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This paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> Section 3 presents some auxiliary results and key estimates to show our mains theorems. The proof of Theorem 1.3 is given in Section 4. <|MaskedSetence|> | **A**: Also, in Section 2.4, we collect the general degenerate elliptic theory concerning elliptic measures and Green functions which can be found in [17, 23, 24].
**B**: Section 2 contains some preliminaries, definitions, and tools that will be used throughout.
**C**: Section 5 is devoted to proving Theorem 1.4.
.
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2. Splitting zeros and connected sums
We recall relevant material on strata of holomorphic 1111-forms and the GL+(2,ℝ)superscriptGL2ℝ\operatorname{GL}^{+}(2,\mathbb{R})roman_GL start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( 2 , blackboard_R )-action on strata. We then discuss the surgeries of splitting zeros and forming a connected sum with a torus. <|MaskedSetence|> <|MaskedSetence|> For additional background material, we refer to [Bai], [Wri3], [Zor]. <|MaskedSetence|> | **A**: For our purposes, we will need to treat these surgeries as globally defined operations on strata, which requires passing to a certain finite cover by marking a prong.
**B**: We also set some notation for the rest of the paper.
**C**:
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<|MaskedSetence|> First, a tutorial overview of contraction theory is presented to generalize and simplify Lyapunov-based stability methods for incremental exponential stability analysis of nonlinear non-autonomous systems. The use of differential dynamics and its similarity to an LTV system allow for LMI and convex optimization formulations that are useful for systematic nonlinear control and estimation synthesis. Second, various methods of machine learning-based control using contraction theory are presented to augment the existing learning frameworks with formal robustness and stability guarantees, extensively using the results of the first part of the paper. <|MaskedSetence|> It is also emphasized that, especially in situations where ISS and uniform asymptotic stability-based arguments render nonlinear stability analysis unnecessarily complicated, the use of exponential stability and the comparison lemma in contraction theory helps to achieve significant conceptual and methodological simplifications. <|MaskedSetence|> Examples are elucidated to provide clear guidelines for its use in deep learning-based stability analysis and its associated control design for various nonlinear systems.
. | **A**: A connection to the KYP and bounded real lemmas is also shown in the context of contraction-based incremental stability analysis.
Considering the promising outcomes on its utilization for model-based learning in Sec. 5–8 and for model-free data-driven learning in Sec. 9, the methods of contraction theory that are surveyed in this paper provide important mathematical tools for formally providing safety and stability guarantees of learning-based and data-driven control, estimation, and motion planning techniques for high-performance robotic and autonomous systems.
**B**: The main contribution of this paper is twofold.
**C**: Such formal guarantees are essential for their real-world applications but could be difficult to obtain without accounting for a contracting property.
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6. <|MaskedSetence|> In this paper, we develop a framework for efficient parameter estimation in a model for dependent network data with binary outcomes and high-dimensional covariates. The model combines the classical high-dimensional logistic regression with the Ising model from statistical physics to simultaneously capture dependence from the underlying network and the effect of high-dimensional covariates. This dependence makes the model different and the analysis more challenging compared to existing results based on independent samples. <|MaskedSetence|> <|MaskedSetence|> Towards this, we show that using the PMPL method the regression parameters can be estimated at the classical high-dimensional rate, despite the presence of dependence, in the entire high-dimensional regime.. | **A**: Conclusion
Understanding the effect of dependence in high-dimensional inference tasks for non-Gaussian models is an emerging research direction.
**B**: To understand which of the covariates have an effect on the outcome under the presence of network dependence, we also consider the problem of estimation given a fixed (known) level of dependence.
**C**: In the this paper we develop an efficient algorithm for jointly estimating the effect of dependence and the high-dimensional regressions parameters using a penalized maximum pseudo-likelihood (PMPL) method and derive its rate of consistency.
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<|MaskedSetence|> To see that Conjecture 1.1 is a generalization of CHC, given a directed graph G𝐺Gitalic_G, for every vertex u𝑢uitalic_u let S(u)={n(uv)∣uv∈E(G)}𝑆𝑢conditional-set𝑛𝑢𝑣𝑢𝑣𝐸𝐺S(u)=\{n(uv)\mid uv\in E(G)\}italic_S ( italic_u ) = { italic_n ( italic_u italic_v ) ∣ italic_u italic_v ∈ italic_E ( italic_G ) } be the star of edges leaving u𝑢uitalic_u, with their direction removed. We claim that an undirected rainbow cycle v1v2…vksubscript𝑣1subscript𝑣2…subscript𝑣𝑘v_{1}v_{2}\ldots v_{k}italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT for the sets S(v)𝑆𝑣S(v)italic_S ( italic_v ) gives rise to a directed cycle in G𝐺Gitalic_G. Otherwise there exists a vertex visubscript𝑣𝑖v_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT such that {vi,vi+1}=n(vivi+1)subscript𝑣𝑖subscript𝑣𝑖1𝑛subscript𝑣𝑖subscript𝑣𝑖1\{v_{i},v_{i+1}\}=n(v_{i}v_{i+1}){ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT } = italic_n ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) and {vi,vi−1}=n(vivi−1)subscript𝑣𝑖subscript𝑣𝑖1𝑛subscript𝑣𝑖subscript𝑣𝑖1\{v_{i},v_{i-1}\}=n(v_{i}v_{i-1}){ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT } = italic_n ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ). <|MaskedSetence|> Thus the
CHC is equivalent to the case of Conjecture 1.1 in which
the color classes are n𝑛nitalic_n stars, each centered at a different vertex. <|MaskedSetence|> | **A**: Hence the sharpness of CHC implies that of Conjecture 1.1.
.
**B**: But this contradicts the fact that only one edge is chosen from S(vi)𝑆subscript𝑣𝑖S(v_{i})italic_S ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) for participation in the rainbow cycle.
**C**: For a directed edge e=uv𝑒𝑢𝑣e=uvitalic_e = italic_u italic_v let n(e)𝑛𝑒n(e)italic_n ( italic_e )
be the undirected pair {u,v}𝑢𝑣\{u,v\}{ italic_u , italic_v }.
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Acknowledgements. Irving Dai was supported by National Science Foundation grant DGE-1148900. Jennifer Hom was supported by National Science Foundation grants DMS-1552285 and DMS-2104144. <|MaskedSetence|> Linh Truong was supported by National Science Foundation Grants DMS-1606451, DMS-2005539, and DMS-2104309. <|MaskedSetence|> We also thank the MATRIX Institute for hosting three of us during the January 2019 workshop: Topology of manifolds: Interactions between high and low dimensions. We are grateful to Artem Kotelskiy and Hugo Zhou for helpful conversations. <|MaskedSetence|> | **A**: The authors would like to thank the Park City Mathematics Institute for hosting us during the July 2019 Research Program: Quantum Field Theory and Manifold Invariants, where part of this work was completed.
**B**: Matthew Stoffregen was supported by National Science Foundation grant DMS-1952762.
**C**: Lastly, we thank the referee for many helpful comments.
.
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<|MaskedSetence|> In order to show that Ψ(Tb1)∈H1,jΨsubscript𝑇subscript𝑏1subscript𝐻1𝑗\Psi(T_{b_{1}})\in H_{1,j}roman_Ψ ( italic_T start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ italic_H start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT (for j≠1𝑗1j\neq 1italic_j ≠ 1), we will show that H1,jsubscript𝐻1𝑗H_{1,j}italic_H start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT will contain an element of gi,j′Hsubscript𝑔𝑖superscript𝑗′𝐻g_{i,j^{\prime}}Hitalic_g start_POSTSUBSCRIPT italic_i , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_H for some i≥2𝑖2i\geq 2italic_i ≥ 2. Suppose that g1,jsubscript𝑔1𝑗g_{1,j}italic_g start_POSTSUBSCRIPT 1 , italic_j end_POSTSUBSCRIPT corresponds to the primitive vector (1,y2,y3,⋯,y2g)1subscript𝑦2subscript𝑦3⋯subscript𝑦2𝑔(1,y_{2},y_{3},\cdots,y_{2g})( 1 , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , ⋯ , italic_y start_POSTSUBSCRIPT 2 italic_g end_POSTSUBSCRIPT ). <|MaskedSetence|> If y¯nyn≡1(modk)subscript¯𝑦𝑛subscript𝑦𝑛annotated1pmod𝑘{\bar{y}}_{n}y_{n}\equiv 1\pmod{k}over¯ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≡ 1 start_MODIFIER ( roman_mod start_ARG italic_k end_ARG ) end_MODIFIER, then the matrix M=I2g+(y¯n−1)E(1,1)+(yn−1)E(2,2)∈H𝑀subscript𝐼2𝑔subscript¯𝑦𝑛1𝐸11subscript𝑦𝑛1𝐸22𝐻M=I_{2g}+({\bar{y}}_{n}-1)E(1,1)+(y_{n}-1)E(2,2)\in Hitalic_M = italic_I start_POSTSUBSCRIPT 2 italic_g end_POSTSUBSCRIPT + ( over¯ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - 1 ) italic_E ( 1 , 1 ) + ( italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - 1 ) italic_E ( 2 , 2 ) ∈ italic_H. <|MaskedSetence|> | **A**: Now setting Q2=Ψ(Ta2)subscript𝑄2Ψsubscript𝑇subscript𝑎2Q_{2}=\Psi(T_{a_{2}})italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = roman_Ψ ( italic_T start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ), consider the following matrices for 1≤i≤g−11𝑖𝑔11\leq i\leq g-11 ≤ italic_i ≤ italic_g - 1:
.
**B**: Hence, Ψ(Tb1)∈Hi,jΨsubscript𝑇subscript𝑏1subscript𝐻𝑖𝑗\Psi({T_{b_{1}}})\in H_{i,j}roman_Ψ ( italic_T start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ italic_H start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT for each i≥3𝑖3i\geq 3italic_i ≥ 3.
Finally, we consider the case i=1𝑖1i=1italic_i = 1.
**C**: Choose a minimum n𝑛nitalic_n such that yn≠0subscript𝑦𝑛0y_{n}\neq 0italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≠ 0.
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There is little doubt that a general Tate-Sen formalism (such as that of [2]) does exist in the multivariable case, and that could be applied to families of multivariable representations (as for [6, §3] and [6, Proposition 5.2.1]). This said, we proceed here with Tate-Sen descent by hand (this already contains most of the necessary ideas).
B. <|MaskedSetence|> Mazzari are supported by the grant MIUR-PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”. <|MaskedSetence|> <|MaskedSetence|> | **A**: Brinon thanks the Department of Mathematics of the University of Padua for organizing his visits during a pandemic lull..
**B**: O.
**C**: Chiarellotto and N.
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3.3. <|MaskedSetence|> <|MaskedSetence|> The rigorous verification that the functions we found satisfy the properties we want was done in SageMath [The21]. <|MaskedSetence|> These are included as ancillary files with the arXiv version of this paper, at. | **A**: The proofs, like many others in this paper, are computer-assisted.
**B**: The specific functions
The specific functions we use to prove Theorem 3.1 using Lemma 3.2 are described in the following two lemmas.
**C**: For each computer-assisted proof, there is a corresponding using Sage file performing the verification as well as a text file containing its output.
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<|MaskedSetence|> Even though it is known that such an approach suffers from the so-called curse-of-dimensionality, it seems interesting, at least, to develop it for theoretical purposes. Similar approximate solving methods to those that can be used for standard optimal control problems can be considered in the present context. For this reason, they will only be briefly mentioned here without entering into details. The optimal control of Markov chains and Markov processes has also been addressed but essentially only in the finite-state space setting [42, 43, 47]; an approach that has also been adapted to the infinite state-space case through either its discretization or its truncation [46], as often done in the Reinforcement Learning and Control of Markov Decision Processes [49]. The approach we consider here keeps the structure of the problem intact and deals with the actual system and its state-space structure directly.
We first address the continuous-time finite-horizon optimal control of stochastic reaction networks and provide the characterization of the optimal continuous-time control law in terms of a Hamilton-Jacobi-Bellman (HJB) equation taking, in this case, the form of a differential-difference equation. This has to be contrasted with the usual partial differential equations obtained in optimal control and the fact that viscosity solutions do not play an as important role as in previous works [50]. <|MaskedSetence|> In the special case of unimolecular networks, the HJB equation exactly reduces to a nonstandard Riccati differential equation which does not seem to have been previously obtained in the literature; see e.g. [51]. <|MaskedSetence|> for the establishing the existence of solutions. Unfortunately, it is not immediate to see any connection with a ”Hamiltonian matrix” as in previous works on the optimal control of deterministic or diffusion processes.. | **A**: However, its structure is sufficiently similar to existing ones so that some of the existing analysis tools can be considered; e.g.
**B**: In fact, it should only be necessary to consider mild-solutions instead.
**C**:
We propose in this paper to keep the original Markov jump process formulation and to address the optimal control problem using Dynamic Programming directly.
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<|MaskedSetence|> <|MaskedSetence|> Hamiltonian) spectral invariants serve as the minmax critical value selectors, the Lagrangian (resp. <|MaskedSetence|> degenerate symplectic fixed points), and could be of independent interests. In the following concluding remarks we summarize some directions of further study of this class of objects.
. | **A**:
6.
**B**: Hamiltonian periodic orbits and Concluding remarks
In Floer theory the Lagrangian (resp.
**C**: Hamiltonian) Ljusternik–Schnirelman theory seems to be a very useful tool towards the Arnold conjecture for degenerate Lagrangian intersections (resp.
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<|MaskedSetence|> The dynamics of coupled vdP oscillators have been extensively studied due to their nonlinear damping propertiesKashchenko (2021); Wang and Yang (2024). Generalizations of vdP oscillators, such as Duffing van der Pol (DvdP) and Rayleigh van der Pol (RvdP) oscillators, offer richer dynamics, including bifurcation and chaosZhang et al. (2023); Szemplińska-Stupnicka and Rudowski (1997), and are particularly useful in modeling the chaotic and complex coordinated behaviors in real-world systemsBusłowicz and Makarewicz (2014); Haken, Kelso, and Bunz (1985) and an electronic Central Pattern Generator that produces biped gait patterns for robotic systems de Pina Filho and Dutra (2009). Using contraction theory, we analyze the conditions under which these coupled generalized vdP oscillators achieve complete synchronization. We also verify our results using numerical simulations.
To address the issue of networks without virtual systems, we introduced the concept of a virtual network. This allows for constructing a higher-dimensional “virtual system", enabling the identification of synchronization conditions for the original network. A critical challenge in applying contraction theory is ensuring that all oscillator trajectories in a network remain within the contraction region of the corresponding virtual system. For specific coupled-oscillator networks (e.g. <|MaskedSetence|> <|MaskedSetence|> coupled RvdP oscillators), the contraction regions are finite regardless of coefficients. Our approach involves finding a trapping region within the contraction region of the virtual system, ensuring that trajectories starting within the trapping region remain within the contraction region.
. | **A**: Van der Pol (vdP) oscillators, initially developed to model electrical circuits employing vacuum tubes, are renowned for their ability to exhibit self-sustained oscillationsVan Der Pol (1927).
**B**: coupled vdP and DvdP oscillators), adjusting coefficients can extend the contraction region to cover the entire phase plane, ensuring all trajectories remain within this region.
**C**: However, for most complex networks (e.g.
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[Arn98]. He remarked that even simple examples provide new interesting phenomena. Gussein-Zade, Luengo, and Melle [GZLMH98, GZLMH99a, GZLMH99b, GZLMH01] pursued a systematic research of the topology and monodromy of germs of complex meromorphic functions (see also works by Tibar and Siersma
[Tib02, ST04], Bodin, Pichon, and Seade [BP07, PS08, BPS09]). Raibaut [Rai12, Rai13] studied motivic zeta functions and motivic Milnor fibers for meromorphic germs, also considered by Libgober, Maxim, and the second author [GVLM16, GVLM18]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: Zúñiga-Galindo and the third author treated p𝑝pitalic_p-adic zeta functions for Laurent polynomials [LCZG13].
**B**: Veys and Zúñiga-Galindo [VZG17] investigated meromorphic continuation and poles of zeta functions and oscillatory integrals for meromorphic functions defined over local fields of characteristic zero (see also [LC17, BGZG18, BG20]).
Nguyen and Takeuchi [NT19] introduced meromorphic nearby cycle functors, and studied some of its applications to the monodromy of meromorphic germs.
.
**C**: Lemahieu and the second author treated the case of topological zeta functions of meromorphic germs, and proved a generalization of the monodromy conjecture in the two variable case [GVL14].
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<|MaskedSetence|> Acknowledgements
I heartily thank J. Anschütz, A. Ducros, W. Gubler, K. Künnemann, Q. <|MaskedSetence|> Scholze and M. Temkin for valuable discussions and correspondence during the preparation of the present article. I am especially thankful for comments by W. <|MaskedSetence|> Scholze on earlier versions of this text. I am similarly grateful to the two referees for providing numerous valuable suggestions for improvement.. | **A**:
1.6.
**B**: Li, P.
**C**: Gubler and P.
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Although micro-macro type models
give an elegant description of the origin of the macroscopic stress tensor for various complex fluids
[4, 9, 13], directly simulating micro-macro models has been a long-standing challenge.
