Given a family $$\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$$ of d distinct nonconstant polynomials, positive integer $$k\le d$$ and real parameter $$\rho$$ , we consider the mean value $$M_{k, \rho} (\varphi, N) \int_{{\rm x} \in [0,1]^k} \sup_{{\rm y} [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; |^\rho \,d{\rm $$ of exponential sums $$S_{\varphi}({\rm \sum_{n=1}^{N} \...

‎it is well known that every (real or complex) normed linear space $l$ is isometrically embeddable into $c(x)$ for some compact hausdorff space $x$‎. ‎here $x$ is the closed unit ball of $l^*$ (the set of all continuous scalar-valued linear mappings on $l$) endowed with the weak$^*$ topology‎, ‎which is compact by the banach--alaoglu theorem‎. ‎we prove that the compact hausdorff space $x$ can ...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.

1. Differentiability Versus Continuity 2 2. Differentiation Rules 3 3. Optimization 7 3.1. Intervals and interior points 7 3.2. Functions increasing or decreasing at a point 7 3.3. Extreme Values 8 3.4. Local Extrema and a Procedure for Optimization 10 3.5. Remarks on finding roots of f ′ 12 4. The Mean Value Theorem 12 4.1. Statement of the Mean Value Theorem 12 4.2. Proof of the Mean Value Th...

In the present paper, we study some properties of fuzzy norm of linear operators. At first the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

The proof of the intermediate value theorem for power series on a LeviCivita field will be presented. After reviewing convergence criteria for power series [19], we review their analytical properties [18]. Then we state and prove the intermediate value theorem for a large class of functions that are given locally by power series and contain all the continuations of real power series: using iter...

A general uniform convergence theorem for numerical integration of Cauchy principal value integrals is proved. Seven special instances of this theorem are given as corollaries.

This article deals with a class of discrete type boundary value problems. Sufficient conditions guaranteeing the existence of at least three positive solutions of this class of boundary value problems are established by using a fixed point theorem in cones in Banach spaces. An example is given to illustrate the main theorem.

convexity theory and duality theory are important issues in math- ematical programming. within the framework of credibility theory, this paper rst introduces the concept of convex fuzzy variables and some basic criteria. furthermore, a convexity theorem for fuzzy chance constrained programming is proved by adding some convexity conditions on the objective and constraint functions. finally,...