with a positive constant λ. The equation Lu = 0 is then a second-degree elliptic equation. We also require the aij to be bounded and measurable, satisfying ‖aij‖L∞(B4) ≤ Λ with another constant Λ > 0. (In case you wonder, the radius ’4’ of the ball is to avoid fractions. Most of our estimates will be of the kind “some expression on B1 ≤ another expression on B4 ” and it would be unconvenient to...

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form Lu(x) = − ∑ aij∂iju+ PV ∫ Rn (u(x)− u(x+ y))K(y)dy. These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case aij ≡ 0 and for ...

carum copticum is a grassy and annual plant which is believed to have several therapeutic effects, including anti-asthmatic properties. we have therefore studied the bronchodilatory and anticholinergic effects of 4 rnl aqueous extract (ae), 0.0 5 ml ethanol extract (ee), and 0.0 0 5 ml essential oil (eo) of carum copticum and 1 rnl of a 5 mm solution of its main ingredient, thymol (t), in compa...

Given an n× n complex matrix A, let μA(x, y) := 1 n |{1 ≤ i ≤ n,Reλi ≤ x, Imλi ≤ y}| be the empirical spectral distribution (ESD) of its eigenvalues λi ∈ C, i = 1, . . . n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD μ 1 √ n An of a random matrix An = (aij)1≤i,j≤n where the random variables aij − E(aij) are iid copie...

The main feature of this paper concerns extensions of the Liouville theorem to the following class of elliptic equations in non-divergence form: aij(x)∂iju + bi(x)∂iu + c(x)u = 0 in R N , with c ≤ 0. We show that the Liouville property holds (that is, the space of bounded solutions has at most dimension one) if the coefficients aij , bi and c are periodic, with the same period, and it does not ...

We derive an algorithm for estimating the largest p ≥ 1 values aij or |aij | for an m×n matrix A, along with their locations in the matrix. The matrix is accessed using only matrix– vector or matrix–matrix products. For p = 1 the algorithm estimates the norm ‖A‖M := maxi,j |aij | or maxi,j aij . The algorithm is based on a power method for mixed subordinate matrix norms and iterates on n × t ma...

Given an n× n complex matrix A, let μA(x, y) := 1 n |{1 ≤ i ≤ n,Reλi ≤ x, Imλi ≤ y}| be the empirical spectral distribution (ESD) of its eigenvalues λi ∈ C, i = 1, . . . n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD μ 1 √ n An of a random matrix An = (aij)1≤i,j≤n where the random variables aij − E(aij) are iid copie...