Class Of Real Functions Research Articles

W-Invexity and optimality problem

As the development of a family of preinvex functions, we define the w-invex set and a class of real functions. These are functions with the names w-preinvex, w-strictly preinvex, w-prequasi-invex, w-strictly prequasi-invex, w-semi-strictly prequasi-invex, and w-pre-pseudo-invex. The fundamental characteristics of these functions are thoroughly examined. Several examples are provided here that serve to clarify the concept in this context. Several of these preinvex functions are addressed to investigate the optimization problems further.

On Logarithmic Holder Condition and Local Extrema of Power Takagi Functions

This paper studies one class of real functions, which we call Takagi power functions. Such functions have one positive real parameter; they are continuous, but nowhere differentiable, and are given on a real line using functional series. These series are similar to the series defining the continuous, nowhere differentiable Takagi function described in 1903. For each parameter value, we derive a functional equation for functions related to Takagi power functions. Then, using this equation, we obtain an accurate two-sides estimate for the functions under study. Next, we prove that for parameter values not exceeding 1, Takagi power functions satisfy the Holder logarithmic condition, and find the smallest value of the constant in this condition. As a result, we get the usual Hölder condition, which follows from the logarithmic Hölder condition. Moreover, for parameter values ranging from 0 to 1, we investigate the behavior of Takagi power functions in the neighborhood of their global maximum points. Then we show that the functions under study reach a strict local minimum on the real axis at binary-rational points, and only at them. Finally, we describe the set of points at which our functions reach a strict local maximum. The benefit of our research lies in the development of methods applicable to continuous functions that cannot be differentiated anywhere. This can significantly expand the set of functions being studied.

Parameterized transformations and truncation: When is the result a copula?

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.

Formula omitted]-lineability: A general method and its application to Darboux-like maps

We describe a simple class of c-dimensional linear subspaces of RR and show that many natural classes of real functions contain subspaces of this form, proving their c-lineability. The examples include all non-empty classes in the algebra of all Darboux-like functions and of their restrictions to the Baire class 2 maps.

Different notions of Sierpiński–Zygmund functions

A function $$f:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$$ is Sierpinski–Zygmund, $$f\in {{\,\mathrm{SZ}\,}}(\mathrm {C})$$ , provided its restriction $$f{\restriction }M$$ is discontinuous for any $$M\subset {{\mathbb {R}}}$$ of cardinality continuum. Often, it is slightly easier to construct a function $$f:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$$ , denoted as $$f\in {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$$ , with a seemingly stronger property that $$f{\restriction }M$$ is not Borel for any $$M\subset {{\mathbb {R}}}$$ of cardinality continuum. It has been recently noticed that the properness of the inclusion $${{\,\mathrm{SZ}\,}}(\mathrm {Bor})\subseteq {{\,\mathrm{SZ}\,}}(\mathrm {C})$$ is independent of ZFC. In this paper we explore the classes $${{\,\mathrm{SZ}\,}}(\Phi )$$ for arbitrary families $$\Phi $$ of partial functions from $${{\mathbb {R}}}$$ to $${{\mathbb {R}}}$$ . We investigate additivity and lineability coefficients of the class $${{\mathbb {S}}}:={{\,\mathrm{SZ}\,}}(\mathrm {C}){\setminus } {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$$ . In particular we show that if $${{\mathfrak {c}}}=\kappa ^+$$ and $${{\mathbb {S}}}\ne \emptyset $$ , then the additivity of $${{\mathbb {S}}}$$ is $$\kappa $$ , that $${{\mathbb {S}}}$$ is $${{\mathfrak {c}}}^+$$ -lineable, and it is consistent with ZFC that $${{\mathbb {S}}}$$ is $${{\mathfrak {c}}}^{++}$$ -lineable. We also construct several examples of functions from $${{\,\mathrm{SZ}\,}}(\mathrm {C}){\setminus } {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$$ that belong also to other important classes of real functions.

