FUNCTIONAL INEQUALITIES AND BEST APPROXIMATION

We investigate the relationship between best approximations by elements of closed convex cones and the estimation of functionals on an inner product space (X,<·,·>)in terms of the inner product on X.

Corresponding Prime Number Distribution Equation

The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for corresponding prime number, proved that the conjecture of twin prime numbers and obtain the corresponding prime distribution equation. According to the distribution rate of corresponding prime numbers, the distribution pattern of twin prime numbers was proved the distribution rate theorem. This is the distribution rate of prime numbers corresponding to composite numbers, which approaches the distribution rate of prime numbers corresponding to integers. Based on the corresponding prime distribution equation, obtain the twin prime inequality function. Then, the formula for calculating twin prime numbers was discussed. There is also the Hardy Littlewood conjecture. This provides a practical and feasible approach for studying the distribution of twin prime numbers.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

ISOMORPHISMS AND DERIVATIONS IN C*-ALGEBRAS

In this article, we prove the Hyers-Ulam-Rassias stability of the following Cauchy-Jensen functional inequality：‖f （x） ＋ f （y） ＋ 2f （z） ＋ 2f （w）‖ ≤‖ 2f x ＋ y2 ＋ z ＋ w ‖（0.1）This is applied to investigate isomorphisms between C＊-algebras, Lie C＊-algebras and JC＊-algebras, and derivations on C＊-algebras, Lie C＊-algebras and JC＊-algebras, associated with the Cauchy-Jensen functional equation 2f （x ＋ y/2 ＋ z ＋ w） = f（x） ＋ f（y） ＋ 2f（z） ＋ 2f（w）.

Global exponential stability of mixed discrete and distributively delayed cellular neural network

This paper concernes analysis for the global exponential stability of a class of recurrent neural networks with mixed discrete and distributed delays. It first proves the existence and uniqueness of the balance point, then by employing the Lyapunov-Krasovskii functional and Young inequality, it gives the sufficient condition of global exponential stability of cellular neural network with mixed discrete and distributed delays, in addition, the example is provided to illustrate the applicability of the result.

CONSTRUCTION OF SOME KIESSWETTER-LIKE FUNCTIONS-THE CONTINUOUS BUT NON-DIFFERENT-IABLE FUNCTION DEFINED BY QUINARY DECIMAL

In this paper, we construct some continuous but non-differentiable functions defined by quinary dec-imal, that are Kiesswetter-like functions. We discuss their properties, then investigate the Hausdorff dimensions of graphs of these functions and give a detailed proof.

Modified approximation and error estimation for King’s type(p,q)-BBH operators

In this paper,the King’s type modification of(p,q)-Bleimann-Butzer and Hahn operators is defined.Some results based on Korovkin’s approximation theorem for these new operators are studied.With the help of modulus of continuity and the Lipschitz type maximal functions,the rate of convergence for these new operators are obtained.It is shown that the King’s type modification have better rate of convergence,flexibility than classical(p,q)-BBH operators on some subintervals.Further,for comparisons of the operators,we presented some graphical examples and the error estimation in the form of tables through MATLAB(R2015a)

Functional Inequalities in Non-Archimedean Normed Spaces

In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in non-Archimedean Banach spaces.

A KIND OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM FOR VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATION

In this paper, we study a kind of boundary value problem for volterra functional differential equation:ε x″(t)=f(t,ε)x′(t)+g(t,x(t),(t),x(t-τ),ε), t∈(0,1) x(t)=(t,ε), t∈, x(1)=ψ(ε) Using the theory of differential inequality, we prove the existence of the solution and give a uniformly valid asympototic expansions of the solution. Meanwhile, an estimation of the derivative solution is given as well.

Functional inequalities for time-changed symmetric α-stable processes

We establish sharp functional inequalities for time-changed symmetric α-stable processes on Rd with d≥1 and α∈(0,2), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x)=(1+|x|)β with β>α we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.

FORCED OSCILLATION AND APPLICATION OF FUNCTIONAL DIFFERENTIAL INEQUALITIES

In this paper we study the oscillations for a class of functional differential inequalities. By using these properties some forced oscillations to the boundary value problems of functional partial differential equations are established.

Operator Jensen's Inequality on C*-algebras

A real continuous function which is defined on an interval is said to beA-convex if it is convex on the set of self-adjoint elements,with spectra in the interval,in all matrix algebras of the unital C-algebra A.We give a general formation of Jensen’s inequality for A-convex functions.

A Note on the Stability of Jordan Triple Higher Ring Derivations

In the present paper, we establish the stability and the superstability of a functional inequality corresponding to the functional equation fn（xyx） = ∑i＋j＋k=n fi（x）fj （y）fk（x）. In addition, we take account of the problem of Jacobson radical ranges for such functional inequality.

Parametric Duality Models for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η,ρ)-Invex Functions

A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.

Global Parametric Sufficient Optimality Conditions for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η,ρ)-invex Functions

In this paper, we discuss a large number of sets of global parametric sufficient optimality conditions under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.

Kloosterman sums with multiplicative coefficients

Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+ε/2)N^(2/1)(log(6N))^(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.

Global Parametric Sufficient Efficiency Conditions for Semiinfinite Multiobjective Fractional Programming Problems Containing Generalized (α, η, ρ)-V-Invex Functions

Abstract In this paper, we discuss numerous sets of global parametric sufficient efficiency conditions under various generalized （a,n, p）-V-invexity assumptions for a semiinfinite multiobjective fractional programming problem.

A comprehensive revisit of the ρ meson with improved Monte-Carlo based QCD sum rules

We improve the Monte-Carlo based QCD sum rules by introducing the rigorous Hoolder-inequalitydetermined sum rule window and a Breit-Wigner type parametrization for the phenomenological spectral function.In this improved sum rule analysis methodology, the sum rule analysis window can be determined without any assumptions on OPE convergence or the QCD continuum. Therefore, an unbiased prediction can be obtained for the phenomenological parameters（the hadronic mass and width etc.）. We test the new approach in the ρ meson channel with re-examination and inclusion of αs corrections to dimension-4 condensates in the OPE. We obtain results highly consistent with experimental values. We also discuss the possible extension of this method to some other channels.

An Integral Representation for the Weighted Geometric Mean and Its Applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.