Showing papers on "Operator (computer programming) published in 2004"

A unified treatment of some iterative algorithms in signal processing and image reconstruction

Abstract: Let T be a (possibly nonlinear) continuous operator on Hilbert space . If, for some starting vector x, the orbit sequence {Tkx,k = 0,1,...} converges, then the limit z is a fixed point of T; that is, Tz = z. An operator N on a Hilbert space is nonexpansive?(ne) if, for each x and y in , Even when N has fixed points the orbit sequence {Nkx} need not converge; consider the example N = ?I, where I denotes the identity operator. However, for any the iterative procedure defined by converges (weakly) to a fixed point of N whenever such points exist. This is the Krasnoselskii?Mann (KM) approach to finding fixed points of ne operators. A wide variety of iterative procedures used in signal processing and image reconstruction and elsewhere are special cases of the KM iterative procedure, for particular choices of the ne operator N. These include the Gerchberg?Papoulis method for bandlimited extrapolation, the SART algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms, simultaneous and sequential methods for solving the convex feasibility problem, the ART and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Dolidze's procedure for the variational inequality problem for monotone operators.

The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability

Abstract: In this work we review recent progress in four-dimensional conformal quantum field theories and scaling dimensions of local operators. Here we consider the example of maximally supersymmetric gauge theory and present techniques to derive, investigate and apply the dilatation operator which measures the scaling dimensions. We construct the dilatation operator by purely algebraic means: Relying on the symmetry algebra and structural properties of Feynman diagrams we are able to bypass involved, higher-loop field theory computations. In this way we obtain the complete one-loop dilatation operator and the planar, three-loop deformation in an interesting subsector. These results allow us to address the issue of integrability within a planar four-dimensional gauge theory: We prove that the complete dilatation generator is integrable at one loop and present the corresponding Bethe ansatz. We furthermore argue that integrability extends to three loops and beyond. Assuming that it holds indeed, we finally construct a novel spin chain model at five loops and propose a Bethe ansatz which might be valid at arbitrary loop order!

Generalized OWA Aggregation Operators

Abstract: We extend the ordered weighted averaging (OWA) operator to a provide a new class of operators called the generalized OWA (GOWA) operators. These operators add to the OWA operator an additional parameter controlling the power to which the argument values are raised. We look at some special cases of these operators. One important case corresponds to the generalized mean and another special case is the ordered weighted geometric operator.

The Factorized S-Matrix of CFT/AdS

Abstract: We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations. This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix. To illustrate the idea, we focus on the closed fermionic su(1|1) sector of the N=4 gauge theory. We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2|3) sector. We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix. Analyzing the available information, we find an intriguing relation between the three associated S-matrices. Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop. The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.

OWA aggregation over a continuous interval argument with applications to decision making

Abstract: We briefly describe the ordered weighted average (OWA) operator. We discuss its role in decision making under uncertainty. We provide an extension of the OWA operator to the case in which our argument is a continuous valued interval rather than a finite set of values. We look at some examples of this type of aggregation. We show how it can be used in some tasks that arise in decision making. We consider the extension of the continuous interval argument OWA operator to the more general case in which the argument values have importance weights. We use this to introduce the idea of an attitudinal-based expected value associated with a continuous random variable.

Eowa and eowg operators for aggregating linguistic labels based on linguistic preference relations

Abstract: In this paper, we define two types of linguistic preference relations (multiplicative linguistic preference relation and additive linguistic preference relation), and study some of their desirable properties. We introduce the extended geometric mean (EGM) operator, extended arithmetical averaging (EAA) operator, extended ordered weighted averaging (EOWA) operator and extended ordered weighted geometric (EOWG) operator. An approach based on the EGM and EOWG operators and multiplicative linguistic preference relations and an approach based on the EAA and EOWA operators and additive linguistic preference relations are proposed to ranking the alternatives in the group decision-making problems. Finally, we give a numerical example to illustrate the developed approaches.

A constraint-handling mechanism for particle swarm optimization

Abstract: This work presents a simple mechanism to handle constraints with a particle swarm optimization algorithm. Our proposal uses a simple criterion based on closeness of a particle to the feasible region in order to select a leader. Additionally, our algorithm incorporates a turbulence operator that improves the exploratory capabilities of our particle swarm optimization algorithm. Despite its relative simplicity, our comparison of results indicates that the proposed approach is highly competitive with respect to three constraint-handling techniques representative of the state-of-the-art in the area.

Higher-order equation-of-motion coupled-cluster methods.

Abstract: The equation-of-motion coupled-cluster (EOM-CC) methods truncated after double, triple, or quadruple cluster and linear excitation operators (EOM-CCSD, EOM-CCSDT, and EOM-CCSDTQ) have been derived and implemented into parallel execution programs. They compute excitation energies, excited-state dipole moments, and transition moments of closed- and open-shell systems, taking advantage of spin, spatial (real Abelian), and permutation symmetries simultaneously and fully (within the spin–orbital formalisms). The related Λ equation solvers for coupled-cluster (CC) methods through and up to connected quadruple excitation (CCSD, CCSDT, and CCSDTQ) have also been developed. These developments have been achieved, by virtue of the algebraic and symbolic manipulation program that automated the formula derivation and implementation altogether. The EOM-CC methods and CC Λ equations introduce a class of second quantized ansatz with a de-excitation operator (Ŷ), a number of excitation operators (X), and a physical (e.g....

