We also include a proof for the conjecture that in any 2D code with unique ground state, all excitations are mobile and can be created with string operators. In Section 6, we specialize to codes with only mobile excitations**C**: It is shown that in this case charges may be described by cohomology classes of a certain Čech complex. We show how to obtain interesting operators and physical excitations from Čech cocycles. Moreover, we define braiding in terms of a cup product in the Čech complex and show that our proposal reduces to what is expected for D=2𝐷2D=2italic_D = 2. Several examples are worked out in Section 7. Some known mathematical definitions and facts used in the main text are reviewed in appendices: Gorenstein rings in Appendix A, local cohomology in Appendix B and Čech cohomology in Appendix C.
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**A**: This section is organized as follows. In Subsection 5.1, we establish terminology and notation**B**: In Subsection 5.3 we prove that for connected polyhedral complexes imbedded in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, all cells are pointed or all cells are unpointed. This allows us to extend the deformation retraction to the unpointed setting. In Subsection 5.4, we then show that, with respect to any map that is affine-linear on cells, these deformation retractions are well-behaved enough to guarantee that sublevel sets are homotopy equivalent within transversal intervals.
**C**: In particular, we recall the classical decomposition of polyhedral sets as a Minkowski sum (Theorem 5). In Subsection 5.2 we use this decomposition in the pointed case (see Definition 5.1) to construct a deformation retraction to a canonical polytopal complex
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**A**:
For a syntomic k𝑘kitalic_k-scheme X𝑋Xitalic_X, we refer to Corollary 2.1.6, [BL22, Warning 4.6.2, Prop. 5.1.1]111Strictly speaking, [BL22, Prop. 5.1.1] constructs the Nygaard filtration only in the affine case**B**: for the definition of the Nygaard filtration**C**: However, similarly to [BL22, Not. 5.5.23], this can be formally extended to an arbitrary k𝑘kitalic_k-scheme
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**A**:
We note here that the model above does not include a latent period, however since the results in this paper relate to the final outcome of an SIR epidemic they are insensitive to quite general assumptions concerning a latent period (see e.g**B**: Moreover, the results of the present paper (suitably modified) carry over to a model in which very general two-type point processes, representing the times they make global and local contacts, are assigned to infectious individuals; so long as each of the contacts is made with an individual chosen uniformly from the population or community. In particular, all results in this paper apply without change to corresponding SEIR epidemics. Adjusting our results to allow for**C**: Ball, 1986)
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**A**: Besides, when the interval of mesh becomes larger, the helicity fails to conserve as well as the refining one.
**B**: Experiments show that our scheme preserves the helicity orders of magnitude better with a simple modification in the definition of the vorticity**C**: Figure 8 show that the helicity of our model conserves better
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**A**: Persistent homology, an important tool in TDA, captures the evolution of the homology of a family of topological spaces associated to the given dataset**B**:
Traditionally, large topological features are emphasized over small ones**C**: In the traditional setup, homology classes represented by geometrically larger cycles tend to persist longer, or equivalently, have longer persistences. They are given more emphasis because, on one hand, they tend to describe global structures of the dataset, and on the other, random variation of the data points tend to give rise to a large number of artificial small cycles.
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**A**: In contrast, the upper bound confidence (UCB) method is used in Xiong et al. (2021) for adaptive exploration. However, they require strictly positive state transition and observation emission kernels to ensure fast convergence to the stationary distribution. The more related work is Jin et al. (2020a), which considers undercomplete POMDPs, in other words, the observations are more than the latent states. Their proposed algorithm can attain the optimal policy without estimating the exact model, but an observable component (Jaeger, 2000; Hsu et al., 2012), which is the same for our algorithm design, while only applies to tabular POMDPs.
**B**: (2021) establish sample complexity guarantees for searching the optimal policy in POMDPs whose models are identifiable and can be estimated by spectral methods. However, Azizzadenesheli et al. (2016) and Guo et al. (2016) add extra assumptions such that efficient exploration of the POMDP can always be achieved by running arbitrary policies**C**: Our work is related to a line of recent work on the sample efficiency of reinforcement learning for POMDPs. In detail, Azizzadenesheli et al. (2016); Guo et al. (2016); Xiong et al
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**A**: As we have seen before the papers on perfect edge domination are less frequent**B**: There is some more bibliography to add to the already vast literature [3, 10, 23, 31, 33, 35] on dominating induced matchings**C**: There is a paper [16] where the authors describe ILP formulations for the PED problem, together with some experimental results.
