Some test length bounds for nonrepeating functions in the {&;, ∨} basis

Testing relative to a nonrepeating alternative in a conjunction-disjunction basis is considered. A lower bound on the test length is established for all nonrepeating functions in this basis. A subsequence of easily testable functions is constructed and the corresponding tests are described. Individual lower test length bounds are proved for functions of a special form; minimality of the tests is established for the functions of the constructed subsequence.     

Separately <Emphasis Type="Italic">F</Emphasis>σ-measurable functions are close to functions of the first Baire class

We prove that a Borel separately F -measurable function f: X×YR on the product of Polish spaces is a function of the first Baire class on the complement X×Y\M of a certain projectively meager set MX×Y.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 573–576, April, 2004.     

<Emphasis Type="Italic">ψ</Emphasis>-continuous functions

Let be a ψ-density topology for a fixed function ψ. This paper is concerned with the family of ψ-continuous functions, that means continuous functions from (ℝ, ) into (ℝ, ). The family of such functions forms a lattice and is not closed under addition and uniform convergence. There exist functions ψ for which even linear functions are not ψ-continuous.      

Weighted inequalities for real-analytic functions in ℝ<Superscript>2</Superscript>

Let ƒ and g be real-analytic functions near the origin in ℝ². Given 1 < p < ∞, we obtain a characterization of the set of positive numbers ∈ and δ that ensures

Strongly (λ, θ)-continuous functions

In the paper, we introduce and investigate basic properties of a new class of functions between topological spaces called strongly (λ, θ)-continuous functions which is stronger than λ-continuous functions. This work was completed while the second author was visiting the Department of Mathematical Sciences, Unisa, Pretoria, South Africa partially supported by the National Research Foundation of South Africa.     

Gleason’s problem,rational functions and spaces of left-regular functions: The split-quaternion setting

We study Gleason’s problem, rational functions and spaces of regular functions in the setting of split-quaternions. There are two natural symmetries in the algebra of split-quaternions. The first symmetry allows to define positive matrices with split-quaternionic entries, and also reproducing Hilbert spaces of regular functions. The second leads to reproducing kernel Krein spaces.     

Definable functions in the ℵ<Subscript>0</Subscript>-categorical weakly circularly minimal structures

We continue studying the analogs of o-minimality and weak o-minimality for circularly ordered sets. We present a complete characterization of the behavior of unary definable functions in an ?₀-categorical 1-transitive weakly circularly minimal structure. Using it, we describe the ?₀-categorical 1-transitive nonprimitive weakly circularly minimal structures of convexity rank greater than 1 up to binarity.     

On ?-optimal points of convex, closed functions

The paper gives an estimate for the Hilbert space distance from a ?-optimal point to the minimum point of a convex, closed function, the subdifferential of which is a strongly monotone operator in its definition domain. Also, the Hausdorff distance between the ?-optimal points of the Tikhonov functions in the non-correct problems of mathematical programming is estimated.     

On weakly (τ, m)-continuous functions

In this paper we introduce a new notion of weakly (τ, m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [20], almosts-continuity [33],p(θ)-continuity [10] andp-continuity [41].     

Uniform approximation by harmonic functions on compact subsets of ℝ<Superscript>3</Superscript>

We consider uniform approximation by harmonic functions on compact subsets in ℝ³. Under an additional assumption that the approximated function is Dini-continuous, we prove a natural analog of well-known Vitushkin’s uniform approximation lemma for an individual analytic function. Bibliography: 14 titles.     

Supremizers of inner <Emphasis Type="Italic">γ</Emphasis>-convex functions

A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a such that holds for all satisfying ||x ₀ − x ₁|| = νγ and . This kind of roughly generalized convex functions is introduced in order to get some properties similar to those of convex functions relative to their supremum. In this paper, numerous properties of their supremizers are given, i.e., of such satisfying lim . For instance, if an upper bounded and inner γ-convex function, which is defined on a convex and bounded subset D of some inner product space, has supremizers, then there exists a supremizer lying on the boundary of D relative to aff D or at a γ-extreme point of D, and if D is open relative to aff D or if dim D ≤ 2 then there is certainly a supremizer at a γ-extreme point of D. Another example is: if D is an affine set and is inner γ-convex and bounded above, then for all , and if 2 ≤ dim D < ∞ then each is a supremizer of f.      

Bernstein functions, complete hyperexpansivity and subnormality—I

Special classes of functions on the classical semigroupN of non-negative integers, as defined using the classical backward and forward difference operators, get associated in a natural way with special classes of bounded linear operators on Hilbert spaces. In particular, the class of completely monotone functions, which is a subclass of the class of positive definite functions ofN, gets associated with subnormal operators, and the class of completely alternating functions, which is a subclass of the class of negative definite functions onN, with completely hyper-expansive operators. The interplay between the theories of completely monotone and completely alternating functions has previously been exploited to unravel some interesting connections between subnormals and completely hyperexpansive operators. For example, it is known that a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {1/_n}(n0). The present paper discovers some new connections between the two classes of operators by building upon some well-known results in the literature that relate positive and negative definite functions on cartesian products of arbitrary sets using Bernstein functions. In particular, it is observed that the weight sequence of a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {_n+1/_n}(n0). It is also established that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. Further, the connection of Bernstein functions with Stieltjes functions and generalizations thereof is exploited to link certain classes of subnormal weighted shifts to completely hyperexpansive ones.     

Properties of (θ,s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of (θ,s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of (θ,s)-continuous functions are obtained and relationships between (θ,s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.     

<Emphasis Type="Italic">M</Emphasis>-affine functions composing Sturm–Liouville families

Given an n-variable mean M defined on a real interval I, an M-affine function is a solution to the functional equation

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When M is a quasilinear mean, the set of continuous M-affine functions is a Sturm–Liouville family on every compact interval \(\left[ a,b\right] \subseteq I\); i.e., for every \(\alpha ,\beta \in \left[ a,b\right] \), there exists an M-affine function f such that \(f\left( a\right) =\alpha \) and \( f\left( b\right) =\beta \). The validity of the converse statement is explored in this paper and several consequences are derived from this study. New characterizations of quasilinear means and the solution to Eq. (1) under suitable conditions are among the more important ones.

     

A unified theory of<Emphasis Type="Italic">θ</Emphasis>-continuity for functions

We introduce a new notion ofθ-M-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory ofθ-continuity [15],θ-semi-continuity [4], quasi-irresoluteness [9],θ-preirresoluteness [29], and weakβ-irresoluteness [30].     

Correlation functions of gauged linear <Emphasis Type="Italic">σ</Emphasis>-model

This is the second paper in a series following Tian and Xu (2015), on the construction of a mathematical theory of the gauged linear σ-model (GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.     

On Haar series of <Emphasis Type="Italic">A</Emphasis>–integrable functions

In this paper we obtain a necessary and sufficient condition on the sequence of natural numbers {q _n } such that the almost everywhere convergence of the cubic partial sums S _qn (x) of the multiple Haar series Σ_n a _nχ_n(x) and the condition lim inf \(\lambda \cdot mes\left\{ {x:\begin{array}{*{20}{c}} {\sup } \\ n \end{array}\left| {S{}_{qn}\left( x \right)} \right| \succ \lambda } \right\} = 0\), imply that the coefficients a _n can be uniquely determined by the sum of the series. Also, we have obtained a necessary and sufficient condition for the series \(\sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}} {\chi _n}\left( x \right)\) with an arbitrary bounded sequence {ε _n} to be a Fourier-Haar series of an A-integrable function.