We study the representations of extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ where $q$ is $N$-th primitive root of unity ($({{\mathbb{C}}_{q}}$ is the quantum torus in two variables). We first prove that $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ for a suitable number of copies is a quotient of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$. Thus any finite dimensional irreducible module for $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$lifts to a representation of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$. Conversely, we prove that any finite dimensional irreducible module for $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ comes from above. We then construct modules for the extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)\oplus \mathbb{C}{{d}_{1}}\oplus \mathbb{C}{{d}_{2}}$ which is integrable and has finite dimensional weight spaces.

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.

Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.

The problem of finding the largest set of nodes in a d-cube of side 3 such that no three nodes are collinear was proposed by Moser. Small values of d (viz., d ≤,3) resulted in elegant symmetric solutions. It is shown that this does not remain the case in 4 dimensions where at most 43 nodes can be chosen, and these must not include the center node.

Moser and Abramson [4] proved: given m≥2 and 0≤kl, k2, …, kp<m, the number of p-combinations

satisfying

is

1

([x] denotes the greatest integer ≤x).

Let $\chi $ be a locally compact metric space and let $\mathcal{A}$ be any of the Lipschitz algebras $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, or $\text{lip}_{\alpha }^{0}\,\chi $. In this paper, we show, as a consequence of rather more general results on Banach algebras, that $\mathcal{A}$ is $C$-character amenable if and only if $\chi $ is uniformly discrete.

This note proves another special case of a conjecture of U. Oberst. Oberst considered [1], for any small category , the X abelian category of abelian group-valued functors on , and the X functor Colim: which takes each diagram to its colimit. The question is, when is Colim exact? For its relationships, see [1], It is a sufficient condition that each component of is upward filtered. Oberst conjectured that it is also necessary, and proved this under some conditions. He mentioned particularly the case that is a monoid, i.e. a category with one object. We shall verify the conjecture in that case.

All finite simple self 2-distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self 2-distance graph.

We give a simple description of boundary sets which may be ignored in calculating the maximum of subharmonic or plurisubharmonic functions.

We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance.

The Levitzki radical, which is fundamental in the study of algebras satisfying a polynomial identity, has been shown to exist in the varieties of alternative and Jordan algebras (see Zhevlakov [8], Zwier [9], and Tsai [7]— for an important application of this radical to alternative algebras satisfying a polynomial identity, see Slater [6]). In fact, Hartley [4] even investigated local nilpotence for Lie algebras, though this property can not be radical in the sense of Kurosh-Amitsur [3] for these algebras.

In this note, a certain generalization of the Cauchy distribution is obtained, using the result of Malik [2].

The generalized gamma distribution having the density

(1)

is introduced by Stacy [1], who studied some of its properties. As remarked by Stacy [1], the familiar gamma, chi, chi-squared, exponential and Weibull distributions are special cases of (1), as are the distributions of certain functions of a normal variable - viz.

Let (X, Σ, μ) be a probability space, let f1, f2, ..., Fk be k σ-subalgebras of Σ, and let p ∊ R be such that 1 < p < + ∞. Let Pi :LP(X, Σ, μ) → LP(X, Σ, μ) be the conditional expectation operator corresponding to fi for every i = 1,2,…, k, and set T = P1 . . . Pk. Our goal in the note is to give a new and simpler proof of the fact that for every f ∊ LP(X, Σ, μ), the sequence (Tnf)n∊N converges in the norm topology of LP(X, Σ, μ), and that its limit is the conditional expectation of f with respect to f1 ∩ f2 ∩ … ∩ Fk.

We consider graded connected Gorenstein algebras with respect to the evaluation map $\text{e}{{\text{v}}_{G\,}}\,=\,\text{Ex}{{\text{t}}_{G}}(k,\varepsilon )\,::\,\text{Ex}{{\text{t}}_{G}}(k,G)\,\to \,\text{Ex}{{\text{t}}_{G}}(k,k)$. We prove that if $e{{v}_{G}}\,\ne \,0$, then the global dimension of $G$ is finite.

We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange’s method we obtain the main term and smaller order terms which contain periodic fluctuations.

The splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree $b\,\ge \,4$, where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of degree $b$ and three of its total inflectional tangents.

Using Krasnoselskii’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. The results of this paper are new for the continuous and discrete time scales.

Let $k$ be a field of characteristic not 2, 3. Let $G$ be an exceptional simple algebraic group over $k$ of type ${{\text{F}}_{4}},$$^{1}{{\text{E}}_{6}}$ or ${{\text{E}}_{7}}$ with trivial Tits algebras. Let $X$ be a projective $G$-homogeneous variety. If $G$ is of type ${{\text{E}}_{7}},$ we assume in addition that the respective parabolic subgroup is of type ${{\text{P}}_{7}}.$ The main result of the paper says that the degree map on the group of zero cycles of $X$ is injective.

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad and the Assouad dimension. For common models of random fractal sets, we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition, and we compute the threshold for the Gromov boundary of Galton–Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton–Watson tree result.

Let $R$ be an arbitrary ring and let ${{\left( - \right)}^{+}}\,=\,\text{Ho}{{\text{m}}_{\mathbb{Z}}}\left( -,\,{\mathbb{Q}}/{\mathbb{Z}}\; \right)$, where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers. Let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that ${{X}^{+}}\,\in \,\mathcal{D}$ for any $X\,\in \,\mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f:\,A\to \,C$ of left $R$-modules with $C\,\in \,\mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided ${{f}^{+}}:\,{{C}^{+}}\,\to \,{{A}^{+}}$ is a $\mathcal{D}$-(pre)cover of ${{A}^{+}}$. Some applications of this result are given.

We provide normal forms and the global phase portraits on the Poincaré disk for all Abel quadratic polynomial diòerential equations of the second kind with ${{\mathbb{Z}}_{2}}$-symmetries.