Various computational techniques have been developed to solve the micro-macro model (1) [14, 15, 16, 17, 18, 19]. The two main approaches are Langevin-equation-based stochastic simulation methods and direct simulation methods based on the microscopic Fokker-Planck equation [18]. One of the earliest Langevin-equation-based numerical methods is the CONNFFESSIT (Calculation of Non-Newtonian Flow: Finite Elements and Stochastic Simulation Technique) algorithm, which couples a finite element discretization to the macroscopic flow with a numerical solver for the microscopic SDE (3) [17, 19]. Along this direction, other stochastic approaches, such as the Lagrangian particle method (LPM) [15] and the Brownian configuration field (BCF) method [16], were proposed to reduce the variance and computational cost of the original CONNFFESSIT algorithm. Several extensions and corresponding numerical experiments have been extensively investigated in recent years [4, 20, 21, 22, 23, 24, 25]. Although stochastic approaches have been the dominant simulation methods for micro-macro models, they suffer from several shortcomings, including high computational costs and stochastic fluctuations. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: However, such methods are well suited only for polymeric models having low-dimensional configuration spaces, and the computational cost of Fokker–Planck–based methods increases rapidly for simulations in strong flows (with highly localized distribution function) or involving high-dimensional configuration spaces [18]..
**B**: An alternative approach is to simulate the Fokker-Planck equation in the configuration space directly.
**C**: Examples include Galerkin spectral element technique [26, 27, 28, 29, 30] and the lattice Boltzmann technique [31, 32].
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The approach we shall use is the geometric decomposition, which leads to explicit bases for finite elements. <|MaskedSetence|> For example, it is used in [26] to construct a local and bounded co-chain projection to the discrete de Rham complexes. <|MaskedSetence|> <|MaskedSetence|> Recently geometric decomposition has been extended to smooth finite elements and smooth finite element de Rham and Stokes complexes [16, 19].
. | **A**: The finite element system in [22] also originates from the geometric decomposition.
**B**: The geometric decomposition of standard finite element de Rham complexes is well-studied in [6, 5, 27], and in [21] for nodal finite element de Rham complexes.
**C**: The geometric decomposition is an important tool for finite element analysis.
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<|MaskedSetence|> <|MaskedSetence|> As we illustrate in §6.1, such constraints may arise because the designer faces an environment where compromising agents’ privacy leads to undesirable outcomes. In §6.2, we define our constrained information design problem and characterize a family of designer objectives where the optimal information structure lies on the Blackwell-Pareto frontier. In a special case, we give a more explicit characterization of the optimal structure and show that it involves very few signal realizations. <|MaskedSetence|> | **A**: Finally, in §6.3, we provide a general recipe for solving information design problems with privacy across multiple receivers..
**B**:
Information Design with Privacy Across Receivers
In this section, we consider a multi-receiver interpretation of our privacy constraint.
**C**: We study information design problems in which the designer supplies information to several receivers and is constrained to using private private information structures.
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<|MaskedSetence|> Section 3 lists some notations and preliminaries for this paper, and in particular introduces the concept of Le Cam’s distance. Section 4 introduces a sufficiency-based procedure for sample amplification, with asymptotic properties for general exponential families and non-asymptotic performances in several specific examples. <|MaskedSetence|> Section 6 presents the general idea of establishing lower bounds for sample amplification, with a universal result specializing to product models. Section 7 discusses more examples in sample amplification and learning, and shows that these tasks are in general non-comparable. <|MaskedSetence|> | **A**: More concrete examples of both the upper and lower bounds, auxiliary lemmas, and proofs are relegated to the appendices.
.
**B**:
Organization.
The rest of this paper is organized as follows.
**C**: Section 5 is devoted to a learning-based procedure for sample amplification, with a general relationship between sample amplification and the χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT estimation error, as well as its applications in several examples.
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<|MaskedSetence|> <|MaskedSetence|> We will show the independence of such a limit from the shape of the domain although we do not need it. The proof of Theorem 2.1 is concluded in Section 4. In Appendix A, we prove the boundedness of the projector onto the finite-dimensional lowest Landau level. This will be used together with elliptic estimates to compute the thermodynamic energy in the low density regime. <|MaskedSetence|> | **A**: Appendix B contains the estimates between the GP and LLL energies.
.
**B**:
Organization of the paper.
We start, in Section 2, by reducing the proof of Theorem 1.1 to the same result under a more stringent, sub-optimal, condition on the parameters.
**C**: In Section 3, we prove the existence of the thermodynamic limit of the homogeneous energy at fixed density.
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.\xi.\overline{e}^{*}D=(\xi^{2}+\overline{q}^{*}c_{2}(V)).\overline{e}^{*}D=kH%
^{2}.D+\overline{q}^{*}c_{2}(V).\overline{e}^{*}D=2c_{2}(V)italic_b italic_H . <|MaskedSetence|> <|MaskedSetence|> italic_D = over¯ start_ARG italic_q end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_V ) . <|MaskedSetence|> over¯ start_ARG italic_e end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_D = ( italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + over¯ start_ARG italic_q end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_V ) ) . over¯ start_ARG italic_e end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_D = italic_k italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . italic_D + over¯ start_ARG italic_q end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_V ) . over¯ start_ARG italic_e end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_D = 2 italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_V )
. | **A**: italic_H .
**B**: italic_ξ .
**C**: italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = over¯ start_ARG italic_e end_ARG start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT over¯ start_ARG italic_q end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_V ) .
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<|MaskedSetence|> This model takes its roots in Statistical Physics, where it usually goes under the name Grand-Canonical [82]. It has recently been used for Coulomb and Riesz costs where it naturally occurs in the large-N𝑁Nitalic_N limit of the multi-marginal problem [59, 26, 60, 61]. A truncated version has been studied in [9, 8]. Its entropic regularization is well known in the literature [17, 16, 48] and plays a central role in the density functional theory of inhomogeneous classical fluids at positive temperature [35, 36]. <|MaskedSetence|> <|MaskedSetence|> | **A**:
In this paper, we study a further generalization where the number of marginals N𝑁Nitalic_N is not fixed.
**B**: We will come back to the entropic model at the end of the paper in Section 6.
.
**C**: An entropic grand canonical problem has also recently appeared in [2], where it is interpreted as a relative entropy minimization with respect to branching Brownian motion, in the framework of regularized unbalanced optimal transport.
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The above analysis was possible because the system is trivial: the state has dimension 3333 and, both the dynamics and the output have a very simple expression. We wish to provide the analytic tool able to automatically perform such analysis, i.e., by following a systematic analytical procedure. <|MaskedSetence|> <|MaskedSetence|> Here, in Chapter 3, we provide a more efficient solution. <|MaskedSetence|> All these analytical systematic procedures use the algebraic operations summarized in section 2.2.
. | **A**: Finally, in Chapters
4 and 5,
we provide the general solution (i.e., for driftless systems, with multiple unknown inputs and that holds even in the case TV).
**B**: Note that, in the absence of unknown inputs, this procedure is the observability rank condition introduced in [4] and summarized in Section 2.4.
**C**: In the presence of a single unknown input and for driftless systems, the solution has very recently been obtained in [46].
| ABC | BCA | BCA | BCA | Selection 4 |
We study the simultaneous actuator selection and controller design problem for Linear Quadratic Regulation (LQR) [3]. The goal is to select a sequence of sets of actuators each with a cardinality constraint, while minimizing the accumulative quadratic cost over a time horizon. <|MaskedSetence|> We study two settings of the problem: episodic and non-episodic settings. In the episodic setting, the interaction with the system breaks into subsequences, each of which starts from a given initial condition and ends at a terminal time step. <|MaskedSetence|> <|MaskedSetence|> We provide online algorithms to solve the problem, and leverage the notion of regret [10, 4, 29] to characterize their performance.
. | **A**: We assume that the system model is unknown, and conside solving the problem by interacting with the system in an online manner.
**B**: Both the episodic and non-episodic settings are widely studied in general reinforcement learning problems, and capture different scenarios in practice [44, 32].
**C**: In the non-episodic setting, the interaction with the system goes on continuously.
| CBA | ACB | ACB | ACB | Selection 4 |
<|MaskedSetence|> Correctly classifying different carcinomas based on their primary anatomical site (e.g. prostate, liver, etc.) is an important problem in medicine, since this allows clinicians to formulate optimal treatment plans for cancer patients [73]. <|MaskedSetence|> This dataset contains a total of n=174𝑛174n=174italic_n = 174 samples from 11 different carcinoma types: prostate, bladder/ureter, breast, colorectal, gastroesophagus, kidney, liver, ovary, pancreas, lung adenocarcinomas, and lung squamous cell carcinoma. <|MaskedSetence|> Collectively, these carcinomas account for about 70% of all cancer-related deaths in the United States [73].. | **A**: In this section, we analyze the carcinomas dataset (U95a GeneChip) in [73].
**B**: The distribution of the n𝑛nitalic_n samples across these 11 respective categories is as follows: 26, 8, 26, 23, 12, 11, 7, 27, 6, 14, and 14.
**C**:
5.3 Application with very large p𝑝pitalic_p: Carcinomas data
Carcinoma is a type of cancer that originates in cells that make up the skin or tissue lining organs.
| CAB | CAB | CAB | CAB | Selection 3 |
diagonal (t,−t,t)𝑡𝑡𝑡(t,-t,t)( italic_t , - italic_t , italic_t ). The blue shaded area is the joint region of r1,r2,r3subscript𝑟1subscript𝑟2subscript𝑟3r_{1},r_{2},r_{3}italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and the red shaded region is the joint region of q1,q2,q3subscript𝑞1subscript𝑞2subscript𝑞3q_{1},q_{2},q_{3}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.
By the first part of Claim 5 it follows that any triangle spanned by the rays r1,r2,r3subscript𝑟1subscript𝑟2subscript𝑟3r_{1},r_{2},r_{3}italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or by the rays q1,q2,q3subscript𝑞1subscript𝑞2subscript𝑞3q_{1},q_{2},q_{3}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT will contain every point in a small neighborhood of the origin.
Therefore a line in the direction of u𝑢uitalic_u which is sufficiently close to the origin will intersect any triangle spanned by the rays R1,R2,R3subscript𝑅1subscript𝑅2subscript𝑅3R_{1},R_{2},R_{3}italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or by the rays Q1,Q2,Q3subscript𝑄1subscript𝑄2subscript𝑄3Q_{1},Q_{2},Q_{3}italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> So by the second part of Claim 5 the origin is in the interior of the joint regions of the ri′superscriptsubscript𝑟𝑖′r_{i}^{\prime}italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and of the qi′superscriptsubscript𝑞𝑖′q_{i}^{\prime}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. <|MaskedSetence|> | **A**: Now consider a unit vector u′superscript𝑢′u^{\prime}italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT which is very close to u𝑢uitalic_u and define the rays ri′=πu′(Ri)subscriptsuperscript𝑟′𝑖subscript𝜋superscript𝑢′subscript𝑅𝑖r^{\prime}_{i}=\pi_{u^{\prime}}(R_{i})italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_π start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) and qi′=πu′(Qi)superscriptsubscript𝑞𝑖′subscript𝜋superscript𝑢′subscript𝑄𝑖q_{i}^{\prime}=\pi_{u^{\prime}}(Q_{i})italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_π start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ).
**B**: Therefore any line in the direction u′superscript𝑢′u^{\prime}italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT which is sufficiently close to the origin will intersect every triangle spanned by the rays R1,R2,R3subscript𝑅1subscript𝑅2subscript𝑅3R_{1},R_{2},R_{3}italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or by the rays Q1,Q2,Q3subscript𝑄1subscript𝑄2subscript𝑄3Q_{1},Q_{2},Q_{3}italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT..
**C**: As long as u′superscript𝑢′u^{\prime}italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is sufficiently close to u𝑢uitalic_u, the rays ri′superscriptsubscript𝑟𝑖′r_{i}^{\prime}italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and qi′superscriptsubscript𝑞𝑖′q_{i}^{\prime}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT will be sufficiently small perturbations of the rays risubscript𝑟𝑖r_{i}italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and qisubscript𝑞𝑖q_{i}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, respectively.
| ACB | ACB | ACB | ACB | Selection 2 |
<|MaskedSetence|> In general, for mechanical systems in absence of irreversible process, should they be relativistic or not, the most elegant and physically justified variational principle is the Hamilton principle. <|MaskedSetence|> It however requires to work in the material (or Lagrangian) representation, which corresponds to the description of each individual material particle. <|MaskedSetence|> It is thus desirable to formulate the variational principle directly in terms of the variables associated to those representations. Several types of Eulerian variational formulations have been developed for relativistic fluids and solids, with various types of constraints and from various point of views, see below. This makes hard the study of the relation between them and their derivation from a unified point of view for both general relativistic fluids and solids.
. | **A**: Depending on the problem at hands, this may not be the most practical description, one reason being that the equations in material coordinates often take a complicated form compared to their Eulerian (spatial) or convective (body) versions.
**B**: This principle is free from any constraints on the field variations, and can be constructed directly from the knowledge of the Lagrangian density of the system.
**C**:
As we will review below, variational principles have played a main role in the derivation of relativistic continuum models, and still form today an essential modelling tool in this area on the theoretical side.
| CBA | CBA | CBA | BCA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Obviously,
{0}∈cl((IntF)∩L)0clInt𝐹𝐿\{0\}\in\operatorname{\mathrm{cl}\,}((\operatorname{Int}F)\cap L){ 0 } ∈ start_OPFUNCTION roman_cl end_OPFUNCTION ( ( roman_Int italic_F ) ∩ italic_L ). Thus,
F∩L⊆cl((IntF)∩L)𝐹𝐿clInt𝐹𝐿F\cap L\subseteq\operatorname{\mathrm{cl}\,}((\operatorname{Int}F)\cap L)italic_F ∩ italic_L ⊆ start_OPFUNCTION roman_cl end_OPFUNCTION ( ( roman_Int italic_F ) ∩ italic_L ). <|MaskedSetence|> Taking into account (7.1). | **A**: The inverse inclusion
holds trivially.
**B**: xn:=(1−1n)x∈(IntF)∩Lassignsubscript𝑥𝑛11𝑛𝑥Int𝐹𝐿x_{n}:=(1-\frac{1}{n})x\in(\operatorname{Int}F)\cap Litalic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT := ( 1 - divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ) italic_x ∈ ( roman_Int italic_F ) ∩ italic_L, for all n∈ℕ𝑛ℕn\in\mathbb{N}italic_n ∈ blackboard_N.
**C**: Since
xn→x→subscript𝑥𝑛𝑥x_{n}\to xitalic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_x, we obtain x∈cl((IntF)∩L)𝑥clInt𝐹𝐿x\in\operatorname{\mathrm{cl}\,}((\operatorname{Int}F)\cap L)italic_x ∈ start_OPFUNCTION roman_cl end_OPFUNCTION ( ( roman_Int italic_F ) ∩ italic_L ).
| BCA | BCA | BCA | BCA | Selection 3 |
Therefore we often seek to understand the global structure of a category via the behaviour of certain types of subcategories. <|MaskedSetence|> The natural extension of this to the big setting, is that of localising tensor ideals, however, it is not known whether there is a set of these. <|MaskedSetence|> Moreover, definable classes also encompass many subcategories of substantial interest: for example, the set of smashing subcategories is in bijection with the set of triangulated definable subcategories.
Just as the Zariski spectrum of a commutative ring R𝑅Ritalic_R provides a local-to-global approach for understanding the structure and properties of R𝑅Ritalic_R-modules, there is a categorification of this to tensor-triangulated categories. In fact, associated to a tensor-triangulated category, there are many ‘spectra’ of interest, each isolating a particular feature. <|MaskedSetence|> | **A**: For example, there is a set of thick tensor ideals of compacts so we can understand the structure of the compact objects via this, but to understand big objects we have to go further.
**B**: On the other hand, there is only a set of definable subcategories, so they provide a way to extend the paradigm from small objects to big objects.
**C**: For example, the Balmer spectrum determines the collection of thick tensor ideals [7], and the Ziegler spectrum determines the definable subcategories [16, 51].
In this vein, our main application to tensor-triangulated categories determines the interaction between localisations and definability, and hence implicitly between localisation and the Ziegler spectrum..
| ABC | ABC | BAC | ABC | Selection 4 |
For the general quadratic measurements model, there is very little work except for an algorithm in [25, 26], which comprises a spectral initialization, followed by iterative gradient descent updates. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Gaussian and noiseless. One of the motivations of this paper is to design a direct second-order method for solving the general quadratic measurements model that starts from arbitrary initial points, has good numerical performance and its convergence properties do not depend on the assumption on Gaussian measurements.
. | **A**: Moreover, we note that almost all aforementioned algorithms for phase retrieval problems and the algorithm of general quadratic measurements need a common assumption that requires the entries of the measurement matrices to be i.i.d.
**B**: Other than that, to the best of our knowledge, no other viable numerical algorithm has been proposed for solving the general quadratic measurements model.
**C**: It was proven that when the number of measurements is large enough, a global optimal solution can be recovered from complex quadratic measurements with a high probability.
| BAC | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> Moreover, let G=SL(d,q),G=SU(d,q)formulae-sequence𝐺SL𝑑𝑞𝐺SU𝑑𝑞G=\operatorname{SL}(d,q),G=\operatorname{SU}(d,q)italic_G = roman_SL ( italic_d , italic_q ) , italic_G = roman_SU ( italic_d , italic_q ) or G=Sp(d,q)𝐺Sp𝑑𝑞G=\operatorname{Sp}(d,q)italic_G = roman_Sp ( italic_d , italic_q ) and we denote these three cases by 𝖫,𝖴𝖫𝖴\mathsf{L},\mathsf{U}sansserif_L , sansserif_U and 𝖲𝖲\mathsf{S}sansserif_S, respectively.
