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**A**: italic_c ( italic_x ) ∈ italic_B → caligraphic_P ( 1 ) end_ARG end_CELL end_ROW start_ROW start_CELL E-Fc) divide start_ARG italic_b ∈ italic_B italic_f ∈ italic_B → caligraphic_P ( 1 ) end_ARG start_ARG sansserif_Ap ( italic_f , italic_b ) ∈ caligraphic_P ( 1 ) end_ARG end_CELL start_CELL italic_β C-Fc) divide start_ARG italic_b ∈ italic_B italic_c ( italic_x ) ∈ caligraphic_P ( 1 ) [ italic_x ∈ italic_B ] italic_B italic_s italic_e italic_t end_ARG start_ARG sansserif_Ap ( italic_λ italic_x start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT **B**: italic_c ( italic_x ) , italic_b ) = italic_c ( italic_b ) ∈ caligraphic_P ( 1 ) end_ARG end_CELL end_ROW end_ARRAY end_CELL end_ROW start_ROW start_CELL end_CELL end_ROW start_ROW start_CELL italic_η C-Fc) divide start_ARG italic_f ∈ italic_B → caligraphic_P ( 1 ) end_ARG start_ARG italic_λ italic_x start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT . sansserif_Ap ( italic_f , italic_x ) = italic_f ∈ italic_B → caligraphic_P ( 1 ) end_ARG ( italic_x italic_n italic_o italic_t italic_f italic_r italic_e italic_e italic_i italic_n italic_f ) end_CELL end_ROW end_ARRAY
**C**: }(1)}}(x\ not\ free\ in\ f)\end{array}start_ARRAY start_ROW start_CELL Function collection to caligraphic_P ( 1 ) end_CELL end_ROW start_ROW start_CELL start_ARRAY start_ROW start_CELL F-Fc) divide start_ARG italic_B italic_s italic_e italic_t end_ARG start_ARG italic_B → caligraphic_P ( 1 ) italic_c italic_o italic_l end_ARG end_CELL start_CELL I-Fc) divide start_ARG italic_c ( italic_x ) ∈ caligraphic_P ( 1 ) [ italic_x ∈ italic_B ] italic_B italic_s italic_e italic_t end_ARG start_ARG italic_λ italic_x start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT | CAB | CBA | BCA | ACB | Selection 3 |
**A**: ℱℱ{\mathcal{F}}caligraphic_F).
**B**: ε𝜀\varepsilonitalic_ε, N𝑁Nitalic_N) and growth functions (e.g**C**: The language of ultrafilters adds one more layer of notational complexity to an already notationally-intensive paper; however, there are gains to be made elsewhere, most notably in eliminating many quantitative constants (e.g | CBA | BAC | ABC | ACB | Selection 1 |
**A**: In Subsection 2.2, we show how to pass from a 2-automaton to a decorated graph. Finally in Subsection 2.3 we show how to turn a decorated graph into an admissible tree.
**B**: The reader interested in the motivation can jump directly to Subsection 2.2 and refer back as needed**C**: The structure of this section is as follows: in Subsection 2.1, we introduce decorated graphs, admissible trees and their associated topological spaces | ACB | ABC | CBA | BCA | Selection 3 |
**A**: On a given domain of the Euclidean space, the problem can be reduced to inverse coefficient problems for elliptic equations which were solved in [62].
We are concerned with the stability of the inverse problem.**B**: Generalizations and alternative methods to solve the problem have been studied in [3, 17, 42, 47, 49], and the related determination of smooth structure was recently studied in [22, 23]**C**: [1, 13, 50] | ACB | ABC | BCA | CBA | Selection 4 |
**A**: Other rewriting algorithms also exist, for example Cohen et al. [26] present algorithms to compute with elements of finite Lie groups.
**B**: One important task in this context is writing elements of classical groups as words in standard generators using SLPs**C**: This is done in Magma [14] using the results of Elliot Costi [6] and in GAP using the results of this paper see Section 6 | BCA | CBA | CAB | ACB | Selection 3 |
**A**: A polynomial P∈R𝑃𝑅P\in Ritalic_P ∈ italic_R is said to have a Jacobian mate**B**: Furthermore, by the sake of
simplicity we set R:=K[x,y]assign𝑅𝐾𝑥𝑦R:=K[x,y]italic_R := italic_K [ italic_x , italic_y ]**C**: 𝐰:=(w1,w2)assign𝐰subscript𝑤1subscript𝑤2\mathbf{w}:=(w_{1},w_{2})bold_w := ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), we write |𝐰|:=w1+w2assign𝐰subscript𝑤1subscript𝑤2|\mathbf{w}|:=w_{1}+w_{2}| bold_w | := italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | CAB | CBA | BCA | BCA | Selection 2 |
**A**: This paper is a result of Thomas Geisser’s suggestion to consider a motivic version of Murre and H. Saito’s work on regular homomorphisms. As mentioned above, the referee’s contribution was enormous**B**: We would like to thank him or her for the careful reading and insightful suggestions. In the very first version of this work, Theorem 1.1 was stated under resolution of singularities. We thank Shuji Saito for pointing out that the assumption was unnecessary. We also thank Bruno Kahn for helpful discussions and writing an appendix. We would also like to thank Federico Binda, Ryo Horiuchi, Shane Kelly, Amalendu Krishna, Takashi Maruyama, Hiroyasu Miyazaki and Rin Sugiyama for helpful conversations.
**C**: Acknowledgements. This paper is based on the author’s dissertation. The author wishes to thank his advisors Thomas Geisser and Hiroshi Saito for their constant advice and encouragement | CAB | ACB | ABC | BCA | Selection 4 |
**A**:
Acknowledgements**B**: We are also very grateful to Davoud Cheraghi for pointing out a gap in an earlier version of the paper and providing many invaluable comments and suggestions. The second author was indebted to Institut de Mathématiques de Toulouse for its hospitality during his visit in 2014/2015 where and when partial of this paper was written. He would also like to thank Davoud Cheraghi and Arnaud Chéritat for persistent encouragements.**C**: We would like to thank Xavier Buff and Arnaud Chéritat for helpful discussions and offering the manuscript [BCR09] | ACB | CBA | CAB | BAC | Selection 1 |
**A**: 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. Specifically,**B**:
The paper is organized as follows**C**: Section 2 contains the necessary background material | ACB | ABC | CAB | BAC | Selection 3 |
**A**: Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e. they satisfy the same Lie algebra and by virtue of Stone-von Neumann, may be realized in some way as harmonic oscillators. If one takes γ=−δ𝛾𝛿\gamma=-\deltaitalic_γ = - italic_δ, then the coupled SUSY equations become
**B**: 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**C**: This is not the only manner in which the two are connected | CAB | BCA | ACB | ABC | Selection 1 |
**A**: Within a row, tiles are unordered**B**: For counting purposes, we view them as ‘balls’ placed arbitrarily into ‘boxes’ labelled with the available markings**C**: The binomial coefficient (14) accounts for the remaining factors in the formula. The factors involving l𝑙litalic_l and n𝑛nitalic_n reduce to 1111 if the corresponding eigenspace exists but is unmarked (l𝑙litalic_l or n𝑛nitalic_n =0absent0=0= 0).
