Constructive Set Function and Extraction of a k-dimensional Element

Nonadditive Set Functions on a Finite Set and Linear Inequalities

When fuzzy production rules are used to approximate reasoning, interaction exists among rules that have the same consequent. Due to this interaction, the weighted average model frequently used in approximate reasoning does not work well in many real-world problems. In order to model and handle this interaction, this paper proposes to use a nonadditive nonnegative set function to replace the weights assigned to rules having the same consequent, and to draw the reasoning conclusion based on an integral with respect to the nonadditive nonnegative set function, rather than on the weighted average model. Handling interaction in fuzzy production rule reasoning in this way can lead to a good understanding of the rules base and an improvement of reasoning accuracy. This paper also investigates how to determine from data the nonadditive set function that cannot be specified by a domain expert.

Nonlinear nonnegative multiregressions based on Choquet integrals

We have seen that the distortion-rate function of a source with respect to a distortion measure provides an unbeatable, yet approximately achievable, bound on the performance of a fixed-rate data compression system with unconstrained block length or delay. The goal of this chapter is to provide the tools for actually computing and bounding the distortion rate functions for particular sources and distortion measures. Toward this end we begin by developing variational equations for inverse distortion-rate functions or rate-distortion functions. The focus on rate-distortion functions instead of distortion-rate functions is both for consistency with the literature and for simplicity. It is usually easier to find the rate as a function of distortion rather than vice-versa.

In this paper, non-additivity of a set function is interpreted as a method to express relations between sets which are not modeled in a set theoretic way. Drawing upon a concept called “quasi-analysis” of the philosopher Rudolf Carnap, we introduce a transform for sets, functions, and set functions to formalize this idea. Any image-set under this transform can be interpreted as a class of (quasi-)components or (quasi-)properties representing the original set. We show that non-additive set functions can be represented as signed σ-additive measures defined on sets of quasi-components. We then use this interpretation to justify the use of non-additive set functions in various applications like for instance multi criteria decision making and cooperative game theory. Additionally, we show exemplarily by means of independence, conditioning, and products how concepts from classical measure and probability theory can be transfered to the non-additive theory via the transform.

We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtained from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.

The membership function of a general type-2 fuzzy set is three-dimensional in order to incorporate its vagueness through the third dimension. Similarly, Circular intuitionistic fuzzy sets (CIFSs) have been recently introduced by Atanassov (2020) as a new extension of intuitionistic fuzzy sets, which are represented by a circle representing the vagueness of the membership function. CIFSs allow decision-makers to express their judgments including this vagueness. In this study, the TOPSIS method, which is one of the most used multi-criteria decision-making methods is extended to its CIF version. The proposed CIF-TOPSIS methodology is applied to the supplier selection problem. Then, a sensitivity analysis based on criteria weights is conducted to check the robustness of the proposed approach. A comparative analysis with single-valued intuitionistic fuzzy TOPSIS method is also performed to verify the developed approach and to demonstrate its effectiveness

The Decision making has been a major research topic in the computing literature for so long due to its vast significance in many real-world applications. Traditional fuzzy decision making (FDM) approaches have limitations due to the inability of the type-1 fuzzy sets (T1 FSs) in modeling higher order uncertainties. Since, the membership function (MF) of an interval type-2 fuzzy set (IT2 FS) is also fuzzy, as superior to T1 FSs, researchers considered IT2 FSs to model higher level of uncertainties in FDM and proposed a number of IT2 FDM methods. However, unlike IT2 FSs, general type-2 fuzzy sets (GT2 FSs) do not consider equal secondary membership values for all its primary membership functions. Hence, GT2 FSs offer more suitability in modelling uncertainties that exist in real-world scenarios. Thus, this paper proposes a more efficient decision making method called the “GT2 Fuzzy Decision Making (GT2 FDM)”, which considers GT2 FSs to model the fuzzy goals and fuzzy constraints in a problem. The working of the proposed approach is demonstrated using an example of room temperature selection. Then we have applied it to the problem of convenient travel time selection using a real-time traffic data set. It is observed that the proposed GT2 FDM approach offers more flexibility to the decision makers in choosing an optimal solution from a much wider solution space and hence is found to be more efficient than the IT2 FDM and classical FDM approaches.

