Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

Journal of Algebraic Combinatorics - Let $$\Gamma $$ denote a finite, simple and connected graph. Fix a vertex x of $$\Gamma $$ and let $$T=T(x)$$ denote the Terwilliger algebra of $$\Gamma $$ with...     

Smallest graphs with given automorphism group

Journal of Algebraic Combinatorics - For a finite group G, denote by $$\alpha (G)$$ the minimum number of vertices of any graph $$\Gamma $$ having $$\mathrm {Aut}(\Gamma )\cong G$$ . In this paper,...     

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.     

On triviality of the reduced Whitehead group over Henselian fields

Let F be a Henselian field of q-cohomological dimension 3, where q is a prime. Let $$\Gamma _F$$ be the totally ordered Abelian value group of F and let D be a central division algebra over F of index a power of q such that the characteristic of the residue field $$\overline{F}$$ is coprime to q. We show that when $$1\le \dim _{{\mathbb {F}}_q}(\Gamma _F/q\Gamma _F)\le 3$$ , the reduced Whitehead group of D is trivial.     

Logarithmic girth expander graphs of $$SL_n({\mathbb {F}}_p)$$

Journal of Algebraic Combinatorics - We provide an explicit construction of finite 4-regular graphs $$(\Gamma _k)_{k\in {\mathbb {N}}}$$ with $$\text {girth}\, \Gamma _k\rightarrow \infty $$ as...     

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space $$\Gamma ^{\infty }$$, in which $$\Gamma$$ is the Bohr compactification of the additive group $${\mathbb {R}}$$ of real numbers. In this way, we need to show that $$\Gamma ^\infty$$ is isomorphic to the spectrum of $$W({\mathbb {R}})$$, the norm closure of the invariant algebra generated by the maps $$\exp q(t)$$, where q(t) is a real polynomial on $${\mathbb {R}}$$.     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

The pure cactus group is residually nilpotent

We show that for each $$n\in {\mathbb {N}}$$ the pure cactus group $$\Gamma _{n+1}$$ embeds into a right-angled Coxeter group and, therefore, is residually nilpotent. In addition, we construct an explicit residually torsion-free nilpotent subgroup of finite index in $$\Gamma _{n+1}$$ .     

On the Behavior of the Spectrum of a Perturbed Steklov Boundary Value Problem with a Weak Singularity

Differential Equations - For the Laplace equation in a bounded domain of the space $$\mathbb {R}^n $$ , $$n\geq 3$$ , with a boundary that contains a piece $$\Gamma $$ of a hyperplane $$L $$ , we...     

On the distribution of the zeros of Eisenstein series for $$\Gamma _0^*(5)$$ Γ 0 ∗ ( 5 ) and $$\Gamma _0^*(7)$$ Γ 0 ∗ ( 7 )

The Ramanujan Journal - The goal of the present paper is to show that all the zeros of the Eisenstein series for $$\Gamma _0^*(5)$$ and $$\Gamma _0^*(7)$$ on the standard fundamental domain lie on...     

Congruences involving the $$U_{\ell }$$Uℓ operator for weakly holomorphic modular forms

The Ramanujan Journal - Let $$\lambda $$ be an integer, and $$f(z)=\sum _{n\gg -\infty } a(n)q^n$$ be a weakly holomorphic modular form of weight$$\lambda +\frac{1}{2}$$ on $$\Gamma _0(4)$$ with...     

On Ivády’s bounds for the gamma function and related results

Periodica Mathematica Hungarica - We prove that the inequality $$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$ holds for all $$x\in [0,1]$$ , $$\beta \ge {\beta...     

Normal subgroup based power graph of finite groups

Ricerche di Matematica - Let G be a finite group and $$N \unlhd G$$ . The normal subgroup based power graph of G, $$\Gamma _N(G)$$ , is an undirected graph with vertex set $$(G{\setminus }N) \cup...     

Minimal time function and viscosity solutions

Two theorems in Ref. 1 are generalized. It is proved that, ifV(A,Γ) is the set of points that can be steered to the origin along a solution of the control systemx′=Ax?c, ifc(t)∈Γ, Γ is a compact subset ofR ⁿ , 0∈ intrelco Γ, and if a rank condition holds, then the minimal time functionT(·) is a viscosity solution of the Bellman equation $$\max \{ \left\langle {DT(x),\gamma - Ax} \right\rangle :\gamma \varepsilon co\Gamma \} - 1 = 0,x\varepsilon V(A,\Gamma )\backslash \{ 0\} ,$$ and of the Hàjek equation $$1 - \max \{ \left\langle {DT(x),\exp [ - AT(x)]} \right\rangle :\gamma \varepsilon co\Gamma \} = 0,x\varepsilon V(A,\Gamma ).$$      

Projections of modular forms on Eisenstein series and its application to Siegel’s formula

The Ramanujan Journal - Let $$k \ge 2$$ and N be positive integers and let $$\chi $$ be a Dirichlet character modulo N. Let f(z) be a modular form in $$M_k(\Gamma _0(N),\chi )$$ . Then we have a...     

Limiting Behavior of Infinite Products Scaled by Pisot Numbers

Journal of Fourier Analysis and Applications - For $$\theta >1$$ , the infinite product $$\Gamma _{\theta }(x)=\prod _{n=0}^{\infty }\cos (\pi \theta ^{-j}x)$$ is the Fourier transform of...     

Derivation of Freund - Witten adelic formula for four-point Veneziano amplitudes

On the basis of an analysis on the adelic group (Tate's formula) a regularization is proposed for the divergent infinite product ofp-adic Γ functions: $$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{1 - p^{ - \alpha } }}, p = 2,3,5...,$$ and the adelic formula $$reg\prod\limits_{p = 2}^\infty {\Gamma p(\alpha ) = \frac{{(\zeta \alpha )}}{{\zeta (1 - \alpha )}},} $$ where ζ(α) is the Riemann ζ function, is proved.     

Breaking up simplicial homology and subadditivity of syzygies

Journal of Algebraic Combinatorics - We consider the following question: If a simplicial complex $$\Gamma $$ has d-homology, then does the corresponding d-cycle always induce cycles of smaller...     

Integrable extensions of the Adler map via Grassmann algebras

Theoretical and Mathematical Physics - We study certain extensions of the Adler map on Grassmann algebras $$\Gamma(n)$$ of order $$n$$ . We consider a known Grassmann-extended Adler map and under...     

Semisymmetric prime-valent graphs of order $$2p^3$$ 2 p 3

Journal of Algebraic Combinatorics - A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime and $$\Gamma $$ a...