There are two isomorphism types of extraspecial groups of order r1+2msuperscript𝑟12𝑚r^{1+2m}italic_r start_POSTSUPERSCRIPT 1 + 2 italic_m end_POSTSUPERSCRIPT, see [Gor68, Thm. <|MaskedSetence|> <|MaskedSetence|> For r=2𝑟2r=2italic_r = 2, the extraspecial groups of minus type are central products of a quaternion group of order 8888 with any number of dihedral groups of order 8888; taking a central product of such an extraspecial group of isomorphism type 2−1+2msuperscriptsubscript212𝑚2_{-}^{1+2m}2 start_POSTSUBSCRIPT - end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 + 2 italic_m end_POSTSUPERSCRIPT with a cyclic group of order 4444 yields a 2222-group of symplectic type. Under the condition r∣q−1conditional𝑟𝑞1r\mid q-1italic_r ∣ italic_q - 1 we have imposed, both the extraspecial groups of order r1+2msuperscript𝑟12𝑚r^{1+2m}italic_r start_POSTSUPERSCRIPT 1 + 2 italic_m end_POSTSUPERSCRIPT and the symplectic type groups of order 22+2msuperscript222𝑚2^{2+2m}2 start_POSTSUPERSCRIPT 2 + 2 italic_m end_POSTSUPERSCRIPT act on the vector space 𝔽qdsuperscriptsubscript𝔽𝑞𝑑\mathbb{F}_{q}^{d}blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT and this action is absolutely irreducible.
. | **A**: Let q=pe𝑞superscript𝑝𝑒q=p^{e}italic_q = italic_p start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT be a prime power, r𝑟ritalic_r a prime dividing q−1𝑞1q-1italic_q - 1 and d=rm𝑑superscript𝑟𝑚d=r^{m}italic_d = italic_r start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT for some integer m⩾1𝑚1m\geqslant 1italic_m ⩾ 1.
**B**: For odd r𝑟ritalic_r, we only consider the isomorphism type which has exponent r𝑟ritalic_r and is denoted r+1+2msuperscriptsubscript𝑟12𝑚r_{+}^{1+2m}italic_r start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 + 2 italic_m end_POSTSUPERSCRIPT since the normaliser in GL(d,q)GL𝑑𝑞\operatorname{GL}(d,q)roman_GL ( italic_d , italic_q ) of an extraspecial group of the other isomorphism type is properly contained in the normaliser of an extraspecial group of type r+1+2msuperscriptsubscript𝑟12𝑚r_{+}^{1+2m}italic_r start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 + 2 italic_m end_POSTSUPERSCRIPT.
**C**: 5.2].
| ABC | ACB | ACB | ACB | Selection 3 |
Many authors consider neural network attacks and robustness properties in a Euclidean input space. Yet, it is commonly admitted that to learn from high dimensional data, data must lie in a low dimensional manifold ([12]). Such manifold has in general non-zero curvature and Riemannian geometry should therefore be a more appropriate setting to analyze distances and sensitivity from an attack point of view. Furthermore, to analyze neural network model separation capabilities and its robustness, it is critical to understand not only the topology of the decision boundaries in the input space but also the topology of iso-information regions induced by the neural network. Again, there is no reason to believe that these sub-manifolds have zero curvature in general. The Fisher information metric (FIM) is a valid metric for such purpose. <|MaskedSetence|> The FIM may then be used as a Riemannian metric at the output and the pullback metric of the Fisher information as a metric for the input manifold ([1]). <|MaskedSetence|> The FIM with respect to data (also called local data matrix) instead of network parameters has also been investigated from a geometric perspective in [14]. The authors have shown that training data are organized on sub-manifolds (leaves) of a foliation of the data domain. A few authors have also tried to exploit this geometric knowledge to construct adversarial attacks or get some form of immunization from them. In [1], the direction of eigenvector corresponding to the maximum eigenvalue of the pullback FIM metric is used as a direction of attack where as in [15], similar developments are proposed to robustify the model by regularizing the neural network loss function by the trace of the FIM.
In [16, 17], the authors directly use the geodesic distance associated with the FIM as a regularisation term during the training of the neural network. They show that exploiting the geometry of the FIM makes the network more robust to adversarial attacks. <|MaskedSetence|> | **A**: The importance of the FIM in the context of deep neural networks has already been pointed out by several authors.
In [13], it is shown that the FIM defines the landscape of the parameter space and the maximum FIM eigenvalue defines an approximation of the appropriate learning rate for gradient methods.
**B**: In [17], they also suggest that some notion of curvature of the statistical manifold is linked to the robustness of the network to adversarial attacks, as we will investigate in this article by looking at the foliated input space.
.
**C**: Indeed, the network output is seen as a discrete probability that lies on a statistical manifold.
| CAB | CAB | ABC | CAB | Selection 2 |
Organization
The rest of the paper is organized as follows. In Section 2 we compute the limiting distribution in various examples. <|MaskedSetence|> <|MaskedSetence|> The universality of the limiting distribution is discussed in Section 5. In Section 6, we discuss open problems and directions for future research. <|MaskedSetence|> | **A**: The proofs of Theorem 1.5 and Theorem 1.8 are given in Section 3.
**B**: Theorem 1.9 and Proposition 1.11 are proved in Section 4.
**C**: A few technical lemmas are proved in Appendix A.
.
| ABC | BAC | ABC | ABC | Selection 4 |
Our proof combines a construction of Zalgaller [Zal00] for the isometric embedding of so-called long tori with very recent works by Tsuboi [Tsu20] and Arnoux et al. [ALM21] for embedding flat tori. <|MaskedSetence|> <|MaskedSetence|> A flat torus is called long or short depending on whether its aspect ratio is respectively large or small. We construct explicit universal triangulations for long and short tori, see Propositions 5 and 13.
Moreover, we show in Section 5.3 (Lemma 12) that every short torus has a geometric realization with 76 triangles. <|MaskedSetence|> To prove the first part of the theorem we finally overlay the universal triangulations for long and short tori to obtain a universal triangulation for all flat tori.. | **A**: We use the latter construction for the isometric embedding of short tori.
**B**: Together with Proposition 5 this implies that every flat torus has a geometric realization with at most 270 triangles.
**C**: Define the aspect ratio of a flat torus as the ratio of its area by the square of the length of its shortest closed geodesic.
| CBA | ACB | ACB | ACB | Selection 3 |
Acknowlegements. The author would like to thank Prof. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This work was done when the author was a Research Associate at the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore. The author gratefully acknowledges the financial support and the excellent working conditions provided by the institute.
. | **A**: Eswara Rao for suggesting the problem and for some fruitful discussions.
**B**: He also thanks Prof.
**C**: Apoorva Khare for going through the whole paper and for his helpful suggestions regarding its exposition.
| ABC | ABC | ABC | CAB | Selection 3 |
Nevertheless, these approaches presume certain structural assumptions about the dynamical systems and costs, which restrict their applications.
With the power of neural networks, deep RL has been proven to be a promising technique for high-dimensional optimal control tasks [24, 25, 26], where a policy is derived to maximize the accumulated reward at each time step. It has been shown in [27] that neural networks could be leveraged to approximate the value function in high-dimensional optimal control problems and therefore alleviate the curse of dimensionality of Hamilton-Jacobi reachability analysis. For example, sinusoidal neural networks is proven to be a good functional approximator for learning the value function of backward-reachable-tube problem for high-dimensional dynamical systems [28]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: In particular, the paper [29] designs a contractive Bellman backup and learns a conservative approximation to the viability kernel.
**B**: A deep reinforcement learning-based method is proposed in [29] to learn a neural network value function for viability kernel, the set of initial states from which a state trajectory could be maintained to satisfy pre-specified constraints.
**C**: This method is further extended to solve infinite-horizon reach-avoid problem in [15].
.
| CBA | BAC | BAC | BAC | Selection 3 |
with deg(ai(x))<deg(ϕ(x))degreesubscript𝑎𝑖𝑥degreeitalic-ϕ𝑥\deg\ (a_{i}(x))<\deg\ (\phi(x))roman_deg ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ) < roman_deg ( italic_ϕ ( italic_x ) ). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We shall write Nϕ(F)=S1+S2+⋯+Sgsubscript𝑁italic-ϕ𝐹subscript𝑆1subscript𝑆2⋯subscript𝑆𝑔N_{\phi}(F)=S_{1}+S_{2}+\cdots+S_{g}italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_F ) = italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ⋯ + italic_S start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT. The polygon determined by the sides of negative slopes of Nϕ(F)subscript𝑁italic-ϕ𝐹N_{\phi}(F)italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_F ) is called the ϕitalic-ϕ\phiitalic_ϕ-principal Newton polygon of F(x)𝐹𝑥F(x)italic_F ( italic_x ) with respect to νpsubscript𝜈𝑝\nu_{p}italic_ν start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and will be denoted by Nϕ+(F)superscriptsubscript𝑁italic-ϕ𝐹N_{\phi}^{+}(F)italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_F ). Note that the length of Nϕ+(F)superscriptsubscript𝑁italic-ϕ𝐹N_{\phi}^{+}(F)italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_F ) is l(Nϕ+(F))=νϕ¯(F(x)¯)𝑙superscriptsubscript𝑁italic-ϕ𝐹subscript𝜈¯italic-ϕ¯𝐹𝑥l(N_{\phi}^{+}(F))=\nu_{\overline{\phi}}(\overline{F(x)})italic_l ( italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_F ) ) = italic_ν start_POSTSUBSCRIPT over¯ start_ARG italic_ϕ end_ARG end_POSTSUBSCRIPT ( over¯ start_ARG italic_F ( italic_x ) end_ARG ); the highest power of ϕ(x)italic-ϕ𝑥\phi(x)italic_ϕ ( italic_x ) dividing F(x)𝐹𝑥F(x)italic_F ( italic_x ) modulo p𝑝pitalic_p.
Let S𝑆Sitalic_S be a side of Nϕ+(F)superscriptsubscript𝑁italic-ϕ𝐹N_{\phi}^{+}(F)italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_F ). Then the length of S𝑆Sitalic_S, denoted l(S)𝑙𝑆l(S)italic_l ( italic_S ) is the length of its projection to the horizontal axis and its height, denoted h(S)ℎ𝑆h(S)italic_h ( italic_S ) is the length of its projection to the vertical axis. Let λ=−h(S)l(S)=−he𝜆ℎ𝑆𝑙𝑆ℎ𝑒\lambda=-\frac{h(S)}{l(S)}=-\frac{h}{e}italic_λ = - divide start_ARG italic_h ( italic_S ) end_ARG start_ARG italic_l ( italic_S ) end_ARG = - divide start_ARG italic_h end_ARG start_ARG italic_e end_ARG its slope, where e𝑒eitalic_e and hℎhitalic_h are two positive coprime integers. The degree of S𝑆Sitalic_S is d(S)=gcd(h(S),l(S))=l(S)e𝑑𝑆ℎ𝑆𝑙𝑆𝑙𝑆𝑒d(S)=\gcd(h(S),l(S))=\frac{l(S)}{e}italic_d ( italic_S ) = roman_gcd ( italic_h ( italic_S ) , italic_l ( italic_S ) ) = divide start_ARG italic_l ( italic_S ) end_ARG start_ARG italic_e end_ARG; it is equal to the the number of segments into which the integral lattice divides S𝑆Sitalic_S. More precisely, if (s,us)𝑠subscript𝑢𝑠(s,u_{s})( italic_s , italic_u start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) is the initial point of S𝑆Sitalic_S, then the points with integer coordinates lying in S𝑆Sitalic_S are exactly
. | **A**: For every 0≤i≤n,0𝑖𝑛0\leq i\leq n,0 ≤ italic_i ≤ italic_n , let ui=νp(ai(x))subscript𝑢𝑖subscript𝜈𝑝subscript𝑎𝑖𝑥u_{i}=\nu_{p}(a_{i}(x))italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_ν start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ).
**B**: The polygon Nϕ(F)subscript𝑁italic-ϕ𝐹N_{\phi}(F)italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_F ) is the union of different adjacent sides S1,S2,…,Sgsubscript𝑆1subscript𝑆2…subscript𝑆𝑔S_{1},S_{2},\ldots,S_{g}italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_S start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT with increasing slopes λ1,λ2,…,λgsubscript𝜆1subscript𝜆2…subscript𝜆𝑔\lambda_{1},\lambda_{2},\ldots,\lambda_{g}italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_λ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT.
**C**: The ϕitalic-ϕ\phiitalic_ϕ-Newton polygon of F(x)𝐹𝑥F(x)italic_F ( italic_x ) with respect to νpsubscript𝜈𝑝\nu_{p}italic_ν start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT (or to p𝑝pitalic_p, briefly) is the lower convex hull of the points {(i,ui)| 0≤i≤n,ai(x)≠0}conditional-set𝑖subscript𝑢𝑖formulae-sequence 0𝑖𝑛subscript𝑎𝑖𝑥0\{(i,u_{i})\,|\,0\leq i\leq n\,,a_{i}(x)\neq 0\}{ ( italic_i , italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) | 0 ≤ italic_i ≤ italic_n , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ≠ 0 } in the euclidean plane, which we denote by Nϕ(F)subscript𝑁italic-ϕ𝐹N_{\phi}(F)italic_N start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_F ).
| ACB | ACB | BCA | ACB | Selection 2 |
<|MaskedSetence|> In this paper we make an effort toward the understanding of LDP for heat kernels and Brownian motions on 𝖱𝖢𝖣(K,∞)𝖱𝖢𝖣𝐾{\sf RCD}(K,\infty)sansserif_RCD ( italic_K , ∞ ) spaces. The main motivation comes from the central role played by the heat semigroup in conjunction with curvature bounds. <|MaskedSetence|> <|MaskedSetence|> The study of LDP for heat kernels and Brownian motions within the 𝖱𝖢𝖣𝖱𝖢𝖣{\sf RCD}sansserif_RCD setting thus appears a rather natural question to be addressed.
. | **A**: This connection, already highlighted in [27] in the setting of Alexandrov spaces, is even stronger when it comes to lower Ricci bounds.
**B**: The current knowledge around (1.4) outside the Riemannian realm is closer to (1.1) rather than (1.2), in the sense that –except for the case of Gaussian spaces– to the best of our knowledge there are no results in genuinely infinite-dimensional spaces, i.e. without local compactness and local doubling.
Main contributions.
**C**: Indeed, the heat flow is at the very heart of the definition of 𝖱𝖢𝖣𝖱𝖢𝖣{\sf RCD}sansserif_RCD spaces [6, 25] and is intimately linked to the geometry of such spaces, as it appears clearly e.g. in Bakry–Émery gradient estimate.
| BCA | BAC | BAC | BAC | Selection 2 |
<|MaskedSetence|> Their structure is well understood, as is their deformation theory. <|MaskedSetence|> As noted above, there is a close connection between isolated threefold canonical Gorenstein singularities and one parameter smoothings of minimally elliptic singularities. In the case of simple elliptic and cusp singularities, such smoothings are in turn closely related to degenerations
of K3𝐾3K3italic_K 3 surfaces (cf. for example [FM83]). This leads us to define divisors of Type II, Type III1, and Type III2 (Definition 2.6 and Figures 1, 2, 3). <|MaskedSetence|> | **A**:
In dimension 2222, Gorenstein canonical singularities are du Val singularities, also called rational double point (RDP), simple, or ADE singularities.
**B**: We then obtain a partial classification of the threefold singularities admitting good crepant resolutions in the special case of the total space of a one parameter smoothing of a simple elliptic or cusp singularity:
.
**C**: The purpose of Sections 2 and 3 is to give a partial classification of the isolated Gorenstein canonical threefold singularities (X,x)𝑋𝑥(X,x)( italic_X , italic_x ) which admit good crepant resolutions, 3333-dimensional analogues of the ADE case, and to discuss their associated deformation theory.
| ACB | BCA | ACB | ACB | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Ber. (1931) p. 238).
Since the discovery of the position-momentum energy relations by Heisenberg [2], it was assumed that similar relations should hold between time and energy, due to the similar role played by space and time in relativity.. | **A**: We, therefore, conclude that the introduction of an operator t𝑡titalic_t is basically forbidden and the time t𝑡titalic_t must necessarily be considered as an ordinary number […] in Quantum Mechanics (cf.
**B**: for this E.
**C**: Schrödinger, Berl.
| CAB | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> In Section 2, we present a technical perturbation result that is useful to guarantee the invertibility of a certain operator related to the curvature. In Section 3 we prove Theorem 1.1 and address the problem of prescribing the curvatures of surfaces with boundary in a conformal class, depending on the sign of the Euler characteristic. We also study a higher-order analogue in Section 4, proving Theorems 1.2 and 1.3, and establishing some existence and nonexistence of conformal metrics depending on the sign of the lowest eigenvalues of equations (1.3) and (1.4).
Acknowledgements.
The authors would like to thank R. <|MaskedSetence|> Mari for indicating some references and related discussions. <|MaskedSetence|> | **A**: Let us conclude this introduction with a brief description of the structure of this work.
**B**: López-Soriano for theinterest in this work and to help them to improve an earlier version of this paper, as well as to L.
