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Acknowledgements. <|MaskedSetence|> He is very grateful to the Radcliffe Institute for providing excellent working conditions. TT is supported by NSF Research Award DMS-0649473, the NSF Waterman award and a grant from the MacArthur Foundation. <|MaskedSetence|> All three authors are very grateful to the University of... | **A**: Basic notation.
**B**: TZ is supported by ISF grant 557/08, an Alon fellowship and a Landau fellowship of the Taub foundation.
**C**: BG was, for some of the period during which this work was carried out, a fellow of the Radcliffe Institute at Harvard.
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<|MaskedSetence|> <|MaskedSetence|> Doing this directly using (1.10) amounts
to solving a system of 123123123123 equations and 123 variables. Due to this we take an
alternative strategy. <|MaskedSetence|> We do not provide proofs for. | **A**: The first part of this procedure is similar to the one used
in [GGV1]*Section 8, and is inspired by [M].
**B**: been already be verified independently in [H] and [GGV2].
**C**: We will verify the
case (m,n)=(50,75)𝑚𝑛5075(m,n)=(50,75)( italic_m , italic_n ) = ( 50 , 75 ).
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To deal with the case where G=F4𝐺subscript𝐹4G=F_{4}italic_G = italic_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT and ⟨v⟩delimited-⟨⟩𝑣\langle v\rangle⟨ italic_v ⟩ has stabiliser of type D4subscript𝐷4D_{4}italic_D start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, we perform a similar calculation with 𝔤[t]𝔤delimited-[]𝑡\ma... | **A**: Thus the rank of the matrix is 24242424 in all characteristics for all choices of t≠0,1𝑡01t\neq 0,1italic_t ≠ 0 , 1, and we are done.
∎
.
**B**: It turns out that 28282828 of these rows are identically zero and so the rank will not change after they are removed.
**C**: Form M𝑀Mitalic_M as before to get a 26×... | CBA | CAB | CBA | CBA | Selection 4 |
The referee’s contribution is significant in this paper. Some explicitly appear in the text, but others got hidden in the revision process. The contributions of the latter nature include the following. (1) The author originally worked exclusively with motivic cohomology with compact supports. It was the referee who ex... | **A**: Most importantly, Definition 2.3 and Axiom 2.6 are due to the referee.
**B**: The use of the symmetric power construction and [Mil86, Theorem 3.13] in the proof of Proposition 3.4 is also due to the referee.
**C**: (3) The definition of unpointed regular homomorphisms is due to the referee (Definition 3.7).
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1.4. Some observations
There are several applications of Inou-Shishikura’s invariant class. The first remarkable application is that Buff and Chéritat used it as one of the main tools to prove the existence of Julia sets of quadratic polynomials with positive area [BC12]. <|MaskedSetence|> <|MaskedSetence|> <|Mask... | **A**: In [Che13] and [Che19], Cheraghi developed several elaborate analytic techniques based on Inou-Shishikura’s results.
**B**: Recently, Cheraghi and his collaborators have found several other important applications.
**C**: The tools in [Che13] and [Che19] have led to part of the recent major progresses on the dy... | BAC | ABC | BAC | BAC | Selection 1 |
The paper is organized as follows. <|MaskedSetence|> In Section 3 we prove the local index theorem for families of ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operators on Riemann orbisurfaces that are factors of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups. <|MaskedSetence|> In Se... | **A**: Finally, in Section 4.3 we give a simple example of a relation between the elliptic metric and special values of Selberg zeta
function for Fuchsian groups of signature (0;1;2,2,2)..
**B**: Section 2 contains the necessary background material.
**C**: Specifically,
we show that the contribution to the local inde... | BCA | BCA | ABC | BCA | Selection 1 |
<|MaskedSetence|> Realizing Harmonic Oscillators With Coupled Supersymmetry
As previously noted, the quantum mechanical harmonic oscillator is a specific instance of a coupled supersymmetry, albeit a somewhat trivial case in which the two coupled SUSY equations are identical. This is not the only manner in which the ... | **A**:
7.
**B**: they satisfy the same Lie algebra and by virtue of Stone-von Neumann, may be realized in some way as harmonic oscillators.
**C**: Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e.
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Throughout the text G𝐺Gitalic_G will denote a group, usually abelian and often ordered, ℭℭ\mathfrak{C}fraktur_C will denote a sufficiently saturated model of Th(G)Th𝐺\mathrm{Th}(G)roman_Th ( italic_G ). <|MaskedSetence|> We will need a few results from [27]. Since this text is not readily available, we try to keep ... | **A**: The next sub-section is dedicated to a quick overview of (parts) of the language we are using, and to the basic properties of definable sets.
2.1.
**B**: By definable we will mean definable with parameters.
**C**: In particular, for the study of ordered abelian groups we chose the language of [4], rather th... | CBA | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Here in this paper, in the presence of rough coefficients, spectral techniques are employed to overcome such hurdle, and by solving local eigenvalue problems we define a space where the exponential decay of solutions is insensitive to high-contrast coefficients. Additionally, the s... | **A**: We note the proposal in [CHUNG2018298] of generalized multiscale finite element methods based on eigenvalue problems inside the macro elements, with basis functions with support weakly dependent of the log of the contrast.
**B**: When LOD or VMS methods are considered, high-contrast coefficients might slow down... | CBA | BCA | CBA | CBA | Selection 4 |
The seminal work of Artzner et al. (1999) has bestowed upon the field of risk assessment a set of four pivotal axioms that stand as the cornerstones of coherence for any reputable risk measure. Building upon this foundational framework, Föllmer and Schied (2002), in tandem with the pioneering efforts of Frittelli and R... | **A**: Additionally, the introduction of dynamic convex risk measures by Detlefsen and Scandolo (2005) further enriched the field, providing insights into the time consistency properties of risk measures over different time horizons..
**B**: (2022) defined a new multivariate conditional Value-at-Risk
risk measure base... | ACB | CBA | CBA | CBA | Selection 2 |
Let q𝑞qitalic_q be any point in π−1(p)superscript𝜋1𝑝\pi^{-1}(p)italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_p ) of ramification index eqsubscript𝑒𝑞e_{q}italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. <|MaskedSetence|> If any σ∈Γq𝜎subscriptΓ𝑞\sigma\in\Gamma_{q}italic_σ ∈ roman_Γ sta... | **A**: Therefore, by the condition (5.1), we can choose a formal coordinate system z′,z′′superscript𝑧′superscript𝑧′′z^{\prime},z^{\prime\prime}italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_z start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT around the nodal point q𝑞qitalic_q such that 𝒪^Σ,q≃ℂ[[z′,z′′]]... | CBA | BAC | CBA | CBA | Selection 3 |
The second question regards how a subsystem of anyons in a many-anyon system may be characterized by certain good quantum numbers. This is an important question because often (e.g., in topological quantum computing) we may be concerned about only the entanglement between such subsystems and ignore what is inside each ... | **A**: Recall that the Hilbert space of a fermionic or bosonic system is taken as a Fock space, which is the tensor product of the local Hilbert spaces of single-particle states.
**B**: To study the Hilbert space structure of a system of non-Abelian anyons, however, there demands a new formulation of the basis, as man... | ABC | ABC | ABC | ACB | Selection 2 |
<|MaskedSetence|> We may assume that both Wλ,n0subscriptsuperscript𝑊0𝜆𝑛W^{0}_{\lambda,n}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT and Wλ,n0subscriptsuperscript𝑊0𝜆𝑛W^{0}_{\lambda,n}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTS... | **A**: Let ϵ∈{0,1}italic-ϵ01\epsilon\in\{0,1\}italic_ϵ ∈ { 0 , 1 } be such that sup(Wλ,nϵ)=max{sup(Wλ,n0),sup(Wλ,n1)}supsubscriptsuperscript𝑊italic-ϵ𝜆𝑛supsubscriptsuperscript𝑊0𝜆𝑛supsubscriptsuperscript𝑊1𝜆𝑛\operatorname{sup}(W^{\epsilon}_{\lambda,n})=\max\{\operatorname{sup}(W^{0}_{%
\lambda,n}),\operatorna... | CAB | BCA | CAB | CAB | Selection 1 |
1.6. <|MaskedSetence|> This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’. <|MaskedSetence|> The work of both authors has also been supported in part by the Simons Foundation. <|MaskedSetence|> | **A**: Acknowledgements.
