Eigenvalues of the k‐th power of a graph

The k‐th power of a graph G, denoted by , is a graph with the same set of vertices as G such that two vertices are adjacent in if and only if their distance in G is at most k. In this paper, we give the bounds on the spectral radius of and . The Nordhaus–Gaddum‐type inequality for the spectral radius of the graph is also presented. Moreover, we obtain an upper bound on the energy of the second power of graphs.     

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two‐weight Hardy, Hardy–Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.     

Chemometric analysis of disubstituent effects on the 13C chemical shifts of the carboxyl carbons (deltaCO) of benzoic acids. A comparative study of the substituent effects on the strength of benzoic acids in apolar aprotic media

The results of measurements of substituent induced chemical shifts of carboxyl carbons (deltaCO) of dichloro- and difluorobenzoic acids, including the monosubstituted ones with substituents at meta- and/or ortho- positions, in chloroform-d and strengths of these acids (log K) in chlorobenzene show an anomalous reverse trend between deltaCO and log K, while the electron density at carboxyl carbons should influence similarly both deltaCO and log K. A detailed chemometric analysis of comparison of disubstituent effects between deltaCO and log K on the basis of Fujita-Nishioka's multiparameter approach and assumption of additivity of substituent effects shows a dominance of the localized pi-polarization mechanism relative to simple electrostatic effects upon deltaCO. Further, steric factors play a significant role in determining deltaCO whereas with respect to log K they were insignificant. The overall anomaly has been rationalized keeping in mind that, while log K is a gross measure of energy differences between the ionized and unionized forms of the acids, deltaCO is a very sensitive probe for determining changes in electron density at the carboxyl carbon of the unionized acid.     

On the norm of a Hilbert's type linear operator and applications

In this paper, the norm of a Hilbert's type linear operator is given. As applications, a new operator inequality and the equivalent forms with the norm are obtained, and particularly some new extended Hilbert's type inequalities and the equivalent forms with the best constant factors are established.     

Jensen's inequality for spectral order and submajorization

Let A be a C^∗-algebra and be a positive unital map. Then, for a convex function defined on some open interval and a self-adjoint element a∈A whose spectrum lies in I, we obtain a Jensen's-type inequality f(?(a))??(f(a)) where ? denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered, i.e., monotone convex or arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.     

Support-type properties of convex functions of higher order and Hadamard-type inequalities

It is well known that every convex function (where I⊂R is an interval) admits an affine support at every interior point of I (i.e. for any x₀∈IntI there exists an affine function such that a(x₀)=f(x₀) and a?f on I). Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result—Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.     

On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces

In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for Euler-Lagrange type cubic functional equations

f(ax+y)+f(x+ay)=(a+1)²(a−1)[f(x)+f(y)]+a(a+1)f(x+y)     

An extension of a Bourgain-Lindenstrauss-Milman inequality

Let ‖⋅‖ be a norm on Rn. Averaging ‖(ε₁x₁,…,εnxn)‖ over all the n² choices of , we obtain an expression |||x||| which is an unconditional norm on Rn. Bourgain, Lindenstrauss and Milman [J. Bourgain, J. Lindenstrauss, V.D. Milman, Minkowski sums and symmetrizations, in: Geometric Aspects of Functional Analysis (1986/1987), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44-66] showed that, for a certain (large) constant η>1, one may average over ηn (random) choices of and obtain a norm that is isomorphic to |||⋅|||. We show that this is the case for any η>1.     

Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces

It is proved that the inequality δX(ε)?cεp, p?2, where δX is the modulus of convexity of X, is sufficient and necessary for the inequality