Extensions of the sine addition law on groups

Aequationes mathematicae - Let G be a group, and let $$\chi $$ and $$\mu $$ be characters of G. We find the solutions of the functional equation $$f(xy) = f(x)\chi (y) + \mu (x)f(y)$$ , $$x,y \in...     

Extension of Hardy–Littlewood–Sobolev Inequalities for Riesz Potentials on Hypergroups

Mediterranean Journal of Mathematics - Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let $$\theta $$ be a G-invariant irreducible character of N. In Rizo (J...     

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

Periodica Mathematica Hungarica - Let G be an additive finite abelian group. For a sequence T over G and $$g\in G$$ , let $$\mathrm {v}_{g}(T)$$ denote the multiplicity of g in T. Let $$\mathcal...     

Random Schreier graphs and expanders

Journal of Algebraic Combinatorics - Let the group G act transitively on the finite set $$\Omega $$ , and let $$S \subseteq G$$ be closed under taking inverses. The Schreier graph $$Sch(G...     

Fundamental groups of moduli of principal bundles on curves

Geometriae Dedicata - Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \, 2$$ , and let $${\text {G}}$$ be a connected semisimple affine algebraic group defined over...     

Exponent of a finite group of odd order with an involutory automorphism

Let $$w = w(x_1, \ldots , x_n)$$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $$\widetilde{w}: G^n\rightarrow G$$ where $$\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$$ for every sequence $$(g_1, \ldots , g_n)$$ . Let $$\mathcal G$$ be a simple and simply connected group which is defined and split over an infinite field K and let $$G =\mathcal G(K)$$ . For the case when $$w = w_1w_2 w_3 w_4$$ and $$w_1, w_2, w_3, w_4$$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $$G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$$ where Z(G) is the center of the group G and $${{\text { Im}}}\, {\widetilde{w}}$$ is the image of the word map $$\widetilde{w}$$ . For the case when $$G = {{\text {SL}}}_n(K)$$ and $$n \ge 3$$ , in the same paper of Hui et al. (2015) it was shown that the inclusion $$G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$$ holds for a product $$w = w_1w_2 w_3$$ of any three non-trivial words $$ w_1, w_2, w_3$$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $$B_2, G_2$$ .     

On nilpotency of higher commutator subgroups of a finite soluble group

Archiv der Mathematik - Let G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with...     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

Trace and Differences of Idempotents in C*-Algebras

Mathematical Notes - Let φ be atrace on aunital C*-algebra $$\mathcal{A}$$ , let $$\mathfrak{M}_\varphi$$ be the ideal of definition of the trace φ, and let $$P,Q\in\mathcal{A}$$ be...     

Menon-type identities with respect to sets of units

The Ramanujan Journal - Let $$n\ge 1$$ and let $$\mathbb {Z}_n^\star $$ be the group of units in the ring of residual classes modulo n, let $$m\ge 0$$ , $$k\ge 0$$ , $$m+k\ge 1$$ , $$u_1,\ldots...     

D-optimal Designs for Seven Objects and a Large Number of Weighings

Let m ? 7 be an integer and let M m ,7 (0,1) be the set of all m × 7 (0,1)-matrices. Define $$ G(m,7) = \max \{ \det X^T X|X \in M_{m,7} (0,1)\} . \eqno (1) $$ A matrix X for which the maximum is achieved is said to be D- optimal . We determine G ( m , 7) for sufficiently large m and exhibit D-optimal matrices. For each m , the Gram matrices, X T X for D-optimal matrices X ε M m ,7 (0,1), are shown to be essentially unique.     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

Mathematical Notes - Let $$f(x)$$ be a function belonging to the Lebesgue class $$L^p({\mathbb R}_+)$$ on the semiaxis $${\mathbb R}_+=[0,+\infty)$$ , $$1\le p\le 2$$ , and let $$\widehat{f}$$ be...     

A sufficient condition for random zero sets of Fock spaces

Archiv der Mathematik - Let $$(r_n)_{n=1}^\infty $$ be a non-decreasing sequence of radii in $$(0, \infty )$$ , and let $$(\theta _n)_{n=1}^\infty $$ be a sequence of independent random arguments...     

On conditional expectations in $$L^p(\mu ;L^q(\nu ;X))$$ L p ( μ ; L q ( ν ; X ) )

Positivity - Let $$(A,{\mathscr {A}},\mu )$$ and $$(B,{\mathscr {B}},\nu )$$ be probability spaces, let $${\mathscr {F}}$$ be a sub- $$\sigma $$ -algebra of the product $$\sigma $$ -algebra...     

Finite groups of even orders all of whose fourth maximal subgroups are weakly $$s_{2}$$ s 2 -permutable

Acta Mathematica Hungarica - Let p be a prime number. A positive integer m is said to be a $$p'$$ -number provided $$p\nmid m$$ . Let G be a finite group, and let H be a subgroup of G. We say...     

The solvability cardinal of the class of polynomials

Aequationes mathematicae - Let G be an Abelian group, and let $${{\mathbb {C}}}^G$$ denote the set of complex valued functions defined on G. A map $$D: {{\mathbb {C}}}^G \rightarrow {{\mathbb...     

A note on the Lebesgue–Ljunggren–Nagell equation $$ax^2+b^{2m}=4y^n$$ a x 2 + b 2 m = 4 y n

Periodica Mathematica Hungarica - Let a, b be fixed positive integers such that $$2\not \mid {ab}$$ , $$\gcd (a,b)=1$$ and a is square-free, and let $$h(-a)$$ denote the class number of...     

On the structure of partition which the difference of their representation function is a constant

Periodica Mathematica Hungarica - Let $$\mathbb {N}$$ be the set of nonnegative integers. For any set $$A \subset \mathbb {N}$$ , let $$R_1(A, n)$$ , $$R_2(A, n)$$ and $$R_3(A, n)$$ be the number...     

An asymptotic formula for the logarithm of generalized partition functions

The Ramanujan Journal - Let $$\Lambda =\{ \lambda _j\}_{j=1}^\infty $$ be a set of positive integers, and let $$p_\Lambda (n)$$ be the number of ways of writing n as a sum of positive integers...