Conformally invariant powers of the Laplacian — A complete nonexistence theorem

Abstract: Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observation dates back to at least the very early part of the last century when it was shown that the equations of massless particles on curved space-time exhibit conformai invariance. In this setting a key operator is the con formally invariant wave operator which has leading term a pseudo-Laplacian. The Riemannian signature variant of this operator is a fundamental tool in the Yam abe problem on compact manifolds. Here one seeks to find a metric, from a given conformai class, that has constant scalar curvature. Recently it has become clear that higher order analogues of these operators, viz., conformally invariant operators on weighted functions (i.e., conformai densities) with leading term a power of the Laplacian, have a central role in generating and solving other curvature prescription problems as well as other problems in geometric spectral theory and mathematical physics [2, 5, 15]. In the flat setting, the existence of such operators dates back to [16], where it is shown that, on 4-dimensional Minkowski space, for k G N = {1,2,...}, the kth power of the flat wave operator Ak, acting on densities of the appropriate weight, is invariant under the action of the conformai group. More generally, if ?[w] denotes the space of conformai densities of weight uiGl, then on a flat conformai manifold of dimension n > 3 (and any signature) there exists, for each k E N, a unique conformally invariant operator

Second order periodic differential operators. Threshold properties and homogenization

Abstract: The vector periodic differential operators (DO’s) A admitting a factorization A = X ∗X , where X is a first order homogeneous DO, are considered in L2(R). Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral expansion of A in a small neighborhood of zero are called threshold effects at the point λ = 0. An example of a threshold effect is the behavior of a DO in the small period limit (the homogenization effect). Another example is related to the negative discrete spectrum of the operator A−αV , α > 0, where V (x) ≥ 0 and V (x) → 0 as |x| → ∞. “Effective characteristics”, such as the homogenized medium, effective mass, effective Hamiltonian, etc., arise in these problems. The general approach to these problems proposed in this paper is based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. Most of the arguments are carried out in abstract terms. As to applications, the main attention is paid to homogenization of DO’s.

Approximations of effective coefficients in stochastic homogenization

Abstract: This note deals with localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of “periodization” and other “cut-off” procedures. For instance in the case of periodic approximation, we consider a cubic sample [0 ,ρ ] d of the random medium, extend it periodically in R d and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that this approximation converges a.s., as ρ →∞ , and gives back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of ρ which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample [0 ,ρ ] d .

Discrete Laplace--Beltrami operators and their convergence

Abstract: The convergence property of the discrete Laplace-Beltrami operators is the foundation of convergence analysis of the numerical simulation process of some geometric partial differential equations which involve the operator. In this paper we propose several simple discretization schemes of Laplace-Beltrami operators over triangulated surfaces. Convergence results for these discrete Laplace-Beltrami operators are established under various conditions. Numerical results that support the theoretical analysis are given. Application examples of the proposed discrete Laplace-Beltrami operators in surface processing and modelling are also presented.

Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations

Abstract: We consider AdS_5 x S^5 string states with several large angular momenta along AdS_5 and S^5 directions which are dual to single-trace Super-Yang-Mills (SYM) operators built out of chiral combinations of scalars and covariant derivatives. In particular, we focus on the SU(3) sector (with three spins in S^5) and the SL(2) sector (with one spin in AdS_5 and one in S^5), generalizing recent work hep-th/0311203 and hep-th/0403120 on the SU(2) sector with two spins in S^5. We show that, in the large spin limit and at the leading order in the effective coupling expansion, the string sigma model equations of motion reduce to matrix Landau-Lifshitz equations. We then demonstrate that the coherent-state expectation value of the one-loop SYM dilatation operator restricted to the corresponding sector of single trace operators is also effectively described by the same equations. This implies a universal leading order equivalence between string energies and SYM anomalous dimensions, as well as a matching of integrable structures. We also discuss the more general 5-spin sector and comment on SO(6) states dual to non-chiral scalar operators.

Riesz transform, Gaussian bounds and the method of wave equation

Abstract: For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL−α on Lp for some α > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature.

Semiclassical relativistic strings in S 5 and long coherent operators in N=4 SYM theory

Abstract: We consider the low energy effective action corresponding to the 1-loop, planar, dilatation operator in the scalar sector of = 4 SU(N) SYM theory. For a general class of non-holomorphic ``long'' operators, of bare dimension L >> 1, it is a sigma model action with 8-dimensional target space and agrees with a limit of the phase-space string sigma model action describing generic fast-moving strings in the S 5 part of AdS 5 × S 5 . The limit of the string action is taken in a way that allows for a systematic expansion to higher orders in the effective coupling = λ/L 2. This extends previous work on rigid rotating strings in S 5 (dual to operators in the SU(3) sector of the dilatation operator) to the case when string oscillations or pulsations in S 5 are allowed. We establish a map between the profile of the leading order string solution and the structure of the corresponding coherent, ``locally BPS'', SYM scalar operator. As an application, we explicitly determine the form of the non-holomorphic operators dual to the pulsating strings. Using action-angle variables, we also directly compute the energy of pulsating solutions, simplifying previous treatments.