**C**: Finally, the authors thank the anonymous referee for the valuable suggestions and comments..
| BAC | ABC | ABC | ABC | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> In section 3, we introduce and study generalized Drazin-Riesz invertible elements in a semi-simple Banach algebra by extending [32, Theorem 2.3]. <|MaskedSetence|> In particular, we generalize many results of [1]. Section 5 is devoted to show that the class of generalized Drazin-Riesz elements forms a regularity in the sense of Müller, which is a generalization of [2]. Finally, section 6 holds results that link two generalized Drazin-Riesz invertible elements, most of these results extend those of [13].
. | **A**:
In this paper we extend to concept of generalized Drazin-Riesz inverse to Banach algebras.
**B**: Preliminary results are presented in section 2.
**C**: In section 4, we focus our study to generalized Drazin-Riesz inverses.
| ABC | ACB | ABC | ABC | Selection 1 |
<|MaskedSetence|> Acknowledgements
It is a pleasure to thank the following people: Aaron Pollack and Shrenik Shah for pointing out a mistake in our arguments in a first version of this article as well as for various instructive discussions; Colette Moeglin for many useful conversations concerning discrete series and Fourier coefficients. <|MaskedSetence|> <|MaskedSetence|> Finally, we thank Michael Harris, Régis De la Bretèche, Taku Ishii, Tadashi Ochiai, Juan Esteban Rodríguez Camargo, Giovanni Rosso, Benoit Stroh, Jun Su and Sarah Zerbes for useful correspondence or conversations related to this article. Finally, we would like to thank the anonymous referee for the valuable corrections and comments that helped us to considerably improve this article.
. | **A**:
1.4.
**B**: Guido Kings and David Loeffler for their useful remarks on some of our calculations.
**C**: The third named author would like to thank Pierre Colmez for his interest and suggestions on this article and for his constant support.
| ABC | CBA | ABC | ABC | Selection 3 |
<|MaskedSetence|> A few methods for transferring localization statements for Fourier pairs to corresponding statements about localization of time-frequency distributions were given in [10]. In some cases, the extra rotational symmetry of the plane provides more refined information than does separate consideration of f𝑓fitalic_f and f^^𝑓\hat{f}over^ start_ARG italic_f end_ARG. The motivation to study quantitative uncertainty principles for the STFT on the phase space ℤn×𝕋nsuperscriptℤ𝑛superscript𝕋𝑛{\mathbb{Z}}^{n}\times{\mathbb{T}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT arises from the classical uncertainty principles for the STFT and the remarkable contribution of this transform in time-frequency analysis (see [9]). <|MaskedSetence|> The role of these operators is to localize a signal simultaneously in time and frequency domains, this can be seen as the uncertainty principle. Recent works in localization operators on ℤnsuperscriptℤ𝑛{\mathbb{Z}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT (see [6]) motivated us to study uncertainty principles for the STFT on the phase space ℤn×𝕋nsuperscriptℤ𝑛superscript𝕋𝑛{\mathbb{Z}}^{n}\times{\mathbb{T}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. The study of operators defined by phase space functions is sometimes called microlocal analysis. We hope that the study of uncertainty principles for the STFT on the phase space ℤn×𝕋nsuperscriptℤ𝑛superscript𝕋𝑛{\mathbb{Z}}^{n}\times{\mathbb{T}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT makes a significant impact in microlocal analysis.
The main aim of this paper is to study a few uncertainty principles related to the STFT on ℤn×𝕋nsuperscriptℤ𝑛superscript𝕋𝑛{\mathbb{Z}}^{n}\times{\mathbb{T}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. More preciously, we prove the uncertainty principle for orthonormal sequences, Donoho–Stark’s uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle and local uncertainty inequality for the STFT on ℤn×𝕋nsuperscriptℤ𝑛superscript𝕋𝑛{\mathbb{Z}}^{n}\times{\mathbb{T}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. We study the version of Donoho–Stark’s uncertainty principle and show that the STFT cannot be concentrated in any small set. Also, we obtain an estimate for the size of the essential support of this transform under the condition that the STFT of a non-zero function is time-frequency concentrated on a measurable set. Then, we investigate the Benedicks-type uncertainty principle and show that the STFT cannot be concentrated inside a set of measures arbitrarily small. <|MaskedSetence|> Finally, we obtain the Heisenberg-type uncertainty inequality using the k𝑘kitalic_k-entropy and study the localization of the k𝑘kitalic_k-entropy of the STFT on ℤn×𝕋nsuperscriptℤ𝑛superscript𝕋𝑛{\mathbb{Z}}^{n}\times{\mathbb{T}}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.. | **A**: Time-frequency localization operators are a mathematical tool to define a restriction of functions to a region in the time-frequency plane to extract time-frequency features.
**B**: Further, we study the Heisenberg-type uncertainty inequality for a general magnitude and provide the result related to the L2(ℤn×𝕋n)superscript𝐿2superscriptℤ𝑛superscript𝕋𝑛L^{2}(\mathbb{Z}^{n}\times{\mathbb{T}}^{n})italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )-mass of the STFT outside sets of finite measure.
**C**:
It is natural to ask whether uncertainty principles involving f𝑓fitalic_f and f^^𝑓\widehat{f}over^ start_ARG italic_f end_ARG possess analogous statements in terms of the STFT.
| CAB | BAC | CAB | CAB | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> In Example 9, we picked simplices, and we retrieved the even values of ζ(s)𝜁𝑠\zeta(s)italic_ζ ( italic_s ) from a spectral representation of their volumes. Perhaps more can be done for Problem 4 by picking more complex sets and computing their volumes in eq. (6), or eq. <|MaskedSetence|> | **A**: 4.93).
**B**: (10).
.
**C**:
We note that there are already some fascinating geometric interpretations for the odd special values ζ(2n+1)𝜁2𝑛1\zeta(2n+1)italic_ζ ( 2 italic_n + 1 ), following the work of Ed Witten on volumes of certain moduli spaces of flat connections ([23], eq.
| CAB | CBA | CAB | CAB | Selection 1 |
The uniqueness of U𝑈Uitalic_U-invariant probability measures on Y𝑌Yitalic_Y projecting to Haar measure on X𝑋Xitalic_X has been proved by C. Bonatti, A. <|MaskedSetence|> Wilkinson [4] when ΓΓ\Gammaroman_Γ has finite covolume. <|MaskedSetence|> <|MaskedSetence|> | **A**: Eskin and A.
**B**: However, a strictly ergodic approach can be also applied.
Details will be discussed elsewhere.
.
**C**: Here we use Nielsen’s condition (N) to deduce a similar property, both for finite and infinite measures, from the existence of a topological attractor.
| ACB | ACB | ACB | ACB | Selection 3 |
2. Computably locally compact subtrees of ℕ∗superscriptℕ{\mathbb{N}}^{*}blackboard_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT
Definition 2.4 of this section introduces computably locally compact trees. <|MaskedSetence|> For basics on computability theory see, e.g., the first two chapters of [47], or the first chapter of [37] which also contains notation on strings and trees. <|MaskedSetence|> They also serve for basic concepts such as Turing programs, computable functions, as well as partial computable (or partial recursive) functions, which will be needed from Section 6 onwards. <|MaskedSetence|> | **A**: Our paper is mostly consistent with the terminology of these two sources.
**B**: This purely computability-theoretic concept will matter for the whole paper.
**C**: In this section we will review some more specialized concepts related to computability.
.
| BAC | CAB | BAC | BAC | Selection 3 |
<|MaskedSetence|> The medial axis transform is one of the most popular examples of such applications (see [1, 25]).
The Centre Symmetry Set of a curve can be viewed also as the set of all singular points of a family of affine λ𝜆\lambdaitalic_λ-equidistants of the curve (see for instance [9, 10, 13, 16, 21, 29, 31] and the references given there. See also [5, 7] for the Centre Symmetry Set in a non-smooth, polygonal case). Generically, the Centre Symmetry Set of a planar curve can admit at most cusp singularities. It was studied by many authors, also in higher dimensions, and in Minkowski spaces. For instance, authors in [20, 21, 22, 26] analyse the local properties of the css based on the theory of Lagrange and Legendre singularities. In [18, 19] the authors study the Centre Symmetry Set of families of plane curves and of families of surfaces. <|MaskedSetence|> To see art created from the Centre Symmetry Set (and the Wigner caustic) see [8]. <|MaskedSetence|> One of the few global properties is the result in [17] that the number of cusps of css of a generic oval is odd and not smaller than 3.33.3 . The results in [2, 12] show that the number of singularities of the Centre Symmetry Set is greater than the number of singularities of the Wigner caustic of a planar oval (in a generic situation). Moreover, it is known that the Wigner caustic is contained in a region bounded by the Centre Symmetry Set of an oval ([2]). In this article we will prove many global geometrical properties of the Centre Symmetry Set of a planar curve, not necessarily an oval. These theorems will concern the number of the so-called smooth branches of the css, the number of singular points, and the number of asymptotes of this set. Before we come to these theorems, in the next section we will precisely state the conditions for specific types of singularities and directly define the term of generic curves, used throughout the article (see Theorem 2.13).
. | **A**: In the articles cited, the authors obtained properties of the css that are inherently local.
**B**: The css is a part of the Global Centre Symmetry Set studied in [12].
**C**: The various kinds of symmetries are a vital part of the study of manifolds in Euclidean spaces and in their geometry applications.
| CBA | CBA | CAB | CBA | Selection 1 |
(2.1)
for all G⊆{1,…n}𝐺1…𝑛G\subseteq\{1,\ldots n\}italic_G ⊆ { 1 , … italic_n }. It follows from the definition that the existence of a regular unitary dilation implies the existence of a unitary dilation for a commuting tuple of contractions. The study of Brehmer was further continued by Halperin in [40] and [41]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Timotin’s approach had thrown some new lights on the geometric and combinatorial parts of the model theory of Agler, Curto and Vasilescu. In [23], Binding, Farenick and Li proved that for every m𝑚mitalic_m-tuple of operators on a Hilbert space, one can simultaneously dilate them to normal operators on the same Hilbert space such that the dilating operators have finite spectrums. On the other hand, there are nontrivial results on dilation of a contractive but not necessarily commuting tuples. In [32], Davis started studying such tuples and then Bunce [25] and Frazho [38] provided a wider and concrete form to this analysis. An extensive research in the direction of non-commuting dilation has been carried out by Popescu in [53]-[57] and also in collaboration with Arias in [6], [7]. In [22], Bhat, Bhattacharyya and Dey proved that for a commuting contractive tuple the standard commuting dilation is the maximal commuting dilation sitting inside the standard non-commuting dilation.
. | **A**: The positivity condition introduced by Brehmer attracted considerable attentions, e.g see the novel works due to Agler [2], Curto and Vasilescu [28],[29].
**B**: An alternative approach to the results due to Agler, Curto and Vasilescu was provided by Timotin in [63].
**C**: Indeed, Curto and Vasilescu generalized the original theorem of Brehmer together with Agler’s results on hypercontractivity by general model theory for multi-operators which satisfy certain positivity conditions.
| BCA | ACB | ACB | ACB | Selection 2 |
<|MaskedSetence|> From the point of view of the approach in [5] this leads to two essential complications. First, the non-abelian ray transform associated to the latter system is simply the parallel transport given by the Yang-Mills connection, whereas for the former it is given by a coupled system of transport equations, see (18)-(19) below. <|MaskedSetence|> Here we recover it using a new argument based on choosing sources for the Yang-Mills field that take values in the center Z(𝔤)𝑍𝔤Z(\mathfrak{g})italic_Z ( fraktur_g ). <|MaskedSetence|> | **A**: The coupled system (2)-(3) is more intricate than the pure Yang-Mills system.
**B**: Second, the algebraic computations related to principal symbols of the terms generated by the threefold linearization become more challenging than in [5].
Nonetheless, we show that the Yang-Mills field in the coupled system can be recovered via a reduction to the pure Yang-Mills case, on the level of principal symbols.
The Higgs field is not present in [5].
**C**: This can be viewed as a key novelty in the paper as it makes the algebraic computations related to the the principal symbols manageable..
| ABC | ABC | ABC | BAC | Selection 3 |
The maximal number r𝑟ritalic_r satisfying the assumptions of Theorem 1.1 is called in [22] the (holomorphic) discrete degree of symmetry of X𝑋Xitalic_X. More generally, in [22] I. Mundet i Riera defines and studies this invariant for continuous group actions on topological manifolds. In some cases, the discrete degree of symmetry can be compared to the maximal dimension of a torus acting effectively on a manifold [22, Theorem 1.7]. <|MaskedSetence|> Mundet i Riera also asks whether the same bound on r𝑟ritalic_r holds also for birational automorphism groups. In fact, this invariant has implicitly appeared in the study of p𝑝pitalic_p-subgroups of birational automorphism groups. <|MaskedSetence|> <|MaskedSetence|> | **A**: For instance, in [34, Theorem 2.9] J.
**B**: Xu proved the following result for non-uniruled algebraic varieties.
Theorem 1.2..
**C**: In connection with Theorem 1.1 I.
| CAB | CAB | BAC | CAB | Selection 4 |
where I𝐼Iitalic_I is the set of all (p−1)𝑝1(p-1)( italic_p - 1 )th roots of −11-1- 1 in F𝐹\mathbb{F}italic_F. We have V(p2)=⋃r∈IV(X−rY)𝑉subscript𝑝2subscript𝑟𝐼𝑉𝑋𝑟𝑌V(p_{2})=\bigcup_{r\in I}V(X-rY)italic_V ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ⋃ start_POSTSUBSCRIPT italic_r ∈ italic_I end_POSTSUBSCRIPT italic_V ( italic_X - italic_r italic_Y ) and this gives the irreducible components of V(p2)𝑉subscript𝑝2V(p_{2})italic_V ( italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). On the other hand, dimW=1=dimVdimension𝑊1dimension𝑉\dim W=1=\dim Vroman_dim italic_W = 1 = roman_dim italic_V. Then W=⋃r∈I′V(X−rY)𝑊subscript𝑟superscript𝐼′𝑉𝑋𝑟𝑌W=\bigcup_{r\in I^{\prime}}V(X-rY)italic_W = ⋃ start_POSTSUBSCRIPT italic_r ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_V ( italic_X - italic_r italic_Y ) for some ∅≠I′⊆Isuperscript𝐼′𝐼\emptyset\neq I^{\prime}\subseteq I∅ ≠ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_I. <|MaskedSetence|> We have I={λ1,λ3,…,λ2p−3}𝐼superscript𝜆1superscript𝜆3…superscript𝜆2𝑝3I=\{\lambda^{1},\lambda^{3},\dots,\lambda^{2p-3}\}italic_I = { italic_λ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_λ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , … , italic_λ start_POSTSUPERSCRIPT 2 italic_p - 3 end_POSTSUPERSCRIPT } and {λ2,λ4,…,λ2p−2}=Fp×superscript𝜆2superscript𝜆4…superscript𝜆2𝑝2superscriptsubscript𝐹𝑝\{\lambda^{2},\lambda^{4},\dots,\lambda^{2p-2}\}=\mathbb{F}_{p}^{\times}{ italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_λ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , … , italic_λ start_POSTSUPERSCRIPT 2 italic_p - 2 end_POSTSUPERSCRIPT } = italic_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT. <|MaskedSetence|> <|MaskedSetence|> | **A**: Let λ𝜆\lambdaitalic_λ be a primitive (2p−2)2𝑝2(2p-2)( 2 italic_p - 2 )th root of 1 in F𝐹\mathbb{F}italic_F where ⟨λ⟩delimited-⟨⟩𝜆\langle\lambda\rangle⟨ italic_λ ⟩ is a subgroup of Fp2×superscriptsubscript𝐹superscript𝑝2\mathbb{F}_{p^{2}}^{\times}italic_F start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT.
**B**: By Lemma 2.10, for any r=λ2j+1∈I𝑟superscript𝜆2𝑗1𝐼r=\lambda^{2j+1}\in Iitalic_r = italic_λ start_POSTSUPERSCRIPT 2 italic_j + 1 end_POSTSUPERSCRIPT ∈ italic_I, we have
.
**C**: Let r′=λ2i+1∈I′superscript𝑟′superscript𝜆2𝑖1superscript𝐼′r^{\prime}=\lambda^{2i+1}\in I^{\prime}italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_λ start_POSTSUPERSCRIPT 2 italic_i + 1 end_POSTSUPERSCRIPT ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.
| ACB | ACB | ACB | BAC | Selection 1 |
to non-regular iterated line graphs with the property ρ𝜌\rhoitalic_ρ. <|MaskedSetence|> In this paper, we also present hyperenergetic iterated line graphs and their complements. The software sage sage is used to verify some of the results.
This paper is organized as follows: Section 2222 introduces basic definitions, equitable partitions and known results on the spectra and energy of graphs. In Section 3333, it is proven that the two distinct quotient matrices defined for an equitable partition of the H𝐻Hitalic_H-join of regular graphs produces the same partial spectrum for the adjacency matrix, Laplacian matrix and signless Laplacian matrix. <|MaskedSetence|> <|MaskedSetence|> Section 6666 provides an upper bound for the independence number of iterated line graphs and their complements along with a theorem regarding the minimum order of a connected graph whose complement is also connected and considers line graphs satisfying the property ρ𝜌\rhoitalic_ρ.
. | **A**: Section 5555 discusses the spectra and energy of the complements of the graphs covered in Section 4444.
**B**: Section 4444 presents findings on the spectra and energy of iterated line graphs that satisfy the property ρ𝜌\rhoitalic_ρ.
**C**: The energy of line graphs is well studied in Kinkar_Das ; Gutman_line ; Hyper ; HSRamane ; OscarRojo .
| CBA | ACB | CBA | CBA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> See also [9] for some partial results.