A characterization of the finiteness of perpetual integrals of Lévy processes

We derive a criterium for the almost sure finiteness of perpetual integrals of Lévy processes for a class of real functions including all continuous functions and for general one-dimensional Lévy processes that drifts to plus infinity. This generalizes previous work of Döring and Kyprianou, who considered Lévy processes having a local time, leaving the general case as an open problem. It turns out, that the criterium in the general situation simplifies significantly in the situation, where the process has a local time, but we also demonstrate that in general our criterium can not be reduced. This answers an open problem posed in (J. Theoret. Probab. 29 (2016) 1192–1198).

A Generalized Palais-Smale Condition in the Fr\'{e}chet space setting

The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting. Despite the importance of Fr\'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr\'{e}chet spaces and Fr\'{e}chet-Finsler manifolds of class $C^1$. The difficulty in the Fr\'{e}chet setting is the lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However, Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences. We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr\'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as solvability theory for differential equations) with Fr\'{e}chet spaces. The idea of this approach is to represent a Fr\'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr\'{e}chet space and therefore can be used to obtain deformation results. This method would be the proper framework for further development of critical point theory in the Fr\'{e}chet setting.

An Orthonormal System of Modified Spherical Harmonics

The motivation for the theory of modified spherical harmonics is the desire to find a class of real functions on the half-sphere $$S_{+}=\left\{ (x,y,t): x^2 + y^2 + t^2 =1,\right. \left. t > 0 \right\} $$ in $$\mathbb {R}^3 = \{(x,y,t)\}$$ which is naturally adapted to $$S_{+}$$ in the sense that it behaves like the classical spherical harmonics on the full unit sphere. Such a system is at hand if we replace the Laplace equation $$\Delta h =0$$ by the equation $$t\Delta v+\frac{\partial v}{\partial t}=0$$ and consider the restrictions of the polynomial solutions $$v=v(x,y,t)$$ to the half-sphere $$S_{+}$$ . Elements of this class of functions are called “modified spherical harmonics”. In order to get similar results as in case of the classical spherical harmonics one has however to replace the Euclidean scalar product on the unit sphere in $$\mathbb {R}^3$$ by a non-Euclidean one defined on $$S_{+}$$ . All this has already been worked out in an earlier paper entitled “Modified Spherical Harmonics”. In the present paper we deduct an explicit orthonormal system of modified spherical harmonics. Such a system was missing.

Relations between two classes of real functions and applications to boundedness and compactness of operators between analytic function spaces

Relations between two classes of real functions and applications to boundedness and compactness of operators between analytic function spaces

On pexiderized Jensen–Hosszú functional equation on the unit interval

We solve the functional equation of the form 2f(x+y2)=g(x+y−xy)+h(xy) in the class of real functions defined on the unit interval [0,1]. We prove that it is not stable, but if two functions from the triple {f,g,h} coincides the analogue equation is stable in the Hyers–Ulam sense.

OSCILLATORY PROPERTIES OF REAL FUNCTIONS WITH WEAKLY BOUNDED MELLIN TRANSFORM

We show that the error term of the counting function of sums of two squares has oscillations of logarithmic frequency and size x1/2(log x)−3/2−ε. We do that by showing a theorem on oscillations for error terms of a rather general class of real functions. If the Mellin transform of f(x) has a singularity at ρ=β+iγ, γ≠0, with the principal summand of the form (s−ρ)−b(log(s−ρ))ch(s), h(ρ)≠0, we obtain oscillations of logarithmic frequency and size xβ(log x)b−1−ε. We also apply this result to other arithmetical functions related to sums of two squares.

Dark solitons of the Qiao's hierarchy

We obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of works by Qiao. The soliton solutions are in the class of real functions approaching constant value fast enough at infinity, the so-called “dark solitons.”