We thank the referees for the careful reading of the first version of the text and for many helpful remarks.
**B**: The results of Section 4 has been obtained under support of the RSF grant 19-11-00056.
**C**: The first author is a Young Russian Mathematics award winner and would like to tha... | ABC | ABC | CBA | ABC | Selection 2 |
Let now nk+1=msubscript𝑛𝑘1𝑚n_{k+1}=mitalic_n start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = italic_m. We show that all conditions (0),(1),(2),(3),(4) will be valid for (rnk+1)subscriptsuperscript𝑟𝑘1𝑛(r^{k+1}_{n})( italic_r start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n... | **A**: The first step was to include ak+1subscript𝑎𝑘1a_{k+1}italic_a start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT hence (4) is valid too.
.
**B**: (2) is valid because we keep all previous ni(1≤i≤k)subscript𝑛𝑖1𝑖𝑘n_{i}\ (1\leq i\leq k)italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 ≤ italic_i ... | ABC | CBA | CBA | CBA | Selection 3 |
In [7], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [3]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structur... | **A**: In section 3, we define the asymptotic inductive dimension of coarse proximity spaces and show that it is an invariant within the category of coarse proximity spaces.
**B**: In [9], the authors construct a functor from the category of coarse proximity spaces to the category of compact Hausdorff spaces that assi... | BCA | BCA | BCA | BCA | Selection 2 |
point in the domain (i.e.,formulae-sequence𝑖𝑒i.e.,italic_i . <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> italic_e . , Σn=1Nnϕn(𝐫)=Σn=1Nnψn(𝐫)=1subscript𝑁𝑛𝑛1Σsubscriptitalic-ϕ𝑛𝐫subscript𝑁𝑛𝑛1Σsubscript𝜓𝑛𝐫1\overset{N_{n}}{\underset{n=1}{\Sigma}}\phi_{n}(\textbf{$\mathbf{r}$})=%
\overset{N_{... | **A**: , at non-nodal locations, as well as at
nodal locations), is equal to one.
**B**: This property also hold for the
pyramid side functions, i.e.,formulae-sequence𝑖𝑒i.e.,italic_i .
**C**: italic_e .
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<|MaskedSetence|> <|MaskedSetence|> Let x∈ℳ×∩𝒱1𝑥superscriptℳsubscript𝒱1x\in{\mathscr{M}}^{\times}\cap{\mathscr{V}}_{1}italic_x ∈ script_M start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT ∩ script_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Then xv1v2=1𝑥subscript𝑣1subscript𝑣21xv_{1}v_{2}=1italic_x italic_v start_POS... | **A**: Going to Λ/Λ1ΛsubscriptΛ1\Lambda/\Lambda_{1}roman_Λ / roman_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we see that the image of v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in Λ/Λ1ΛsubscriptΛ1\Lambda/\Lambda_{1}roman_Λ / roman_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is 1111.
**B**: Clai... | BAC | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> Such refinement of nefness was also given in a recent work [T22] by a different method. <|MaskedSetence|> We note that the approach of the current paper is not ‘similar’, as opposed to a mention in [T22]. . | **A**:
Remark.
**B**: Theorem 1.1 in particular recovers nefness of J𝐽Jitalic_J in [Ka98] by Proposition 2.1.
**C**: 222We mention that the current paper first appeared on arXiv and [T22] was received by a journal, both in October 2019, whereas a manuscript of the current paper was brought to the attention of th... | BAC | ABC | ABC | ABC | Selection 4 |
(2)⇒(3)⇒23(2)\Rightarrow(3)( 2 ) ⇒ ( 3 ). Fix M𝑀Mitalic_M and b𝑏bitalic_b as in (3). <|MaskedSetence|> <|MaskedSetence|> As N𝑁Nitalic_N is prime over Mb𝑀𝑏Mbitalic_M italic_b, we may assume N⪯N∗precedes-or-equals𝑁superscript𝑁N\preceq N^{*}italic_N ⪯ italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. <|Mas... | **A**: For the final sentence, let N∗superscript𝑁N^{*}italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT be any model dominated by b𝑏bitalic_b over M𝑀Mitalic_M.
**B**: But, as N∗superscript𝑁N^{*}italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is minimal over
Mb𝑀𝑏Mbitalic_M italic_b by Proposition 5.14(2)... | ACB | CAB | CAB | CAB | Selection 4 |
Outside of the finite type settings, the singularities of finite-dimensional Schubert varieties in the affine Grassmannian, in type A𝐴Aitalic_A as well as all types, have been widely studied, see e.g., [MOV05, KL+09, BM10, HR20]. For Schubert varieties in general Kac-Moody flag varieties, a criterion for smoothness is... | **A**: In the thesis [Bru14], Brunson investigated (set-theoretic) equations for finite-dimensional Schubert conditions in certain matrix coordinates for the affine Grassmannian, extending the work of [KLMW04].
**B**: The authors would like to thank Shiliang Gao, Allen Knutson, Mark Shimozono, and Alex Yong for helpfu... | ACB | ACB | ABC | ACB | Selection 4 |
Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> On the other hand, in ou... | **A**: In the usual story of moduli spaces of wild harmonic bundles over a punctured compact Riemann surface, one fixes the singular part of the Higgs field and the parabolic structure at the punctures.
**B**: Furthermore, we will have the additional data of a “framing”.
**C**: Under certain stability conditions, one... | ACB | CBA | ACB | ACB | Selection 3 |
Broadly speaking, our work is related to a vast body of work on value-based reinforcement learning in tabular (Jaksch et al., 2010; Osband et al., 2014; Osband and Van Roy, 2016; Azar et al., 2017; Dann et al., 2017; Strehl et al., 2006; Jin et al., 2018) and linear settings (Yang and Wang, 2019b, a; Jin et al., 2019;... | **A**: In comparison, we focus on policy-based reinforcement learning, which is significantly less studied in theory.
**B**: Also, our setting is related to the low-Bellman-rank setting studied by Jiang et al.
**C**: (2017); Dong et al.
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The conjecture has only been proven for various special cases in very specific settings. <|MaskedSetence|> <|MaskedSetence|> In this paper, we study the Ehrhart volume conjecture. <|MaskedSetence|> The main idea that goes into the disprove pertains to a certain construction of a ball in ℝnsuperscriptℝ𝑛\mathbb{R}^{... | **A**: For instance, Ehrhart proved the conjecture in the two dimensional case and for simplices [11].
**B**: We show that the claimed inequality fails for some convex bodies, providing a counter example to the Ehrhart volume conjecture.
**C**: The conjecture has also been settled for a large class of rational polyto... | ACB | ABC | ACB | ACB | Selection 4 |
Plan of the paper
In §2, we discuss mirror symmetry for Fano varieties and give the definitions needed to state it precisely and recall basic definitions and constructions for quiver flag varieties. In §3, we introduce the class of Y𝑌Yitalic_Y-shaped quiver flag varieties, and describe a SAGBI basis for sections of... | **A**: Finally, we give one example of a degeneration and mirror outside of the family of Y𝑌Yitalic_Y-shaped quiver flag varieties: here, the degeneration is to a bound ladder quiver.
**B**: In §4, we define ladder diagrams and ladder quivers for Y𝑌Yitalic_Y-shaped quiver flag varieties, and prove that the SAGBI deg... | BAC | BAC | BAC | BAC | Selection 2 |
For the curvature equations in classical geometry, the existence of hypersurfaces with prescribed Weingarten curvature was studied by Pogorelov [40], Caffarelli-Nirenberg-Spruck [4, 5], Guan-Guan [18], Guan-Ma [19] and the later work by Sheng-Trudinger-Wang [42]. The Hessian equation on Riemannian manifolds was also st... | **A**: For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition, we refer to the book [15].
