Search

The chord log-Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-chord measure of a convex body. The chord log-Minkowski problem has been extensively studied by Guo, Xi, and Zhao [Math. Ann. (2023), DOI 10.1007/s00208-023-02721-8]; Lutwak, Xi, Yang, and Zhang [Commun. Pure Appl. Math. (2023), DOI 10.1002/cpa.22190]; Qin [Adv. Math. 427 (2023), Paper No. 109132]. In this paper, we solve the chord log-Minkowski problem when q\in (0,1), without symmetry assumptions. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of p-adic groups. The input for this construction is a pair (\lambda, \chi) where \lambda is a stable vector in a certain representation coming from a Moy–Prasad filtration, and \chi is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet. [ABSTRACT FROM AUTHOR]

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, using the method of blow-up analysis, we establish a trace Trudinger-Moser inequality involving L^p-norm and obtain the corresponding extremal on a compact Riemann surface with a smooth boundary. The result generalizes those of Li-Liu [Math. Z. 250 (2005), pp. 363–686] and Zhang [Commun. Pure Appl. Anal. 20 (2021), pp. 1721–1735]. [ABSTRACT FROM AUTHOR]

Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone–Čech compactification \beta Y is scattered. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

In Theorem 0.1 of the paper "A Severi type theorem for surfaces in \mathbb {P}^6" [Proc. Amer. Math. Soc. 149 (2021), pp. 591–605], we claimed to have given a complete classification of smooth surfaces in \mathbb {P}^6 with one 4-secant plane through the general point of \mathbb {P}^6, but the classification is still incomplete. [ABSTRACT FROM AUTHOR]