∎ | BAC | CBA | ABC | BAC | Selection 3 |
**A**: The first syzygy of Hibi rings is discussed in Section 3. Explicit expression for the first Betti number for planar distributive lattices has been discussed in Section 4.
**B**: In Section 2, we collect basic notations, terminology, and results that will be used in the paper**C**: The paper is structured as follows | CBA | ABC | BCA | BAC | Selection 1 |
**A**: This is done by first calculating the dp-rank of a certain 1-based reduct of the group, and then studying the effect of re-introducing the order into that structure.
**B**: Theorem 1 is proved in Section 4**C**: The proof proceeds by showing that strongly dependent ordered abelian groups have finite spines and explicitly calculating the dp-rank of the latter | ACB | ACB | CAB | ABC | Selection 3 |
**A**: To this end, one constructs from an atlas a proper étale Lie groupoids (see [MM03, Proposition 5.29])**B**: Recall from [Poh17, MM03] that every paracompact, smooth and effective orbifold can be represented by a so called atlas groupoid**C**: Following this procedure for a locally finite orbifold atlas, one sees that the atlas groupoid will even be source proper. Note that an atlas which is not locally finite yields a non source proper Lie groupoid. Since all atlas groupoids of a fixed orbifold are Morita equivalent, source properness is not stable under Morita equivalence.
| BAC | ABC | BCA | CBA | Selection 1 |
**A**: The assumption made by Zeidler that the action of the group is free leads to the simplification that the controlled Hilbert space obtained from the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-sections of the bundle on which the Dirac operator acts is already ample**B**: For proper actions this is not the case and a stabilization is necessary**C**: In the present paper we in particular also provide the generalization of the construction of the index class from free to proper actions.
Further applications are given in [BL, Sec. 6] | ABC | CBA | BCA | BAC | Selection 1 |
**A**: We remark that in this case, our method is similar to that of [MR3591945], with some differences. First we consider that T~~𝑇\tilde{T}over~ start_ARG italic_T end_ARG can be nonzero**B**: Also, our scheme is defined by a sequence of elliptic problems, avoiding the annoyance of saddle point systems. We had to reconsider the proofs, in our view simplifying some of them.
**C**: Of course, the numerical scheme and the estimates developed in Section 3.1 hold. However, several simplifications are possible when the coefficients have low-contrast, leading to sharper estimates | BAC | BCA | BAC | CBA | Selection 2 |
**A**: Now if T′superscript𝑇′T^{\prime}italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is another element**B**: A∈I(M)𝐴𝐼𝑀A\in I(M)italic_A ∈ italic_I ( italic_M ) if
and only if FT(A⋇A)=0subscript𝐹𝑇superscript𝐴⋇𝐴0F_{T}(A^{\divideontimes}A)=0italic_F start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_A start_POSTSUPERSCRIPT ⋇ end_POSTSUPERSCRIPT italic_A ) = 0**C**: multiplicative left-ideal generated by this functional i.e | CBA | CAB | BCA | BCA | Selection 1 |
**A**: I would also like to thank the anonymous referees for their many corrections and suggestions**B**: In particular for a suggested simplification of the proof of Lemma 4.2.**C**:
I would like to thank Mohammed Abouzaid, Marcel Bökstedt, Sylvain Courte, Tobias Ekholm, Yasha Eliashberg, Søren Galatius, and John Rognes for conversations surrounding the material in this paper | BCA | ACB | ABC | ACB | Selection 1 |
**A**:
Acknowledgement: The author would like thank Michael Finkelberg and Tatsuyuki Hikita for discussions, David Anderson and Hiroshi Iritani for helpful correspondences, and Thomas Lam for pointing out inaccuracies in a previous version of this paper**B**: This research was supported in part by JSPS KAKENHI Grant Number JP26287004 and JP19H01782.**C**: The author would also like to thank Daisuke Sagaki and Daniel Orr for their collaborations | ABC | ACB | BAC | BCA | Selection 2 |
**A**:
In general, the left- and right-cohomological dimensions of a monoid are not equal**B**: One immediate corollary of the above result is that if M𝑀Mitalic_M is a finitely presented special monoid with left- and right-cohomological dimensions both at least equal to 2222, then the left cohomological dimension of M𝑀Mitalic_M is equal to its right cohomological dimension.**C**: In fact they are completely independent of each other; see [27] | ACB | CAB | BCA | BAC | Selection 1 |
**A**:
Recently, many external factors including changes in international situations, increase of war risk and significant environmental changes, all cause the financial market becomes much more volatile than before, and the financial market also has different orders of risk data mixed over a short period of time**B**: Therefore, it is valuable to study the risk measures on variable exponent Bochner–Lebesgue spaces**C**: Under this position space, the order risk position is no longer a fixed positive number, but a measurable function. The characteristics of this space are able to characterise risk positions in the above volatile financial market. | ABC | BCA | BCA | BCA | Selection 1 |
**A**: This section is preparatory for Section 7.**B**:
In Section 6, we prove the independence of parameters for integrable highest weight representations of twisted affine Kac-Moody algebras over a base**C**: We also prove that the Sugawara operators acting on the integrable highest weight representations of twisted affine Kac-Moody algebras are independent of the parameters up to scalars | BAC | CAB | CBA | CBA | Selection 2 |
**A**: This statistical weight is clearly a special case of Eq**B**: (8) when there is only one type of real anyons, one type of boundaries, the number of boundaries is one, and only one pseudo-species. Hence, in light of Eqs. (2)-(5), the second binomial, i.e., the reduced statistical weight, in the statistical weight above leads to the following matrix of exclusion statistics.**C**:
where N1subscript𝑁1N_{1}italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT counts the anyons of the pseudo-species, and there is only one pseudo-species on a disk | CBA | CBA | BCA | ACB | Selection 3 |
**A**:
IB and JPdS wish to thank the “Laboratoire International Associé” and the “Indo-French Program for Mathematics” of the CNRS as well as the International Research Staff Exchange Scheme “MODULI”, projet 612534 from the Marie Curie Actions of the European commission**B**: Finally, we are grateful for the referee’s contributions which helped us to give proper explanations to a certain number of arguments.**C**: All three authors profit to thank the “Équipe de Théorie de Nombres” of the Institut de Mathématiques de Jussieu–Paris Rive Gauche for financing the participation of IB and PHH in the jury of JPdS’s “Habilitation”; this is where this collaboration began | ABC | BCA | ACB | CAB | Selection 3 |
**A**: In that case the conclusion follows easily from the induction hypothesis and the fact that for every α<β𝛼𝛽\alpha<\betaitalic_α < italic_β,**B**: We only need to argue for the conclusion in the case that β𝛽\betaitalic_β is a nonzero limit ordinal**C**:
The proof is by induction on β𝛽\betaitalic_β | ACB | CBA | CAB | BAC | Selection 2 |
**A**: We thank the referees for the careful reading of the first version of the text and for many helpful remarks**B**: This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’**C**: The results of Section 4 has been obtained under support of the RSF grant 19-11-00056. The work of both authors has also been supported in part by the Simons Foundation. The first author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.
| ABC | CAB | ACB | BAC | Selection 1 |
**A**:
Any tubing of S𝑆Sitalic_S is isotopic to a tubing where each tube has one foot on the component of S𝑆Sitalic_S containing K𝐾Kitalic_K, and one foot on a closed component of S𝑆Sitalic_S**B**: In particular, it suffices to change tubes one at a time.**C**: Since tubes are boundaries of 3-dimensional 1-handles, we may assume that, after an isotopy, any two such tubings have disjoint tubes | ACB | BAC | BAC | CAB | Selection 1 |
**A**: I would like to thank Christopher Dodd, Michael Groechenig and Tamas Hausel for helpful conversations**B**: I would like to thank Tsao-Hsien Chen and Siqing Zhang for useful comments on an earlier version of this paper.