**B**: Also, we recommend the recent book written by Lieberman [33] for the Neumann and the oblique derivative problems of linear and quasilinear elliptic equations.
**C**: Xu got th... | ABC | ABC | BAC | ABC | Selection 4 |
2.2 Injective (Hyperconvex) metric spaces
A hyperconvex metric space is one where any collection of balls with non-empty pairwise intersections forces the non-empty intersection of all balls. <|MaskedSetence|> Isbell [52] proved that every metric space admits a smallest hyperconvex hull (cf. <|MaskedSetence|> <|Ma... | **A**: These were studied by Aronszajn and Panitchpakdi [8] who showed that every hyperconvex space is an absolute 1-Lipschitz
retract.
**B**: the definition of tight span below).
**C**: Dress rediscovered this concept in [31] and subsequent work provided much development in the context of phylogenetics [77, 32].
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The subtlety which is often glossed over in the context of classical differential geometry, where all the mappings are assumed
smooth, or at least twice or thrice continuously differentiable, is that, due to nonlinearity, the passage from say a given set of first order equations, e.g. <|MaskedSetence|> <|MaskedSetenc... | **A**: the system of isometric immersion equations in our example, to the higher order equations requires a minimum of regularity.
**B**: There is no guarantee that in the absence of this regularity the geometric information hidden in the higher order equations would be accessible.
**C**: To quote Picard111[53, Page ... | ABC | BAC | ABC | ABC | Selection 3 |
<|MaskedSetence|> Take V⊂Xm𝑉superscript𝑋𝑚V\subset X^{m}italic_V ⊂ italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT an open subset. Then we can write
V=U1×⋯×Um𝑉subscript𝑈1⋯subscript𝑈𝑚V=U_{1}\times\dots\times U_{m}italic_V = italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × ⋯ × italic_U start_POSTSUBS... | **A**: 3 φσsubscript𝜑𝜎\varphi_{\sigma}italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT is an open map.
**B**: Take
Vσ=Uσ(1)×⋯×Uσ(m)subscript𝑉𝜎subscript𝑈𝜎1⋯subscript𝑈𝜎𝑚V_{\sigma}=U_{\sigma(1)}\times\dots\times U_{\sigma(m)}italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_U start_POSTSU... | ABC | ABC | ABC | CBA | Selection 3 |
<|MaskedSetence|> only zero in this case) for a single choice of
parameters. In particular, in the case s1=12subscript𝑠112s_{1}=\tfrac{1}{2}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG, the requirement −4V22−(1+2V3)2≥04superscriptsubscript𝑉22superscript12subscri... | **A**: Therefore, in this
algebra in fact there exist only isolated real square root of
𝖡=−𝐞3+𝐞12𝖡subscript𝐞3subscript𝐞12\mathsf{B}=-\mathbf{e}_{3}+\mathbf{e}_{12}sansserif_B = - bold_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT..
**B**: made non-negative (i.e.
**C... | BCA | ABC | BCA | BCA | Selection 4 |
Two major tools for studying stochastic controlled systems are Pontryagin’s maximum principle and Bellman’s dynamic programming. <|MaskedSetence|> Indeed, except for a very few specific cases, the determination of an optimal control (either exact or near) is a highly nontrivial problem to tackle.
In the Markovian c... | **A**: While these two methods are known to be very efficient for establishing some key properties (e.g existence of optimal controls, smoothness of the value functional, sufficiency of subclasses of controls, etc), the problem of solving explicitly or numerically a given stochastic control problem remains a critical i... | ABC | BCA | ABC | ABC | Selection 1 |
There is a quite interesting evolution of constructions to present free groups in a self-similar way or even as automaton groups (see [15] for an overview). This culminated in constructions to present free groups of arbitrary rank as automaton groups where the number of states coincides with the rank [18, 17]. <|Mas... | **A**: While these constructions and the involved proofs are generally deemed quite complicated, the situation for semigroups turns out to be much simpler.
**B**: Here, the main difference is that the free monoid in one generator can indeed be generated by an automaton: it is generated by the adding machine (see 1), w... | ACB | ACB | BCA | ACB | Selection 4 |
<|MaskedSetence|> In Appendix B we recall Knerr’s non-standard parabolic Schauder estimates. In Appendix C we prove that mean curvature flows with bounded curvature and controlled area ratios are unique in the class of Brakke flows. We prove Ilmanen’s localized avoidance principle in Appendix D. <|MaskedSetence|> In ... | **A**: Finally, in Appendix H we localize certain topological monotonicity results..
**B**: Appendix E recalls the non-compact Ecker-Huisken maximum principle.
**C**:
We apply this construction to the study of the mean curvature flow of generic low entropy hypersurfaces in Section 10 and to the study of the first no... | BAC | CBA | CBA | CBA | Selection 2 |
<|MaskedSetence|> In [GiuntiHoefer19], the authors considered zero particle veolcities. <|MaskedSetence|> <|MaskedSetence|> This allows for many clusters of overlapping particles. A corresponding result for the Poisson equation has been obtained in [GiuntiHoferVelazquez18].
. | **A**: with only a (1+β)1𝛽(1+\beta)( 1 + italic_β ) moment bound.
**B**: The particle positions can be distributed to rather general stationary processes, and the radii are i.i.d.
**C**: For a more complete list and discussion of this literature, we refer the reader to [GiuntiHoferVelazquez18, GiuntiHoefer19].
In ... | CBA | CBA | ACB | CBA | Selection 2 |
<|MaskedSetence|> Yau for the constant support and encouragement. <|MaskedSetence|> <|MaskedSetence|> The calculations in Example 7.5 is done together with P. Bousseau at the time.
. | **A**: The author particularly want to thank Pierrick Bousseau for the discussions back in 2016 at MSRI.
**B**: The author would like to thank Man-Wai Cheung, Paul Hacking, Siu-Cheong Lau, Tsung-Ju Lee, Cheuk-Yu Mak for helpful discussions.
**C**: Acknowledgment
The author would like to thank S.T.
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<|MaskedSetence|> In Section 3, we review Devoto’s equivariant elliptic cohomology. In Section 4, we recall the definition of twisted equivariant elliptic cohomology. In Section 5, we construct twisted quasi-elliptic cohomology. <|MaskedSetence|> <|MaskedSetence|> | **A**: In Section 5.3, we define a model of twisted loop space, with which we can construct twisted quasi-elliptic cohomology.
**B**: The theories are computed by applying the properties of quasi-elliptic cohomology theories and equivariant K-theories, especially the conclusions in Appendix A, which are corollaries of... | ACB | BAC | BAC | BAC | Selection 4 |
<|MaskedSetence|> It actually implies DF⟨1,1⟩=∑F˙2subscript𝐷𝐹11superscript˙𝐹2D_{F}\langle 1,1\rangle=\sum\dot{F}^{2}italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ⟨ 1 , 1 ⟩ = ∑ over˙ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [24, Corollary 6.5 (2), p. <|MaskedSetence|> Indeed... | **A**: For a formally real quasi-Pythagorean field F𝐹Fitalic_F, Ware [32, Corollary 1] proved that GF(2)subscript𝐺𝐹2G_{F}(2)italic_G start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( 2 ) is the free pro-2222 product of a free pro-2222 group and a pro-2222 group generated by involutions, provided conditions that hold... | BAC | CBA | CBA | CBA | Selection 3 |
To address such an issue of divergence, nonlinear gradient TD (Bhatnagar et al., 2009) explicitly linearizes the value function approximator locally at each iteration, that is, using its gradient with respect to the parameter as an evolving feature representation. Although nonlinear gradient TD converges, it is unclear... | **A**: Moreover, in contrast to the NTK regime, the induced feature representation is able to deviate from the initial one and subsequently evolve into the globally optimal one, which corresponds to the global minimizer of the MSPBE.
