robench-2024b
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48 items • Updated
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Acknowledgements. BG was, for some of the period during which this work was carried out, a fellow of the Radcliffe Institute at Harvard. <|MaskedSetence|> TT is supported by NSF Research Award DMS-0649473, the NSF Waterman award and a grant from the MacArthur Foundation. <|MaskedSetence|> <|MaskedSetence|> This work... | **A**: TZ is supported by ISF grant 557/08, an Alon fellowship and a Landau fellowship of the Taub foundation.
**B**: He is very grateful to the Radcliffe Institute for providing excellent working conditions.
**C**: All three authors are very grateful to the University of Verona for allowing them to use classrooms at... | BAC | BAC | CBA | BAC | Selection 2 |
been already be verified independently in [H] and [GGV2]. We will verify the
case (m,n)=(50,75)𝑚𝑛5075(m,n)=(50,75)( italic_m , italic_n ) = ( 50 , 75 ). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We do not provide proofs for. | **A**: Due to this we take an
alternative strategy.
**B**: Doing this directly using (1.10) amounts
to solving a system of 123123123123 equations and 123 variables.
**C**: The first part of this procedure is similar to the one used
in [GGV1]*Section 8, and is inspired by [M].
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Using the calculations in [Law95], one sees by inspection that the number of Jordan blocks of unipotent elements on the adjoint module is independent of good characteristic; this is reflecting the fact that the centralisers of unipotent elements Gusubscript𝐺𝑢G_{u}italic_G start_POSTSUBSCRIPT italic_u end_POSTSUBSCRI... | **A**: Most of the time Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT is irreducible and the theory of high weights identifies Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT uniquely up to isomorphism (possibly after twisting with ... | CBA | CAB | CAB | CAB | 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**: The referee also directed the author to the paper [ACMV17a].
**B**: Finally, Subsection 3.5 would have been nonexistent, had it not been for the referee’s explicit question on the topic.
**C**: (3) The definition of unpointed regular homomorphisms is due to the referee (Definition 3.7).
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<|MaskedSetence|> <|MaskedSetence|> 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]. Recently, Cheraghi and his collaborators have found several other important applications. In [Che13] ... | **A**:
1.4.
**B**: The tools in [Che13] and [Che19] have led to part of the recent major progresses on the dynamics of quadratic polynomials.
**C**: Some observations
There are several applications of Inou-Shishikura’s invariant class.
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The paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In Section 4.1 we find a simple formula for a local Kähler potential of the elliptic metric, and in Section 4.2 we show that in the limit when the order of the elliptic element
tends to ∞\infty∞ the elliptic metric coincides wit... | **A**: Section 2 contains the necessary background material.
**B**: Specifically,
we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces.
Since the cases of smooth (both compact and punctu... | ABC | ACB | ACB | ACB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> This is not the only manner in which the two are connected. <|MaskedSetence|> 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 ... | **A**:
7.
**B**: Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e.
**C**: 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 tr... | ABC | ACB | ACB | ACB | Selection 2 |
<|MaskedSetence|> By definable we will mean definable with parameters. We will need a few results from [27]. <|MaskedSetence|> <|MaskedSetence|> 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. Ordered abelian groups... | **A**: Since this text is not readily available, we try to keep the present work as self contained as possible, referring to more accessible sources whenever we are aware of such.
**B**: Throughout the text G𝐺Gitalic_G will denote a group, usually abelian and often ordered, ℭℭ\mathfrak{C}fraktur_C will denote a suffi... | BAC | BAC | BAC | BAC | Selection 2 |
One difficulty that hinders the development of efficient methods is the presence of high-contrast coefficients [MR3800035, MR2684351, MR2753343, MR3704855, MR3225627, MR2861254]. <|MaskedSetence|> <|MaskedSetence|> Additionally, the spectral techniques remove
macro-elements corner singularities that occur in LOD me... | **A**: 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.
**B**: We note the proposal in [CHUNG2018298] of ... | CAB | CAB | ABC | CAB | 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**: The conventional depiction of these theories operates within a fixed temporal frame, offering a foundational understanding of risk.
Over the past two decades, not only has the study of static risk measures flourished, but also dynamic theories of risk measurement have developed into a thriving and mathematical... | CAB | CAB | CAB | CAB | Selection 1 |
Proof.
Decompose σ=τϵadh𝜎𝜏superscriptitalic-ϵadℎ\sigma=\tau\epsilon^{{\rm ad}h}italic_σ = italic_τ italic_ϵ start_POSTSUPERSCRIPT roman_ad italic_h end_POSTSUPERSCRIPT as in (3). <|MaskedSetence|> By Lemma 2.1, since m𝑚mitalic_m divides s¯c¯𝑠𝑐\bar{s}cover¯ start_ARG italic_s end_ARG italic_c (by assumption), ... | **A**: In this case Gτ=superscript𝐺𝜏absentG^{\tau}=italic_G start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT = SO(2n+1)2𝑛1(2n+1)( 2 italic_n + 1 ) and following the notation of the identity (6),
.
**B**: To prove that V(λ)𝑉𝜆V(\lambda)italic_V ( italic_λ ) integrates to a Gσsuperscript𝐺𝜎G^{\sigma}italic_G s... | BCA | BCA | BCA | BAC | Selection 2 |
Both HaldaneHaldane (1991) and WuWu (1994) considered ideal gases with a fixed boundary condition, which is in fact the periodic boundary condition, which renders the anyon systems being on spaces without actual boundaries. Neither did Ref. Hu et al. (2014) consider boundaries.
Nevertheless, materials with boundaries... | **A**: A recent workHung and Wan (2015) has shown how gapped boundary conditions of a topological order dictate the ground state degeneracy of the topological order and how certain anyons in the bulk may connect to the gapped boundary.
**B**: Hence we expect that the boundary conditions of a topological order would af... | CAB | BCA | CAB | CAB | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> This would have the intended effect. However, such a condition cannot be expressed without circularity. Fortunately, the intended content can indeed be expressed (cf. the previous footnote); our device for doing so is to phrase this content by reference to a well-defined suborder ℚ... | **A**: Our way around this difficulty is to incorporate, in our definition, a clause which stipulates that the above operation can be carried out.
**B**: This will ensure that the generic specializing function for T∼μ(α)subscriptsimilar-to𝑇𝜇𝛼\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{T}}}_{\mu(\alph... | ACB | ACB | ACB | ABC | Selection 2 |
<|MaskedSetence|> Acknowledgements.
We thank the referees for the careful reading of the first version of the text and for many helpful remarks. <|MaskedSetence|> 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 Sim... | **A**: This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’.
**B**: The first author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.
.
**C**:
1.6.
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<|MaskedSetence|> If cn=cn+1subscript𝑐𝑛subscript𝑐𝑛1c_{n}=c_{n+1}italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT then clearly An(ck)<An+1(ck)subscriptsuperscript𝐴subscript𝑐𝑘𝑛subscriptsuperscript𝐴subscript𝑐𝑘𝑛1A^{(c_{k})}_{n}<A^{(c_{k})}_{n... | **A**: We show that (cn)subscript𝑐𝑛(c_{n})( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is balanced.
**B**: These two statements gives the claim.
We show that (cn2)subscriptsuperscript𝑐2𝑛(c^{2}_{n})( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT... | ACB | ABC | ABC | ABC | Selection 2 |
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