**C**: I would like to thank my advisor Tom Nevins for many helpful discussions on this subject, and for his comments on this paper | CBA | ABC | BCA | CAB | Selection 3 |
**A**: The proofs of Propositions**B**: We will prove them in a certain order to arrive at Corollaries 25 and 37, thus concluding by the optimality of the weight and the interpretation in terms of dual nonlinear semigroup**C**:
This section is devoted to the proofs of the results of Section 3 | BAC | CAB | BAC | CBA | Selection 4 |
**A**: This section is similar to that in [8]**B**: The differences focus on the quantitative hypothesis on the phase function which is introduced in [2] and will play
a crucial role in the induction argument via multilinear oscillatory estimates of Bennett, Carbery and Tao [3]. In particular, we emphasize the role played among other things by the parabolic rescaling.**C**: In Section 2, we introduce the implements that will be used in the subsequent context | ABC | BCA | CBA | CBA | Selection 2 |
**A**: To be precise, we are interested in the following stochastic nonlinear wave equation on a smooth bounded domain 𝒪⊂ℝ2𝒪superscriptℝ2{\mathscr{O}}\subset\mathbb{R}^{2}script_O ⊂ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT,**B**:
Our aim here is to extend the existing studies to the wave equation with exponential nonlinearity subject to randomness**C**: In this way, we generalise the above mentioned results of Ondreját for two dimensional domains, by allowing the exponential nonlinearites, as well as the results of Ibrahim, Majdoub, and Masmoudi and others by allowing randomness | CBA | BCA | CAB | BCA | Selection 3 |
**A**:
In characteristic zero, symplectic cohomology SHSH{\operatorname{SH}}roman_SH is known to have the structure of a BV-algebra (see e.g**B**: the BV operator is compatible with a product and a bracket, both given by counting pairs of pants but with asymptotic markers treated differently.**C**: [1]), i.e | ACB | CBA | CBA | BAC | Selection 1 |
**A**:
We close this paper with the consideration that we used the notion of up-adjacency to define our Markov chain**B**: One may define a similar process using the notion of down-adjacency**C**: But it is not trivial to relate this new walk to the spectrum of the Laplacian. If it can be done, one is one step closer to understand the relation of the irreducibility of this new Markov chain with the orientability of the digraph, in analogy to what is done in eidi2023irreducibility to simplicial complexes. | BCA | BCA | CAB | ABC | Selection 4 |
**A**: Those**B**: point ⟨X⟩delimited-⟨⟩𝑋\langle X\rangle⟨ italic_X ⟩ determines the 17171717-space 𝕁17cABsuperscriptsubscript𝕁17𝑐𝐴𝐵\mathbb{J}_{17}^{cAB}blackboard_J start_POSTSUBSCRIPT 17 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c italic_A italic_B end_POSTSUPERSCRIPT**C**: We also note that
X𝑋Xitalic_X is stabilised by the actions of the elements Mxsubscript𝑀𝑥M_{x}italic_M start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, Lxsubscript𝐿𝑥L_{x}italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, Mx′superscriptsubscript𝑀𝑥′M_{x}^{\prime}italic_M start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and Lx′′superscriptsubscript𝐿𝑥′′L_{x}^{\prime\prime}italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT | BCA | CAB | BCA | CBA | Selection 2 |
**A**:
This topology is referred to as the induced topology of δ𝛿\deltaitalic_δ**B**: It is always completely regular, and is Hausdorff if and only if the proximity δ𝛿\deltaitalic_δ is separated**C**: Every separated proximity space admits a unique (up to a proximity isomorphism) compactification, which we describe briefly below. | ABC | ACB | BAC | BAC | Selection 1 |
**A**: italic_g **B**: , C¯¯=A¯¯∗B¯¯¯¯𝐶¯¯𝐴¯¯𝐵\overline{\overline{C}}=\overline{\overline{A}}\,*\,\overline{\overline{B}}over¯ start_ARG over¯ start_ARG italic_C end_ARG end_ARG = over¯ start_ARG over¯ start_ARG italic_A end_ARG end_ARG ∗ over¯ start_ARG over¯ start_ARG italic_B end_ARG end_ARG)
implies regular matrix multiplication. The superscript T𝑇Titalic_T implies**C**: The symbol ∗*∗ between two matrices (e.g.,formulae-sequence𝑒𝑔e.g.,italic_e | BCA | CAB | BAC | ACB | Selection 1 |
**A**: Both type 2 and type 3 (whose precise descriptions are given below) are log smooth extensions of rational type**B**: In type 2, the residue extension is of transcendence degree 1 and the extension of the value group is finite**C**: In type 3, the residue extension is trivial.
| CBA | ABC | CBA | ACB | Selection 2 |
**A**: a𝑎aitalic_a if p=apc𝑝𝑎subscript𝑝𝑐p=ap_{c}italic_p = italic_a italic_p start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT**B**: The**C**: By contrast, we conjecture that pcsubscript𝑝𝑐p_{c}italic_p start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is sharp in
Conjecture 6.3 | CBA | BAC | ACB | CAB | Selection 3 |
**A**: 2.8] for log canonical pairs (for a smooth complex projective variety X𝑋Xitalic_X).**B**: 4.2] and [D15, Thm**C**:
Using Theorem 6.2, we establish a precise comparison between two L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT extension theorems of [K10, Thm | ACB | ABC | ACB | CBA | Selection 4 |
**A**: This is because the randomized data is not localized and has the same amplitude at each point of the domain. In comparison, the deterministic scaling threshold does not depend on the geometry because it corresponds to the data zoomed out at a point, which is localized.**B**: For compact manifold the canonical randomization would be based on the spectral expansion of the Laplacian, in which case the probabilistic scaling depends on the global geometry of the underlying manifold**C**:
Considering more general geometries in addition to 𝕋dsuperscript𝕋𝑑\mathbb{T}^{d}blackboard_T start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, will lead to different scenarios | ACB | CAB | CBA | CAB | Selection 3 |
**A**: The fine structure to a large extent revolves around understanding the leaves of classifying trees**B**: The leaves are controlled by depth zero types and so we remind the reader of the definition (for more definitions and classical results consult the Appendix).**C**:
Since the proof of the structure theorem, there has been a considerable amount of work analyzing the ‘fine structure’ of classifiable theories | ACB | ACB | BCA | CAB | Selection 3 |
**A**: the inclusion need not be full.**B**:
As 𝒜𝒜\mathcal{A}caligraphic_A is semi-simple, any tensor functor on 𝒜𝒜\mathcal{A}caligraphic_A is automatically faithful**C**: We are not asking that 𝒜𝒜\mathcal{A}caligraphic_A is embedded in 𝒞𝒞\mathcal{C}caligraphic_C, ie | BCA | CAB | ABC | ACB | Selection 2 |
**A**: The geometry of Kazhdan-Lusztig varieties in (Kac–Moody) flag varieties is also closely related to formulas that compute the restriction of torus-equivariant
cohomology/K𝐾Kitalic_K-theoretic Schubert classes to fixed points**B**: Since Kazhdan-Lusztig varieties degenerate to subword complexes, their Hilbert series can be computed explicitly using combinatorics of reduced words**C**: Under an appropriate grading induced by the torus action, their multidegree/K𝐾Kitalic_K-polynomials can recover the Andersen–Jantzen–Soergel/Billey [AJS94, Bil99] and Graham/Willems [Gra02, Wil04] restriction formulas. Our Gröbner basis completes the story that explicitly interprets these restriction formulas using Gröbner geometry in affine type A𝐴Aitalic_A. For details, see Remark 5.4. | ACB | ABC | BCA | CAB | Selection 2 |
**A**: As we can see from the description so far, there is a heavy emphasis on the the fact that all of our objects are framed**B**: Without the framing, we would need to take the Stokes data up to conjugation by diagonal matrices to get something well defined on isomorphism classes. We should also remark that the idea of using these types of framed wild harmonic bundles, and using Stokes data to build the twistor coordinates, is heavily inspired by the observations made in [GMN13, Section 9.4.3].
