Let V be the m×m upper-left corner of an n× n Haar-invariant unitary matrix. Let λ1, · · · , λm be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ1, · · · , λm goes to the circular law, that is, the uniform distribution on {z ∈ C; |z| ≤ 1} as m → ∞ with m/n → 0. We also prove that the empirical distribution of λ1, · · · , λm goes to the arc law, that is, t...

In this paper, we consider the circular Cauchy distribution μx on the unit circle S with index 0 ≤ |x| < 1 and we study the spectral gap and the optimal logarithmic Sobolev constant for μx, denoted respectively by λ1(μx) and CLS(μx). We prove that 1 1+|x| ≤ λ1(μx) ≤ 1 while CLS(μx) behaves like log(1 + 1 1−|x| ) as |x| → 1.

We construct three groups Λ1, Λ2, Λ3, which can all be decomposed as amalgamated products F9∗F81F9 and have very few normal subgroups of finite or infinite index. Concretely, Λ1 is a simple group, Λ2 is not simple but has no non-trivial normal subgroup of infinite index, and Λ3 is not simple but has no proper subgroup of finite index.

The λ-dilate of a set A is λ · A = {λa : a ∈ A}. We give an asymptotically sharp lower bound on the size of sumsets of the form λ1 ·A+ · · ·+λk ·A for arbitrary integers λ1, . . . , λk and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plünnecke’s inequality.

Let λ1, . . . , λ4 be non-zero with λ1/λ2 irrational and negative, and let S be the set of values attained by the form λ1x 3 1 + · · · + λ4x4 when x1 has at most 3 prime divisors and the remaining variables are prime. We prove that most real numbers are close to an element of S.

A simple, specific, accurate and precise Q-Absorbance ration spectrophotometric method was developed validated for estimation of Zonisamide Aripirazole in Synthetic Mixture. showed iso-absorptive point at 230.50nm Distilled water. The second wavelength used 240.60nm which is λmax distilled concentration the drugs determined by using ratio absorbance (λ1 =240.60 nm) (λ2 =230.50 nm). This linear ...

We prove the following estimate for the spectrum of the normalized Laplace operator ∆ on a finite graph G, 1− (1− k[t]) t ≤ λ1 ≤ · · · ≤ λN−1 ≤ 1 + (1− k[t]) 1 t , ∀ integers t ≥ 1. Here k[t] is a lower bound for the Ollivier-Ricci curvature on the neighborhood graph G[t] (here we use the convention G[1] = G), which was introduced by Bauer-Jost. In particular, when t = 1 this is Ollivier’s esti...

C. A. Rodger, M. A. Tiemeyer∗, Auburn University Let K = K(a, p;λ1, λ2) be the multigraph with: the number of vertices in each part equal to a; the number of parts equal to p; the number of edges joining any two vertices of the same part equal to λ1; and the number of edges joining any two vertices of different parts equal to λ2. This graph was of interest to Bose and Shimamoto in their study o...