**B**: Going beyond the NTK regime, we prove that, when the value function approximat... | BAC | BCA | BAC | BAC | Selection 4 |
1.7. Acknowledgements
During the writing of the paper, I was supported by the starter grant “Categorified Donaldson–Thomas theory” No. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Finally, I offer my heartfelt gratitude to Paul, Sophia, Sacha, Kristin and Nina, for their help and support throughout the wri... | **A**: I would like to thank Andrei Okounkov and Olivier Schiffmann for helpful conversations, Tristan Bozec for patiently explaining his work on crystals to me, Lucien Hennecart and Shivang Jindal for helpful comments regarding an earlier version of the paper, and an anonymous referee for a careful reading of the pape... | CBA | CBA | ACB | CBA | Selection 2 |
furthermore B→C→𝐵𝐶B\to Citalic_B → italic_C. Apply [33, Corollary
5.14] to A𝐴Aitalic_A and B𝐵Bitalic_B. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This shows that φ𝜑\varphiitalic_φ is closed. | **A**: Finally, B~→B→C→~𝐵𝐵→𝐶\widetilde{B}\to B\to Cover~ start_ARG italic_B end_ARG → italic_B → italic_C, thus C⊧φmodels𝐶𝜑C\models\varphiitalic_C ⊧ italic_φ because φ𝜑\varphiitalic_φ is
closed under homomorphisms.
**B**: Then A~⊧φmodels~𝐴𝜑\widetilde{A}\models\varphiover~ start_ARG italic_A end_ARG ⊧ italic_φ ... | ACB | BCA | BCA | BCA | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For instance, employing Busemann functions on CAT(−1)1(-1)( - 1 ) spaces, Foertsch and Radke [10] characterized complete CAT(κ𝜅\kappaitalic_κ) spaces with κ<0𝜅0\kappa<0italic_κ < 0, with geodesic Hamenstädt boundary up to isometry. Moreover, Foertsch and Schroe... | **A**: The class of Hamenstädt metrics are defined by using Busemann functions, for related definitions and properties see [7, Section 3.3].
**B**: Roughly speaking, a Busemann function on a Gromov hyperbolic space is defined to be the distance function from a point on
the Gromov boundary.
**C**: This notion is very ... | ABC | ABC | ABC | ABC | Selection 1 |
<|MaskedSetence|> Within this generalised framework, the existence of multidimensional scale functions, known as ‘scale matrices’, was first discussed in [19] and were used to derive fluctuation identities and first passage results for continuous-time MAPs. <|MaskedSetence|> Further studies on MAPs and their exit/pas... | **A**: More recently, [17], derive and compare results for continuous-time MAPs with lattice (discrete-space) and non-lattice support.
**B**:
A natural generalisation of the above processes are the broad family of Markov Additive Processes (MAPs), which incorporate an externally influencing Markov environment, provi... | BCA | BAC | BCA | BCA | Selection 1 |
<|MaskedSetence|> has also stimulated substantial mathematical analysis of competition models involving two species. We mention the work of [30, 34, 42, 45] for passive dispersal, and [6, 9, 17, 16, 38, 39] for conditional dispersal. An interesting application concerns the evolution of dispersal in stream populations,... | **A**: In the following, we will address two conjectures of Dockery et al.
**B**: concerning a model involving N𝑁Nitalic_N competing species, which are identical except for the passive dispersal rates..
**C**: The work of Hastings and Dockery et al.
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<|MaskedSetence|> <|MaskedSetence|> Diagrammatically defined chains of algebras appear to have not been considered as objects whose representation category can be studied through the lens of representation stability. <|MaskedSetence|> The chain with respect to which one is considering representation stability there ... | **A**: It appears that much of the work in representation stability has focussed on algebraic objects which are either close to symmetric groups [5] [14] [8] (Wilson, Putman, Sam, Gunturkun, Snowden ) or are close to Lie groups [14] [17] (Sam, Snowden, Putman).
**B**:
To our knowledge, Temperley-Lieb algebras have n... | BAC | BAC | BAC | ABC | Selection 2 |
<|MaskedSetence|> It states that the analysis of sites ("analysis situs") proposed by Gottfried Wilhelm Leibniz (1646-1716) is required to answer the question whether the seven bridges between the four city districts of Königsberg allow for a walk in which each bridge is passed exactly once. <|MaskedSetence|> A city ... | **A**: A tour, or round trip, in which each bridge, or arrow, is passed once and only once has become known as an Eulerian tour.
**B**: The article renowned as the first graph theoretical article [43] is the 1736 article by Leonhard Euler (1707-1783) on a walk over the seven bridges of Königsberg [45].
**C**: It can ... | CBA | BAC | BAC | BAC | Selection 4 |
<|MaskedSetence|> Fmsubscript𝐹𝑚F_{m}italic_F start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT) is associated to the Chow group (or singular cohomology) (resp. K-theory). <|MaskedSetence|> We assume the equivariant cohomology theory 𝕙Tsubscript𝕙𝑇\mathbb{h}_{T}blackboard_h start_POSTSUBSCRIPT italic_T end_POSTSUBSCR... | **A**:
For example, Fasubscript𝐹𝑎F_{a}italic_F start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT (resp.
**B**: In particular, this includes the completed equivariant Chow ring, the completed equivariant K-theory and equivariant algebraic cobordism.
Let S𝑆Sitalic_S be the formal group algebra defined in [4]:.
**C**... | ACB | ACB | CAB | ACB | Selection 2 |
Surprisingly, this is sharper than both our estimate in Corollary 3 and Sarnak and Xue’s estimate (but still weaker than Marshall’s estimate) for the compact case. The improvement results from our sharper injectivity radius estimates in Subsection 10.3. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The probl... | **A**: In the real hyperbolic case, our estimates extend Yeung’s estimates for cocompact lattices [Yeu94, Theorem 2.4.1] to noncocompact lattices.
Finally, we study the de Rham cohomology of complete finite volume hyperbolic manifolds with cusps along a cofinal tower.
**B**: [Zuc82]), and on an estimate of the numbe... | ACB | ACB | CAB | ACB | Selection 2 |
We shall present in §3 a proof of Theorem 0.9, which is essentially a recast of the arguments in [9, §5]. <|MaskedSetence|> <|MaskedSetence|> Our intention here is to elucidate some issues on local vs. global isometric immersions in the literature. <|MaskedSetence|> | **A**: Note also that Theorem 0.9 (2) is a new result elusive in the existing literature.
.
**B**: Mardare [29], as well as applying Theorem 0.1 proved earlier in this note.
**C**: We utilise ideas from Tenenblat [44] and S.
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However, if a positive IRE is not a IET, i.e., its scheme contains at least one twisted cycle, then some of the beginning intervals in that cycle necessarily overlap, as well as do some of the ending intervals in this cycle (every point of the real axis is covered with the same number of beginning and ending intervals... | **A**: The overlapping intervals will be joined at these points and disjoint beyond them.
**B**: In order to obtain a tree from every particular cycle of a positive IRE, one has to fix a set of special points numbered by the twist number of this cycle.
**C**: This configuration does not determine a dynamical system u... | CBA | CBA | ACB | CBA | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> We conclude the paper giving the natural generalisation of the Lie model Theorem for rough approximate subgroups. Applications of this result to the case of metric groups will be studied in a future paper.
We study in general piecewise hyperdefinable sets in Section 1, focussing ... | **A**: The aim of this paper is to give the abstract basis for that kind of results, to find in the end possibly interesting applications to combinatorics.
**B**:
Hrushovski already indicated in unpublished works that, using piecewise hyperdefinable groups, it should be possible to extend some of the results of [Hru1... | ABC | BAC | BAC | BAC | Selection 3 |
•
In order to obtain a theory that more closely encapsulates some of the geometric features of finitary affine hyperplane arrangements, in Section 3 we state axioms for Finitary Affine Oriented Matroids (FAOMs). <|MaskedSetence|> A main theoretical feature of this restricted setting is that FAOMs are “orientations of... | **A**: These are AOMs with some local cardinality restrictions.