**C**: One of the reasons for taking framed objects, is so that the non-trivial Stokes matrix entries are actual coordinates on the isomorphism classes of our objects | ACB | CBA | BCA | BCA | Selection 1 |
**A**: We refer to the introduction of the latter article for further**B**: Some models in financial mathematics and econometrics are threshold diffusions, for instance continuous-time versions of SETAR (self-exciting threshold auto-regressive) models, see e.g. [15, 41]**C**: SBM and OBM and their local time have been recently investigated in the context of option pricing, as for instance in [20] and [16].
In [37] it is shown that a time series of threshold diffusion type captures leverage and mean-reverting effects | ABC | ACB | BAC | CAB | Selection 4 |
**A**: However, such computational efficiency guarantees rely on the regularity condition that the state space is already well explored**B**: Such a condition is often implied by assuming either the access to a “simulator” (also known as the generative model) (Koenig and Simmons, 1993; Azar et al., 2011, 2012a, 2012b; Sidford et al., 2018a, b; Wainwright, 2019) or finite concentratability coefficients (Munos and Szepesvári, 2008; Antos et al., 2008; Farahmand et al., 2010; Tosatto et al., 2017; Yang et al., 2019b; Chen and Jiang, 2019), both of which are often unavailable in practice.**C**:
A line of recent work (Fazel et al., 2018; Yang et al., 2019a; Abbasi-Yadkori et al., 2019a, b; Bhandari and Russo, 2019; Liu et al., 2019; Agarwal et al., 2019; Wang et al., 2019) answers the computational question affirmatively by proving that a wide variety of policy optimization algorithms, such as policy gradient (PG) (Williams, 1992; Baxter and Bartlett, 2000; Sutton et al., 2000), natural policy gradient (NPG) (Kakade, 2002), trust-region policy optimization (TRPO) (Schulman et al., 2015), proximal policy optimization (PPO) (Schulman et al., 2017), and actor-critic (AC) (Konda and Tsitsiklis, 2000), converge to the globally optimal policy at sublinear rates of convergence, even when they are coupled with neural networks (Liu et al., 2019; Wang et al., 2019) | BCA | CBA | ABC | CBA | Selection 1 |
**A**: The first three authors would also like to thank Eric Stucky for his edits to this paper and Amal Mattoo for his contributions to this research.
The fourth author was supported by the Trond Mohn Foundation project “Algebraic and Topological Cycles in Complex and Tropical Geometries”.**B**: The first three authors are grateful for the support of NSF RTG grant DMS-1745638 and making the program possible, as well as to thank Professor Victor Reiner for providing both guidance and independence in their research efforts**C**: A portion of this research was carried out as part of the 2018 REU program at the School of Mathematics at University of Minnesota, Twin Cities | CBA | ACB | BCA | ABC | Selection 1 |
**A**: The Erdós-Straus conjecture can also be rephrased as a problem of an inequality. That is to say, the conjecture can be restated as saying that for all n≥3𝑛3n\geq 3italic_n ≥ 3 the inequality holds**B**:
Despite its apparent simplicity, the problem still remain unresolved. However there has been some noteworthy partial results**C**: For instance it is shown in [2] that the number of solutions to the Erdós-Straus Conjecture is bounded poly-logarithmically on average. The problem is also studied extensively in [3] and [4] | BCA | CAB | BAC | ACB | Selection 2 |
**A**: Many of the remaining did not have a convex partition supporting the choice of line bundles. The next example is an example of this; in this case (but not usually), one is able to find a degeneration of the complete intersection to a toric variety; we find the Laurent polynomial associated to this toric variety and after taking the rigid maximally mutable Laurent polynomial on its polytope, find a mirror.
**B**: Of the 141 Fano fourfolds found in [18], all can be modeled as quiver flag zero loci in a Y𝑌Yitalic_Y-shaped quiver**C**: For 99 of these, the methods above produce a candidate Laurent polynomial mirror | ABC | BAC | CAB | ABC | Selection 3 |
**A**: Recently in [39], the second author and J.J. Xu got the boundary gradient estimates and the corresponding existence theorem for the Neumann boundary value problem on mean curvature equation.
