Showing papers on "Operator (computer programming) published in 2006"

Some geometric aggregation operators based on intuitionistic fuzzy sets

Abstract: The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.

Extremum Problems for Eigenvalues of Elliptic Operators

Abstract: Eigenvalues of elliptic operators.- Tools.- The first eigenvalue of the Laplacian-Dirichlet.- The second eigenvalue of the Laplacian-Dirichlet.- The other Dirichlet eigenvalues.- Functions of Dirichlet eigenvalues.- Other boundary conditions for the Laplacian.- Eigenvalues of Schrodinger operators.- Non-homogeneous strings and membranes.- Optimal conductivity.- The bi-Laplacian operator.

The Functional Calculus for Sectorial Operators

Abstract: In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

A Primitive Operator for Similarity Joins in Data Cleaning

Abstract: Data cleaning based on similarities involves identification of "close" tuples, where closeness is evaluated using a variety of similarity functions chosen to suit the domain and application. Current approaches for efficiently implementing such similarity joins are tightly tied to the chosen similarity function. In this paper, we propose a new primitive operator which can be used as a foundation to implement similarity joins according to a variety of popular string similarity functions, and notions of similarity which go beyond textual similarity. We then propose efficient implementations for this operator. In an experimental evaluation using real datasets, we show that the implementation of similarity joins using our operator is comparable to, and often substantially better than, previous customized implementations for particular similarity functions.

Efficient exact set-similarity joins

Abstract: Given two input collections of sets, a set-similarity join (SSJoin) identifies all pairs of sets, one from each collection, that have high similarity. Recent work has identified SSJoin as a useful primitive operator in data cleaning. In this paper, we propose new algorithms for SSJoin. Our algorithms have two important features: They are exact, i.e., they always produce the correct answer, and they carry precise performance guarantees. We believe our algorithms are the first to have both features; previous algorithms with performance guarantees are only probabilistically approximate. We demonstrate the effectiveness of our algorithms using a thorough experimental evaluation over real-life and synthetic data sets.

Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality

Abstract: We study operators in four-dimensional gauge theories which are localized on a straight line, create electric and magnetic flux, and in the UV limit break the conformal invariance in the minimal possible way. We call them Wilson-'t Hooft operators, since in the purely electric case they reduce to the well-known Wilson loops, while in general they may carry 't Hooft magnetic flux. We show that to any such operator one can associate a maximally symmetric boundary condition for gauge fields on AdSE2×S2. We show that Wilson-'t Hooft operators are classified by a pair of weights (electric and magnetic) for the gauge group and its magnetic dual, modulo the action of the Weyl group. If the magnetic weight does not belong to the coroot lattice of the gauge group, the corresponding operator is topologically nontrivial (carries nonvanishing 't Hooft magnetic flux). We explain how the spectrum of Wilson-'t Hooft operators transforms under the shift of the θ-angle by 2π. We show that, depending on the gauge group, either SL(2,[openface Z]) or one of its congruence subgroups acts in a natural way on the set of Wilson-'t Hooft operators. This can be regarded as evidence for the S-duality of N=4 super-Yang-Mills theory. We also compute the one-point function of the stress-energy tensor in the presence of a Wilson-'t Hooft operator at weak coupling.

Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry

Abstract: One of the challenges in geometry processing is to automatically reconstruct a higher-level representation from raw geometric data. For instance, computing a parameterization of an object helps attaching information to it and converting between various representations. More generally, this family of problems may be thought of in terms of constructing structured function bases attached to surfaces. In this paper, we study a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator. When applied to a sphere, this function basis corresponds to the classical spherical harmonics. On more general objects, this defines a function basis well adapted to the geometry and the topology of the object. Based on physical analogies (vibration modes), we first give an intuitive view before explaining the underlying theory. We then explain in practice how to compute an approximation of the eigenfunctions of a differential operator, and show possible applications in geometry processing.

Multilinear operators for higher-order decompositions

Abstract: We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to concisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outer-products of the columns of N matrices and allows a divorce from a matricized representation and a very concise expression of the PARAFAC decomposition. We explore the properties of the Tucker and Kruskal operators independently of the related decompositions. Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.

An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations

Abstract: In this paper, we define the concept of uncertain multiplicative linguistic preference relation, and introduce some operational laws of uncertain multiplicative linguistic variables. We propose some new aggregation operators including the uncertain linguistic geometric mean (ULGM) operator, uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic ordered weighted geometric (ULOWG) operator, and induced uncertain linguistic ordered weighted geometric (IULOWG) operator. The IULOWG operator is a more general type of aggregation operator, which is based on the ULGM and ULOWG operators. Moreover, based on the ULOWG and IULOWG operators and the formula for the comparison between two uncertain multiplicative linguistic variables, we develop an approach to group decision making with uncertain multiplicative linguistic preference relations, and, finally, an application of the approach to group decision-making problem with uncertain multiplicative linguistic preference relations is pointed out.

