Showing papers on "Operator (computer programming) published in 2009"

Proximal Splitting Methods in Signal Processing

Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.

Wavelets on Graphs via Spectral Graph Theory

Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $\L$. Given a wavelet generating kernel $g$ and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(t\L)$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $\L$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.

Dynamics of Linear Operators

Abstract: Introduction 1. Hypercyclic and supercyclic operators 2. Hypercyclicity everywhere 3. Connectedness and hypercyclicity 4. Weakly mixing operators 5. Ergodic theory and linear dynamics 6. Beyond hypercyclicity 7. Common hypercyclic vectors 8. Hypercyclic subspaces 9. Supercyclicity and the angle criterion 10. Linear dynamics and the weak topology 11. Universality of the Riemann zeta function 12. About 'the' Read operator Appendices Notations Index Bibliography.

Multi-operator media retargeting

Abstract: Content aware resizing gained popularity lately and users can now choose from a battery of methods to retarget their media. However, no single retargeting operator performs well on all images and all target sizes. In a user study we conducted, we found that users prefer to combine seam carving with cropping and scaling to produce results they are satisfied with. This inspires us to propose an algorithm that combines different operators in an optimal manner. We define a resizing space as a conceptual multi-dimensional space combining several resizing operators, and show how a path in this space defines a sequence of operations to retarget media. We define a new image similarity measure, which we term Bi-Directional Warping (BDW), and use it with a dynamic programming algorithm to find an optimal path in the resizing space. In addition, we show a simple and intuitive user interface allowing users to explore the resizing space of various image sizes interactively. Using key-frames and interpolation we also extend our technique to retarget video, providing the flexibility to use the best combination of operators at different times in the sequence.

Novel quark-field creation operator construction for hadronic physics in lattice QCD

Abstract: A new quark-field smearing algorithm is defined which enables efficient calculations of a broad range of hadron correlation functions. The technique applies a low-rank operator to define smooth fields that are to be used in hadron creation operators. The resulting space of smooth fields is small enough that all elements of the reduced quark propagator can be computed exactly at reasonable computational cost. Correlations between arbitrary sources, including multihadron operators can be computed a posteriori without requiring new lattice Dirac operator inversions. The method is tested on realistic lattice sizes with light dynamical quarks.

Uncertain linguistic hybrid geometric mean operator and its application to group decision making under uncertain linguistic environment

Abstract: In this paper, we propose an uncertain linguistic hybrid geometric mean (ULHGM) operator, which is based on the uncertain linguistic weighted geometric mean (ULWGM) operator and the uncertain linguistic ordered weighted geometric (ULOWG) operator proposed by Xu [Z. S. Xu, "An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations", Decision Support Systems41 (2006) 488–499] and study some desirable properties of the ULHGM operator. We have proved both ULWGM and ULOWG operators are the special case of the ULHGM operator. The ULHGM operator generalizes both the ULWGM and ULOWG operators, and reflects the importance degrees of both the given arguments and their ordered positions. Based on the ULWGM and ULHGM operators, we propose a practical method for multiple attribute group decision making with uncertain linguistic preference relations. Finally, an illustrative example demonstrates the practicality an...

On the numerical evaluation of Fredholm determinants

Abstract: Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painleve transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method for the solution of Fredholm equations of the second kind. Using Gauss—Legendre or Clenshaw—Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk scaling limit and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy 1 processes.

An axiomatic approach of fuzzy rough sets based on residuated lattices

Abstract: Rough set theory was developed by Pawlak as a formal tool for approximate reasoning about data Various fuzzy generalizations of rough approximations have been proposed in the literature As a further generalization of the notion of rough sets, L-fuzzy rough sets were proposed by Radzikowska and Kerre In this paper, we present an operator-oriented characterization of L-fuzzy rough sets, that is, L-fuzzy approximation operators are defined by axioms The methods of axiomatization of L-fuzzy upper and L-fuzzy lower set-theoretic operators guarantee the existence of corresponding L-fuzzy relations which produce the operators Moreover, the relationship between L-fuzzy rough sets and L-topological spaces is obtained The sufficient and necessary condition for the conjecture that an L-fuzzy interior (closure) operator derived from an L-fuzzy topological space can associate with an L-fuzzy reflexive and transitive relation such that the corresponding L-fuzzy lower (upper) approximation operator is the L-fuzzy interior (closure) operator is examined

Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory

Abstract: We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters as eigenvalues; (ii) they can be represented as differential operators of the $W$-type (in particular, acting on the time-variables in the Hurwitz-Kontsevich tau-function). The operators have the simplest form if expressed in terms of the matrix Miwa-variables. They form an important commutative associative algebra, a Universal Hurwitz Algebra, generalizing all group algebra centers of particular symmetric groups which are used in description of the Universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams, evaluated at the product of all diagrams, which characterize particular ramification points of the covering.

Induced aggregation operators in decision making with the Dempster-Shafer belief structure

Abstract: In this study, we analyze the induced aggregation operators. The analysis begins with a revision of some basic concepts such as the induced ordered weighted averaging operator and the induced ordered weighted geometric operator. We then analyze the problem of decision making with Dempster-Shafer (D-S) theory of evidence. We suggest the use of induced aggregation operators in decision making with the D-S theory. We focus on the aggregation step and examine some of its main properties, including the distinction between descending and ascending orders and different families of induced operators. Finally, we present an illustrative example in which the results obtained with different types of aggregation operators can be seen. © 2009 Wiley Periodicals, Inc.

Compressed Imaging With a Separable Sensing Operator

Abstract: Compressive imaging (CI) is a natural branch of compressed sensing (CS). Although a number of CI implementations have started to appear, the design of efficient CI system still remains a challenging problem. One of the main difficulties in implementing CI is that it involves huge amounts of data, which has far-reaching implications for the complexity of the optical design, calibration, data storage and computational burden. In this paper, we solve these problems by using a two-dimensional separable sensing operator. By so doing, we reduce the complexity by factor of 106 for megapixel images. We show that applying this method requires only a reasonable amount of additional samples.

Hidden Grassmann Structure in the XXZ Model II: Creation Operators

Abstract: In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space \({\mathcal W_\alpha}\) of all quasi-local operators, which are products of the disorder field \({q^{\alpha \sum_{j=-\infty}^0\sigma ^3_j}}\) with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b*(ζ), c*(ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t*(ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b*(ζ), c*(ζ), t*(ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants.

Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point

Abstract: We extend the Rome-Southampton regularization independent momentum-subtraction renormalization scheme (RI/MOM) for bilinear operators to one with a nonexceptional, symmetric subtraction point. Two-point Green’s functions with the insertion of quark bilinear operators are computed with scalar, pseudoscalar, vector, axial-vector and tensor operators at one-loop order in perturbative QCD. We call this new scheme RI/SMOM, where the S stands for “symmetric.” Conversion factors are derived, which connect the RI/SMOM scheme and the MS? scheme and can be used to convert results obtained in lattice calculations into the MS? scheme. Such a symmetric subtraction point involves nonexceptional momenta implying a lattice calculation with substantially suppressed contamination from infrared effects. Further, we find that the size of the one-loop corrections for these infrared improved kinematics is substantially decreased in the case of the pseudoscalar and scalar operator, suggesting a much better behaved perturbative series. Therefore it should allow us to reduce the error in the determination of the quark mass appreciably.