There are a number of classical theorems in combinatorial convexity that admit colorful generalizations, originating with the colorful Carathéodory theorem due to Bárány [3] (and the colorful Helly theorem due to Lovász). These generalizations are of significant interest because they often have important applications that can not be obtained from their classical “black-and-white” versions. See [12, chapters 8–10] and [4] for further discussions, examples, and references.
. | **A**: This was originally conjectured by Arocha, Bracho, and Montejano [2], who proved the planar case, i.e.
**B**: The purpose of this note is to prove a so-called colorful generalization of the Goodman–Pollack–Wenger theorem.
**C**: a colorful version of Hadwiger’s theorem.
| BAC | BAC | BAC | ACB | Selection 1 |
The idea of proof as indicated in [Gro86, pg. <|MaskedSetence|> 254]. In [EM02, pg. 136], the authors showed that the proof indeed goes through if the distribution 𝒟𝒟\mathcal{D}caligraphic_D is contact. In fact, their argument remains valid for any strongly bracket-generating distribution (or fat distribution, see [Mon02] for a definition). Moreover, they also planned out a strategy that could work for an arbitrary bracket-generating distribution as well. <|MaskedSetence|> Their argument heavily depends on estimating the codimension of certain semi-analytic sets in the jet bundle. <|MaskedSetence|> | **A**: 84] contained an error that Gromov later acknowledged in [Gro96, pg.
**B**: It was also conjectured in the same article that Gromov’s original statement should hold for a smooth bracket-generating distribution if certain higher order jet calculations are performed.
.
**C**: In [AdPS20], the authors carried out the ideas of [EM02] and proved the hℎhitalic_h-principle for smooth maps transverse to real analytic bracket-generating distributions on a real analytic manifold.
| ACB | ACB | ACB | CBA | Selection 1 |
We discuss now the organization of this paper. <|MaskedSetence|> We discuss locally compact quantum groups, 𝔾𝔾\mathbb{G}blackboard_G-boundaries, closed quantum subgroups, idempotent states, and (co)amenability. We reserve Section 3333 for our main theorems. <|MaskedSetence|> <|MaskedSetence|> | **A**: Section 2222 is reserved for preliminary concepts.
**B**: We recall the construction of the Furstenberg boundary, the Kac and unimodularity properties, the construction of the canonical Kac quotient, and the basics of 𝔾𝔾\mathbb{G}blackboard_G-invariant states.
**C**: We spend the remainder of the paper proving our main theorems highlighted above.
.
| ACB | ABC | ABC | ABC | Selection 2 |
<|MaskedSetence|> This pairing allows us to give explicit formulas for the coefficients of the exponential and logarithm functions of Anderson A𝐴Aitalic_A-modules (see Def. <|MaskedSetence|> Our formulas apply in a very general setting, and give large improvements over previously known formulas for the coefficients of the exponential and logarithm. For example, they recover the powerful results contained in [3, Prop. <|MaskedSetence|> | **A**:
The major innovation in this paper is a new technique we develop, which we call a motivic pairing.
**B**: 2.2] in the case of A𝐴Aitalic_A-finite t𝑡titalic_t-modules.
.
**C**: 2.7; original definition in [1]).
| ACB | ACB | ACB | BCA | Selection 3 |
∑k=0∞|ak||z|ksuperscriptsubscript𝑘0subscript𝑎𝑘superscript𝑧𝑘\sum_{k=0}^{\infty}|a_{k}|\,|z|^{k}∑ start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | italic_z | start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT is bounded by 1111 for |z|≤1/3𝑧13|z|\leq 1/3| italic_z | ≤ 1 / 3 and the constant 1/3131/31 / 3 is optimal. This result, although looks simple, not only motivates many but also
generates intensive research activities to study analogues of this result in various setting–which we call Bohr’s phenomena in modern language. <|MaskedSetence|> <|MaskedSetence|> This investigation led to Bohr type questions in different settings. <|MaskedSetence|> [8], Defant and Frerick [19], and
Djakov and Ramanujan [27] have established further results on Bohr’s phenomena for multidimensional power series.. | **A**: In a series of papers, Aizenberg [5, 6, 7], Aizenberg et al.
**B**: This topic is interesting in its own right from the point of view
of analysis.
**C**: In fact, the idea of Bohr that relates to power series was revived by many with great interest in the nineties due to the extensions to holomorphic functions of several complex variables and to more abstract settings.
For example in 1997, Boas and Khavinson [15] defined n𝑛nitalic_n-dimensional Bohr radius for the family of holomorphic functions bounded by 1111 on the unit polydisk.
| CAB | BCA | BCA | BCA | Selection 2 |
Hardy-Littlewood-Pólya operator is related to some important topics in analysis and there have been many results about this operator and its analogous and generalizations. <|MaskedSetence|> In the past three decades, the so-called Hilbert-type operators, including Hardy-Littlewood-Pólya-type operators, have been extensively studied by Yang and his coauthors, see the survey [21] and Yang’s book [22]. <|MaskedSetence|> Fu et al. have studied in [9] some p𝑝pitalic_p-adic Hardy-Littlewood-Pólya-type operators.Very recently, in the work [3], Brevig established some norm estimates for certain Hardy-Littlewood-Pólya-type operators in terms of the Riemann zeta function. <|MaskedSetence|> | **A**: Some further results have been obtained in [4].
.
**B**: The classical results of this topic can be founded in the famous monograph [13].
**C**: For more recent results see for example [20] and [23].
| ABC | BCA | BCA | BCA | Selection 3 |
I would like to convey my deepest gratitude to J. <|MaskedSetence|> <|MaskedSetence|> Without their instruction, this article never have seen the light of the world.
I also thank to every members in Algebra and Number Theory group at the University of Maine, especially, J. Buttcane, B. Hanson, and A. <|MaskedSetence|> The author would like to convey our sincere appreciation to anonymous referee
for reading our paper and invaluable comments which significantly improve the exposition of this manuscript.
. | **A**: Shahidi for incredible suggestions to apply
Cogdell, Shahidi, and Tsai’s theory [CST17] to other situation.
**B**: Cogdell and F.
**C**: Knightly for providing vibrant environment and their constant encouragement, while the paper was made.
| BAC | BAC | BAC | BAC | Selection 1 |
Starting from [AG90], several results on finite topological type have been published (e.g. <|MaskedSetence|> In general, these results follow a similar approach. First, a growth condition on geometric quantities (e.g. Theorem 1.1 (1)) is required to ensure a small excess estimate, which is obtained using Abresch-Gromoll’s excess estimate (Theorem 2.1 below). <|MaskedSetence|> Theorem 1.1 (2)) is needed to guarantee a triangle comparison of Toponogov type. By combining the small excess estimate and the triangle comparison, it can be concluded that there are no critical points of distance functions, in the sense of Grove-Shiohama, outside a compact subset. <|MaskedSetence|> | **A**: [SW93, She96]).
**B**: Then the conclusion of finite topological type is derived from Grove-Shiohama’s critical point theory ([Che91]).
.
**C**: Additionally, a regularity assumption on metrics (e.g.
| ACB | BAC | ACB | ACB | Selection 1 |
<|MaskedSetence|> Therefore, if a line in the direction u(θ)𝑢𝜃u(\theta)italic_u ( italic_θ ) intersects K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, then it must intersect K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT before it intersects K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. <|MaskedSetence|> If ℓℓ\ellroman_ℓ intersects K3subscript𝐾3K_{3}italic_K start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, then ℓℓ\ellroman_ℓ is contained in the open halfspace y<0𝑦0y<0italic_y < 0, since the y𝑦yitalic_y-coordinate of u(θ)𝑢𝜃u(\theta)italic_u ( italic_θ ) equals 00. If ℓℓ\ellroman_ℓ also intersects K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, then ℓℓ\ellroman_ℓ meets K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in the intersection of the open halfspaces y<0𝑦0y<0italic_y < 0 and x<0𝑥0x<0italic_x < 0, while ℓℓ\ellroman_ℓ meets K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in the intersection of the open half spaces y<0𝑦0y<0italic_y < 0 and x>0𝑥0x>0italic_x > 0. Since u(θ)⋅(1,0,0)=sinθ>0⋅𝑢𝜃100𝜃0u(\theta)\cdot(1,0,0)=\sin\theta>0italic_u ( italic_θ ) ⋅ ( 1 , 0 , 0 ) = roman_sin italic_θ > 0, the line ℓℓ\ellroman_ℓ must intersect K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT before it intersects K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. <|MaskedSetence|> | **A**: On the other hand, consider a line ℓℓ\ellroman_ℓ in the direction u(θ)𝑢𝜃u(\theta)italic_u ( italic_θ ) for any 0<θ<π0𝜃𝜋0<\theta<\pi0 < italic_θ < italic_π.
**B**:
As in the previous case, we note that for 0<θ<β0𝜃𝛽0<\theta<\beta0 < italic_θ < italic_β we have u(θ)⋅n>0⋅𝑢𝜃𝑛0u(\theta)\cdot n>0italic_u ( italic_θ ) ⋅ italic_n > 0.
**C**: It follows that for 0<θ<β0𝜃𝛽0<\theta<\beta0 < italic_θ < italic_β the vector u(θ)𝑢𝜃u(\theta)italic_u ( italic_θ ) is a non-transversal direction for F𝐹Fitalic_F, and the proof of the claim is complete.
∎
.
| CAB | BAC | BAC | BAC | Selection 4 |
<|MaskedSetence|> OE is supported by the ERC Consolidator Grant 772466 “NOISE”. AP was supported by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/R022615/1, Isaac Newton Trust (INT) grant G101121, European Research Council (ERC) starting grant 804166 (SPRS) and the Swiss NSF. <|MaskedSetence|> FV was supported by the Knut and Alice Wallenberg Foundation, the Swedish research council, and the Ruth and Nils-Erik Stenbäck Foundation. <|MaskedSetence|> | **A**: JT is supported by the Swedish research council.
**B**: We are grateful to Juhan Aru and Titus Lupu for helpful discussions..
**C**:
Acknowledgements
The research of AD has been supported by the Deutsche Forschungsgemeinschaft (DFG) grants DR 1096/1-1 and DR 1096/2-1.
| CAB | CAB | CAB | BCA | Selection 3 |
<|MaskedSetence|> In 1996 and 1997, [5] and [6] generalized the result to the cases that the stabilizer H𝐻Hitalic_H is not contained in any proper parabolic ℚ−limit-fromℚ\mathbb{Q}-blackboard_Q -subgroup of G𝐺Gitalic_G.
Let us focus on an example which we will further investigate in this paper. Consider Vn,k:={A∈Matn×n(ℝ)|det(A)=k}assignsubscript𝑉𝑛𝑘conditional-set𝐴subscriptMat𝑛𝑛ℝ𝐴𝑘V_{n,k}:=\{A\in\mathrm{Mat}_{n\times n}(\mathbb{R})|\det(A)=k\}italic_V start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT := { italic_A ∈ roman_Mat start_POSTSUBSCRIPT italic_n × italic_n end_POSTSUBSCRIPT ( blackboard_R ) | roman_det ( italic_A ) = italic_k }. Take G:=SLn×SLnassign𝐺𝑆subscript𝐿𝑛𝑆subscript𝐿𝑛G:=SL_{n}\times SL_{n}italic_G := italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT × italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT with an action on Vn,ksubscript𝑉𝑛𝑘V_{n,k}italic_V start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT given by (g1,g2)x=g1xg2−1subscript𝑔1subscript𝑔2𝑥subscript𝑔1𝑥superscriptsubscript𝑔21(g_{1},g_{2})x=g_{1}xg_{2}^{-1}( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_x = italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT. Notice that SLn(ℝ)/SLn(ℤ)𝑆subscript𝐿𝑛ℝ𝑆subscript𝐿𝑛ℤSL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_R ) / italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_Z ) is of finite volume. Thus all the finiteness conditions hold. Over ℝℝ\mathbb{R}blackboard_R, the action is transitive on Vn,ksubscript𝑉𝑛𝑘V_{n,k}italic_V start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT. For any x∈Vn,k𝑥subscript𝑉𝑛𝑘x\in V_{n,k}italic_x ∈ italic_V start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT, the stabilizer is H:={(xgx−1,g)|g∈SLn(ℝ)}assign𝐻conditional-set𝑥𝑔superscript𝑥1𝑔𝑔𝑆subscript𝐿𝑛ℝH:=\{(xgx^{-1},g)|g\in SL_{n}(\mathbb{R})\}italic_H := { ( italic_x italic_g italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_g ) | italic_g ∈ italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_R ) }. Consider the involution σ(g1,g2)=(xg2x−1,x−1g1x)𝜎subscript𝑔1subscript𝑔2𝑥subscript𝑔2superscript𝑥1superscript𝑥1subscript𝑔1𝑥\sigma(g_{1},g_{2})=(xg_{2}x^{-1},x^{-1}g_{1}x)italic_σ ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_x italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x ) on G(ℝ)𝐺ℝG(\mathbb{R})italic_G ( blackboard_R ). <|MaskedSetence|> Thus we can take arbitrary x∈Vn,k(ℤ)𝑥subscript𝑉𝑛𝑘ℤx\in V_{n,k}(\mathbb{Z})italic_x ∈ italic_V start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT ( blackboard_Z ) to be our w→→𝑤\vec{w}over→ start_ARG italic_w end_ARG in the above theorem. <|MaskedSetence|> | **A**: By taking the union of all SLn(ℤ)×SLn(ℤ)−limit-from𝑆subscript𝐿𝑛ℤ𝑆subscript𝐿𝑛ℤSL_{n}(\mathbb{Z})\times SL_{n}(\mathbb{Z})-italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_Z ) × italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_Z ) -orbits, we have the following asymptotic behavior as Example 1.6 in [3] shows:.
**B**: Then H𝐻Hitalic_H consists of points fixed by σ𝜎\sigmaitalic_σ.
**C**:
In the above theorem, the symmetric condition that H𝐻Hitalic_H consists of the fixed points of an involution plays a central role.
| CBA | CBA | CBA | ACB | Selection 1 |
The computational results of Algorithm 2, FRBS, MFBS and AGR for the instances randomly generated above are presented in Table 1. In detail, the value of (n,m,l,q)𝑛𝑚𝑙𝑞(n,m,l,q)( italic_n , italic_m , italic_l , italic_q ) is listed in the first four columns. For each instance, the number of gradient evaluations and the CPU time (in seconds) are given in the rest of the columns. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: One can observe that our method, namely Algorithm 2, substantially outperforms the other three methods in terms of number of gradient evaluations and CPU time.
**B**: Notice that our method uses both primal and dual extrapolation schemes, while FRBS only uses a dual extrapolation scheme, and MFBS and AGR do not use any of them.
**C**: The numerical results in Table 1 demonstrate that primal and dual extrapolation schemes have an acceleration effect.
.
| ABC | ACB | ABC | ABC | Selection 3 |
and let H𝐻Hitalic_H be the graph induced by vertices V1∪⋯∪Vdsubscript𝑉1⋯subscript𝑉𝑑V_{1}\cup\dotsm\cup V_{d}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ ⋯ ∪ italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. <|MaskedSetence|> Set d=⌈n/n1⌉𝑑𝑛subscript𝑛1d=\lceil n/n_{1}\rceilitalic_d = ⌈ italic_n / italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⌉, so that H𝐻Hitalic_H has at least n𝑛nitalic_n vertices, and let m=dn1−n𝑚𝑑subscript𝑛1𝑛m=dn_{1}-nitalic_m = italic_d italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_n, which satisfies m<n1𝑚subscript𝑛1m<n_{1}italic_m < italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> Then H′superscript𝐻′H^{\prime}italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT has at least δn/4𝛿𝑛4\mathsf{\delta}n/4italic_δ italic_n / 4 edges.
∎
. | **A**: The remaining graph H′superscript𝐻′H^{\prime}italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT has n𝑛nitalic_n
vertices, and at least (d−1)n1≥n−n1>n/2𝑑1subscript𝑛1𝑛subscript𝑛1𝑛2(d-1)n_{1}\geq n-n_{1}>n/2( italic_d - 1 ) italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_n - italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_n / 2 vertices of degree at least δ𝛿\mathsf{\delta}italic_δ.
**B**: Then H𝐻Hitalic_H has dn1𝑑subscript𝑛1dn_{1}italic_d italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT
vertices, each of degree at least δ𝛿\mathsf{\delta}italic_δ, since each v∈Vi𝑣subscript𝑉𝑖v\in V_{i}italic_v ∈ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is adjacent to δ𝛿\mathsf{\delta}italic_δ other vertices in Visubscript𝑉𝑖V_{i}italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.
**C**: Remove any m𝑚mitalic_m vertices of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
| BCA | BAC | BCA | BCA | Selection 3 |
Of particular interest to this paper is the topic of satellite constellation reconfiguration in the context of Earth observations (EO). <|MaskedSetence|> Near-polar orbits enable EO satellite systems to scan different parts of the globe in each orbit, making them ideal for detecting changes in the Earth’s land cover, vegetation, and civil infrastructures. <|MaskedSetence|> Of latest attention to the EO community has been the concept of agile satellites that assume attitude control capability, which is deemed to enhance the overall system responsiveness and scheduling efficiency [10]. Recently, several works have explored the concept of maneuverable satellites in the domain of EO satellite systems as a new paradigm to bolster the system observation capacity by directly manipulating the orbits [11, 12, 1, 13, 14]. <|MaskedSetence|> The former deals with the optimal design of a (destination) constellation configuration that satisfies a set of mission requirements; the latter is concerned with the minimum-cost transportation of satellites from one configuration to another provided the knowledge of both end states. Although we may approach these two interdependent problems independently in a sequential manner (i.e., a destination configuration is first designed and followed by the optimal assignment of satellites to new orbital slots), the outcome of such an open-loop procedure may result in a suboptimal reconfiguration process as a whole [6, 16]. Without taking into account the satellite transportation aspect in design, the optimized new configuration may be too costly or, in fact, infeasible to achieve. This background motivates us to concurrently consider constellation design and transfer aspects in satellite constellation reconfiguration.
