let h be an infinite--dimensional hilbert space and k(h) be the set of all compact operators on h. we will adopt spectral theorem for compact self-adjoint operators, to investigate of higher derivation and higher jordan derivation on k(h) associated with the following cauchy-jencen type functional equation 2f(frac{t+s}{2}+r)=f(t)+f(s)+2f(r) for all t,s,rin k(h).

A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (xn) statistically quasiCauchy if st − limn→∞ d(xn+1, xn) = 0, and lacunary statistically quasi-Cauchy if Sθ − limn→∞ d(xn+1, xn) = 0.Weprove that a subset E of ametric space is totally bounded if and only if any sequence of points in E has a subsequence...

The crease set of an event horizon or a Cauchy horizon is an important object which determines qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat’s principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a class...

The concept of I–convergence is an important generalization of statistical convergence which depends on the notion of an ideal I of subsets of the set N of positive integers. In this paper we introduce the ideas of I–Cauchy and I∗–Cauchy sequences in cone metric spaces and study their properties. We also investigate the relation between this new Cauchy type condition and the property of complet...

We establish a generalization of the inequality introduced by Mitrinović and Pečarić in 1988. We prove mean value theorems of Cauchy type for that new inequality by taking its difference. Furthermore, we prove the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity. Finally, we define new means...

In this paper, the Bregman and Burbea-Rao divergences for matrices are investigated. Two mean-value theorems for the divergences induced by C-functions are derived. As application, certain Cauchy type means of the entries of the matrices are constructed. By utilizing three classes of parametrized convex functions, the exponential convexity of the divergences, thought as a function of the parame...