S-convex set and s-convex functions on Heisenberg group

In this paper, we characterize a vector-valued convex set function by its epigraph. The concepts of a vector-valued set function and a vector-valued concave set function are given respectively. The definitions of the conjugate functions for a vector-valued convex set function and a vector-valued concave set function are introduced. Then a Fenchel duality theorem in multiobjective programming problem with set functions is derived.

The convex function is one of the topics in mathematics that is closely related to the theory of inequality. Furthermore, the definition of convex function has an extension, which is the first and second kind of s-convex function, for fixed s ∈ (0,1. Convex function has a relation to the Hermite-Hadamard-Fejèr inequality, which is an integral inequality involving a convex function. Further development of these inequalities involves the s-convex function and through the concept of fractional integral. In this study, we discuss the Hermite-Hadamard-Fejèr type inequality that applies to the second kind of s-convex function via the Riemann-Liouville fractional integral. From these results, the relationship between these inequalities with the same type of inequality for convex function, are obtained.

Set functions are a feature of functional logic programming to encapsulate all results of a non-deterministic computation in a single data structure. Given a function f of a functional logic program written in Curry, we describe a technique to synthesize the definition of the set function of f. The definition produced by our technique is based on standard Curry constructs. Our approach is interesting for three reasons. It allows reasoning about set functions, it offers an implementation of set functions which can be added to any Curry system, and it has the potential of changing our thinking about the implementation of non-determinism, a notoriously difficult problem.

A set function (which is not necessarily additive) on a measurable space I is called orderable if for each measurable order $\mathcal{R}$ there is a measure $\varphi ^\mathcal{R} v$ on I such that for all initial segments J, $(\varphi ^\mathcal{R} v)(J) = v(J)$. Properties of orderable set functions v which have infinitely many null points are investigated in this paper. We show that such set functions are continuous and that a set A is v-null if and only if $| {\varphi ^\mathcal{R} v} |(A) = 0$ for all measurable orders $\mathcal{R}$ . A characterization of orderable nonatomic set functions as well as a characterization of weakly continuous set functions which have a mixing value are given. It is also shown that if a set function is weakly continuous with respect to a measure, then it is weakly equivalent to some measure.

Rewarding Maps: On Greedy Optimization of Set Functions

On the nonparametric integral over a bv surface

This paper is the first of the author’s three articles on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness $$\sqrt {K - 1} $$ in the uniform norm and order of closeness K − 1 in the Sobolev norm L p 1 for all $$p < \tfrac{C}{{K - 1}}$$ . In the present article we study integrability of mappings with bounded specific oscillation on spaces of homogeneous type. As an example, we consider mappings with bounded distortion on the Heisenberg group. We prove that a mapping with bounded distortion belongs to the Sobolev class W ,loc 1 , where p → ∞ as the distortion coefficient tends to 1.

This present paper is concerned with set functions related to {0, 1} two valued measures. These set functions are either outer measures or have many of the same characteristics. We investigate their properties and look at relations among them. We note in particular their association with the semi‐separation of lattices.To be more specific, we define three set functions μ″, μ′, and related to μ ϵ I(L) the {0, 1} two valued set functions defined on the algebra generated by the lattice of sets L st μ is a finitely additive monotone set function for which μ(ϕ) = 0. We note relations among them and properties they possess.ln particular necessary and sufficient conditions are given for the semi‐separation of lattices in terms of equality of set functions over a lattice of subsets.Finally the notion of I‐lattice is defined, we look at some properties of these with certain other side conditions assume, and end with an application involving semi‐separation and I‐lattices.