**B**: Moreover, we derive some order-theoretic properties of the geometric parallelism relation in FAOMs (§3.6).
**C**: We prove that order complexes of covector posets of FAOMs are shellable (§3.2) and explicitly describe their homeomorphism type (§3.3... | ACB | ACB | BAC | ACB | Selection 1 |
The first-named author was supported by the DFG project AN 1545 “Equivariant and weak orientations in the motivic homotopy theory”. The first-named and the second-named authors of the article were supported by the SPP 1786 “Homotopy theory and algebraic geometry” (DFG). <|MaskedSetence|> A part of this work was writt... | **A**: Petersburg University supported by the BASIS foundation grants “Young Russia Mathematics” and the Ministry of Science and Higher Education of the Russian Federation, agreement No.
**B**: D.
**C**: The third-named author was supported by the BASIS foundation grant “Young Russia Mathematics”.
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There are comprehensive survey papers that review the research on consensus protocols [19, 20, 21, 22]. <|MaskedSetence|> There is a large number of papers that propose consensus protocols with switching network topologies and convergence proofs of these algorithms are provided under various assumptions [27, 28, 29, 3... | **A**: In many scenarios, the network topology of the consensus protocol is a switching topology due to failures, formation reconfiguration, or state-dependence.
**B**: This assumption means that the union of graphs over an infinite interval is strongly connected.
**C**: The convergence of the algorithm is provided u... | ABC | ABC | ABC | BAC | Selection 3 |
Acknowledgement. <|MaskedSetence|> Walter Gubler for providing us with the main idea and helping us to see the possibility of the generalisation to normal bases over arbitary global fields. We are also indebted to Prof. Joseph Silverman for pointing the connection with [SilCall]. The first author is grateful to Pro... | **A**: Qing Liu for answering questions regarding a generalisation.
**B**: The first author would also like to thank Prof.
**C**: We are indebted to Prof.
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<|MaskedSetence|> See, e.g.,
Detommaso et al. <|MaskedSetence|> (2018); Liu et al. (2019); Gong et al. (2019); Wang et al. (2019); Zhang et al. (2020); Ye et al. (2020)
and the references therein.
Departing from MCMC where independent stochastic particles are used, it leverages interacting deterministic particles to ... | **A**: (2018); Han and Liu (2018); Chen et al.
**B**: In addition to gradient-based MCMC, variational transport also shares similarity with Stein variational gradient descent (SVGD) (Liu and Wang, 2016), which is a more recent particle-based algorithm for Bayesian inference.
Variants of SVGD have been subsequently pro... | BAC | CBA | BAC | BAC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> Moreover, in this paper we assume that the parameters of the model are known, but in many practical situations one is given a realization of the graph and the task is estimating the unknown parameters, see [10, 20, 21]. <|MaskedSetence|> This is an interesting open problem.
The ... | **A**:
In practice, not all nodes that enter the network have the same degree, and thus it would be interesting to extend our result to the case of a random initial degree distribution.
**B**: If we consider a more general class of preferential attachment graphs, for which a model-free approach is used and therefore ... | ACB | ACB | ACB | ABC | Selection 3 |
<|MaskedSetence|> This paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> In Section 4, we present the lower complexity bounds for saddle point problems without individual variables. Finally in Section 5, we show how the proposed algorithm can be applied to the problem computing Wasserstein barycente... | **A**:
Paper organization.
**B**: In Section 3, we provide the main algorithm of the paper to solve such kind of problems.
**C**: Section 2 presents a saddle point problem of interest along with its decentralized reformulation.
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<|MaskedSetence|> <|MaskedSetence|> This problem was formulated by Stepanec [7] and Zykov [8] for general graphs and by Hubicka and Syslo [9] in the strictly fundamental class context. In more concrete terms this problem is equivalent to finding the cycle basis with the sparsest cycle matrix. In [5] a unified perspec... | **A**:
The length of a cycle is its number of edges.
**B**: The minimum cycle basis (MCB) problem is the problem of finding a cycle basis such that the sum of the lengths (or edge weights) of its cycles is minimum.
**C**: Some applications of the MCB problem are described in [5, 11, 10, 12].
.
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<|MaskedSetence|> When H=1/2𝐻12H=1/2italic_H = 1 / 2, one recovers standard Brownian motion. It is well known that fractional Brownian motion has stationary nonnegatively correlated increments for H≥1/2𝐻12H\geq 1/2italic_H ≥ 1 / 2 - see e.g. [16]. <|MaskedSetence|> <|MaskedSetence|> | **A**:
and variance function is V(t)=|t|2H𝑉𝑡superscript𝑡2𝐻V(t)=|t|^{2H}italic_V ( italic_t ) = | italic_t | start_POSTSUPERSCRIPT 2 italic_H end_POSTSUPERSCRIPT.
**B**: Therefore Proposition 2.3 applies and the optimal measure for BHsubscript𝐵𝐻B_{H}italic_B start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT on th... | ABC | ABC | BAC | ABC | Selection 4 |
∎
It is quite interesting, at this point, to ask whether there exist classes of surface bundles for which there is always virtual excessive homology, and hence virtual algebraic fibrations. <|MaskedSetence|> <|MaskedSetence|> In particular, it would be interesting to decide this case for the class of Kodaira fibra... | **A**: (The surface bundles discussed in Theorem 1 cannot be Kodaira fibrations, as these have strictly positive signature, see e.g.
**B**: [BHPV04].
**C**: For instance, this is the case when the fibration is a holomorphic bundle, see e.g.
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<|MaskedSetence|> In particular, the results of this paper are used in work of the author that uses the cohomology of Shimura varieties to deduce new formulas for the cohomology of Rapoport-Zink spaces ([Ber21]) and related work of the author and K.H. <|MaskedSetence|> In §2 we develop the abstract theory of the coho... | **A**:
The normalization of transfer factors for non-strongly regular elements is needed in the analysis of the trace formula for the cohomology of Shimura varieties.
**B**: We also use §2 to prove (before Corollary 3.10) that for a fixed pair (γ𝐇,γ)∈𝐇(F)(𝐆,𝐇)−reg×𝐆(F)superscript𝛾𝐇𝛾𝐇subscript𝐹𝐆𝐇reg𝐆𝐹(... | ACB | ACB | ACB | ACB | Selection 4 |
Proof.
Since 𝔐𝔐\mathfrak{M}fraktur_M is a Breuil-Kisin module over W(κ)𝑊𝜅W(\kappa)italic_W ( italic_κ ), 𝔐𝔐\mathfrak{M}fraktur_M is projective module over W(κ)[[u]]𝑊𝜅delimited-[]delimited-[]𝑢W(\kappa)[\![u]\!]italic_W ( italic_κ ) [ [ italic_u ] ]. <|MaskedSetence|> Further, 𝔐/u𝔐𝔐𝑢𝔐\mathfrak{M}/u\... | **A**: For the given ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-algebra ℛℛ\mathscr{R}script_R, the (ℛ⊗ℤpW(κ))[[u]]subscripttensor-productsubscriptℤ𝑝ℛ𝑊𝜅delimited-[]delimited-[]𝑢(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[\![u]\!]( script_R ⊗ start_POSTSUBSCRIPT black... | ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> A symmetric function is a formal power series f(𝒙)∈ℚ[[x1,x2,…]]𝑓𝒙ℚdelimited-[]subscript𝑥1subscript𝑥2…f(\bm{x})\in\mathbb{Q}[[x_{1},x_{2},\dots]]italic_f ( bold_italic_x ) ∈ blackboard_Q [ [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_... | **A**: Symmetric functions
Let 𝒙=(x1,x2,…)𝒙subscript𝑥1subscript𝑥2…\bm{x}=(x_{1},x_{2},\dots)bold_italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … ) be a set of formal variables, and 𝒙n=(x1,…,xn)subscript𝒙𝑛subscript𝑥1…subscript𝑥𝑛\bm{x}_{n}=(x... | CAB | CAB | CBA | CAB | Selection 2 |
<|MaskedSetence|> italic_μ and thus ℱ4(μ)=μsuperscriptℱ4𝜇𝜇\mathcal{F}\hskip 0.5pt^{4}(\mu)=\mucaligraphic_F start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_μ ) = italic_μ due
to I2=idsuperscript𝐼2idI^{2}=\mathrm{id}italic_I start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_id. This implies λ4=1superscript𝜆... | **A**: hence ℱ2(μ)=I.μformulae-sequencesuperscriptℱ2𝜇𝐼𝜇\mathcal{F}\hskip 0.5pt^{2}(\mu)=I.\hskip 0.5pt\mucaligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_μ ) = italic_I .