. | **A**: Many present-day EO satellite systems are monolithic or small-scale constellation systems distributed in near-polar low Earth orbits (LEO), mostly in sun-synchronous orbits to leverage consistent illumination conditions.
**B**: In this paper, we investigate the concept of reconfiguration as a means for system adaptability and responsiveness that adds a new dimension to the operation of next-generation EO satellite constellation systems.
The problem of satellite constellation reconfiguration consists of two different, yet coupled, problems: the constellation design problem and the constellation transfer problem [6, 15, 16].
**C**: However, the long revisit time for a particular target makes near-polar orbits unsuitable for missions that require rapid adaptive mission planning and enhanced coverage, such as satellite-based emergency mapping, surveillance, and reconnaissance missions, to name a few [8, 9].
| ACB | ACB | ACB | ACB | Selection 2 |
In Section 2, we briefly recall relevant background on semiclassical pseudodifferential operators. Section 3 contains our analysis of the spectral asymptotics for systems exhibiting a potential well. In Section 4, we then apply the spectral asymptotics derived in the previous section to the chiral Hamiltonian of the pseudodifferential Harper model (1.11) and of the low-energy model (1.13).
The article also contains an appendix which consists of Section A where we prove auxiliary results used in the proofs of Section 3,
and Section B where we for comparison discuss the anti-chiral limits of models (1.11) and (1.13). In the former model, there are various quasimodes at potential wells located at different energy levels, but not necessarily at zero, see Theorem B.2. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: The bands near zero energy of the anti-chiral limits of each corresponding model are shown in the right panels of Figures 3 and 4 for comparison.
Spectral aspects of these models will be discussed in the forthcoming article [BGW22].
.
**B**: The gap-condition (1.2) fails, but the operator is diagonalizable so the scalar results of Theorem 1.1 apply directly.
**C**: In the latter model, there are no wells at all, see Remark B.3.
| BCA | ACB | BCA | BCA | Selection 1 |
<|MaskedSetence|> His work became available very late in our writing. <|MaskedSetence|> <|MaskedSetence|> We do not know whether it can be used to prove that entropic representability of matroids is undecidable (and thus be applied to show that it is undecidable whether an access structure admits an ideal secret scheme). It also does not cover approximate results like almost-multilinear representability.
Multilinear representations of Dowling geometries were studied by Beimel, Ben-Efraim, Padró, and Tyomkin in [BBEPT14].
This work was extended by Ben-Efraim and Matúš to entropic matroids [MBE20] building on Matúš’ earlier work on these matroids in [Mat99].. | **A**: The methods used in both papers are related to each other, and also to the methods of [KY22]: all three papers reduce a representability problem to the uniform word problem for finite groups.
**B**: The similarity in methods seems to end there: the proof in [Li23] uses different (though related) combinatorial configurations of random variables, and is significantly shorter than ours.
**C**: Very recently Cheuk Ting Li proved that the conditional independence implication problem is undecidable, as well as that the networking coding problem is undecidable [Li23].
| CAB | CAB | CAB | ABC | Selection 3 |
4.3 Practical aspects of using the flag algebra approach
flagmatic, developed by Emil Vaughan and hosted on github.com/jsliacan/flagmatic, calculates graph densities and passes SDP formulations to solvers like CSDP[9] and SDPA[93]. In order to obtain rigorous mathematical proofs, the floating point-based solutions from the SDPs need to be rounded to (fractional) values while ensuring the matrices remain positive semi-definite. <|MaskedSetence|> <|MaskedSetence|> We do not supply certificates of stability and symmetry from [65], as they are incompatible with our ad-hoc arguments in Lemma 4.7 and Theorem 4.9. <|MaskedSetence|> | **A**: flagmatic handles this rounding process and produces verifiable certificates for the proofs, allowing anyone with access to the software to verify the results independently.
**B**: Below we describe some specifics regarding the bounds for each problem.
.
**C**: The certificates for our lower bounds can be found at doi.org/10.5281/zenodo.6364588.
| ACB | ABC | ACB | ACB | Selection 3 |
Let D∈Der(𝖵)𝐷Der𝖵D\in\operatorname{{\texttt{Der}\,}(\operatorname{\mathsf{V}})}italic_D ∈ start_OPFUNCTION Der ( sansserif_V ) end_OPFUNCTION, let δ=−iDℂ𝛿𝑖superscript𝐷ℂ\delta=-iD^{\mathbb{C}}italic_δ = - italic_i italic_D start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT where Dℂsuperscript𝐷ℂD^{\mathbb{C}}italic_D start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT is the complexification of D𝐷Ditalic_D, i.e. <|MaskedSetence|> Then it is easy to check that δ𝛿\deltaitalic_δ is a derivation in 𝖵ℂsuperscript𝖵ℂ\operatorname{\mathsf{V}}^{\mathbb{C}}sansserif_V start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT. Moreover, it is a ∗*∗-derivation i.e. <|MaskedSetence|> <|MaskedSetence|> | **A**: δ(w∗)=−(δ(w))∗𝛿superscript𝑤superscript𝛿𝑤\delta(w^{*})=-(\delta(w))^{*}italic_δ ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = - ( italic_δ ( italic_w ) ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, where (x+iy)∗=x−iysuperscript𝑥𝑖𝑦𝑥𝑖𝑦(x+iy)^{*}=x-iy( italic_x + italic_i italic_y ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_x - italic_i italic_y for x,y∈V𝑥𝑦𝑉x,y\in Vitalic_x , italic_y ∈ italic_V is the usual involution of the complexification.
**B**: Dℂ(x+iy)=Dx+iDysuperscript𝐷ℂ𝑥𝑖𝑦𝐷𝑥𝑖𝐷𝑦D^{\mathbb{C}}(x+iy)=Dx+iDyitalic_D start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ( italic_x + italic_i italic_y ) = italic_D italic_x + italic_i italic_D italic_y.
**C**: Then by [33, Corollary 10], we have that δ∈Her𝖡(Vℂ)𝛿𝐻𝑒𝑟𝖡superscript𝑉ℂ\delta\in Her\operatorname{\mathsf{B}}(V^{\mathbb{C}})italic_δ ∈ italic_H italic_e italic_r sansserif_B ( italic_V start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ) that is φ(δ)⊂ℝ𝜑𝛿ℝ\varphi(\delta)\subset\mathbb{R}italic_φ ( italic_δ ) ⊂ blackboard_R for all φ∈(𝖡(Vℂ))∗𝜑superscript𝖡superscript𝑉ℂ\varphi\in(\operatorname{\mathsf{B}}(V^{\mathbb{C}}))^{*}italic_φ ∈ ( sansserif_B ( italic_V start_POSTSUPERSCRIPT blackboard_C end_POSTSUPERSCRIPT ) ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT such that φ(Id)=1=‖φ‖𝜑𝐼𝑑1norm𝜑\varphi(Id)=1=\|\varphi\|italic_φ ( italic_I italic_d ) = 1 = ∥ italic_φ ∥.
.
| BAC | BAC | BAC | CBA | Selection 3 |
One particular case of this conjecture is to prove that when mEsubscript𝑚𝐸m_{E}italic_m start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT is odd, then rankℤ(E(ℚ))=0subscriptrankℤ𝐸ℚ0\operatorname{rank}_{\mathbb{Z}}(E({\mathbb{Q}}))=0roman_rank start_POSTSUBSCRIPT blackboard_Z end_POSTSUBSCRIPT ( italic_E ( blackboard_Q ) ) = 0. In this direction, there is great progress. <|MaskedSetence|> <|MaskedSetence|> Section 4).
When the elliptic curves has a rational point of order 2222, using Selmer groups an upper bound for r𝑟ritalic_r can be given in terms of ω(N)𝜔𝑁\omega(N)italic_ω ( italic_N ). Here, N𝑁Nitalic_N is the conductor of E𝐸Eitalic_E, and ω(N)𝜔𝑁\omega(N)italic_ω ( italic_N ) is the number of distinct prime factors of N𝑁Nitalic_N (cf. Section 2.3). <|MaskedSetence|> | **A**: From this, it may be deduced (see Proposition 4.3) that Watkins’s conjecture holds for elliptic curves with conductor a prime power and nontrivial rational 2222-torsion, (cf.
**B**: This upper bound allowed Esparza-Lozano & Pasten [11] to prove that Watkins’s conjecture holds for a quadratic twist E(D)superscript𝐸𝐷E^{(D)}italic_E start_POSTSUPERSCRIPT ( italic_D ) end_POSTSUPERSCRIPT by D𝐷Ditalic_D with ω(D)𝜔𝐷\omega(D)italic_ω ( italic_D ) large enough, whenever E𝐸Eitalic_E has a rational point of order 2222..
**C**: For example, Kazalicki and Kohen [14],[15] proved that if the congruence number δEsubscript𝛿𝐸\delta_{E}italic_δ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT of E𝐸Eitalic_E (which is a multiple of the modular degree mEsubscript𝑚𝐸m_{E}italic_m start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT) is odd, then rankℤ(E(ℚ))=0subscriptrankℤ𝐸ℚ0\operatorname{rank}_{\mathbb{Z}}(E({\mathbb{Q}}))=0roman_rank start_POSTSUBSCRIPT blackboard_Z end_POSTSUBSCRIPT ( italic_E ( blackboard_Q ) ) = 0.
| CAB | BAC | CAB | CAB | Selection 4 |
Organization
In Section 2, we prove the promised conjugation symmetry, Theorem 2.1, for the decomposition of cobordism maps in framed instanton homology. <|MaskedSetence|> In Section 4, we reinterpret ν♯(K)superscript𝜈♯𝐾\nu^{\sharp}(K)italic_ν start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT ( italic_K ) in terms of the decomposition of the map associated to the trace of zero-surgery on K𝐾Kitalic_K (Proposition 4.3), and refine the slice genus bound for ν♯(K)superscript𝜈♯𝐾\nu^{\sharp}(K)italic_ν start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT ( italic_K ) (Proposition 4.5). We then use these results in Section 5 to prove our main result, Theorem 1.2, regarding the parity of ν♯(K)superscript𝜈♯𝐾\nu^{\sharp}(K)italic_ν start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT ( italic_K ). In Sections 6 and 7, we apply these results to technical questions about the framed instanton homology of surgeries, proving Theorems 1.12 and 1.13. <|MaskedSetence|> The last two sections give broader topological applications of all of this technical work. In Section 9, we prove that either τ♯(K)>0superscript𝜏♯𝐾0\tau^{\sharp}(K)>0italic_τ start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT ( italic_K ) > 0 or ν♯(K)>0superscript𝜈♯𝐾0\nu^{\sharp}(K)>0italic_ν start_POSTSUPERSCRIPT ♯ end_POSTSUPERSCRIPT ( italic_K ) > 0 implies h(S−13(K))<0ℎsubscriptsuperscript𝑆31𝐾0h(S^{3}_{-1}(K))<0italic_h ( italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( italic_K ) ) < 0, and deduce Theorem 1.4. <|MaskedSetence|> | **A**: Finally, in Section 10, we study knots with r0(K)subscript𝑟0𝐾r_{0}(K)italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_K ) small, proving Theorem 1.3; recall from the discussion above that this theorem then implies our bound on denominators of surgery slopes in Theorem 1.7.
.
**B**: We use this apparatus in Section 8 to prove Theorem 1.15 and to define our instanton analogue of Hom’s epsilon invariant.
**C**: In Section 3, we establish some notation and recall various results about these cobordism maps.
| CBA | CBA | CBA | CAB | Selection 1 |
<|MaskedSetence|> 87 and end of p. 89] and independently by Vázquez [49]. We have not been able to access Vázquez’s paper. Araki’s definition is based on Serre’s construction of (3.10) using the cubical singular complex; see [42]. Araki then proved a number of properties of these operations and listed them in [4, Summary p. <|MaskedSetence|> In particular, [4, (K) p. 90] yields (i). It is possible to prove (ii) and (iii) for (3.10) using Araki’s definition, but there seems to be no reference in the literature. Multiple authors affirm without proof that the operations defined using Dress’ and Serre’s constructions of (3.10) agree; see e.g. [44, p. 328], [39, p. 737] and [33, §5]. This is certainly true, but we could not locate a proof in the literature. <|MaskedSetence|> | **A**:
Steenrod operations on the Serre spectral sequence (3.10) were first defined by Araki [4, Definition 2 p.
**B**: This is why we do not rely on Araki’s construction for the proof of 3.3.
.
**C**: 89].
| ACB | ACB | ACB | CBA | Selection 3 |
Several learning-based UA heuristics are proposed in [27] which do not assume perfect channel knowledge. The authors compare their proposals to two baseline (distributed) heuristics: connecting to all available BSs and connecting to as many BSs as necessary to satisfy the users’ quality-of-service constraints.
Another study is [34] which compares four simple distributed UA policies in a setting where users join and leave the network. They show that UA should be based on the highest throughput. <|MaskedSetence|> In this work, the authors showed that out of these three heuristics, the SNR-based approach performed best in terms of spectral efficiency. However, some of these UA schemes require knowledge of the entire network. Contrary to most of these works that only verify their results on small instances, we aim to design a UA scheme that is scalable and performs well also in larger settings. Lastly, a framework for UA is proposed in [16] based on minimizing the transmission time and the power budget of a BS, by using a clustering algorithm to cluster users in the same beam. However, this research does not take into account the possibility for users to have MC. <|MaskedSetence|> <|MaskedSetence|> | **A**: Several heuristics considering MC are proposed that optimize energy efficiency [10, 11] or throughput [35].
**B**: However, MC and beamforming are not considered.
Three heuristics for UA are studied in [14]: (1) connect to the least-loaded BS, (2) connect to the highest instantaneous rate, and (3) connect to the highest SNR.
**C**: However, unlike our study, these devised heuristics overlook beam directionality.
.
| BAC | ABC | BAC | BAC | Selection 4 |
<|MaskedSetence|> We then study aperiodic SHS in §3 and the contact non-degenerate examples with finitely many periodic orbits in §4, proving Theorems C and D respectively. In §5 we give the proof for the first two parts of Theorem A, starting with the construction of helix boxes in §5.1 that establishes a very general result for Birkhoff sections in integrable regions. <|MaskedSetence|> Finally, in §7, we establish the existence of Birkhoff splittings for SHS whose Reeb vector field admits an analytic integral.
Acknowledgements: We are grateful to Patrice Le Calvez, for discussions concerning surface dynamics and many ideas to prove Theorem 4.1, and to Francisco Torres de Lizaur for useful discussions. We also thank Pierre Dehornoy, for pointing out a detail of helix boxes that allowed us to simplify the statements of Theorems 5.20 and 5.17. <|MaskedSetence|> | **A**: In §6, we explain why the last statement in Theorem A holds, discuss different genericity results for the existence of Birkhoff sections and establish Theorem B.
**B**: We thank the referees for all the corrections and comments that allowed us to improve the quality of this work.
.
**C**: We start presenting known facts about stable Hamiltonian structures and surface dynamics in §2, in particular, we present the notion of structural decomposition of a SHS and Definition 2.5 of contact non-degenerate SHS.
| CAB | CAB | CAB | CAB | Selection 3 |
As described above, taking advantage of the combination of massive MIMO and RIS for THz communications is an indispensable trend to enhance the network capacity and to reduce the energy consumption [14]. To achieve this, one of the most critical solutions is to apply the beamforming technique to the THz system with massive array elements. However, differing from conventional systems without RIS, the RIS-aided THz system requires to jointly optimize the passive beamforming at the RIS and the active beamforming at the transceivers [15, 16, 17]. To be specific, the work in [16] initially proposes the concept of jointly passive and active beamforming design for RIS-aided multiple-input single-output (MISO) wireless network, and both semidefinite relaxation (SDR) and alternating optimization methods are employed to get a high-quality approximate solution as well as a lower bound. In terms of the point-to-point RIS-assisted MIMO system, the authors in [17] consider the spectral efficiency maximization problem by jointly optimizing the reflection coefficients, the precoder and the combiner. In addition, unlike the works that use continuous phase shifts of RIS, the authors in [18] propose a pragmatic RIS model with a limited number of discrete phase shifts to meet the requirement of practical applications. Then the sum-rate maximization problem is investigated by solving digital beamforming at the BS and discrete analog beamforming at the RIS. <|MaskedSetence|> Hence, it is more practical for massive MIMO and RIS techniques to adopt the subarray architecture and each radio frequency (RF) chain only connects to a subset of antennas [23]. Given the subarray structure, selecting analog beams for transceiver and RIS is an efficient way to guarantee the high sum-rate performance [24]. It should be noted that conventional MIMO systems only carry out the beam selection procedure at the transceiver sides [25, 26]. By contrast, the RIS-aided THz multi-user MIMO (MU-MIMO) system requires to achieve analog beam selection at the base station (BS), the RIS and users, respectively. In other words, there are three different kinds of beam selection targets that need to be considered simultaneously. <|MaskedSetence|> <|MaskedSetence|> | **A**: To cope with this complex combinatorial optimization (CO) problem, the heuristic beam search algorithms are time-consuming and computationally infeasible in real-time communications.