The generalization of the extended Stevens operators to higher ranks and spins, and a systematic review of the tables of the tensor operators and their matrix elements

Abstract: Spherical?(S) and tesseral?(T) tensor operators?(TOs) have been extensively used in, for example, EMR and optical spectroscopy of transition ions. To enable a systematic review of the published tables of the operators and their matrix elements?(MEs) we have generated the relevant tables by computer, using Mathematica programs. Our review reveals several misprints/errors in the major sources of TTOs?the conventional Stevens operators (CSOs?components ) and the extended ones (ESOs?all q) for rank k = 2,4, and?6?as well as of some STOs with . The implications of using incorrect operators and/or MEs for the reliability of EMR-related programs and interpretation of experimental data are discussed. Studies of high-spin complexes like Mn12 (S = 10) and Fe19 (S = 33/2) require operator and ME listings up to k = 2S, which are not presently available. Using the algorithms developed recently by Ryabov, the generalized ESOs and their MEs for arbitrary rank k and spin S are generated by computer, using Mathematica. The extended tables enable simulation of the energy levels for arbitrary spin systems and symmetry cases. Tables are provided for the ESOs not available in the literature, with odd k = 3,5, and?7 for completeness; however, for the newly generalized ESOs with the most useful even rank k = 8,10, and?12 only, in view of the large listings sizes. General source codes for the generation of the ESO listings and their ME tables are available from the authors.

Harnack inequalities for non-local operators of variable order

Abstract: We consider harmonic functions with respect to the operator formula math Under suitable conditions on n(x,h) we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator L is allowed to be anisotropic and of variable order.

Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations

Abstract: In this article, we introduce the induced ordered weighted geometric (IOWG) operator and its properties. This is a more general type of OWG operator, which is based on the induced ordered weighted averaging (IOWA) operator. We provide some IOWG operators to aggregate multiplicative preference relations in group decision-making (GDM) problems. In particular, we present the importance IOWG (I-IOWG) operator, which induces the ordering of the argument values based on the importance of the information sources; the consistency IOWG (C-IOWG) operator, which induces the ordering of the argument values based on the consistency of the information sources; and the preference IOWG (P-IOWG) operator, which induces the ordering of the argument values based on the relative preference values associated with each one of them. We also provide a procedure to deal with “ties” regarding the ordering induced by the application of one of these IOWG operators. This procedure consists of a sequential application of the aforementioned IOWG operators. Finally, we analyze the reciprocity and consistency properties of the collective multiplicative preference relations obtained using IOWG operators. © 2004 Wiley Periodicals, Inc.

Exploring the Capability of Immune Algorithms: A Characterization of Hypermutation Operators

Abstract: In this paper, an important class of hypermutation operators are discussed and quantitatively compared with respect to their success rate and computational cost We use a standard Immune Algorithm (IA), based on the clonal selection principle to investigate the searching capability of the designed hypermutation operators We computed the parameter surface for each variation operator to predict the best parameter setting for each operator and their combination The experimental investigation in which we use a standard clonal selection algorithm with different hypermutation operators on a complex “toy problem”, the trap functions, and a complex NP-complete problem, the 2D HP model for the protein structure prediction problem, clarifies that only few really different and useful hypermutation operators exist, namely: inversely proportional hypermutation, static hypermutation and hypermacromutation operators The combination of static and inversely proportional Hypermutation and hypermacromutation showed the best experimental results for the “toy problem” and the NP-complete problem

How topological partitions of the electron distributions reveal delocalization

Abstract: The topological partitions of the electron distributions are based on the gradient field analysis of local functions which carry the relevant physical or chemical information. They yield a set of contiguous non-overlapping volumes called basins which entirely span the geometrical space. The integral of the charge density distribution over a given basin, say ΩA, is the basin population ΩA which is the expectation value of the population operator ΩA. It is shown that the basin population operators are correlated and I propose to study this correlation with the help of the covariance matrix of the basin populations. A covariance operator is derived accordingly. This method is applied to the basin populations calculated in the AIM and ELF topological approaches in order to provide a tool enabling one to discuss the reliability of simplified representations of electron densities in terms of superposition of promolecular densities or of resonant Lewis structures.

Yang-Mills Correlation Functions from Integrable Spin Chains

Abstract: The relation between the dilatation operator of = 4 Yang-Mills theory and integrable spin chains makes it possible to compute the one-loop anomalous dimensions of all operators in the theory. In this paper we show how to apply the technology of integrable spin chains to the calculation of Yang-Mills correlation functions by expressing them in terms of matrix elements of spin operators on the corresponding spin chain. We illustrate this method with several examples in the SU(2) sector described by the XXX1/2 chain.