**B**: This has an immediate consequence as.
**C**: Note that we
also get I.μ=ℱ2(μ)=λ2μformulae-sequence𝐼𝜇superscriptℱ2𝜇... | ACB | ACB | BCA | ACB | Selection 1 |
whose parameters (ρ,𝔲,T)𝜌𝔲𝑇(\rho,\mathfrak{u},T)( italic_ρ , fraktur_u , italic_T ) satisfy the compressible Euler system. <|MaskedSetence|> It was formally derived in Sone’s book [39] that for the Boltzmann equation with the complete diffusive boundary condition (1.3), the limiting Euler system would be impose... | **A**: The goal of the this paper is to rigorously justify this limit by using the method of Hilbert expansion, in the context of short time smooth solutions.
**B**: The precise statement will be given in the later part of Introduction, after we introduce some required notations.
.
**C**: In particular, for the domai... | CAB | CAB | BCA | CAB | Selection 1 |
The organization of the paper is as follows. In Section 2, we introduce our notation and provide the required definitions with the notion of weak solution used in the paper. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We conclude the paper with Appendix A where we state all the required a priori estimates... | **A**: In Section 3, we introduce the concept of a semiflow selection in terms of the two state variables: the velocity and the energy.
**B**: Section 4 is devoted to the existence of a random dynamical system, Theorem 4.4, which is a central result of
this paper.
**C**: We also analyze the properties (compactness, s... | ACB | ACB | CAB | ACB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> (2014)],
[Pfister et al. (2018)], [Chakraborty and Zhang (2019)]), graphical modeling ([Lauritzen (1996)], [Gan, Narisetty and Liang (2019)]), linguistics ([Nguyen and Eisenstein (2017)]), clustering (Székely and Rizzo, 2005), dimension reduction (Fukumizu, Bach and Jordan, 2004; S... | **A**: Inspired by the work of Blum, Kiefer and Rosenblatt (1961) and Dugué (1975), Deheuvels (1981) studied a test of multivariate independence based on the Möbius decomposition, generalized in Bouzebda (2014)..
**B**: Testing independence also has many applications, including causal inference ([Pearl (2009)], [Peter... | ABC | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> We say that ℒℒ\mathcal{L}caligraphic_L is relational if it only contains relation symbols. <|MaskedSetence|> For every A⊆ℕ𝐴ℕA\subseteq\mathbb{N}italic_A ⊆ blackboard_N we denote by ⟨A⟩𝒩subscriptdelimited-⟨⟩𝐴𝒩\langle A\rangle_{\mathcal{N}}⟨ italic_A ⟩ start_POSTSUBSCRIPT caligraphic_N end_POSTSUB... | **A**: Let ℒℒ\mathcal{L}caligraphic_L be a first order language.
**B**: If 𝒩𝒩\mathcal{N}caligraphic_N is ultrahomogeneous, then 𝒦:=Age(𝒩)assign𝒦Age𝒩\mathcal{K}:=\mathrm{Age}(\mathcal{N})caligraphic_K := roman_Age ( caligraphic_N ) satisfies the amalgamation property, i.e., for every two embeddings f:𝒜→ℬ:𝑓→𝒜ℬ... | ACB | ACB | CAB | ACB | Selection 4 |
\emptyset&\textrm{otherwise},\end{cases}italic_p ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × { italic_t } ) = { start_ROW start_CELL bold_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL ( italic_t = italic_t start_POSTSUBSCRIPT 7 end_POST... | **A**: 𝐃1subscript𝐃1\mathbf{D}_{1}bold_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT) is a union of mutually disjoint m𝑚mitalic_m 2222-disks in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT bounded by α1∗~~superscriptsubscript𝛼1\widetilde{\alpha_{1}^{*}}over~ start_ARG italic_α s... | ABC | ABC | ABC | CBA | Selection 1 |
<|MaskedSetence|> Here, the challenge consists in considering nonlinear damping which are distributed everywhere in the domain and acting only on the rotational angles. <|MaskedSetence|> More precisely, we prove that the energy decay rate, as introduced in [2] for a nonlinearly damped hyperbolic system coupled by vel... | **A**: Taking into account the work of Alabau-Boussouira [2, 4], we aim here to establish a general and explicit decay result for the energy associated with the system (1.5)–(1.7) below.
**B**: Indeed, the latter method is originally developed in [1] where the author completed the study carried out in [23] and improve... | ACB | CAB | CAB | CAB | Selection 3 |
This system is far from perfect, and various refinements of it have been proposed.
Still, even in this basic form, we can appreciate how escrows turn the strategy ‘running away with the money/goods’ into a non-rational choice. <|MaskedSetence|> Similarly, in refusing to pay Bogdan, Aki will never get her money back.... | **A**: The basic structure of an escrow is depicted in Figure 2.
.
**B**: In practice, the introduction of time windows that release the escrow to the original owner if certain conditions are met can mitigate the problem.
**C**: In refusing to ship the goods, Bogdan doesn’t gain any money because Aki will never relea... | CBA | CBA | CBA | ACB | Selection 3 |
Self-concordant functions have received strong interest in recent years due to the attractive properties that they allow to prove for many statistical estimation settings [Marteau-Ferey et al., 2019, Ostrovskii & Bach, 2021]. The original definition of self-concordance has been expanded and generalized since its incep... | **A**: [2015], in which more general properties of these
pseudo-self-concordant functions were established.
**B**: This was also the case in Ostrovskii & Bach [2021] and Tran-Dinh et al.
**C**: For example, the logistic loss function used in logistic regression is not strictly self-concordant, but it fits into a clas... | CBA | CBA | ABC | CBA | Selection 1 |
The representation type of tensor product algebras has been studied in various contexts. In the 1970s, Bondarenko and Drozd [BD] considered the representation type of finite groups, while Auslander and Reiten [AR] dedicated their effort to the representation type of triangular matrix rings. Moving into the 1980s, mat... | **A**: The most recent progress in this field can be attributed to Leszczyn´´n\acute{\text{n}}over´ start_ARG n end_ARGski and Skowron´´n\acute{\text{n}}over´ start_ARG n end_ARGski, as clear in their series of papers [L1, LS1, LS2].
**B**: However, it is still open to distinguish representation-finite cases and tame ... | ACB | ACB | ACB | ACB | Selection 3 |
For this case we present Algorithm 2. <|MaskedSetence|> Here, as in Algorithm 1, the proximal operator is computed inexactly. <|MaskedSetence|> <|MaskedSetence|> Hence, the problem (4) is solved by Fast Gradient Descent. Further, we note that the algorithm’s steps in lines 3, 6, and 7 are local and separable on ea... | **A**: The problem (4) is divided into two minimization subproblems, by X𝑋Xitalic_X, and by Y𝑌Yitalic_Y.
**B**: Note that we need to communicate with other devices only when we solve the problem (4) and need to multiply by the matrix W𝑊Witalic_W.
**C**: This algorithm is the Tseng method [44] with a resolvent/prox... | CBA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> A brief history of the formality problem
This formality result has quite an involved history, which we try to summarise here. Firstly, there are many incarnations of the result for the Yoneda algebra of a semisimple representation of a 2CY algebra. As pointed out by Bocklandt, Galluzzi and Vaccarino... | **A**:
4.6.4.
