Matrix valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized differential properties of such functions related to nonsmooth-smoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher’s theorem, B-derivative, directional derivative, and se...

Matrix-valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized differential properties of such functions related to nonsmooth-smoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher’s theorem, -derivative, directional derivative, and sem...

We establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions. A constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling. Moreover, we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of sol...

If the function f : I → R is differentiable on the interval I ⊆ R , then for each x,a ∈ I, according to the mean value theorem, there exists a number c(x) belonging to the open interval determined by x and a , and there exists a real number θ (x) ∈]0,1[ such that f (x)− f (a) = (x−a) f (1) (c(x)) and f (x)− f (a) = (x−a) f (1) (a+(x−a)θ (x)) . In this paper we shall study the differentiability ...

Let Ω be a Cartan domain of rank r and genus p and Bν , ν > p−1, the Berezin transform on Ω; the number Bνf(z) can be interpreted as a certain invariant-mean-value of a function f around z. We show that a Lebesgue integrable function satisfying f = Bνf = Bν+1f = · · · = Bν+rf , ν ≥ p, must be M-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordi...

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 1 2 k(k + 1) − 1 3 k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov’s mean value theorem for 100% of the critical interval 1 6 s 6 1 2 k(k + 1).

We develop a substantial enhancement of the efficient congruencing method to estimate Vinogradov’s integral of degree k for moments of order 2s, thereby obtaining for the first time near-optimal estimates for s > 5 8k . There are numerous applications. In particular, when k is large, the anticipated asymptotic formula in Waring’s problem is established for sums of s kth powers of natural number...

 (b− a)M, for all x ∈ [a, b] . The constant 14 is best possible in the sense that it cannot be replaced by a smaller constant. In [2], the author has proved the following Ostrowski type inequality. Theorem 2. Let f : [a, b] → R be continuous on [a, b] with a > 0 and differentiable on (a, b) . Let p ∈ R\ {0} and assume that Kp (f ) := sup u∈(a,b) { u |f ′ (u)| } < ∞. Then we have the inequality...

Some Ostrowski type inequalities via Cauchy’s mean value theorem and applications for certain particular instances of functions are given.

Let W(k, 2) denote the least number s for which the system of equations ~ _ i x ~ = ~ S = l y i ( 1 <~j~k) has a solution with ~S=lx~+l v e ~ = l y ~ +1. We show that for large k one has W(k, 2) ~< 89 k + logtog k + O(1)), and moreover that when K is large, one has W(k, 2) ~< 89 + 1) + 1 for at least one value k in the interval [K, K 4/3 +~]. We show also that the least s for which the expected...