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**A**: Therefore the proof of Theorem 1.1 cannot be directly applied**B**: Nevertheless, as announced in the introduction, the strong Liouville property still holds, see the statement of Theorem 1.2.
**C**: As for discrete ∞\infty∞-harmonic functions, recall that F∞subscript𝐹F_{\infty}italic_F start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT is not an averaging operator, since property i) in Definition 1.1 and the observation in Remark 3.1 both fail
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**A**: We will give two proofs for our main theorem, each based on these two definitions. Hence we repeat here both definitions.**B**:
One important invariant of a greedoid is the greedoid polynomial. It was defined by Björner, Korte and Lovász, and they give two equivalent definitions (greedoid, , Theorem 6.1)**C**: One of them is topological, and uses the (abstract) dual complex of the greedoid. Another definition uses activities
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**A**: The typically nonlinear and large-scale structure characterizing NNs, however, generally complicate their analysis 26, 21. Therefore, available approaches, despite commonly working well in practice, generally come without formal certificates of closed-loop stability and performance.
**B**: More recently, instead, an algorithm based on output range analysis methodologies was devised in 35 to verify if the closed-loop operation of a linear uncertain system with a neural network controller guarantees safety for a set of initial conditions**C**: Nevertheless, neural network-based methods have historically been used in uncertain system control design as a proxy for suboptimal control actions 52, 28, 38, robust feedback policies based on pole-placement 53, 36, or to solve frequently encountered linear matrix inequalities 13, 23
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**A**: The existence of solutions for arbitrary large initial data is an open problem**B**: This uniqueness result seems new for chemotaxis problems in the context of critical spaces, including the classical Keller-Segel system (1.1). This issue has been raised in the context of Navier-Stokes equations (see [20] and some references therein).**C**: Using the estimates developed in the proof of Theorem
1.3, we prove the following uniqueness theorem without assuming any smallness condition of the initial data
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**A**: Then, using the existence of nondegenerate pairings between these quantum objects, we prove in Subsection 6.3 isomorphisms between the Hopf-cyclic (co)homology of the coordinate algebra of a quantum linear group, and the relative Hopf-cyclic (co)homology of the extended quantum enveloping algebra of the corresponding Lie algebra relative to the quantum enveloping algebra of a maximal compact subalgebra.**B**: We first recall the extended quantum enveloping algebras in Subsection 6.1, and the corresponding coordinate algebras of functions on quantum (linear) groups in Subsection 6.2**C**:
The q𝑞qitalic_q-deformation analogues of the Hopf-cyclic van Est isomorphisms, on the other hand, are proved in Section 6
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BCA
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BAC
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ABC
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CBA
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Selection 4
|
**A**: The alternative strategy will be followed up elsewhere, both in the present context and in the setting of [17].**B**: There is a possibility to modify the approach by invoking Neumann-to-Dirichlet maps instead, which would have two advantages: one could consider all rates of vertex volume decay in (iii), and certain geometric smoothness requirements could be somewhat relaxed.
Nevertheless, in this paper we stick with the DN version of the approach, in order to align the exposition with that of [17]**C**: This is due to the fact that the argument of our paper [17] is based on the Dirichlet-to-Neumann (DN) machinery
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CAB
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BAC
|
ACB
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CBA
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Selection 4
|
**A**: F] (numerical characterization of k𝑘kitalic_k-Du Bois) and [MP22a, Cor. 3.40] (bounds on dimΣdimensionΣ\dim\Sigmaroman_dim roman_Σ in terms of the relevant numerical invariant).
∎**B**: The normality claim is [MP22a, Cor. 5.6]**C**: For hypersurface singularities, various dimension bounds (covering the claim of the theorem) were obtained in both [MOPW23] and [JKSY22a]. The general lci case follows from [MP22a, Thm
|
ACB
|
CAB
|
CBA
|
BCA
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Selection 2
|
**A**: Case 2: large profit**B**: In this case, agents of type θ=0𝜃0\theta=0italic_θ = 0 would choose to run trials: their expected profit from seeking approval is $40 million. On average, 5% of such agents would receive approval, so many of the approved drugs may actually be ineffective. For example, if there were 20 times as many ineffective drugs as effective drugs, a quick calculation shows that 56% of approved drugs would be ineffective ones.