**B**: This observation, along with deformation-theoretic arguments, was used to understand formal neighbourhoods in the coarse moduli space of representations of 2-Calabi–Yau algebras in [BGV16], with fuller details (including but not limited to the étale neighbourhood theorem for the coarse moduli sp... | ABC | ABC | ABC | CAB | Selection 3 |
<|MaskedSetence|> Let xi,jsubscript𝑥𝑖𝑗x_{i,j}italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT be the image of Xi,jsubscript𝑋𝑖𝑗X_{i,j}italic_X start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT in R𝑅Ritalic_R. Also, Fi,j¯=fi,j¯subscript𝐹𝑖𝑗subscript𝑓𝑖𝑗\overline{F_{i,j}}=f_{i,j}over¯... | **A**: Now v(I3)⊆I3+v(S)βm𝑣subscript𝐼3subscript𝐼3𝑣𝑆subscript𝛽𝑚v(I_{3})\subseteq I_{3}+v(S)\beta_{m}italic_v ( italic_I start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ⊆ italic_I start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_v ( italic_S ) italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, so F−Lβm∈I𝐹𝐿su... | CBA | CBA | CBA | CBA | Selection 2 |
It is well known that, the notion of non-positive curvature spaces were mentioned by
J. Hadamard and E. <|MaskedSetence|> Busemann and A.D. Aleksandrov generalized the concept of geodesic metric spaces based on the concept of manifolds with a non-positive sectional curvature. <|MaskedSetence|> <|MaskedSetence|> | **A**: Gromov suggested the notation CAT(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 ) for a non-positive curvature geodesic metric space.
**B**: Cartan in the 1920’s.
In 1950, H.
**C**: The letters C, A and T in CAT(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 ) stand for Cartan, Aleksandrov and Toponogov, respectively.
.
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In this section we validate the efficacy of the proposed algorithm and we verify our theoretical results
on optimization problems that arise in machine learning applications. Specifically, in the first two sections we use the conventional SIGMA (uniform sampling) to solve the maximum likelihood estimation problem base... | **A**: In Remark 4 we discuss how to efficiently compute the reduced Hessian matrix for Generalized Linear Models.
.
**B**: Furthermore, in Section 4.3 we provide comparisons between the conventional SIGMA and SIGMA with the different sampling strategies of Section 2.5.
**C**: Moreover, we provide additional experim... | BCA | BCA | BCA | CAB | Selection 3 |
Acknowledgements. We thank M. Solleveld for his careful reading and many useful comments, particularly regarding Section 11, and G. <|MaskedSetence|> Reeder for their helpful suggestions. We also thank the referees for the thorough checking of the paper, for the corrections and suggestions for improvement. This rese... | **A**: thanks Université Paris Cité and Sorbonne Université for their hospitality while part of this work was completed.
.
**B**: Lusztig and M.
**C**: D.C.
| BCA | BCA | BCA | CAB | Selection 2 |
<|MaskedSetence|> In [FNQ18] and [KRS20] the authors proved the logarithmic Sobolev inequality and the fractional logarithmic Sobolev inequality on the Heisenberg group and on homogeneous groups, respectively. <|MaskedSetence|> In this paper, we prove logarithmic Sobolev inequalities on graded groups and weighted log... | **A**: A fractional weighted version of (1.3) on homogeneous groups was proved in [KS20].
**B**: There are works on the fractional Laplacian [Bec12] as well as on the.
**C**:
In [Mer08], the author obtained a logarithmic Gagliardo-Nirenberg inequality.
| ACB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> That proof was rather complicated and required the extensive use of techniques from the study of iterated reflection. In this note, we present a simpler proof of Theorem 1.2 that uses more traditional and accessible techniques from ordinal analysis, namely, cut-elimination for infi... | **A**: In [4], the focus was on the iterated reflection side; indeed, Theorem 1.2 was derived from a Schmerl-style [6] conservation theorem for iterated reflection principles.
**B**: Moreover, it should make the result more accessible to proof-theorists who are familiar with cut-elimination techniques.
Here is our p... | CAB | CAB | CBA | CAB | Selection 2 |
Our main tools originate in some recent work on the theory of stacks. Alper, Halpern-Leistner and Heinloth [AHLH23] have recently developed a theory which produces moduli spaces for Artin stacks, generalizing results of Keel-Mori on Deligne-Mumford stacks. This can be combined with the theory of ΘΘ\Thetaroman_Θ-stabi... | **A**: If the numerical invariant satisfies certain properties, then the open substack of semistable objects admits a good moduli space (as defined in [Alp13]).
**B**: This is an intrinsic way of constructing the moduli space, in the sense that we do not need to choose a parameter space nor the action of a group.
.
... | CAB | CBA | CAB | CAB | Selection 1 |
Relative boundedness can be understood as an analogue of relative Lipschitz continuity for variational inequalities. <|MaskedSetence|> This fact plays an important role in considering relatively Lipschitz continuous Lagrange saddle point problems and their reduction to corresponding variational inequalities with the ... | **A**: In Sect.
**B**: 4 is devoted to adaptive algorithms for relatively smooth optimization problems.
**C**: It should be noted that the subgradient of a relatively Lipschitz continuous function satisfies the relative boundedness condition.
| CAB | CBA | CAB | CAB | Selection 4 |
<|MaskedSetence|> He would like to thank Professor Bernard Leclerc and Professor Erez Lapid for useful comments and suggestions; He really appreciates Professor Markus Reineke for his instruction and his proof for the set of irreducible components of quiver Grassmannians; He really thanks Bernard Keller for comments o... | **A**: NO.202006040123.
2.
**B**: Premise.
**C**: The author is grateful to Professor Qin Fan for many helpful discussions.
| CAB | CAB | CAB | CBA | Selection 1 |
The paper is organized as follows. In Sections 1, 2 and 3 we give the description of the lattice path model and formulate the main theorem. <|MaskedSetence|> In Section 5 we will reduce the problem of counting paths between the wall and the filter to a problem of counting paths between two lines. <|MaskedSetence|> ... | **A**: In Section 4 we define wall and filter restrictions and recall the reflection principle.
**B**: In Sections 6, 7 we will prove theorems for path counting in the presence of two filters and two filters together with the wall.
**C**: The proof of the main theorem is given in Section 8.
| ABC | CBA | ABC | ABC | Selection 3 |
The first author introduced in [1] a relational framework for developing the key notions and results on hidden variables and non-locality, which can be seen as a relational variant of the probabilistic setting of [10]. He introduced what he called “relational empirical models” and used them to show that the basic resu... | **A**: have a priori nothing to do with quantum mechanics, and hence they apply to any other field where independence plays a role, e.g.
**B**: In fact, the existential-positive-conjunctive fragment suffices.
**C**: Even on the level of atoms no finite axiomatization exists [39].
| BCA | ABC | BCA | BCA | Selection 4 |
However, in the free product case the appeal to surface theory is more delicate and less obviously valid. That is, realising a homotopy equivalence of a graph as a homeomorphism of a surface is a core part of the theory for free group automorphisms going back to [4], but it is not just that this analogue is absent in t... | **A**: In [11] this is called ‘weakly clean’ and in that paper, Proposition B.2, it is shown that this implies clean, which means having a primitive transition matrix.
**B**: (In [2] it is proved that any irreducible automorphism has a locally connected Whitehead graph - see also section 5.
**C**: For instance, any i... | CBA | CBA | BCA | CBA | Selection 1 |
base case Δ=3Δ3\Delta=3roman_Δ = 3. They proved that P∗superscript𝑃P^{*}italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is the only HZ-graph with maximum degree Δ=3Δ3\Delta=3roman_Δ = 3, an alternative proof was given later by Král’, Sereny, and Stiebitz (see [13, pp. <|MaskedSetence|> <|MaskedSetence|> Our mai... | **A**: These developments were used
to prove the Core Conjecture [2].
**B**: 67–63]).
**C**: The next case, Δ=4Δ4\Delta=4roman_Δ = 4, was recently solved by Cranston and Rabern [6], they proved that the only HZ-graph with maximum degree Δ=4Δ4\Delta=4roman_Δ = 4 is obtained from the graph K5subscript𝐾5K_{5}italic_K s... | BCA | CAB | BCA | BCA | Selection 4 |
where C0subscript𝐶0C_{0}italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a universal constant.