Linearized model Fokker-Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests

Abstract: A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a model collision operator satisfying these properties is described. This operator is based on the exact linearized test-particle collision operator, with approximations to the field-particle terms that preserve conservation laws and an H-theorem. It includes energy diffusion, pitch-angle scattering, and finite Larmor radius effects corresponding to classical real-space diffusion. The numerical implementation in the continuum gyrokinetic code GS2 Kotschenreuther et al., Comput. Phys. Comm. 88, 128 1995 is fully implicit and guarantees exact satisfaction of conservation properties. Numerical results are presented showing that the correct physics is captured over the entire range of collisionalities, from the collisionless to the strongly collisional regimes, without recourse to artificial dissipation. © 2009 American Institute of Physics. DOI: 10.1063/1.3155085

Large gauge invariant nonstandard neutrino interactions

Abstract: Theories beyond the standard model must necessarily respect its gauge symmetry. This implies strict constraints on the possible models of nonstandard neutrino interactions, which we analyze. The focus is set on the effective low-energy dimension six and eight operators involving four leptons, decomposing them according to all possible tree-level mediators, as a guide for model building. The new couplings are required to have sizable strength, while processes involving four charged leptons are required to be suppressed. For nonstandard interactions in matter, only diagonal tau-neutrino interactions can escape these requirements and can be allowed to result from dimension six operators. Large nonstandard neutrino interactions from dimension eight operators alone are phenomenologically allowed in all flavor channels and are shown to require at least two new mediator particles. The new couplings must obey general cancellation conditions both at the dimension six and dimension eight levels, which result from expressing the operators obtained from the mediator analysis in terms of a complete basis of operators. We illustrate with one example how to apply this information to model building.

Elastic scaling of data parallel operators in stream processing

Abstract: We describe an approach to elastically scale the performance of a data analytics operator that is part of a streaming application. Our techniques focus on dynamically adjusting the amount of computation an operator can carry out in response to changes in incoming workload and the availability of processing cycles. We show that our elastic approach is beneficial in light of the dynamic aspects of streaming workloads and stream processing environments. Addressing another recent trend, we show the importance of our approach as a means to providing computational elasticity in multicore processor-based environments such that operators can automatically find their best operating point. Finally, we present experiments driven by synthetic workloads, showing the space where the optimizing efforts are most beneficial and a radioastronomy imaging application, where we observe substantial improvements in its performance-critical section.

Connected Operators : A review of region-based morphological image processing techniques

Abstract: Connected operators are filtering tools that act by merging elementary regions called flat zones. Connecting operators cannot create new contours nor modify their position. Therefore, they have very good contour preservation properties and are capable of both low-level filtering and higher-level object recognition. This article gives an overview on connected operators and their application to image and video filtering. There are two popular techniques used to create connected operators. The first one relies on a reconstruction process. The operator involves first a simplification step based on a "classical" filter and then a reconstruction process. In fact, the reconstruction can be seen as a way to create a connected version of an arbitrary operator. The simplification effect is defined and limited by the first step. The examples we show include simplification in terms of size or contrast. The second strategy to define connected operators relies on a hierarchical region-based representation of the input image, i.e., a tree, computed in an initial step. Then, the simplification is obtained by pruning the tree, and, third, the output image is constructed from the pruned tree. This article presents the most important trees that have been used to create connected operators and also discusses important families of simplification or pruning criteria. We also give a brief overview on efficient implementations of the reconstruction process and of tree construction. Finally, the possibility to define and to use nonclassical notions of connectivity is discussed and Illustrated.

The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras

Abstract: We study a twisted version of the Yang–Baxter equation, called the Hom–Yang–Baxter equation (HYBE), which is motivated by Hom–Lie algebras. Three classes of solutions of the HYBE are constructed, one from Hom–Lie algebras and the others from Drinfeld's (dual) quasi-triangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group.

Hidden Grassmann structure in the XXZ model III: introducing the Matsubara direction

Abstract: We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio ρ(ζ) of transfer matrix eigenvalues and a nearest neighbour correlator ω(ζ, ξ). We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integrals.

Right-handed neutrino magnetic moments

Abstract: We discuss the phenomenology of the most general effective Lagrangian, up to operators of dimension five, built with standard model fields and interactions including right-handed neutrinos. In particular, we find there is a dimension five electroweak moment operator of right-handed neutrinos, not discussed previously in the literature, which could have interesting phenomenological consequences.

Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Abstract: Spaces, Actions, Representations and Curvature.- Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory.- Paneitz Operator and Paneitz Curvature.- Intertwining Families.- Conformally Covariant Families.