Hindman conjectured that for any finite partition of ℕ, there exists a monochromatic set of the form {x,y,x + y,xy}. Recently, Bowen proved this conjecture for all 2-partitions. In this paper, we extend Bowen’s result to semirings (S, +, ·) where S · s is piecewise syndetic for every s ∈ S. To achieve this, we provide a combinatorial proof of a piecewise syndetic generalization of the Bergelson–Glasscock IP_r^* van der Waerden’s Theorem. Furthermore, we address the non-commutative case, discussing extensions to semirings with non-commutative operations.

An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form C ∈ ℤ[X₁,…,X_n], provided that n ≥ 14.

A result of K. Hiraga [Compos. Math., 2004, 140(6): 1625–1656] says that endoscopic transfer is compatible with Aubert-Zelevinski involution. In this paper, we generalize Hiraga’s result to metaplectic group setting.

We show conditions on k such that any number x in the interval

can be represented in the form

x_1^{a_1}{x}_2^{a_2}+x_3^{a_3}{x}_4^{a_4}+\cdots +x_{k-1}^{a_{k-1}}{x}_k^{a_k}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x_{1}^{a_{1}}x_{2}^{a_{2}}+x_{3}^{a_{3}}x_{4}^{a_{4}}+\cdots+x_{k-1}^{a_{k-1}}x_{k}^{a_{k}}$$\end{document}

, where the exponents a_2i−1 and a_2i are positive integers satisfying a_2i−1 + a_2i = s for

i=1,2,\dots ,\frac{k}{2}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots,{k\over 2}$$\end{document}

, and each x_i belongs to the generalized Cantor set. Moreover, we discuss different types of non-diagonal polynomials and clarify the optimal results in low-dimensional cases.

The aim of this article is to study the Auslander algebra of any representation-finite string algebra. More precisely, we introduce the notion of gluing algebras and show that the Auslander algebra of a representation-finite string algebra is a quotient of a gluing algebra of

. As applications, the Auslander algebras of two classes of string algebras whose quivers are Dynkin types

\mathbb{A}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{A}$$\end{document}

and

\mathbb{D}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{D}$$\end{document}

are described. Moreover, the representation types of the above Auslander algebras are also given exactly.

In this paper, we consider Whittaker modules for the truncated current Lie algebra where the underlying Lie algebra is of type B₂ and the level is one. More precisely, we give the construction of a series of simple Whittaker modules and show that they classify simple Whittaker modules under some conditions. When the Whittaker function is singular, we show that there might exist simple Whittaker modules with non-trivial Whittaker vectors, and we concretely give the expressions of those non-trivial Whittaker vectors. Our results show that the existence of non-trivial Whittaker vectors depends not only on the Whittaker function, but also on the actions of the Casimir elements.

Let (

) be an n-exangulated category with enough projectives and enough injectives, and

\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{X}$$\end{document}

be a cluster-tilting subcategory of

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

. Liu and Zhou have shown that the quotient category

\mathcal{C}\mathcal{/}\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}/\mathscr{X}$$\end{document}

is an n-abelian category. In this paper, we prove that if

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has Auslander–Reiten n-exangles, then

\mathcal{C}\mathcal{/}\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}/\mathscr{X}$$\end{document}

has Auslander–Reiten n-exact sequences. Moreover, we also show that if a Frobenius n-exangulated category

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has Auslander–Reiten n-exangles, then the stable category

\overline{\mathcal{C}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{\mathscr{C}}$$\end{document}

of

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has Auslander–Reiten (n + 2)-angles.

Inspired by Nakaoka’s series of investigations on the construction of abelian structure from a triangulated category with the help of cotorsion pairs, we want to know whether the “high dimension” ones, i.e., (left) n-cotorsion pairs, also have the same “function”. Through some special constructions, we obtain that the heart of a twin left n-cotorsion pair in a triangulated category is a preabelian category. Moreover, if we only consider a single left n-cotorsion pair, then its heart becomes an abelian category. At last, we give an application of some results of a twin left n-cotorsion pair on the colocalization sequence.

In this paper, we investigate the Dirichlet problem concerning the homogeneous mixed Hessian type equation in the convex cone

with prescribed asymptotic behavior at 0 ∈ Ω, where Ω is a strictly (k − 1)-convex bounded domain. By constructing approximating solutions, we establish the corresponding theorems on the existence and uniqueness of C^1,1 solutions. The key is to construct subsolutions for the approximating non-degenerate mixed Hessian type equation. Moreover, our main technique is to establish uniform gradient estimates and second-order estimates which are independent of the approximation.

In this paper we are interested in the existence and uniqueness of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations

in the setting of the weighted Sobolev spaces.

In this paper, we establish the quantitative weighted endpoint estimates for multilinear pseudo-differential operators. The corresponding conclusions for multilinear square functions are also obtained. These were mainly done by using the Nazarov–Treil–Volberg’s method in conjunction with some ideas from Stockdale.