Comparison theorems for second order nonlinear difference equations

New oscillation results are obtained for the second order nonlinear difference equation

Δ(rnf(Δxn−1))+g(n,xn)=0,     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition

     

Variations on the Drygas equation and its stability

We study the stability of the Drygas functional equation:

g(xy)+g(xy⁻¹)=2g(x)+g(y)+g(y⁻¹)     

Linearized oscillation of odd order nonlinear neutral delay differential equations (I)

In this paper, we proved that the odd order nonlinear neutral delay differential equation

[x(t)−p(t)g(x(t−τ))^](n)+q(t)h(x(t−σ))=0     

Démonstration de la conjecture de la masse positive

On a compact Riemannian manifold (Vn,g)(n>2), when the conformal Laplacian L is invertible, we show, under necessary hypotheses, that if the Green function GL of L is of the form (here r=d(P,Q)) GL(P,Q)=1/(n−2)ωn−1rn−2+H(P,Q) with H(P,Q) bounded on V, then

     

A Turán-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n?1 and real numbers x>0 we have

α?Γ(n−1)(x)Γ(n+1)(x)−Γ(n)²(x),     

On an inverse spectral problem for a quadratic Jacobi matrix pencil

Given two monic polynomials P2n and P2n−2 of degree 2n and 2n−2 (n?2) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n×n Jacobi matrices B and C for which

     

Oscillation and nonoscillation criteria for second order quasilinear difference equations

New oscillation and nonoscillation theorems are obtained for the second order quasilinear difference equation

Δ(|Δxn−1|^ρ−1Δxn−1)+pn|xn|^ρ−1xn=0,     

A note on the total number of cycles of even and odd permutations

We prove bijectively that the total number of cycles of all even permutations of [n]={1,2,…,n} and the total number of cycles of all odd permutations of [n] differ by (−1)ⁿ(n−2)!, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity:

     

Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result

Let (M,g) be an n-dimensional (n?2) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C^∞. This paper is concerned with the study of the wave equation on (M,g) with locally distributed damping, described by

     

The convolution of functions and distributions

The non-commutative convolution f∗g of two distributions f and g in D^′ is defined to be the limit of the sequence {(fτn)∗g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that

     

Front formation and motion in quasilinear parabolic equations

This paper deals with the singular limit for

L?u:=ut−Fx(u,?ux)−?⁻¹g(u)=0,     

An inequality for sums of binary digits, with application to Takagi functions

Let ?(x)=2inf{|x−n|:n∈Z}, and define for α>0 the function

     

Solutions of second-order and fourth-order ODEs on the half-line

We start by studying the existence of positive solutions for the differential equation

u^″=a(x)u−g(u),     

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

We consider random self-adjoint Jacobi matrices of the form

(Jωu)(n)=an(ω)u(n+1)+bn(ω)u(n)+an−1(ω)u(n−1)     

Convex Komlós sets in Banach function spaces

In 1967 Komlós proved that for any sequence n{fn} in L₁(μ), with ‖fn‖?M<∞ (where μ is a probability measure), there exists a subsequence n{gn} of n{fn} and a function g∈L₁(μ) such that for any further subsequence n{hn} of n{gn},

     

Uniqueness theorems for meromorphic mappings with few hyperplanes

Let f,g be linearly nondegenerate meromorphic mappings of Cm into CPn. Let be hyperplanes in CPn in general position, such that

(a)

f−1(Hj)=g−1(Hj), for all 1?j?q,

(b)

dim(f−1(Hi)∩f−1(Hj))?m−2 for all 1?i<j?q, and

(c)

f=g on .

It is well known that if q?3n+2, then f≡g. In this paper we show that for every nonnegative integer c there exists positive integer N(c) depending only on c in an explicit way such that the above result remains valid if q?(3n+2−c) and n?N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n?0. At the same time, a big number of recent uniqueness theorems are generalized considerably.     

On integer matrices obeying certain matrix equations

We discuss integer matrices B of odd order ν which satisfy

B^T = ± B, BB^T = vI − J, BJ = 0

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2 and extended centroid C and let f(x1,..., x _n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that

$\delta (d(f(r_1 , \ldots ,r_n ))f(r_1 , \ldots ,r_n ) - f(r_1 , \ldots ,r_n )g(f(r_1 , \ldots ,r_n ))) = 0$

for all r1,..., r _n ∈ R. Then d and g are both inner derivations on R and one of the following holds: (1) d = g = 0; (2) d = ?g and f(x ₁,..., x _n )² is central-valued on R.