Convexity of chance constrained programming problems with respect to a new generalized concavity notion

In this paper, convexity of chance constrained problems have been investigated. A new generalization of convexity concept, named h-concavity, has been introduced and it has been shown that this new concept is the generalization of the α-concavity. Then, using the new concept, some of the previous results obtained by Shapiro et al. [in Lecture Notes on Stochastic Programming Modeling and Theory, SIAM and MPS, 2009] on properties of α-concave functions, have been extended. Next the convexity of chance constraints with independent random variables is investigated. It will be shown how concavity properties of the mapping related to the decision vector have to be combined with suitable properties of decrease or increase for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels and then sufficient conditions for convexity of chance constrained problems which has been introduced by Henrion and Strugarek [in Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41:263–276, 2008] has been extended in this paper for a wider class of real functions.

On the length of level sets of real functions

In this paper, we obtain some results in the spirit of Gamma-lines theory (complex analysis) for large classes of real functions f ( x,y ). In Section 3 our main result describe the length of solutions of f ( x,y )= A ν , for arbitrary finite sets of values A ν . This leads to some consequences for real algebraic polynomials P ( x,y ) in Section 1 and modules | p ( z )| of complex polynomials p ( z ) in Section 2.

A fixed point result in Menger spaces using a real function

The main result of this paper is a fixed point theorem of self-mappings in Menger spaces which satisfy certain inequality. This inequality involves a class of real functions which we call Φ-functions. As a corollary we obtain a result in the corresponding metric spaces. The result is supported by an example. The class of real functions we have used is the conceptual extension of altering distance functions used in metric fixed point theory.

Argument of outer functions on the real line

A complete description of the modulus of an outer function on the real line is well known. Indeed, this characterization is considered as one of the classical results of the theory of Hardy spaces. However, a satisfactory characterization of the argument of an outer function on the real line is not available yet. In this paper, we define some classes of real functions which can serve as the argument of an outer function. In particular, for any $0 < p \leq \infty$, an increasing bi-Lipschitz function is the argument of an outer function in $H^p(\mathbb{R})$.

Effectively open real functions

A function f is continuous iff the pre-image f - 1 [ V ] of any open set V is open again. Dual to this topological property, f is called open iff the image f [ U ] of any open set U is open again. Several classical open mapping theorems in analysis provide a variety of sufficient conditions for openness. By the main theorem of recursive analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping V ↦ f - 1 [ V ] being effective: given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f - 1 [ V ] . Analogously, effective openness requires the mapping U ↦ f [ U ] on open real subsets to be effective. The present work combines real analysis with algebraic topology and Tarski's quantifier elimination to effectivize classical open mapping theorems and to establish several rich classes of real functions as effectively open.

Quasi-analyticité, o-minimalité et élimination des quantificateurs

In this article, we study using model theory certain classes of real functions: restricted quasi-analytic classes. Let E be such a class of functions, we introduce a natural language L , including E and a theory T in L such that T admits quantifiers elimination, is o-minimal and is equivalent to the complete theory of R in L . To cite this article: A. Rambaud, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

A new “Optimization‐Preserving‐Operator” applied to global optimization

PurposeTo introduce Optimization‐Preserving‐Operators (O‐P‐Os), which are operators that are defined on classes of real functions that depend on a single variable, and allow us to eliminate local optima and to preserve global optima.Design/methodology/approachOutline a new method to build O‐P‐Os. These are introduced as O‐P‐O* and lead to a new approach for solving global optimization problems.FindingsIt was found that classical discretization methods for obtaining optimum of one variable function was too time‐consuming. The simple method introduced provided solutions to the test functions chosen as examples. The solutions were provided in a short time.Research limitations/implicationsProvides new tools for mathematical programming and in particular the global optimization problems. The O‐P‐O* introduced innovative technique for solving such problems.Practical implicationsO‐P‐O* produces solutions to global optimization problems in a much improved time. The algorithm derived, and the steps for its operation proved on implementation, the efficiency of the new method. This was demonstrated by numerical results for selected functions obtained using microcomputer systems.Originality/valueProvides new way of solving global optimization problems.

Global optimization‐preserving operators

In this paper, some kinds of operators, defined on a class of real functions depending on a single real variable, preserving some fundamental properties for solving global optimization problems are introduced. These operators allow the transfer of the search of the global extremum of a determined objective function, f into those of its transformed function, Tf, for which a global extremum is easy to obtain.