A new rate control scheme is used to calculate the target bit rate for each frame based on a quadratic formulation of the rate distortion function. The distortion measure are assumed to be either the average quantization scale of a frame or mean square error. The rate distortion function is modeled as a second order function of the inverse of the distortion measure. We presented a closed form solution for the target bit allocation which applies to both the digital storage media (DSM) applications (e.g., MPEG-2) and very low bit rate (VLBR) applications (e.g., MPEG-4). We perform experiments on a wavelet-based coder to demonstrate the versatility of our scheme. Model parameters are estimated using statistical linear regression analysis. Since the estimation uses the past encoded frames of the same picture prediction type (I, P and B pictures), the proposed approach is a single pass rate control technique. For the MPEG-2 coder the fluctuations of the bit counts are significantly reduced by 20 percent to 65 percent in the standard deviation of the bit count while the picture quality remains the same. For the MPEG-4 VM coder, the PSNR is improved by almost one dB as compared to the best MPEG-4 VM rate control using exhaustive combinations of quantization scale parameters.

We propose a method, denoted P-orthogonalization, for converting a general contracted basis set to a computationally more efficient segmented contracted basis set, while inheriting the full accuracy of the general contracted basis set. The procedure can be used for any general contracted basis set to remove the redundancies between general contracted functions in terms of primitive functions. The P-orthogonalization procedure is used to construct a segmented contracted version of the polarization consistent basis sets, which are optimized for density functional theory calculations. Benchmark calculations show that the new pcs-n basis sets provide uniform error control of the basis set incompleteness for molecular systems composed of atoms from the first three rows in the periodic table (H-Kr) and for different exchange-correlation functionals. The basis set errors at a given zeta quality level are lower than other existing basis sets, and the pcs-n basis sets are furthermore shown to be among the computationally most efficient. The pcs-n basis sets are available in qualities ranging from (unpolarized) double-zeta to pentuple zeta quality and should therefore be well suited for both routine and benchmark calculations using density functional theory methods in general.

Geometry of exclusion principles in discrete systems

In the language of random counting measures, many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology, are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.

We study bond prices in Black–Cox model with jumps in asset value. We assume that the jump size distribution is arbitrary and, if default occurs, following Longstaff and Schwartz [A Simple Approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finance 50 (1995), 789–819] and Zhou [The Term Structure of Credit Spreads with Jump Risk. Journal of Banking & Finance 26 (2001), 2015–2040], the payoff at maturity date depends on a general write-down function. Under this general setting, we propose an integral equation approach for the bond prices. As an application of this approach, we study the analytic properties of the bond prices. Also we derive an infinite series expression for the bond prices.

We study the effect of a shock on firms’ costs in a general setting by considering both perfect and imperfect competition and a general cost function. We show that, counterintuitively, firms’ profits may increase with cost increases. We generalize Seade’s (1985) results by considering the adaptation of firms’ technological process. We find an additional effect that we call and which is determined by the extent to which firms’ marginal and average costs differ as a result of a shock. This effect is broken down into two components: the indirect technology which is related to the elasticity of the demand slope, and the technology which is solely related to technology. We apply this framework to environmental regulations, which provide a good context in which to examine technological process adaptation because they push firms to use abatement technologies and to modify their production processes. We introduce an explicit abatement cost function that is sufficiently flexible to represent the various types of abatement technologies that are found in the literature: end-of-pipe technology, process-integrated technology, and cleaner production (or fuel switching). We show that end-of-pipe abatement technologies induce a positive direct technology effect, that process integrated abatement technologies induce a negative technology effect, and that cleaner production induces a null technology effect.

G. J. M. H. Buskes [2] investigated a series of completeness properties for an archimedean Riesz space E. Each of these properties can be applied also in a more general setting, i.e., for the case when E is a lattice ordered group.If a is one of the properties under consideration, then we denote by Ga the class of all lattice ordered groups G which have the property a.The notion of radical class of lattice ordered groups was introduced in [8]; cf. also [16].The relations between this notion and the classes Ga will be dealt with in the present paper.We are mainly interested in the question whether Ga (or some reasonably large subclass of Ga) is a radical class.This question is related to the problem of existence of a-kernels.For some properties defined by means of sequences similar considerations were established in [10] and [11]. PRELIMINARIESThe standard notation for lattice ordered groups will be applied (cf.[3] and [6]).The group operation will be written additively.We denote by G the class of all lattice ordered groups.For G e G let c(G) be the system of all convex ^-subgroups of G\ this system is partially ordered by inclusion. c c cThe lattice operations in c(G) will be denoted by /\ and V-I n fact.A coincides with the set-theoretical intersection.Let {Ai} ie i be a nonempty subset of c(G) and c let H = V H{.It is well-known that H is the set of all g e G having the property iei that there is a finite subset {i(l),i(2),... ,i(n)} of I such that there exist elements hi e Hi(i),..., h n e Hi( n ) with g = hi + h 2 + ... + h n .