**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. Especially for the mean curvature equation with prescribed contact angle boundary value problem, Ural’tseva [52], Simon-Spruck [43] and Gerhardt [16] got the boundary gradient estimates and the corresponding existence theorem**C**: Meanwhile, the Neumann and oblique derivative problem of partial differential equations were widely studied. For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition, we refer to the book [15] | ACB | BCA | CBA | ABC | Selection 3 |
**A**: The Fourier transform of this function when t=1𝑡1t=1italic_t = 1 is**B**: We will employ as bump functions t∇hPt𝑡subscript∇ℎsubscript𝑃𝑡t\nabla_{h}P_{t}italic_t ∇ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, where ∇hPtsubscript∇ℎsubscript𝑃𝑡\nabla_{h}P_{t}∇ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT stands for the horizontal derivatives
of the Poisson kernel Ptsubscript𝑃𝑡P_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT on ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT**C**: consider scalar ones | CAB | ABC | CBA | ABC | Selection 3 |
**A**: The following statement regarding products of filtrations are obtained at the simplicial level (and in more generality) in [72, Proposition 2.6] and in [42, 73]**B**: These proofs operate at the simplicial level.**C**: The statement about metric gluings appeared in
[7, Proposition 4] and [68, Proposition 4.4] | ACB | CAB | ABC | CAB | Selection 1 |
**A**: A more difficult task would be to consider other types of regularity regimes, e.g**B**: the fractional Sobolev one. For this direction, See [44] and the discussions therein.**C**:
Based on the observations made in this article, the proofs of [14, 15] for the rigidity statements are also valid under little Hölder c1,2/3superscript𝑐123c^{1,2/3}italic_c start_POSTSUPERSCRIPT 1 , 2 / 3 end_POSTSUPERSCRIPT regularity assumption | ACB | BCA | ACB | BAC | Selection 2 |
**A**: Therefore
x∈ϕ−1(ϕ(φσ−1(ψτ→−1(x))))𝑥superscriptitalic-ϕ1italic-ϕsuperscriptsubscript𝜑𝜎1superscriptsubscript𝜓→𝜏1𝑥x\in\phi^{-1}(\phi(\varphi_{\sigma}^{-1}(\psi_{\overrightarrow{\tau}}^{-1}(x))))italic_x ∈ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_ϕ ( italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_ψ start_POSTSUBSCRIPT over→ start_ARG italic_τ end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x ) ) ) ) then x∈ϕ−1(ϕ(V))𝑥superscriptitalic-ϕ1italic-ϕ𝑉x\in\phi^{-1}(\phi(V))italic_x ∈ italic_ϕ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_ϕ ( italic_V ) )**B**: Thus,**C**: by remark 5.14 we have that ϕ(φσ−1(ψτ→−1(x)))=ϕ(x)italic-ϕsuperscriptsubscript𝜑𝜎1superscriptsubscript𝜓→𝜏1𝑥italic-ϕ𝑥\phi(\varphi_{\sigma}^{-1}(\psi_{\overrightarrow{\tau}}^{-1}(x)))=\phi(x)italic_ϕ ( italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_ψ start_POSTSUBSCRIPT over→ start_ARG italic_τ end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x ) ) ) = italic_ϕ ( italic_x ) | CBA | BCA | ACB | CBA | Selection 2 |
**A**: We do not know, but this question can be seen as a test for the condensed wall gadget**B**: If it is proven that ladders with 13 rungs still do not possess the edge-Erdős-Pósa property, then clearly there is a counterexample graph not based on a condensed wall**C**: However, if it is shown that ladders with 13 rungs have the edge-Erdős-Pósa property, then this is yet another strong indication that the condensed wall plays a key role for this property.
| CBA | CAB | ACB | ABC | Selection 4 |
**A**: As a result the expression under square**B**: Indeed, we have bS=−5subscript𝑏𝑆5b_{S}=-5italic_b start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - 5, bI=4subscript𝑏𝐼4b_{I}=4italic_b start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = 4 and
D=bS2−bI2=(3)2𝐷superscriptsubscript𝑏𝑆2superscriptsubscript𝑏𝐼2superscript32D=b_{S}^{2}-b_{I}^{2}=(3)^{2}italic_D = italic_b start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( 3 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT**C**: square roots | CBA | CAB | CAB | BCA | Selection 1 |
**A**: One popular approach is to employ verification arguments to check if a given solution of the HJB equation coincides with the value function at hand, and obtain as a byproduct the optimal control**B**: In the Markovian case, a classical approach in solving stochastic control problems is given by the dynamic programming principle based on Hamilton-Jacobi-Bellman (HJB) equations**C**: Discretization methods also play an important role towards the resolution of the control problem. In this direction, several techniques based on Markov chain discretization schemes [33], Krylov’s regularization and shaking coefficient techniques (see e.g [31, 32]) and Barles-Souganidis-type monotone schemes [2] have been successfully implemented. We also refer the more recent probabilistic techniques on fully non-linear PDEs given by [20] and the randomization approach of [28, 29, 30].
| BAC | CAB | CBA | ABC | Selection 1 |
**A**: The construction used to prove Theorem 6 can also be used to obtain results which are not immediate corollaries of the theorem (or its corollary for automaton semigroups in 8)**B**: The version for automaton semigroups does not follow directly from 8, as the free monogenic semigroup is not a complete automaton semigroup [4, Proposition 4.3] or even a (partial) automaton semigroup (see [8, Theorem 18] or [20, Theorem 1.2.1.4]).
**C**: As an example, we prove in the following theorem that it is possible to adjoin a free generator to every self-similar semigroup without losing the self-similarity property and that the analogous statement for automaton semigroups holds as well | BCA | CBA | ACB | BCA | Selection 3 |
**A**: Introducing a refined variant of the Euler totient function tailored to specific subsets of real numbers, characterized by right continuity while upholding the essence of the original function, paves a seamless path beyond this anticipated obstacle.**B**: The inherent limitation of the Euler totient function, restricted to positive integers and lacking one-sided continuity over the real numbers, posed a formidable challenge, now elegantly surmounted within this paper**C**:
The present study adeptly navigates a significant impediment that might have otherwise hindered previous investigations in this domain | CBA | BCA | ACB | ACB | Selection 1 |
**A**: K.C. was supported by the KIAS Individual Grant MG078902, a POSCO Science Fellowship, an Asian Young Scientist Fellowship, and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00219980). C.M. was supported by the NSF grant DMS-1905165**B**: F.S. was supported by a Leverhulme Trust Research Project Grant RPG-2016-174. We are grateful to Richard Bamler, Costante Bellettini, Robert Haslhofer, Or Hershkovits, Daren Cheng, Ciprian Manolescu, Leon Simon, and Brian White for useful conversations related to this paper as well as to the anonymous referees for their careful reading and many helpful suggestions.
**C**: O.C. was partially supported by a Terman Fellowship, a Sloan Fellowship, and NSF grants DMS-1811059, DMS-2016403, and DMS-2304432. He would also like to acknowledge the MATRIX Institute for its hospitality during the time which some of this article was completed | CAB | ACB | BCA | CBA | Selection 3 |
**A**: Then, it will remain to observe that combinatorially relevant terms cancel and that the remaining terms can be bounded sufficiently well, uniformly in m𝑚mitalic_m. This proof is quite lengthy. Indeed, expanding the square will lead to terms with up to 5555 indices, thus giving rise to a huge number of cases that need to be distinguished.**B**:
This section contains the main technical part of the proof of our main result, the probabilistic estimates stated in Proposition 3.3 and Lemma 4.1**C**: The strategy that we will use to estimate all these terms is to expand the square of sums over the particles and then to use independence of the positions of the particles to calculate the expectations, distinguishing between terms where different particles appear and where one or more particles appear more than once | CAB | BAC | CBA | BCA | Selection 1 |
**A**: Staller, d𝑑ditalic_d means the vertex is already claimed by Dominator**B**: And n𝑛nitalic_n refers to “null” which means that the vertex
is still free — not yet claimed by any player in the current game**C**: While 1111 means that the vertex is already dominated, | BCA | ACB | ABC | ACB | Selection 3 |
**A**: Then in Section 4 we establish the tropical/holomorphic correspondence for Maslov index zero discs with boundary on special Lagrangian fibres. Then we prove the equivalence of counting of Maslov index two discs with the same boundary conditions with the weighted count of broken lines in Section 5. This includes an explanation of the renormalization process of Hori-Vafa [HV].**B**:
The arrangement of the paper is as follows: In Section 2, we review the special Lagrangian fibration constructed in [CJL] and its properties**C**: We review the tropical geometry on the base of the special Lagrangian with the complex affine structure in Section 3 | CAB | CBA | BAC | BCA | Selection 1 |
**A**: The family (6.1) is of type 2+ 2+ 2 whenever the parameter t𝑡titalic_t is a root of unity**B**: Its feature is the presence of a right angle; in fact the spaces of this family all belong to the self-conjugated spaces studied in Section 4.1**C**: Up to homothety they form a family parametrized by a rational number a≠0,±1𝑎0plus-or-minus1a\neq 0,\pm 1italic_a ≠ 0 , ± 1 and a root of unity y≠1𝑦1y\neq 1italic_y ≠ 1, with τ𝜏\tauitalic_τ a purely imaginary root of
| BCA | ACB | CAB | ABC | Selection 4 |
**A**: One choice of the groups that can provide nontrivial twist as well as good physical meaning is the loop groups LG𝐿𝐺LGitalic_L italic_G with G𝐺Gitalic_G a compact Lie group.