Let ϕ(x,t)=ψ(d(x,x0,t)r)italic-ϕ𝑥𝑡𝜓𝑑𝑥subscript𝑥0𝑡𝑟\phi(x,t)=\psi(\frac{d(x,x_{0},t)}{r})italic_ϕ ( italic_x , italic_t ) = italic_ψ ( divide start_ARG italic_d ( italic_x , italic_x start_POSTSUBSCRIPT 0 end... | **A**: Assume that ϕFitalic-ϕ𝐹\phi Fitalic_ϕ italic_F achieves its positive maximum at the point (x1,t1)subscript𝑥1subscript𝑡1(x_{1},t_{1})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ).
**B**: Suppose that the maximum of the function ϕFitalic-ϕ𝐹\phi Fital... | CBA | BAC | BAC | BAC | Selection 3 |
Throughout this paper, we work exclusively with 𝕜𝕜\Bbbkroman_𝕜 a commutative ring with global dimension zero. <|MaskedSetence|> First, it induces that every object is cofibrant and fibrant in chain complexes and simplicial modules, and the projective and injective model structures are equal. In particular, the mode... | **A**: There are several reasons this condition is imposed.
**B**: Moreover, as every module is flat, the tensor product preserves finite limits.
**C**: In [HKRS17], it was shown that the model structures for comodules are left-induced from injective model structures which are in general not monoidal model categories... | CAB | ACB | ACB | ACB | Selection 4 |
<|MaskedSetence|> Continuous limits
The purpose of this section is to prove Theorem 1.5. In Section 4.1 we recall the continuous β𝛽\betaitalic_β-corners processes from Section 1.1 and derive a few of their properties. <|MaskedSetence|> <|MaskedSetence|> Finally, in Section 4.4 we derive our continuous multi-level ... | **A**: In Section 4.3 we derive the continuous measures from Section 4.1 as diffuse limits of the measures in (1.5).
**B**: In Section 4.2 we summarize some useful notation for Jack symmetric functions and explain how the latter relate to the measures in (1.5).
**C**:
4.
| CBA | ABC | CBA | CBA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Furthermore, we extend these results to the n𝑛nitalic_n-tuple saddle point problem in Section 3. <|MaskedSetence|> Generalizations to n𝑛nitalic_n-tuple cases are provided in Section 5. In Section 6, numerical experiments for a 3-field formulation of the Biot model are provided t... | **A**:
The outline of the remainder of this paper is as follows.
**B**: In section 2, we briefly recall the classic saddle point problem and its Schur complement, and introduce the twofold saddle point problem and the form of Schur complement, we then construct and analyze the block-triangular and block-diagonal prec... | ABC | ABC | ABC | ABC | Selection 4 |
The authors would like to thank the handling editor and two referees for their very detailed comments.
Changxin Mo acknowledges support from the National Natural Science Foundation of China (Grant No. 12201092), the Natural Science Foundation Project of CQ CSTC (Grant No. <|MaskedSetence|> KJQN202200512), the Chongqi... | **A**: CSTB2022NSCQ-MSX0896), the Science and Technology Research Program of Chongqing Municipal Education Commission
(Grant No.
**B**: 21XLB040), P.
**C**: R.
| ABC | ABC | ABC | CBA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The city of Kharkiv has been devastated during the ongoing invasion of
Ukraine, and the Institute has been severely damaged [NDGP22]. We dedicate this paper to Profs. Eremenko and Lyubich,
to the people of Kharkiv, and to all victims of the invasion of Ukraine.
... | **A**: At that time, Alex Eremenko was based at the Institute of Low Temperature Physics and Engineering, and it was there that he formulated what is now known as Eremenko’s conjecture.
**B**: Their collaboration, which pioneered the use of approximation theory in complex dynamics, took place in the fall of 1983 in Kh... | CBA | CBA | BCA | CBA | Selection 1 |
3. Correspondence: Parahoric vs. <|MaskedSetence|> A similar correspondence also works for Higgs bundles and local systems [2, 14]. <|MaskedSetence|> <|MaskedSetence|> | **A**: Although the correspondence is only given in characteristic zero, it can be naturally generalized to prime characteristic under some necessary conditions (tame weights).
**B**: In this section, we first give the correspondence of parahoric torsors and equivariant bundles in positive characteristic, which is a d... | BCA | CAB | CAB | CAB | Selection 2 |
This is modified from the conservative Belyi polynomial Bd,1subscript𝐵𝑑1B_{d,1}italic_B start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We may choose a𝑎aitalic_a such that a+dd−1𝑎𝑑𝑑1a+\frac{d}{d-1}italic_a + divide start_ARG italic_d end_ARG start_ARG i... | **A**: Like Bd,1subscript𝐵𝑑1B_{d,1}italic_B start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT, it has just two finite critical points at 00 and 1111.
**B**: The first critical point, 0, is preperiodic, as f(0)=dd−1𝑓0𝑑𝑑1f(0)=\frac{d}{d-1}italic_f ( 0 ) = divide start_ARG italic_d end_ARG start_ARG italic_d - 1 e... | ABC | ABC | BCA | ABC | Selection 1 |
To prove (i) assume that f𝑓fitalic_f is Baire measurable. <|MaskedSetence|> <|MaskedSetence|> Then Player II fixes any point a∈G∩W𝑎𝐺𝑊a\in G\cap Witalic_a ∈ italic_G ∩ italic_W and picks U0:=B(a,1)∩G∩Wassignsubscript𝑈0𝐵𝑎1𝐺𝑊U_{0}:=B(a,1)\cap G\cap Witalic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := italic_B ... | **A**: When the game is finished, one of the two cases is possible:.
**B**: We will describe a winning strategy for Player II in the game GfBairesubscriptsuperscript𝐺Baire𝑓G^{\mathrm{Baire}}_{f}italic_G start_POSTSUPERSCRIPT roman_Baire end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT.
Let G⊆X𝐺𝑋G... | BAC | BCA | BCA | BCA | Selection 2 |
Matching book embeddings of bipartite circulants C𝐶Citalic_C are given where the page number is equal to the vertex degree Δ(C)Δ𝐶\Delta(C)roman_Δ ( italic_C ), supporting the conjecture in [2]. It can be shown that regular dispersable graphs must be bipartite [21]. A nonbipartite circulant is nearly dispersable if o... | **A**: So far,
all nonbipartite circulants have been nearly dispersable and we conjecture here that nonbipartite, vertex-transitive graphs are nearly dispersable.
Previous results support both conjectures.
**B**: Other classes of vertex-transitive graph that are known to be nearly dispersable include the product of ... | ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> We shall establish convergence of the corresponding Laplace transforms and characterize the limit by a second order linear differential equation of singular type, related to the Bessel differential equation. <|MaskedSetence|> We note that a corresponding differential equation for characteristic func... | **A**: This would require the existence of the second moment, which for the inverse ΓΓ\Gammaroman_Γ-distribution is in general not at disposal.
**B**:
Let us turn to the proof of Theorem 2.
**C**: This approach necessitates that the terms 𝖠𝖠\mathsf{A}sansserif_A and 𝖡𝖡\mathsf{B}sansserif_B are non-negative.
| BCA | ABC | BCA | BCA | Selection 3 |
<|MaskedSetence|> They contain all algebraic numbers, as well as their logarithms, and some transcendental numbers like π𝜋\piitalic_π; they are exceedingly commonplace however not well understood.
We do not know how to decide (26), but we point out to some work that might prove to be helpful. One is Conjecture 1 in... | **A**: See [Ayo14] for definitions and a discussion about these two conjectures.
**B**: The class of numbers that can be expressed as integrals of algebraic functions over semialgebraic sets are known as periods [KZ01].
**C**: A more direct conjecture is one made by Grothendieck that predicts the transcendence degree... | CAB | BCA | BCA | BCA | Selection 3 |
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