Chapter 34 - Set Functions over Finite Sets: Transformations and Integrals

The choice of fuzzy set function affects how well the fuzzy systems approximate an unknown function. We first examine several popular fuzzy systems, based on which we generalize the error of the fuzzy system by the mathematical functional. Two classes of extended fuzzy systems with special set function are proposed respectively in one-dimensional and two-dimensional reproducing kernel spaces, and they are in fact the best interpolation operators in the sense of classical mathematical approximation. The extended fuzzy system can be used as easily as any typical fuzzy system but has better characters. Our major contributions are as follows. We prove that the existing fuzzy system with a triangle-shaped set function is by no means the best choice for an uncertain system. On the contrary, the reproducing kernel-shaped set function can give the best approximation in the sense of mathematical functional. According to the result, overlapped symmetric triangles or trapezoid set functions reduce fuzzy systems to piecewise linear systems. Universal Gaussian bell-curve set functions can work well but its learning cost is too expensive to apply under online condition. We explain what our proposed extended fuzzy system is useful for a given uncertain system. This solution is associated with another problem: Is the reproducing kernel condition too specific to limit its applications? We must stress that the reproducing kernel function is easily found in any continuous function space. Consequently, the extended fuzzy system can approximate any continuous function with arbitrary accuracy. Furthermore one may question why we do not directly use an optimal approximation functional instead of the extended fuzzy system. Here we answer this problem by the following point of views: The two methods face different applicable conditions. The optimal approximation functional is used to interpolate those accurate data. These data are needed under certain conditions such as distributions and quantity, etc. When little information is available, an optimal approximation functional will do nothing. But the extended fuzzy system gives a good approximation to an uncertain system and can work efficiently under the condition of vague or poor information. Consequently, the two methods have different optimal criteria and different input and output descriptions. Furthermore, the extended fuzzy systems can be created flexibly by artificial experiences or other ways. Two experiments are used to verify the effectiveness and efficiency of our proposed extended fuzzy systems.

The problem of maximizing a constrained monotone set function has many practical applications and generalizes many combinatorial problems such as k-Coverage, Max-SAT, Set Packing, Maximum Independent Set and Welfare Maximization. Unfortunately, it is generally not possible to maximize a monotone set function up to an acceptable approximation ratio, even subject to simple constraints. One highly studied approach to cope with this hardness is to restrict the set function, for example, by requiring it to be submodular. An outstanding disadvantage of imposing such a restriction on the set function is that no result is implied for set functions deviating from the restriction, even slightly. A more flexible approach, studied by Feige and Izsak [ITCS 2013], is to design an approximation algorithm whose approximation ratio depends on the complexity of the instance, as measured by some complexity measure. Specifically, they introduced a complexity measure called supermodular degree, measuring deviation from submodularity, and designed an algorithm for the welfare maximization problem with an approximation ratio that depends on this measure. In this work, we give the first (to the best of our knowledge) algorithm for maximizing an arbitrary monotone set function, subject to a k-extendible system. This class of constraints captures, for example, the intersection of k-matroids (note that a single matroid constraint is sufficient to capture the welfare maximization problem). Our approximation ratio deteriorates gracefully with the complexity of the set function and k. Our work can be seen as generalizing both the classic result of Fisher, Nemhauser and Wolsey [Mathematical Programming Study 1978], for maximizing a submodular set function subject to a k-extendible system, and the result of Feige and Izsak for the welfare maximization problem. Moreover, when our algorithm is applied to each one of these simpler cases, it obtains the same approximation ratio as of the respective original work. That is, the generalization does not incur any penalty. Finally, we also consider the less general problem of maximizing a monotone set function subject to a uniform matroid constraint, and give a somewhat better approximation ratio for it.

In a weighted sum model such as the Analytic Hierarchy Process, a set function value is constructed from the weights of the model. In this paper, we use relative individual scores to propose a set function that shows the features of an alternative. The set function value for the alternative is calculated by averaging the values of the set function representation of the weights generated when the alternative has the highest comprehensive score. By interpreting the functions, we can understand the features of an alternative. We discuss the properties of the set functions, and extend to Choquet integral models.

In this paper, we introduce four types of generalized convexity for an n -set function and discuss optimality and duality for a multiobjective programming problem involving n -set functions. Under some mild assumption on the new generalized convexity, we present a few optimality conditions for an efficient solution and a weakly efficient solution to the problem. Also we prove a weak duality theorem and a strong duality theorem for the problem and its Mond-Weir and general Mond-Weir dual problems respectively.Keywordsmultiobjective programmingn-set functionoptimalitydualitygeneralized convexity

Families of measures play important roles in statistics and economic decision theory in order to model uncertainty. They also appear as the core of a cooperative game in game theory. The envelopes (supremum or infimum) of the members of the family are set functions and their integrals give bounds for the integrals in the family. The main result is a characterization of submodular set functions by means of envelopes of additive set functions. The method of generating a set function as supremum of a given family of set functions will be employed, too, for proving the Radon-Nikodym Theorem in the next chapter.

Nonadditive Set Functions on a Finite Set and Linear Inequalities