**C**: Suppose that companies who receive approval make $1 billion in profit, 100 times their investment
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ACB
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CAB
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BAC
|
CBA
|
Selection 1
|
**A**:
Here, we discuss basic properties for the boundary of a gradient Einstein-type manifold by means of γ𝛾\gammaitalic_γ**B**: Our proof follows as in the latter two papers. For the sake of clarity, we have adapted the proof to our case.**C**: We observe that such properties has been proved in particular cases by He-Petersen-Wylie [11] and Miao-Tam [15]
|
CAB
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BAC
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CBA
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ACB
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Selection 4
|
**A**: In this case, the group operation is the vector addition**B**: Traditionally, the additive notation is used for the group operation in the theory of mixed lattice groups and semigroups,**C**:
A mixed lattice vector space is defined similarly as a partially ordered vector space V𝑉Vitalic_V with two partial orderings ≤\leq≤ and ≼precedes-or-equals\preccurlyeq≼ such that (V,≤,≼)𝑉precedes-or-equals(V,\leq,\preccurlyeq)( italic_V , ≤ , ≼ ) is a mixed lattice
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BCA
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BAC
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CBA
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CAB
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Selection 1
|
**A**: These results, that are here not mentioned, have a less prominent role in our treatment as they will be deduced from the main structure with which we shall work**B**: In summary, Chevallier proved**C**:
Chevallier also obtains other results connected to the Lie algebra structure of 𝒮𝒮\mathcal{S}caligraphic_S induced by the cross product [4]
|
BCA
|
ABC
|
CAB
|
ABC
|
Selection 1
|
**A**:
Both results in Table 1 and in Figure 2b suggest that neural operators perform mapping of finite-banded functions.888One may object that we considered only smooth data**B**: This is, indeed, true, but as shown in [De ̵+22] both FNO and DeepONet produce Runge-like oscillations when applied to data with discontinuities, so with current versions of these operators non-smooth data is out of question**C**: Bands are fixed by the choice of architecture and training data. Then, the natural direction is to define neural operator with fixed number of harmonics (basis functions) for both input and output function:
|
BAC
|
ABC
|
ACB
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BCA
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Selection 2
|
**A**: Let x:=∑i=1kλiδgi⊗asi∈ℓ1(G)⊗𝒜assign𝑥superscriptsubscript𝑖1𝑘tensor-productsubscript𝜆𝑖subscript𝛿subscript𝑔𝑖subscript𝑎subscript𝑠𝑖tensor-productsuperscriptℓ1𝐺𝒜x:=\sum_{i=1}^{k}\lambda_{i}\delta_{g_{i}}\otimes a_{s_{i}}\in\ell^{1}(G)%
\otimes\mathcal{A}italic_x := ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_a start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_G ) ⊗ caligraphic_A be an arbitrary element, where {gi:1≤i≤k}⊂Gconditional-setsubscript𝑔𝑖1𝑖𝑘𝐺\{g_{i}:1\leq i\leq k\}\subset G{ italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : 1 ≤ italic_i ≤ italic_k } ⊂ italic_G and {si:1≤i≤k}⊂ℕconditional-setsubscript𝑠𝑖1𝑖𝑘ℕ\{s_{i}:1\leq i\leq k\}\subset\mathbb{N}{ italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : 1 ≤ italic_i ≤ italic_k } ⊂ blackboard_N (with possible repetitions) are two infinite sets with possible repitions**B**: n_{t}})})}}{|\psi(\mathcal{T}_{\delta_{h_{n_{t}}}\otimes\rho(e_{n_{t}})})|}italic_c start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT := divide start_ARG over¯ start_ARG italic_ψ ( caligraphic_T start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_ρ ( italic_e start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) end_ARG end_ARG start_ARG | italic_ψ ( caligraphic_T start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_ρ ( italic_e start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ) | end_ARG**C**: Further, let asi=ρ(∑j=1∞xj(si)ej)+vsisubscript𝑎subscript𝑠𝑖𝜌superscriptsubscript𝑗1subscriptsuperscript𝑥subscript𝑠𝑖𝑗subscript𝑒𝑗subscript𝑣subscript𝑠𝑖a_{s_{i}}=\rho(\sum_{j=1}^{\infty}x^{(s_{i})}_{j}e_{j})+v_{s_{i}}italic_a start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_ρ ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) + italic_v start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT (since image of ρ𝜌\rhoitalic_ρ is complemented in 𝒜𝒜\mathcal{A}caligraphic_A).
