We establish a new spectral criterion for Kazhdan’s property (T ) which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property (T ) for the groups ELn(R), where n ≥ 3 and R is an arbitrary finitely generated associative ring. We also strengthen some of the results on property (T ) for Kac-Moody groups from [DJ].

We establish a new spectral criterion for Kazhdan's property (T) which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property (T) for the groups ELn(R), where n ≥ 3 and R is an arbitrary finitely generated associative ring. We also strengthen some of the results on property (T) for Kac-Moody groups from [DJ].

We determine the existence and stability regimes of bright 211-dimensional spatial solitary waves in media with quadratic ~or x ! and focusing cubic nonlinearities. We derive a necessary criterion for linear stability of these solitons, and use it to show that the quadratic nonlinearity enables stable solitons to exist when the cubic nonlinearity is sufficiently weak. We discuss why the Vakhito...

Let G be a reductive affine algebraic group, and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups Hx of G, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-Röhrle criteria, in the caseX = G . Our criterion builds on a close analogue of a theorem of Mundet...

a graph g = (v,e) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : v (g) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ v (g), xy ∈ e(g), and the total number of 0, 1 and 2 are balanced. that is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). in thi...

Grönwall's function \(G\) is defined for all natural numbers \(n>1\) by \(G(n)=\frac{\sigma(n)}{n \cdot \log n}\) where \(\sigma(n)\) the sum of divisors \(n\) and \(\log\) logarithm. We require properties extremely abundant numbers, that to say left right maxima \(n \mapsto G(n)\). also use colossally hyper numbers. There are several statements equivalent famous Riemann hypothesis. state hy...