A fuzzy measure identification method by diamond pairwise comparisons and $${\phi_s}$$ transformation

We propose an identification method of fuzzy measures by diamond pairwise comparisons. Right and left side of the diamond means ordinal weights’ comparison and up and down means interaction degrees. From the comparisons, we estimate a hierarchy diagram with interaction degrees and weights of evaluation items using the agglomerative hierarchical clustering method. From the diagram, to identify the fuzzy measure, conversion method is proposed.     

Analysis of orthogonality and of orbits in affine iterated function systems

We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert space has an orthonormal basis (ONB) of exponentials . We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3. Research supported in part by the National Science Foundation DMS 0457491.     

Redefined fuzzy B-algebras

By two relations belonging to and quasi-coincidence (q) between fuzzy points and fuzzy sets, we define the concept of (α, β)-fuzzy subalgebras where α, β are any two of with . We state and prove some theorems in (α, β)-fuzzy B-algebras.     

Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning

In this paper, we propose a modification of Benson’s algorithm for solving multiobjective linear programmes in objective space in order to approximate the true nondominated set. We first summarize Benson’s original algorithm and propose some small changes to improve computational performance. We then introduce our approximation version of the algorithm, which computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of -nondominated points. This work is motivated by an application, the beam intensity optimization problem of radiotherapy treatment planning. This problem can be formulated as a multiobjective linear programme with three objectives. The constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. With our algorithm we solve the problem approximately within a specified accuracy in objective space. We present results on four clinical cancer cases that clearly illustrate the advantages of our method.     

Loeb measure,finitely additive measures,and a theorem of Banach

The Loeb measure construction from nonstandard analysis is applied to two theorems in standard measure theory. In both cases the essential simplification offered by the approach is the ability to work with a σ-additive measure space, even if the hypotheses only guarantee finite additivity. The key to this simplification is the property of -saturated nonstandard models, that any finitely additive measure on an internal algebra extends immediately to a σ-additive measure.      

Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions

Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the conditionf(t−1)⁻¹dμ(t)<∞. In this paper we study the order of the uniform approximation of the function

Koksma–Hlawka type inequalities of fractional order

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order is in . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.      

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Dedicated to Maurice Dodson on his retirement—finally!     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

Measures on Graphs and Groupoid Measures

The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G ₁ and G ₂ are graph-isomorphic, then the von Neumann algebras L ^∞(μ ₁) and L ^∞(μ ₂) are *-isomorphic, where μ ₁ and μ ₂ are the same kind of our graph measures of G ₁ and G ₂, respectively. Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007.     

A {\mathsf{D}}-induced duality and its applications

This paper attempts to extend the notion of duality for convex cones, by basing it on a prescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone , and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the -induced duality in the paper. We further introduce the notion of -induced polar sets within the same framework, which can be viewed as a generalization of the -induced dual cones and is convenient to use for some practical applications. Properties of the extended duality, including the extended bi-polar theorem, are proven. Furthermore, attention is paid to the computation and approximation of the -induced dual objects. We discuss, as examples, applications of the newly introduced -induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization. Research supported in part by the Foundation ‘Vereniging Trustfonds Erasmus Universiteit Rotterdam’ in The Netherlands, and in part by Hong Kong RGC Earmarked Grants CUHK4174/03E and CUHK418406.     

Optimal Decompositions of Translations of L 2-Functions

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space . Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space , or equivalent the spectral theory of a unitary representation U of the rank-n lattice in . Starting with a non-zero vector , we look for relations among the vectors in the cyclic subspace in generated by ψ. Since these vectors involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L ²-independence. This refers to infinite linear combinations of integral translates of a fixed function with l ²-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals. Work supported in part by the U.S. National Science Foundation.     

Signed Fuzzy-Valued Measures and Radon-Nikodym Theorem of Fuzzy-Valued Measurable Functions

In this paper, we introduce and study the signed fuzzy-valued measure and using the representation theorem of fuzzy numbers, we establish the Radon-Nikodym theorem for fuzzy-valued measurable functions with respect to fuzzy-valued measures.AMS Subject Classification (2000): 28E10, 04A72The research of this work is supposed by a grant of the education committee of Liaoning Province, China #20161049     

Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

In this paper we consider, in dimension d≥ 2, the standard finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L ^∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L ¹(Ω), we prove that the unique solution of the discrete problem converges in (for every q with ) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon measure. In the case where the dimension is d = 2 or d = 3 and where the coefficients are smooth, we give an error estimate in when the right-hand side belongs to L ^r (Ω) for some r > 1.     

A renormalization approach to irrational rotations

We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle , . In particular, we obtain sharp results for the diffusion of the walk on generated by the location of points of the sequence {n α + β} on a binary partition of the unit interval. Finally, we give some applications of our method.      

Characterization of equilibrium measures for critical reversible Nearest Particle Systems

We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than and obeys some natural regularity conditions. The research of this author is partially supported by the FNS Grant #200021-107475/1     

Variational Inequality with Fuzzy Convex Cone

In this study, we discuss one type of variational inequality problem with a fuzzy convex cone , denoted by VI( , f). Different classes of fuzzy convex cones which are considered in different context of the problems will be discussed. According to the existence theorem, an approach derived from the concepts of multiple objective mathematical programming problems for solving the VI( , f) is proposed. An algorithm is developed to find its fuzzy optimal solution set with complexity analysis.     

On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators

Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers holds if the class has an envelope function that is μ-integrable, or if is bounded in L _p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date.     

A geometric analysis of Renegar’s condition number,and its interplay with conic curvature

For a conic linear system of the form Ax ∈K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Among these, Renegar’s condition number is arguably the most prominent for its relation to data perturbation, error bounds, problem geometry, and computational complexity of algorithms. Nonetheless, is a representation-dependent measure which is usually difficult to interpret and may lead to overly conservative bounds of computational complexity and/or geometric quantities associated with the set of feasible solutions. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K; furthermore our bounds highlight the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar’s condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements the literature where only lower bounds have heretofore been developed.     

Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

We introduce the set of bicomplex numbers which is a commutative ring with zero divisors defined by where We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr?dinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr?dinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symmetries. We obtain the standard Born’s formula for the class of bicomplex wave functions having a null hyperbolic angle.