|
BAC
|
CBA
|
BCA
|
BCA
|
Selection 1
|
**A**: Set m:=γap(n−12)assign𝑚𝛾𝑎𝑝binomial𝑛12m:=\gamma ap\binom{n-1}{2}italic_m := italic_γ italic_a italic_p ( FRACOP start_ARG italic_n - 1 end_ARG start_ARG 2 end_ARG )**B**: Fix a≤λn𝑎𝜆𝑛a\leq\lambda nitalic_a ≤ italic_λ italic_n, b≤Ca𝑏𝐶𝑎b\leq Caitalic_b ≤ italic_C italic_a, and
two disjoint sets A𝐴Aitalic_A and B𝐵Bitalic_B of sizes a𝑎aitalic_a and b𝑏bitalic_b, respectively**C**: The
|
CAB
|
CBA
|
ABC
|
BAC
|
Selection 3
|
**A**: A dimer covering or perfect matching is a collection of edges in the graph G𝐺Gitalic_G such that each vertex is covered in exactly one edge.
The set of all dimer coverings of G𝐺Gitalic_G will be denoted as ℳ(G)ℳ𝐺\mathcal{M}(G)caligraphic_M ( italic_G ).**B**: Kasteleyn has shown that every plane graph has a Pfaffian orientation [Kas63]**C**: For example, the orientation in Figure 1 is a Pfaffian orientation
|
ACB
|
BCA
|
BAC
|
CBA
|
Selection 4
|
**A**: In order to do this first we prove the existence of dlt models (local and global) and the ACC property for log canonical thresholds**B**: Note that the ACC for for log canonical thresholds is also proved in [Fuj22a].
**C**: We will prove termination of flips for effective pairs as in [Bir07]
|
CAB
|
BAC
|
BCA
|
CAB
|
Selection 3
|
**A**: The first line of research studies offline RL in standard MDPs without any partial observability**B**: The second line of research studies online RL in POMDPs where the actions are specified by history-dependent policies.
Thus, the actions does not directly depends on the latent states and thus these works do not involve the challenge due to confounded data. The third line of research studies OPE in POMDPs where the goal is to learn the value of the target policy as opposed to learning the optimal policy. As a result, these works do not to need to handle the challenge of distributional shift via pessimism.**C**: Table 1: We compare with most related representative works in closely related lines of research
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CAB
|
BAC
|
ABC
|
BCA
|
Selection 4
|
**A**: Bercu2020Efficient designed an online Newton’s method for logistic regression, and Boyer2023asymptotic generalized that method to general regression problems**B**:
The asymptotics of second-order Newton’s methods for unconstrained problems have recently been investigated**C**: Compared to first-order methods that often consider averaged iterates and/or exclude the stepsize 1/t1𝑡1/t1 / italic_t due to technical challenges, both works showed the normality of the last iterate with 1/t1𝑡1/t1 / italic_t stepsize. However, those analyses are not applicable to our study for two reasons.
|
CBA
|
CAB
|
CAB
|
BAC
|
Selection 4
|
**A**: Another open problem is the analysis of isoparametric generalized Taylor-Hood families in 2D and 3D to cope with curved boundaries. Perturbation arguments similar to those used in [6], [7] for isogeometric generalized Taylor-Hood families seem to be a promising approach for this open problem.**B**:
The present paper suffers from the same rather severe restrictions on hexahedral meshes in 3D as in previous work**C**: The analysis of discrete inf-sup conditions for general hexahedral meshes remains an open problem
|
CBA
|
CAB
|
BAC
|
ABC
|
Selection 2
|
**A**: The detection of columns in B𝐵Bitalic_B at each step can be made
with cost O(ln(s)n)𝑂𝑠𝑛O(\ln(s)n)italic_O ( roman_ln ( italic_s ) italic_n )**B**: This provides a total cost O(ln(s)sn)𝑂𝑠𝑠𝑛O(\ln(s)sn)italic_O ( roman_ln ( italic_s ) italic_s italic_n ).**C**: O(bin)𝑂subscript𝑏𝑖𝑛O(b_{i}n)italic_O ( italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_n )
|
ACB
|
BCA
|
ABC
|
ACB
|
Selection 2
|
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