Local bulk operators in AdS/CFT correspondence : A boundary view of horizons and locality

Abstract: We develop the representation of local bulk fields in anti-de Sitter (AdS) space by nonlocal operators on the boundary, working in the semiclassical limit and using ${\mathrm{AdS}}_{2}$ as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincar\'e patch and inside the Rindler horizon of a two-dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.

Holder estimates for solutions of integra-differential equations like the fractional laplace

Abstract: We provide a purely analytical proof of Holder continuity for harmonic functions with respect to a class of integro-differential equations like the ones associated with purely jump processes. The assumptions on the operator are more flexible than in previous works. Our assumptions include the case of an operator with variable order, without any continuity assumption in that order.

An extension problem related to the fractional Laplacian

Abstract: The operator square root of the Laplacian $(-\lap)^{1/2}$ can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

Solving gapped Hamiltonians locally

Abstract: We show that any short-range Hamiltonian with a gap between the ground and excited states can be written as a sum of local operators, such that the ground state is an approximate eigenvector of each operator separately. We then show that the ground state of any such Hamiltonian is close to a generalized matrix product state. The range of the given operators needed to obtain a good approximation to the ground state is proportional to the square of the logarithm of the system size times a characteristic ``factorization length.'' Applications to many-body quantum simulation are discussed. We also consider density matrices of systems at non zero temperature.

Vector valued reproducing kernel hilbert spaces of integrable functions and mercer theorem

Abstract: We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.

Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces

Abstract: The aim of this paper is twofold First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective Also, some gaps in the current knowledge about those concepts are filled in Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations

Video distribution system

Abstract: Movies are distributed to customer households in a secure format on digital media such as optical disks for playing back via proprietary set-top boxes (14) A system operator produces movie disks in large quantities and delivers the disks to widely geographically dispersed distribution agents Agents produce copies of the disks with unique agent identification codes embedded therein, and distribute the disks to local customers who have compatible playback devices When customers view movies, information identifying the movie and a distribution agent who is responsible for the movie being distributed to that customer are communicated to the central computer of the system operator (20) The content providers are then paid royalties due and responsible distribution agents are compensated

A Note on Linguistic Hybrid Arithmetic Averaging Operator in Multiple Attribute Group Decision Making with Linguistic Information

Abstract: In this paper, we propose a linguistic hybrid arithmetic averaging (LHAA) operator, which is based on linguistic weighted arithmetic averaging (LWAA) operator and extended ordered weighted averaging (EOWA) operator, and study some desirable properties of the LHAA operator. The LHAA operator can not only reflect the importance degrees of both the given argument and its ordered position, but also relieve the influence of unfair arguments on the decision results by weighting these arguments with small values. Based on the LWAA and LHAA operators, we develop a practical approach to multiple attribute group decision making under linguistic environment. The approach first aggregates the individual linguistic preference values into a collective linguistic preference value for each alternative by using the LWAA and LHAA operators (it is worth pointing out that the aggregation process does not produce any loss of linguistic information), and then orders the collective linguistic preference values to obtain the best alternative(s). Finally, an illustrative example is also given to verify the approach and to demonstrate its feasibility and practicality.

A generalized cross‐tabulation matrix to compare soft‐classified maps at multiple resolutions

Abstract: This paper addresses two grand challenges in the development of methods for Geographic Information Science (GIS). First, this paper presents techniques to compute a cross‐tabulation matrix for soft‐classified pixels. Second, it shows how to compute the cross‐tabulation matrix at multiple scales. The traditional approach to construct the cross‐tabulation matrix uses a Boolean operator to analyse pixels that are hard‐classified. For soft‐classified pixels, the contemporary approach uses a Multiplication operator; the fuzzy approach uses a Minimum operator; whereas this paper proposes a multiple‐resolution approach that uses a Composite operator. There are difficulties with the traditional, contemporary, and fuzzy methods of computing the cross‐tabulation matrix. The proposed multiple‐resolution method resolves those difficulties. Furthermore, the proposed method facilitates multiple‐resolution analysis, so it can examine how results change as a function of scale. The paper derives the equations to compute c...