Fortunately, deep learning (DL) can be cast as a promising technology to address high-dimension, complex environment, non-linear, non-convex and other mathematically intractable problems of wireless networks [27, 28, 29].
**B**: It is worth noting that most of the reported works pay more attention to designing active and passive beamforming through conventional optimization methods [19, 20, 21], which are more likely to be restricted by the complex and non-convex issues.
In addition, both active MIMO and passive RIS possess an extremely large number of array elements at THz band, but current research works tend to optimize the phase shifts of RIS elements one by one during the signal processing stage, which definitely leads to high latency and heavy computational complexity for RIS-aided THz communication systems [22].
**C**: More specifically, the DL that belongs to a subclass of machine learning can be used to obtain near-optimal performances for RIS-enabled communication systems by establishing a mapping relationship between channel state information (CSI) and objective function, e.g., sum-rate, energy efficiency, optimal precoders..
| BAC | BAC | BAC | BAC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> By Lemma 3.3, the class of (ISK4𝐼𝑆subscript𝐾4ISK_{4}italic_I italic_S italic_K start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, wheel)-free graphs is 2-cutset-safe. Therefore, by Theorem 2.5, we may assume that G𝐺Gitalic_G does not have a clique cutset or a proper 2-cutset. Now we apply Theorem 3.1. <|MaskedSetence|> Suppose G𝐺Gitalic_G is the line graph of a chordless graph H𝐻Hitalic_H with maximum degree at most three. Then, since G𝐺Gitalic_G does not contain the line graph of a subdivision of the (t×t)𝑡𝑡(t\times t)( italic_t × italic_t )-wall as an induced subgraph, it follows that H𝐻Hitalic_H does not contain a subdivision of the (t×t)𝑡𝑡(t\times t)( italic_t × italic_t )-wall as a subgraph (note that edges become vertices in the line graph). By Lemma 1.1, tw(H)≤f(t)tw𝐻𝑓𝑡\operatorname{tw}(H)\leq f(t)roman_tw ( italic_H ) ≤ italic_f ( italic_t ), and thus by Lemma 3.2, tw(G)≤3f(t)+2tw𝐺3𝑓𝑡2\operatorname{tw}(G)\leq 3f(t)+2roman_tw ( italic_G ) ≤ 3 italic_f ( italic_t ) + 2. If G𝐺Gitalic_G is a complete bipartite graph, then tw(G)≤ttw𝐺𝑡\operatorname{tw}(G)\leq troman_tw ( italic_G ) ≤ italic_t (since G𝐺Gitalic_G is t𝑡titalic_t-clean). Suppose G𝐺Gitalic_G is a long rich square with central square S𝑆Sitalic_S. Then, G∖S𝐺𝑆G\setminus Sitalic_G ∖ italic_S is a forest, so tw(G)≤5tw𝐺5\operatorname{tw}(G)\leq 5roman_tw ( italic_G ) ≤ 5.
This completes the proof.
∎
. | **A**: Let G𝐺Gitalic_G be a t𝑡titalic_t-clean (ISK4𝐼𝑆𝐾4ISK4italic_I italic_S italic_K 4, wheel)-free graph.
**B**: We show that tw(G)≤3f(t)+2tw𝐺3𝑓𝑡2\operatorname{tw}(G)\leq 3f(t)+2roman_tw ( italic_G ) ≤ 3 italic_f ( italic_t ) + 2, where f𝑓fitalic_f is the function from Lemma 1.1.
**C**: If G𝐺Gitalic_G is series-parallel, then tw(G)≤2tw𝐺2\operatorname{tw}(G)\leq 2roman_tw ( italic_G ) ≤ 2 ([6]).
| ABC | BCA | ABC | ABC | Selection 1 |
In this section we work out two toy examples. The special feature of these examples is that explicit necessary and sufficient conditions for finiteness of moments are available. <|MaskedSetence|> Two notable exceptions are Douc et al. (2008) for the ARCH model and Giraitis et al. (2004) for the LARCH model. <|MaskedSetence|> Using our approach, we recover and extend the results of Douc et al. (2008).
4.1. <|MaskedSetence|> | **A**: Note that necessary conditions are rarely discussed in the literature.
**B**: A simple example of order 2..
**C**: The first one has positive coefficients as in the present article.
| ACB | ACB | ACB | ACB | Selection 1 |
In section 2 we describe the basic assumptions and notations of multidimensional Mellin analysis. In section 3
in the setting of the multidimensional Mellin analysis we introduce moduli of continuity and use them to define Besov-Mellin spaces. In section 4 we prove that Besov-Mellin spaces are the interpolation spaces (in the sense of J.Peetre) between two Sobolev-Mellin spaces. <|MaskedSetence|> Sections 7-9 are devoted to the Hilbert case. We introduce and discuss Laplace-Mellin operator in section 7. In section 8 we define relevant Paley-Wiener spaces and in section 9 consider a partition of unity on the spectrum of the Laplace-Mellin operator. In section 10 we describe Besov-Mellin spaces in terms of approximations by Paley-Wiener functions and in terms of some Hilbert frames. The article also contains extensions of the Landau-Kolmogorov-Stein inequality to the Mellin analysis setting. <|MaskedSetence|> <|MaskedSetence|> However, at least some of them are just adaptations and particular realizations of previously obtained (but maybe not very well known) more general results (see [14], [28]-[40], [42]). In this sense the article can be treated as a review paper. Our main goal was to attract attention to multidimensional Mellin analysis, which is currently underdeveloped.. | **A**: To make the paper self contained some related definitions, facts and proofs are collected in Appendices 1-3.
All our statements concerning the multidimensional Mellin analysis seems to be new.
**B**: Throughout the paper we extensively use the Peetre’s theory of interpolation and approximation spaces.
**C**: In section 5 we introduce Bernstein-Mellin spaces and in section 6 prove corresponding direct and inverse approximation theorems including a Jackson-type inequality.
| CBA | CBA | CBA | BAC | Selection 3 |
For the definition of relative bi-exactness, the reader is referred to Definition 15.1.2 of [2]. Note that the ‘if’ direction of Theorem 1.1 is not a weak version of Conjecture 1.2. <|MaskedSetence|> Proposition 2.2) and uses two technologies related to relatively hyperbolic groups. <|MaskedSetence|> <|MaskedSetence|> It is worth noting that the Mineyev-Yaman’s bicombing of relatively hyperbolic groups alone is not sufficient to derive Conjecture 1.2 for the reason explained in Remark 3.7. Therefore, Conjecture 1.2 is still considered open.. | **A**: The second one is based on the notion of separating cosets of hyperbolically embedded subgroups, which was introduced by Hull and Osin in [8] and developed further by Osin in [12].
We will construct two arrays using each of these techniques and combine them to make a proper array.
**B**: The first one is a bicombing of fine hyperbolic graphs, which was constructed by Mineyev and Yaman in [10] using the same idea as in [9].
**C**: Indeed, while the assumption of Theorem 1.1 is stronger, the conclusion is also stronger.
Our proof of Theorem 1.1 is based on a characterization of bi-exactness using exactness and the existence of a proper array (cf.
| CBA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> The existence of root subgroup coordinates is related to presently intractable questions about imaginary roots for 𝔤(A)𝔤𝐴\mathfrak{g}(A)fraktur_g ( italic_A ), such as their multiplicities. <|MaskedSetence|> In the affine case the ordering of imaginary roots is unimportant because the root vectors commute. <|MaskedSetence|> It is possible that commutativity of imaginary root vectors is a characterization of the finite and affine cases.
. | **A**: This is not true in general in the indefinite cases.
**B**:
A preliminary comment: It is not clear whether there are interesting analogues of root subgroup coordinates, and so on, in indefinite cases.
**C**: To produce root subgroup coordinates it is necessary to ‘order’ the roots in some generalized sense, as in the affine case, see (3.3.1).
| BCA | CAB | BCA | BCA | Selection 4 |
<|MaskedSetence|> Since {Qi}i=1qsuperscriptsubscriptsubscript𝑄𝑖𝑖1𝑞\{Q_{i}\}_{i=1}^{q}{ italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT is in N𝑁Nitalic_N-subgeneral position, z𝑧zitalic_z is not zero of more than N𝑁Nitalic_N functions Qi(F)subscript𝑄𝑖𝐹Q_{i}(F)italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_F ). <|MaskedSetence|> <|MaskedSetence|> Choose R1⊂Rsuperscript𝑅1𝑅R^{1}\subset Ritalic_R start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊂ italic_R such that
♯R1=rank{Qi}i∈R1=k+1♯superscript𝑅1ranksubscriptsubscript𝑄𝑖𝑖superscript𝑅1𝑘1\sharp R^{1}=\mathrm{rank}\{Q_{i}\}_{i\in R^{1}}=k+1♯ italic_R start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = roman_rank { italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ italic_R start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_k + 1 and R1superscript𝑅1R^{1}italic_R start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT satisfies Lemma 2.1 v) with respect to numbers {emax{νQi(F)(z)−M,0}}i=1q.\bigl{\{}e^{\max\{\nu_{Q_{i}(F)}(z)-M,0\}}\bigl{\}}_{i=1}^{q}.{ italic_e start_POSTSUPERSCRIPT roman_max { italic_ν start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_F ) end_POSTSUBSCRIPT ( italic_z ) - italic_M , 0 } end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT . Then we have
. | **A**: Put R={1,…,N+1}.𝑅1…𝑁1R=\{1,...,N+1\}.italic_R = { 1 , … , italic_N + 1 } .
**B**: Suppose that z𝑧zitalic_z is zero of Qi(F)subscript𝑄𝑖𝐹Q_{i}(F)italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_F ) for every i=1,…,ℓ(ℓ≤N)𝑖1…ℓℓ𝑁i=1,\ldots,\ell\ (\ell\leq N)italic_i = 1 , … , roman_ℓ ( roman_ℓ ≤ italic_N ) and z𝑧zitalic_z is not zero of Qi(F)subscript𝑄𝑖𝐹Q_{i}(F)italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_F ) for every i>ℓ𝑖ℓi>\ellitalic_i > roman_ℓ.
**C**: Proof.
Fix a point z∈S𝑧𝑆z\in Sitalic_z ∈ italic_S.
| CBA | CBA | ACB | CBA | Selection 4 |
The CLT for the deterministic cocycle is then derived from the corresponding result for the random cocycle by a stopping time argument based in part on an asymptotic estimate due to A. Ancona [Anc90].
We point out that in the setting of products of independent and identically distributed random matrices, the central limit theorem was established, in varying levels of generality, by Bellman in [Bel54], H. <|MaskedSetence|> Kesten in [FK60], Tutubalin in [Tut77], Le Page in [LP82], Y. Guivarc’h and A. Raugi in [GR85], I. Ya. <|MaskedSetence|> A. <|MaskedSetence|> | **A**: Margulis in [GM89], Hennion in [Hen97], Jan in [Jan00], and more recently, and under an optimal finite second moment condition, by Y. Benoist and.
**B**: Golsheid and G.
**C**: Furstenberg and H.
| ACB | CBA | CBA | CBA | Selection 4 |
<|MaskedSetence|> In this case, the quotient is simply (−π2,π2)𝜋2𝜋2(-\frac{\pi}{2},\frac{\pi}{2})( - divide start_ARG italic_π end_ARG start_ARG 2 end_ARG , divide start_ARG italic_π end_ARG start_ARG 2 end_ARG ).
The equivariant eigenfunctions form the standard basis YNm(θ,φ)=2N+1eimθPNm(cosφ)superscriptsubscript𝑌𝑁𝑚𝜃𝜑2𝑁1superscript𝑒𝑖𝑚𝜃superscriptsubscript𝑃𝑁𝑚𝜑Y_{N}^{m}(\theta,\varphi)=\sqrt{2N+1}e^{im\theta}P_{N}^{m}(\cos\varphi)italic_Y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( italic_θ , italic_φ ) = square-root start_ARG 2 italic_N + 1 end_ARG italic_e start_POSTSUPERSCRIPT italic_i italic_m italic_θ end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( roman_cos italic_φ ). <|MaskedSetence|> <|MaskedSetence|> The complex line bundle Lmsubscript𝐿𝑚L_{m}italic_L start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT is the product bundle (−π2,π2)×ℂ𝜋2𝜋2ℂ(-\frac{\pi}{2},\frac{\pi}{2})\times{\mathbb{C}}( - divide start_ARG italic_π end_ARG start_ARG 2 end_ARG , divide start_ARG italic_π end_ARG start_ARG 2 end_ARG ) × blackboard_C. In the. | **A**: Here, PNmsuperscriptsubscript𝑃𝑁𝑚P_{N}^{m}italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT are the standard
associated Legendre polynomials and (3.2) is the m𝑚mitalic_mth Legendre operator.
**B**: The real part of YNmsuperscriptsubscript𝑌𝑁𝑚Y_{N}^{m}italic_Y start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT is 2N+1PNm(cosφ)cosmθ2𝑁1superscriptsubscript𝑃𝑁𝑚𝜑𝑚𝜃\sqrt{2N+1}P_{N}^{m}(\cos\varphi)\cos m\thetasquare-root start_ARG 2 italic_N + 1 end_ARG italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( roman_cos italic_φ ) roman_cos italic_m italic_θ.
Its nodal set is the union of the nodal sets {PNm=0}∪{cosmθ=0}superscriptsubscript𝑃𝑁𝑚0𝑚𝜃0\{P_{N}^{m}=0\}\cup\{\cos m\theta=0\}{ italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT = 0 } ∪ { roman_cos italic_m italic_θ = 0 }, and it has a singular set of mN𝑚𝑁mNitalic_m italic_N points given by the Cartesian product of the zero set of cosmφ𝑚𝜑\cos m\varphiroman_cos italic_m italic_φ and the zero set of PNm(cosφ)superscriptsubscript𝑃𝑁𝑚𝜑P_{N}^{m}(\cos\varphi)italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( roman_cos italic_φ ).
**C**:
The most familiar case n=1𝑛1n=1italic_n = 1 is very special and we pause to contrast it with the case n>1𝑛1n>1italic_n > 1.
| CAB | CAB | CAB | CBA | Selection 2 |
Also known as potential.
Before finishing the section, we mention two main Tauberian theorems needed later on. <|MaskedSetence|> 127] and then Graham & Vaaler’s theorem [11, p. 294] which is just a refinement of the Ikehara-Wiener theorem. <|MaskedSetence|> We know that PNT was proved in the late 19th century. <|MaskedSetence|> 127].. | **A**: The motivation for the Ikehara-Wiener theorem was to provide a simpler proof of the prime number theorem.
**B**: First Ikehara & Wiener’s theorem [38, p.
**C**: However, Ikehara & Wiener used a theorem of Wiener to obtain the following result in the early 1930s that implies PNT [20, p.
| BAC | BAC | ABC | BAC | Selection 2 |
<|MaskedSetence|> This arrangement generates a supercurrent that flows through this device known as the Josephson junction. <|MaskedSetence|> They are candidates for developing quantum computing devices. <|MaskedSetence|> The standard model for describing this coupling is the resistively shunted junction (RSJ) model [37],. | **A**:
The Josephson effect is a phenomenon that occurs when two or more superconductors are placed next to each other, with some potential barrier or restriction between them.
**B**: There are ways to build superconducting circuits based on the Josephson junction.
**C**: As an example of this type, we have the superconducting quantum interference device (SQUID) [37].
In this context, the role of tunneling coordinate across the Josephson junction is played by the phase difference of the Cooper pair wave function.
| ABC | CAB | ABC | ABC | Selection 3 |
Although we have tried to give a self-contained presentation, we need to assume some familiarity with the foundations of non-commutative probability theory as well as the theory of Hopf algebras. The reader who would like to see more details on non-commutative probability and related topics is referred to [6, 18, 20, 29, 30]. <|MaskedSetence|> <|MaskedSetence|> The next section introduces a collection of notations and conventions used throughout the paper. In Section 3, we consider the group structure coming from convolution in the case of classical probability. <|MaskedSetence|> Section 5 recalls the shuffle algebra approach to moment-cumulant relations in non-commutative probability. Sections 6, 7 and 8, discuss the monotone, free and Boolean group structures for states.
. | **A**: Regarding background and details on Hopf and pre-Lie algebras, in particular of combinatorial nature, we refer the reader to the book [8].
The paper is organised as follows.
**B**: Section 4 presents the notion of universal product in the context of non-commutative probability.
**C**: Detailed studies of the notion of universal product in non-commutative probability can be found in [4, 17, 19, 28].
| CAB | ABC | CAB | CAB | Selection 3 |
<|MaskedSetence|> The notion of two-sidedness equally makes sense for skew braces. <|MaskedSetence|> Some of his ideas are pursued further in the present paper, in order to obtain that every two-sided skew brace is an extension of a weakly trivial skew brace by a two-sided brace. Weakly trivial skew braces are a new notion generalizing trivial and almost trivial skew braces and form the main topic of Section 3.