**B**: The problem does not lie in what the elliptic cohomology theory is, but the property of the equivariant groups directly**C**: One question arising with Proposition 6.8 is how we can get a more interesting example of twisted quasi-elliptic cohomology that can detect deeper physical meaning | ABC | CBA | BCA | CAB | Selection 2 |
**A**: This will be bootstrapped to the multi-color case in later sections**B**: We**C**: Note that the 1111-color case with the completeness requirement is not very interesting, and also not useful for the general case: completeness states that every node on
the left must be connected, via the unique edge relation, to every node on the right – regardless of the matrix | ABC | CBA | ACB | BCA | Selection 3 |
**A**: For instance, there is a version of 2222-henselian valuations for fields with non-trivial radicals [9], useful in Galois Theory (see Example 23)**B**: The next problem follows the same idea.**C**:
Cordes also found that many results concerning quadratic forms and related subjects are still valid when replacing the squares group by R(F)𝑅𝐹R(F)italic_R ( italic_F ) | BAC | BCA | ABC | CAB | Selection 2 |
**A**: Szepesvári, 2018; Dalal et al., 2018; Srikant and Ying, 2019) settings. See Dann et al. (2014) for a detailed survey. Also, when the value function approximator is linear, Melo et al. (2008); Zou et al**B**: (2019); Chen et al. (2019b) study the convergence of Q-learning. When the value function approximator is nonlinear, TD possibly diverges (Baird, 1995; Boyan and Moore, 1995; Tsitsiklis and Van Roy, 1997). Bhatnagar et al. (2009) propose nonlinear gradient TD, which converges but only to a locally optimal solution**C**: See Geist and Pietquin (2013); Bertsekas (2019) for a detailed survey. When the value function approximator is an overparameterized multi-layer neural network, Cai et al. (2019) prove that TD converges to the globally optimal solution in the NTK regime. See also the independent work of Brandfonbrener and
Bruna (2019a, b); Agazzi and Lu (2019); Sirignano and Spiliopoulos (2019), where the state space is required to be finite. In contrast to the previous analysis in the NTK regime, our analysis allows TD to attain a data-dependent feature representation that is globally optimal. | BAC | ABC | CAB | ACB | Selection 2 |
**A**: In particular, we would want to rigidify homotopy coherent comultiplications and coactions.
However, we face multiple obstacles**B**: As in algebras, we need to construct homotopy coherent coalgebraic structures**C**: Firstly, a common assumption for algebras and modules is that the monoidal product is closed and preserves colimits. So in order to dualize the results for coalgebras and comodules, we would need a monoidal product that preserves limits, which in practice is never the case. | ACB | ABC | BAC | BCA | Selection 3 |
**A**: }\end{cases}italic_v ( italic_x ) ≔ { start_ROW start_CELL roman_min start_POSTSUBSCRIPT italic_y ∈ italic_I start_POSTSUBSCRIPT italic_π , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f ( italic_y ) end_CELL start_CELL if italic_x ∈ blackboard_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL roman_min start_POSTSUBSCRIPT italic_y ∈ [ 0 , 1 ] end_POSTSUBSCRIPT italic_f ( italic_y ) end_CELL start_CELL otherwise**B**: end_CELL end_ROW
**C**: \min\limits_{y\in[0,1]}f(y)&\text{ otherwise | ABC | BCA | ABC | CAB | Selection 2 |
**A**: We turn to non-zero weak sequential limit points**B**: First of all, for very special spaces, the existence of a non-zero weak sequential limit point does imply hypercyclicity**C**: Recall that a Banach space has the Schur property if every weakly convergent sequence converges; and a Fréchet space is a Montel space if every bounded set is relatively compact. In these spaces, any weakly convergent sequence has (at least) a convergent subsequence. Thus the following is a consequence of Theorem 3.2.
| ACB | ABC | CAB | BCA | Selection 2 |
**A**:
Many aspects of 2222-category theory benefit from a passage to double categories**B**: For example, a good notion of limit for 2222-categories is that of a 2222-limit, where the universal property is expressed by an isomorphism between hom-categories, rather than hom-sets**C**: As clingman and the author show in [clingmanMoser], a 2222-limit cannot be characterized as a 2222-terminal object in the 2222-category of cones, but a passage to double categories allows such a characterization by results of Grandis and Paré [Grandis, GrandisPare]. Indeed, they show that the 2222-limit of a 2222-functor F𝐹Fitalic_F is double terminal in the double category of cones over the corresponding double functor ℍFℍ𝐹\mathbb{H}Fblackboard_H italic_F. This result also holds in the more homotopical case of bi-limits, where the universal property is expressed by an equivalence of hom-categories, as clingman and the author show in [cM2]. | BAC | ABC | CAB | CAB | Selection 2 |
**A**: However, we will denote**B**: Strictly speaking, the above definitions apply only to the case where GG\rm Groman_G is special**C**: When GG\rm Groman_G is not special, the above objects will in fact need to be replaced by the derived push-forward of the above objects viewed as sheaves on the
big étale site of k𝑘kitalic_k to the corresponding big Nisnevich site of k𝑘kitalic_k, as discussed in (8.3.6) | BCA | CBA | BAC | CAB | Selection 4 |
**A**: These were introduced in joint work with Sven Meinhardt in [DM20], as part of a project to realise the cohomological Hall algebras defined by Kontsevich and Soibelman [KS11] as positive halves of generalised Yangians**B**: The construction of the BPS Lie algebra for arbitrary symmetric quivers with potential is recalled in §2.4**C**: Note that the BPS Lie algebra is defined by a quite different perverse filtration, on vanishing cycle cohomology of a different Calabi–Yau category.