The logarithmic norm. History and modern theory

Abstract: In his 1958 thesis Stability and Error Bounds, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems, using differential inequalities that distinguished between forward and reverse time integration. Originally defined for matrices, the logarithmic norm can be extended to bounded linear operators, but the extensions to nonlinear maps and unbounded operators have required a functional analytic redefinition of the concept. This compact survey is intended as an elementary, but broad and largely self-contained, introduction to the versatile and powerful modern theory. Its wealth of applications range from the stability theory of IVPs and BVPs, to the solvability of algebraic, nonlinear, operator, and functional equations.

Kaluza-Klein Holography

Abstract: We construct a holographic map between asymptotically AdS5 × S5 solutions of 10d supergravity and vacuum expectation values of gauge invariant operators of the dual QFT. The ingredients that enter in the construction are (i) gauge invariant variables so that the KK reduction is independent of any choice of gauge fixing; (ii) the non-linear KK reduction map from 10 to 5 dimensions (constructed perturbatively in the number of fields); (iii) application of holographic renormalization. A non-trivial role in the last step is played by extremal couplings. This map allows one to reliably compute vevs of operators dual to any KK fields. As an application we consider a Coulomb branch solution and compute the first two non-trivial vevs, involving operators of dimension 2 and 4, and reproduce the field theory result, in agreement with non-renormalization theorems. This constitutes the first quantitative test of the gravity/gauge theory duality away from the conformal point involving a vev of an operator dual to a KK field (which is not one of the gauged supergravity fields).

Network effects and the choice of mobile phone operator

Abstract: This paper explores the role of network effects in the consumer’s choice of mobile phone operators in the UK. It contributes to the existing literature by taking a new approach to testing for direct network effects and by using individual-level data, which allows to analyse the impact that the immediate social network has on consumer choice in network markets. For our empirical analysis we use two sources of data: market-level data from the British telecommunications regulator OFCOM and micro-level data on consumers’ usage of mobile telephones from the survey, Home OnLine. We estimate two classes of models which illustrate the role of network effects. The first is an aggregate model of the comparative volume of on-net and off-net calls. This finds that the proportion of off-net calls falls as mobile operators charge a premium for off-net calls, but even in the absence of any price differential between on-net and off-net, there is still a form of pure network effect, where a disproportionate number of calls are on-net. The second is a model of the individual consumer’s choice of operator. This finds that individual choice shows considerable inertia, as expected, but is heavily influenced by the choices of others in the same household. There is some evidence that individual choice of operator is influenced by the total number of subscribers for each operator, but a much stronger effect is the operator choice of other household members.

Binary Differential Evolution

Abstract: The ability of differential evolution (DE) to perform well in continuous-valued search spaces is well documented. The arithmetic reproduction operator used by differential evolution is simple, however, the manner in which the operator is defined, makes it practically impossible to effectively apply the standard DE to other problem spaces. An interesting and unique mapping method is examined which will enable the DE algorithm to operate within binary space. Using angle modulation, a bit string can be generated using a trigonometric generating function. The DE is used to evolve the coefficients to the trigonometric function, thereby allowing a mapping from continuous-space to binary-space. Instead of evolving the higher-dimensional binary solution directly, angle modulation is used together with DE to reduce the complexity of the problem into a 4-dimensional continuous-valued problem. Experimental results indicate the effectiveness of the technique and the viability for the DE to operate in binary space.

Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators

Abstract: We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inverse-power law interactions, and hard spheres. The functional spaces of these coercivity estimates depend on the collision kernel of these operators. They cover the spectral gap estimates for the linearized Boltzmann operator with Maxwell molecules, improve these estimates for hard potentials, and are the first explicit coercivity estimates for soft potentials (including in particular the case of Coulombian interactions). We also prove a regularity property for the linearized Boltzmann operator with non locally integrable collision kernels, and we deduce from it a new proof of the compactness of its resolvent for hard potentials without angular cutoff.

Operator Preconditioning

Abstract: Operator preconditioning offers a general recipe for constructing preconditioners for discrete linear operators that have arisen from a Galerkin approach. The key idea is to employ matching Galerkin discretizations of operators of complementary mapping properties. If these can be found, the resulting preconditioners will be robust with respect to the choice of the bases for trial and test spaces. I survey the application of operator preconditioning to finite elements and boundary elements.

On spectral theory for Schrödinger operators with strongly singular potentials

Abstract: We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrodinger operators on [a , ∞), a ∈ ℝ, with a regular finite end point a and the case of Schrodinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl–Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrodinger operators on (a , ∞) with a potential strongly singular at the end point a . We focus on situations where the potential is so singular that the associated maximally defined Schrodinger operator is self-adjoint (equivalently, the associated minimally defined Schrodinger operator is essentially selfadjoint) and hence no boundary condition is required at the finite end point a . For this case we show that the Weyl–Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrodinger operators. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials

Abstract: Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrodinger operator with a complex-valued potential.