In Section 4, we then obtain our main results. For example, we show that aside from trivial and almost trivial skew braces on simple groups, the only simple two-sided skew braces are infinite two-sided braces. Also, we generalize some results of Jacobson radical rings to two-sided skew braces and improve some of the results obtained in [15] on the connection between the additive and multiplicative groups of two-sided skew braces. In [13] and its appendix, the authors introduced (strongly) prime and (strongly) semiprime skew braces. For two-sided braces, both variations coincide with the usual notion for rings. It is then asked by the authors whether these variations always coincide. <|MaskedSetence|> | **A**: So far, the only paper focusing on two-sided skew braces is due to Nasybullov [15].
**B**: In Section 5 we give an affirmative answer to this question in the case of two-sided skew braces.
.
**C**:
Jacobson radical rings are in bijective correspondence with two-sided braces and have been extensively studied; see for example [1, 2, 22, 24].
| BAC | CAB | CAB | CAB | Selection 4 |
This theorem is stated and proved as Theorem 5.3. Its power can be fully appreciated when one deals with a particular invariant f𝑓fitalic_f. In Section 6 we present several results regarding matroid valuative functions, and tools to prove that an invariant is valuative. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Whereas, in some other cases we just indicate the consequence of the main result for paving or sparse paving matroids — although we believe that computations for cuspidal matroids are feasible, we have given priority to illustrate the method, instead of undergoing long calculations here.
. | **A**: Then, in Sections 7 – 9 we discuss applications of Theorem 1.4 for various invariants.
**B**: These are used in subsequent sections of the paper.
Some subsections are a thorough overview of notions and tools that already appeared in the literature under various names.
**C**: In some cases, we perform the full calculation for cuspidal matroids, hence obtaining a counterpart formula for all elementary split matroids.
| ACB | BAC | BAC | BAC | Selection 3 |
Many pressing questions arise from this new perspective on partition rank. <|MaskedSetence|> There are several obstacles to this achievement. <|MaskedSetence|> <|MaskedSetence|> For instance if the coordinates are themselves are vectors then the value of the tensor might depend on the matroid of these vectors. However, we still believe there are a wide variety of settings in which Theorem 7 applies.
. | **A**: Of the most pertinent is how to apply our main theorems, Theorem 7 and Theorem 8 in more general settings than the ones presented.
**B**: In applying Theorem 7, one needs to construct a tensor that is constant on k𝑘kitalic_k-tuples with the same partition.
**C**: For many tensors that identify combinatorial properties, the value on a k𝑘kitalic_k-tuple depends heavily on the combinatorics of the k𝑘kitalic_k-tuple beyond just which coordinates are equal.
| ACB | ABC | ABC | ABC | Selection 4 |
Theorem 12 states that for estimating two parameters, a strong gap persistence theorem holds for the most informative bound and the HCRB. <|MaskedSetence|> <|MaskedSetence|> It has recently been shown that there are scenarios where the NHCRB cannot be saturated by a separable measurement Hayashi and Ouyang (2022); Conlon et al. (2024). <|MaskedSetence|> | **A**: A natural question this opens up is whether this is true for estimating more than two parameters.
**B**: Another open question is whether a gap persistence theorem holds for the most informative bound and the NHCRB.
**C**: Hence, in these settings, it is possible to have a gap between the NHCRB and the most informative bound.
.
| ABC | ACB | ABC | ABC | Selection 4 |
Carathéodory’s approximation theorem for the open unit disc has been proven using the Pick-Nevanlinna interpolation problem, as shown in [19], which extends seamlessly to the matrix-valued case. Recently, this theorem has also been proven using dilation methods [4] and the Herglotz Representation theorem [6]. <|MaskedSetence|> <|MaskedSetence|> Recently, it has been established that if the Carathéodory approximation theorem holds for the matrix-valued case, then the result can be extended to the operator-valued case using approximation arguments. <|MaskedSetence|> | **A**: The dilation theory and Pick-Nevanlinna interpolation theory have not been explored for all quotient domains, but a version of Herglotz representation has been established; see [6].
**B**: However, the method to prove matrix-valued Carathéodory approximation via Herglotz representation theory does not work in several variables.
Nevertheless, Pick-Nevanlinna interpolation is known for a particular quotient domain, namely, the symmetrized bidisc 𝔾2subscript𝔾2{\mathbb{G}}_{2}blackboard_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.
**C**: Therefore, we will focus on discussing the Carathéodory approximation theorem for the matrix-valued case.
.
| ABC | CBA | ABC | ABC | Selection 4 |
Despite the long tradition of studying realizability and its variants from a categorical perspective, a systematic and in-depth analysis of computability-like reducibilities, such as Turing [33, 38], Medvedev [39, 15], Muchnik [15], and Weihrauch reducibilities [4], from a categorical perspective is still lacking.
A categorical presentation of these notions is useful to highlight some of their abstract structural properties. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In a relative realizability topos, the instance degrees correspond to a generalization of (realizer-based) Weihrauch reducibility, called extended Weihrauch degrees, and the “classical” Weihrauch degrees correspond precisely to the ¬¬\lnot\lnot¬ ¬-dense modest instance degrees in Kleene-Vesley realizability. Upon closer inspection, it is not hard to check that realizer-based Weihrauch reducibility is a particular case of Bauer’s notion.. | **A**: These are two fields that traditionally employ very different languages and notations, so a categorical understanding of reducibility in computability with categorical logic descriptions in terms of doctrines has the positive effect of fostering collaborations between the two communities.
A first approach to Medvedev and Muchnik reducibility via hyperdoctrines has been introduced in [23], while Weihrauch reducibility for assemblies (or multi-represented spaces) has been introduced only very recently in [20, 36] through the notion of realizer-based Weihrauch reducibility.
**B**: Moreover, this work provides a common language for the categorical logic community and the computability theorists.
**C**: In [1], Bauer introduced an abstract notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics.
| BAC | BAC | BCA | BAC | Selection 2 |
Spec(𝒫(S))→Spec(R)→Spec𝒫𝑆Spec𝑅\operatorname{Spec}(\mathcal{P}(S))\rightarrow\operatorname{Spec}(R)roman_Spec ( caligraphic_P ( italic_S ) ) → roman_Spec ( italic_R ) given by M↦M∗maps-to𝑀superscript𝑀∗M\mapsto M^{\ast}italic_M ↦ italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is injective. Suppose M∗=N∗superscript𝑀∗superscript𝑁∗M^{\ast}=N^{\ast}italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. <|MaskedSetence|> Then a∗=S∖A∉Msuperscript𝑎∗𝑆𝐴𝑀a^{\ast}=S\setminus A\notin Mitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_S ∖ italic_A ∉ italic_M thus a∈M∗=N∗𝑎superscript𝑀∗superscript𝑁∗a\in M^{\ast}=N^{\ast}italic_a ∈ italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT and so a∗∉Nsuperscript𝑎∗𝑁a^{\ast}\notin Nitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∉ italic_N. <|MaskedSetence|> It follows that M⊆N𝑀𝑁M\subseteq Nitalic_M ⊆ italic_N and so M=N𝑀𝑁M=Nitalic_M = italic_N. This map is also continuous, because the inverse image of D(a)𝐷𝑎D(a)italic_D ( italic_a ) under this map equals V(a∗)=D(1−a∗)𝑉superscript𝑎∗𝐷1superscript𝑎∗V(a^{\ast})=D(1-a^{\ast})italic_V ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = italic_D ( 1 - italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ).
Finally, we show that this map preserves wild primes. If M𝑀Mitalic_M is a wild prime (i.e., non-principal maximal ideal) of 𝒫(S)𝒫𝑆\mathcal{P}(S)caligraphic_P ( italic_S ), then {k}∈M𝑘𝑀\{k\}\in M{ italic_k } ∈ italic_M for all k∈S𝑘𝑆k\in Sitalic_k ∈ italic_S. For the k𝑘kitalic_k-th unit idempotent eksubscript𝑒𝑘e_{k}italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT we have (ek)∗=S∖{k}∉Msuperscriptsubscript𝑒𝑘∗𝑆𝑘𝑀(e_{k})^{\ast}=S\setminus\{k\}\notin M( italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_S ∖ { italic_k } ∉ italic_M thus ek∈M∗subscript𝑒𝑘superscript𝑀∗e_{k}\in M^{\ast}italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT and so ⨁k∈SRk⊆M∗subscriptdirect-sum𝑘𝑆subscript𝑅𝑘superscript𝑀∗\bigoplus\limits_{k\in S}R_{k}\subseteq M^{\ast}⨁ start_POSTSUBSCRIPT italic_k ∈ italic_S end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊆ italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. Then by Corollary 2.3, M∗superscript𝑀∗M^{\ast}italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is a wild prime of R𝑅Ritalic_R. Conversely, let M∗superscript𝑀∗M^{\ast}italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT be a wild prime of R𝑅Ritalic_R where M𝑀Mitalic_M is a maximal ideal of 𝒫(S)𝒫𝑆\mathcal{P}(S)caligraphic_P ( italic_S ). If M𝑀Mitalic_M is a tame prime of 𝒫(S)𝒫𝑆\mathcal{P}(S)caligraphic_P ( italic_S ) then M=𝒫(S∖{k})𝑀𝒫𝑆𝑘M=\mathcal{P}(S\setminus\{k\})italic_M = caligraphic_P ( italic_S ∖ { italic_k } ) for some k∈S𝑘𝑆k\in Sitalic_k ∈ italic_S. <|MaskedSetence|> | **A**: This yields that A∈N𝐴𝑁A\in Nitalic_A ∈ italic_N.
**B**: It follows that M∗=πk−1(𝔭k)superscript𝑀∗subscriptsuperscript𝜋1𝑘subscript𝔭𝑘M^{\ast}=\pi^{-1}_{k}(\mathfrak{p}_{k})italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( fraktur_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) which is a contradiction.
∎
.
**C**: If A∈M𝐴𝑀A\in Mitalic_A ∈ italic_M then consider the element a=(ai)𝑎subscript𝑎𝑖a=(a_{i})italic_a = ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) where ai=1subscript𝑎𝑖1a_{i}=1italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 for all i∈A𝑖𝐴i\in Aitalic_i ∈ italic_A and otherwise ai=0subscript𝑎𝑖0a_{i}=0italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0.
| CAB | BAC | CAB | CAB | Selection 4 |
Different from the existing works which mainly focus on the EE maximization [6] and sum rate maximization [5, 8] of BAC-NOMA schemes, this paper considers the delay minimization problem of a hybrid BAC-NOMA assisted MEC offloading scenario. <|MaskedSetence|> Since the existence of FD interference, those BDs may not be able to finish offloading when the downlink transmission is active. <|MaskedSetence|> <|MaskedSetence|> By comparing with benchmarks, simulation results demonstrate that the proposed protocol can significantly reduce the offloading latency.
. | **A**: Therefore, a second time duration is scheduled to continue the offloading by using the NOMA uplink transmission.
**B**: In particular, the signal of the downlink transmission can excite the circuit of BDs to upload the tasks to the MEC server.
**C**: By applying the Dinkelbach method and quadratic transformation, an iterative based algorithm is proposed to obtain the sub-optimal solution to the original non-convex problem.
| BAC | BAC | BAC | CBA | Selection 2 |
n𝑛nitalic_n. Let a∈𝕂((x))[y]/(f)𝑎𝕂𝑥delimited-[]𝑦𝑓a\in\mathbb{K}((x))[y]/(f)italic_a ∈ blackboard_K ( ( italic_x ) ) [ italic_y ] / ( italic_f ) be represented by A∈𝕂((x))[y]<n𝐴𝕂𝑥subscriptdelimited-[]𝑦absent𝑛A\in\mathbb{K}((x))[y]_{<n}italic_A ∈ blackboard_K ( ( italic_x ) ) [ italic_y ] start_POSTSUBSCRIPT < italic_n end_POSTSUBSCRIPT. We regard 𝕂((x))[y]/(f)𝕂𝑥delimited-[]𝑦𝑓\mathbb{K}((x))[y]/(f)blackboard_K ( ( italic_x ) ) [ italic_y ] / ( italic_f ) as a
𝕂((x))𝕂𝑥\mathbb{K}((x))blackboard_K ( ( italic_x ) )-algebra of dimension n𝑛nitalic_n. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: We define the characteristic
polynomial of a𝑎aitalic_a to be the characteristic polynomial χ∈𝕂((x))[t]𝜒𝕂𝑥delimited-[]𝑡\chi\in\mathbb{K}((x))[t]italic_χ ∈ blackboard_K ( ( italic_x ) ) [ italic_t ] of the multiplication endomorphism by a𝑎aitalic_a in 𝕂((x))[y]/(f)𝕂𝑥delimited-[]𝑦𝑓\mathbb{K}((x))[y]/(f)blackboard_K ( ( italic_x ) ) [ italic_y ] / ( italic_f ), so we have χ(a)=0𝜒𝑎0\chi(a)=0italic_χ ( italic_a ) = 0.
**B**: The minimal polynomial
of a𝑎aitalic_a is the monic polynomial μ∈𝕂((x))[t]𝜇𝕂𝑥delimited-[]𝑡\mu\in\mathbb{K}((x))[t]italic_μ ∈ blackboard_K ( ( italic_x ) ) [ italic_t ] of smallest
degree in t𝑡titalic_t such that μ(a)=0𝜇𝑎0\mu(a)=0italic_μ ( italic_a ) = 0.
**C**: Since 𝕂((x))[y]/(f)𝕂𝑥delimited-[]𝑦𝑓\mathbb{K}((x))[y]/(f)blackboard_K ( ( italic_x ) ) [ italic_y ] / ( italic_f ) is a.
| BAC | BAC | CBA | BAC | Selection 4 |
<|MaskedSetence|> HINTS is fast, accurate, and widely applicable to diverse classes of differential equations. The method is agnostic in terms of the differential equations, computational domains (shape and dimension), and discretization.
We conducted a systematic study on the performance of HINTS-Jacobi, where we examined why and how this hybrid solver functions. <|MaskedSetence|> Such a combination, instead of substitution, incorporates advantages from both ends, improving the performance beyond individual solvers. A hybrid methodology, instead of starting from scratch, retains the advantages from numerical solvers that have been studied for centuries as well as from DeepONet as a powerful operator approximator.
In Section S3.5, we show inferior results where the DeepONet is replaced by the k-nearest neighbor with Gaussian kernel. <|MaskedSetence|> | **A**: While the DeepONet is trained with a finite dataset, the setup of HINTS enables the generalization into infinite test cases with the preservation of convergence to machine zero..
**B**: We have demonstrated the capability of HINTS in solving linear differential equations.
**C**: In summary, the embedded DeepONet solver provides fast, approximate solutions for low-frequency eigenmodes in the solution, while the embedded relaxation method handles the remaining higher-frequency eigenmodes, guaranteeing the convergence of the solver.
| ABC | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> If there is no statistical dependence whatsoever, then the conditional entropy reduces to the usual entropy. However, if there is dependence, then there is a reduction in the uncertainty of the first particle when given access to the second. <|MaskedSetence|> Since conditional entropy is never negative in classical information science, this situation represents a radical departure from the classical case and has been popularly described as “knowing less than nothing” [5].
. | **A**: In the classical case, conditional entropy is meant to capture the uncertainty about the state of one particle given access to a second particle that is potentially statistically dependent on the first [3].
**B**:
Given this peculiar situation, one of the earliest puzzles in quantum information science was to devise a quantum generalization of the concept of conditional entropy.
**C**: When generalizing the definition of conditional entropy in a straightforward way to the quantum case (by means of this reduction), one finds that quantum conditional entropy is negative when evaluated for the EPR state mentioned above [4].
| BAC | BAC | ACB | BAC | Selection 4 |
Concerning the intercritical range on nonlinearities, the energy scattering was proved by Akahora, Kikuchi, and Nawa in [4], and as for the already mentioned papers, the scattering for such equations is proved by means of a Kenig-Merle approach. <|MaskedSetence|> <|MaskedSetence|> The main novelty in [4] is that they replace a usual monotonicity condition (which is in fact not satisfied by combined power nonlinearities) with weaker assumptions. Then, in order to prove a desired lower bound on the Pohozaev functional G𝐺Gitalic_G, see (1.5) below, Akahori, Kikuchi, and Nawa distinguish different cases depending on sign of G𝐺Gitalic_G along minimizing sequences. <|MaskedSetence|> | **A**: Specifically, the tool given by [4, Lemma 3.3] enables them to employ a concentration-compacteness and rigidity scheme.
.
**B**: Among them, one also finds the NLS equation with combined nonlinearities with intercritical exponents.
**C**: In fact, in [4] a wide class of NLS-type equations is treated.
| CBA | CBA | CBA | BCA | Selection 3 |
<|MaskedSetence|> There, the authors showed that a “generic” motion polynomial could be written in finitely many ways as a product of monic linear motion polynomials. <|MaskedSetence|> For instance, using different factorizations of a cubic motion polynomial, the authors in [9] found a new family of paradoxically mobile closed-loop linkages with six revolute axes. Notice that the family contained the well-known Bricard Octahedra of type three [7] as a sub-class. Computational design issues were the topic of [10]. In [6, 20], the factorization of bounded motion polynomials was used for the construction of rational Kempe linkages. <|MaskedSetence|> Moreover, in [24], it was used to synthesize single-loop variable-degree-of-freedom mechanisms.
. | **A**: The potential for applications in architectural design is studied in [15].
**B**:
The first publication on motion polynomial factorization is [9].
**C**: They also related these factorizations to a decomposition of rational motions into coupled rotations and converted it into the construction of mechanical linkages.
| BCA | ABC | BCA | BCA | Selection 3 |