| CBA | ABC | BAC | ACB | Selection 3 |
**A**: It is motivated by Steiner’s formula relating curvatures to the change in area under parallel surfaces**B**: The main result is to prove the correspondence between discrete CMC-1 surfaces and circle patterns (Theorem 0.2).**C**:
In this section, we define a notion of integrated mean curvature on horospherical nets | BCA | ACB | CBA | BAC | Selection 1 |
**A**: exists a finite set F⊆V1𝐹subscript𝑉1F\subseteq V_{1}italic_F ⊆ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT such that ∀x∈V1,∃x′∈F,x≡1x′formulae-sequencefor-all𝑥subscript𝑉1formulae-sequencesuperscript𝑥′𝐹subscript1𝑥superscript𝑥′\forall x\in V_{1},\exists x^{\prime}\in F,x\equiv_{1}x^{\prime}∀ italic_x ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ∃ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_F , italic_x ≡ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT**B**: We define W≜⋂x′∈FV2(x′,y′)≜𝑊subscriptsuperscript𝑥′𝐹superscriptsubscript𝑉2superscript𝑥′superscript𝑦′W\triangleq\bigcap_{x^{\prime}\in F}V_{2}^{(x^{\prime},y^{\prime})}italic_W ≜ ⋂ start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_F end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT, which is an open set since it
is a finite intersection of open sets**C**: We claim that V1×W⊆f−1(U)subscript𝑉1𝑊superscript𝑓1𝑈V_{1}\times W\subseteq f^{-1}(U)italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_W ⊆ italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_U ). To prove this fact notice that | ABC | BAC | CAB | CBA | Selection 1 |
**A**: Figure 2 illustrates both diagrams combined, along with all the ZFC-provable inequalities that we are aware of**B**: See [Bla10, BJ95] for the definitions and the proofs for the inequalities (with the exception of cof(ℳ)≤𝔦cofℳ𝔦{\operatorname{cof}(\mathcal{M})}\leq\mathfrak{i}roman_cof ( caligraphic_M ) ≤ fraktur_i, which was proved in [BHHH04]). In the following, we only give the definitions of the non-Cichoń-characteristics that we will investigate in this paper.
**C**: We also investigate some of the characteristics in the Blass diagram [Bla10, Pg. 481] | BCA | CBA | CBA | CBA | Selection 1 |
**A**: One of them is called a parabolic visual metric based on the vertical geodesic in some negatively curved solvable Lie groups in [24]**B**: It should be noted that this parabolic visual metric was formerly named Euclid–Cygan metric by Hersonsky and Paulin [20] in the study of the rigidity of discrete isometry groups of negatively curved spaces. With the aid of this notion, Dymarz [8, 9] recently studied the quasi-isometric rigidity of mixed type locally compact amenable hyperbolic groups.
**C**: In [24, 26], Shanmugalingam and Xie proved that all self quasi-isometries of these groups are almost isometries | CBA | CAB | ACB | CAB | Selection 3 |
**A**: The reader is referred to [MJT21a, § 3] and [BG21, § 11] for examples of spherical homogeneous spaces for which versions of Proposition 1.1 and Theorem 1.3 are applied to determine their (k,F)𝑘𝐹(k,F)( italic_k , italic_F )-forms**B**: Other results, based in part on a weaker version of Proposition 1.1, concerning the real forms of complex symmetric spaces can be found in [MJT21b].**C**:
Let us mention that forms of spherical homogeneous spaces (see Definition 4.15) over an arbitrary base field of characteristic zero were studied by Borovoi and Gagliardi in [Bor20, BG21] | ACB | BCA | ACB | ACB | Selection 2 |
**A**:
A natural generalisation of the above processes are the broad family of Markov Additive Processes (MAPs), which incorporate an externally influencing Markov environment, providing greater flexibility to the characteristics of the underlying process in terms of its claim frequency and severity distributions, see [1] (Chapter XI). 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**B**: More recently, [17], derive and compare results for continuous-time MAPs with lattice (discrete-space) and non-lattice support. It is worth noting here that the authors in this work do discuss some of the corresponding results for the fully-discrete (time and space) MAP model considered in this paper, however, only a limited number of results are stated and a variety of important steps and proofs were omitted.**C**: [15] extended the initial findings of [19] by providing the probabilistic construction of the scale matrices, identifying their transforms and considering an extensive study of exit problems including one-sided and two-sided exits, as well as exits for reflected processes via implementation of the occupation density formula. Further studies on MAPs and their exit/passage times can be found in [8], [4], [9], among others | ABC | ACB | BCA | ABC | Selection 2 |
**A**: It has been shown that in some circumstances, faster dispersal is sometimes selected for [46, 49]. See also [27, 40, 48]. We also mention the work [36] on the evolution of dispersal in phytoplankton populations, where individuals compete non-locally for sunlight.
**B**: 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, which are subject to a uni-directional drift [51, 56]**C**: The work of Hastings and Dockery et al. has also stimulated substantial mathematical analysis of competition models involving two species | BCA | CBA | ACB | BAC | Selection 2 |
**A**: The chain with respect to which one is considering representation stability there is of course still the chain of symmetric groups. Thus, representation stability with respect to a chain of diagrammatically defined algebras is not considered in [1] (Barter, Entova-Aizenbud, Heidersdorf).**B**: 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. Diagrammatics and representation stability have, however, been uttered in the same breadth, but in a different sense: in [1] (Barter, Entova-Aizenbud, Heidersdorf) the authors produce a functor from the category of FIFI\operatorname{FI}roman_FI-modules modulo finite length FIFI\operatorname{FI}roman_FI-modules to the abelian envelope of the Deligne category**C**:
To our knowledge, Temperley-Lieb algebras have not been studied in the representation stability literature, or within the broader context of representation stability and FIFI\operatorname{FI}roman_FI-modules. 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) | CAB | BCA | CBA | ACB | Selection 3 |
**A**: The 3-regular Cayley graph can be transformed to a 4-regular graph middle pages graph (Fig.8) that offers each branching number an Eulerian tour, in line with classic 3-to-4-regular transformations**B**: René Descartes (1596-1650) knew that packaging an infinite number of cubes with three edges at each node gives a 3-dimensional universe of adjacent cubes. It can be projected to a 4-regular Cartesian coordinate system of adjacent squares.
**C**: Plato (427-347bC) knew that a 3-regular cube with three edges at each node encapsulates a dual 4-regular octahedron with four edges at each node, and vice versa | ACB | BAC | CBA | ABC | Selection 1 |
**A**: Moreover, multiplying the product above by the left-hand side of Theorem 1.4 yields the octuple product in 4.13**B**: Hence under Theorem 1.4, the octuple identity and Proposition 5.4 are equivalent.
**C**: The q𝑞qitalic_q-coefficients in the right-hand side above are the same (up to a sign) as the q𝑞qitalic_q-coefficients on the right of the octuple identity from 4.13 | CBA | CBA | CAB | BCA | Selection 4 |
**A**: One of the main difficulties in proving a contraction theorem in the Kähler category is the lack of a base-point free theorem analogous to that of [KM98, Theorem 3.3] in the projective case**B**: However, there is a base-point free conjecture in the Kähler category involving nef and big cohomology classes which can be thought of as an analogue of [KM98, Theorem 3.3]. This conjecture is stated in [H18, Conjecture 1.1] for manifolds.
**C**: Note that, an exact analogue of [KM98, Theorem 3.3] is impossible on a compact Kähler variety which is not projective, since the existence of a big divisor on a compact Kähler variety with rational singularities implies that it is projective by [Nam02, Theorem 1.6] (see Theorem 2.13) | ACB | ABC | BCA | BAC | Selection 1 |
**A**: 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**B**: In the real hyperbolic case, our estimates extend Yeung’s estimates for cocompact lattices [Yeu94, Theorem 2.4.1] to noncocompact lattices.
**C**: The improvement results from our sharper injectivity radius estimates in Subsection 10.3 | BCA | CBA | ACB | BAC | Selection 3 |
**A**: More specifically, we proved**B**:
In version 1 of this note, which can still be found on the ArXiv, we showed that the analogous version of the conjecture for complex functions on {−1,1}nsuperscript11𝑛\{-1,1\}^{n}{ - 1 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT which have modulus 1111 fails**C**: This solves a question raised by Gady Kozma some time ago (see [K], comment from April 2, 2011) | ACB | ABC | CAB | ACB | Selection 3 |
**A**: This Dyson process of mutually avoiding Brownian motions is a leading player [OY02] in**B**:
The OU dynamics on GUE induces a form of Dyson’s Brownian motion [Dys62] on the eigenvalues**C**: In fact this can be used to also prove the value of the highest eigenvalue starts de-correlating at the same time scale of n−1/3superscript𝑛13n^{-1/3}italic_n start_POSTSUPERSCRIPT - 1 / 3 end_POSTSUPERSCRIPT | BAC | CAB | CBA | BAC | Selection 2 |
**A**:
In the original definition of separation profile, the subset which is removed is a subset of vertices**B**: But for bounded degree graphs, it is easy to see that the two resulting definitions of separation profiles have same asymptotic behaviors**C**: Moreover, the advantage of working with edges instead of vertices is that our main intermediate result, Proposition 3.2, does not require the assumption that the graph has bounded degree. | ACB | BAC | ABC | BCA | Selection 3 |
**A**: However, Floater-Hormann becomes indistinguishable from 5thsuperscript5𝑡ℎ5^{th}5 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT-order splines.
Further, when considering the amount of coefficients/nodes required to determine the interpolant, plotted in the right panel (with logarithmic scales on both axes)**B**: The polynomial convergence rates of Floater-Hormann and all**C**: The observations made in 2D remain valid | ACB | BCA | CAB | ACB | Selection 2 |
**A**:
In the remaining part of this section we shall prove Theorems 1.1 and 1.2 at one stroke**B**: Nevertheless, in this case we need to pay extra care to fix a cohomology class.**C**: The proof of Theorem 1.1 essentially follows from (and is somewhat simpler than) that of its local variant, i.e., Theorem 1.2, as we do not need to tackle with boundary conditions over closed manifolds | BCA | ACB | CAB | ABC | Selection 2 |
**A**: Conditions for such families of maps to define a permutation of the field 𝔽𝔽\mathbb{F}blackboard_F are well studied and established for special classes like Dickson polynomials [20], linearized polynomials [21] and few other specific forms [13, 14] to name a few.
**B**: Some well-studied families of polynomials include the Dickson polynomials and reverse Dickson polynomials, to name a few**C**: There has been extensive study about a family of polynomial maps defined through a parameter a∈𝔽𝑎𝔽a\in\mathbb{F}italic_a ∈ blackboard_F over finite fields | BAC | BCA | CBA | BAC | Selection 3 |
**A**: Now every Rauzy class may contain both untwisted and twisted IRE schemes, however the twist total value is the same for the whole class. Induction can transfer turns between a scheme and its dual, while the twist total remains unchanged.
**B**: The last proposition adds one more argument towards the necessity of considering IREs instead of IETs: generically, for an untwisted IET scheme, its dual is twisted and therefore is not an IET scheme, while the twist total characterizes a pair of mutually dual schemes (σ,ℐσ)𝜎ℐ𝜎(\sigma,{\mathcal{I}}\sigma)( italic_σ , caligraphic_I italic_σ ) as a whole and does not change due to induction**C**: It is natural to generalize the classical notion of Rauzy classes for IETs towards IRE schemes: if one IRE scheme is obtained from another as a result of applying an elementary induction step, then they belong to the same Rauzy class | ABC | BAC | BAC | CAB | Selection 4 |
**A**: Then we define the local minima of such problems, followed by the assumptions we use in our convergence**B**: We start with the definition
of the key geometry property of sets involving sparse and low-rank optimization problems**C**: This section is organized as follows | CAB | BCA | ACB | CBA | Selection 4 |
**A**: This type of weak dissipative structure has been recently observed for several interesting systems, e.g. the Timoshenko system with the Fourier law [23] or the Cattaneo law [24] of heat conductions.**B**:
which exactly coincides with the one in [35, Theorem 1.2]**C**: The estimate (3.21) shows the strict dissipativity of Type (2,3), whose definition is referred to [22, Defintion 1.0.2] | BCA | CBA | CAB | BCA | Selection 3 |
**A**: We consider it interesting to discuss the direct and inverse problems in general. In fact, one may expect that solutions of these questions would yield classification results similar to the one of [BGT12].**B**: Indeed, it can be restated to say that Lie cores of piecewise definable subgroups generated by pseudo\hypfinite definable approximate subgroups are nilpotent [BGT12, Proposition 9.6]**C**:
The classification theorem by Breuillard, Green and Tao for finite approximate subgroups [BGT12] can be interpreted as an answer to the direct problem for the class of piecewise definable subgroups generated by pseudo\hypfinite definable approximate subgroups | BAC | ACB | BAC | CBA | Selection 4 |
**A**: Not every FAOM is realizable as ℒ(𝒜)ℒ𝒜\mathscr{L}(\mathscr{A})script_L ( script_A ) for a finitary arrangement**B**:
Our “Finitary Affine Oriented Matroids” axiomatize properties of the polyhedral stratification of Euclidean space induced by finitary hyperplane arrangements**C**: Still, some familiar geometric and topological features generalize nicely to the non-realizable case as well. | ABC | CAB | BAC | ABC | Selection 3 |
**A**: To fix this, we will show that (28) is actually induced by a pullback map.**B**: However, this argument does not prove that (28) is an isomorphism of rings, not just of A∗(pt)superscript𝐴ptA^{*}(\mathrm{pt})italic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_pt )-modules**C**:
obtained in [DoZh], see also [EdGr, DrTy] | CAB | CAB | BCA | CBA | Selection 4 |
**A**: This technique formulates the objective function and constraints using linear matrix inequalities (LMIs) to synthesize a Markov chain capable of achieving a desired distribution while adhering to specified transition constraints. Notably, this study does not impose any assumptions on the Markov chain, rendering the problem inherently non-convex. The problem is convexified for practical purposes but the optimal convergence rate of the original non-convex problem cannot be attained.**B**: In [8], the Metropolis-Hastings algorithm is extended to incorporate safety upper bound constraints on the probability vector**C**: This paper includes numerical simulations that demonstrate the application of the extension in a probabilistic swarm guidance problem.
In order to enhance convergence rates, [9] introduces a convex optimization-based technique for Markov chain synthesis | CAB | BAC | BCA | CBA | Selection 1 |
**A**: §2.1). Consequently, many of the obstructions involved in Corollary 1.17 follow simply from singular homology considerations (c.f. Remark 6.4).**B**:
We note, however, that Weinstein embeddings have more restricted topology compared to Liouville embeddings**C**: Namely, the complementary cobordism must admit a Morse function with all critical points having index at most half the dimension (see e.g | CBA | CAB | BCA | ACB | Selection 2 |
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