Showing papers on "Operator (computer programming) published in 1992"

On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators

Abstract: This paper shows, by means of an operator called asplitting operator, that the Douglas--Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas--Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

Scale and the differential structure of images

Abstract: Why and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a ‘scale-space representation’, in which a given image is represented by a one-dimensional family of images representing that image on various levels of inner spatial scale. An early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family of scaled partial differential operators: the Gaussian kernel (the lowest order, resettling operator) and its linear partial derivatives. They enable local image analysis through the detection of local differential structure in a robust way, while at the same time capturing global features through the extra scale degree of freedom. In this paper we show why the operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale, cartesian differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles found in mammalian frontend visual systems.

Wide-angle beam propagation using Pade approximant operators.

Abstract: A new beam-propagation method is presented whereby the exact scalar Helmholtz propagation operator is replaced by any one of a sequence of higher-order (n, n) Pade approximant operators. The resulting differential equation may then be discretized to obtain (in two dimensions) a matrix equation of bandwidth 2n + 1 that is solvable by using Standard implicit solution techniques. The final algorithm allows (for n = 2) accurate propagation at angles of greater than 55 deg from the propagation axis as well as propagation through materials with widely differing indices of refraction.

Topological lattice models in four-dimensions

Abstract: We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group G. When G=SU(2), the statistical weight is constructed from the 15j-symbol as well as the 6j-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called BF model. The q-analog of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the q-deformed version of the model would define a new type of invariants of knots and links in four dimensions.

High-speed digital signal processing and control

Abstract: An attempt is made to organize and survey recent work, and to present it in a unified and accessible form. The need for a new approach suitable for high-speed processing is discussed in the context of several applications in control and communications, and a historical perspective of the use of difference operators in numerical analysis is presented. The general systems calculus, based on divided-different operators is introduced to unify the continuous-time and discrete-time systems theories. This calculus is then used as a framework to treat the three problems of system state estimation; system identification and time-series modeling; and control system design. Realization aspects of algorithms based on the difference operator representation, including such issues as coefficient rounding and implementation with standard hardware, are also discussed. >

Applications and extensions of OWA aggregations

Abstract: We discuss the idea of ordered weighting averaging (OWA) operators. These operators provide a family of aggregation operators lying between the “and” and the “or”. We introduce two possible semantics associated with the OWA operator, the first being a kind of generalized logical connective and the second being a new type of probabilistic expected value. We suggest some applications of these operators. Among the applications we discuss are those involving multicriteria decision making under uncertainty and search procedures in games. We provide for a formulation of OWA operators that can be used in environments in which the underlying scale is simply an ordinal one.

Neumann and mixed problems on curvilinear polyhedra

Abstract: We prove regularity results inL p Sobolev spaces. On one hand, we state some abstract results byL p functional techniques: exponentially decreasing estimates in dyadic partitions of cones and dihedra, operator valued symbols and Marcinkievicz's theorem. On the other hand, we derive more concrete statements with the help of estimates about the first non-zero eigenvalue of some Laplace-Beltrami operators on spherical domains.

Reconstruction techniques for classical inverse Sturm-Liouville problems

Abstract: This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

A Mathematica package for computing operator product expansions

Abstract: A general purpose Mathematica™ package for computing Operator Product Expansions of composite operators in meromorphic conformal field theory is described. Given the OPEs for a set of “basic” fields, OPEs of arbitrarily complicated composites can be computed automatically. Normal ordered products are always reduced to a standard form. Two explicit examples are presented: the conformal anomaly for superstrings and a free field realization for the $\widehat{B_2}$ Kac-Moody-algebra.

Operator approach to performance analysis of root-MUSIC and root-min-norm

Abstract: The authors carry out a performance analysis of two eigenstructure-based direction-of-arrival estimation algorithms, using a series expansion of projection operators (or projectors) on the signal and noise subspaces. In the interest of algebraic simplicity, an operator formalism is utilized. A perturbation analysis is performed on the projectors, the results of which are used to determine the effect on the estimated parameters. The approach makes it possible to carry out the analysis to any chosen order of expansion of the projectors by using an original recurrence formula developed for the higher-order terms in the series expansion of the projectors. This method is used to study the root-MUSIC and root-min-norm algorithms and establish the superiority of root-MUSIC in all cases. The analysis has also resulted in insightful asymptotic expressions that describe the statistical behavior of the estimated angles and radii of the signal zeros. >

Spatial Operator Algebra for multibody system dynamics

Abstract: The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.

Non-determinism in functional languages

Abstract: The introduction of a non-deterministic operator in even a very simple functional programming language gives rise to a plethora of semantic questions. These questions are not only concerned with the choice operator itself. A surprisingly large number of different parameter passing mechanisms are made possible by the introduction of bounded non-determinism. The diversity of semantic possibilities is examined systematically using denotational definitions based on mathematical structures called power domains

A null operator analysis of atb gaps

Abstract: Proposition d'une configuration syntaxique faisant intervenir les notions de trou parasite et d'element nul, au lieu d'une transformation de mouvement, pour l'analyse des structures coordonnees. Exemples en anglais

Unitarity and irreversibility in chaotic systems

Abstract: We analyze the spectral properties of the Perron-Frobenius operator [ital U], associated with some simple highly chaotic maps. We obtain a spectral decomposition of [ital U] in terms of generalized eigenfunctions of [ital U] and its adjoint. The corresponding eigenvalues are related to the decay rates of correlation functions and have magnitude less than one, so that physically measurable quantities manifestly approach equilibrium. To obtain decaying eigenstates of unitary and isometric operators it is necessary to extend the Hilbert-space formulation of dynamical systems. We describe and illustrate a method to obtain the decomposition explicitly.

Wheel alignment system

Abstract: The disclosure herein relates to an alignment system and method which provides improved accuracy measurement of vehicle wheel alignment characteristics, guidance for an operator through known yet obscure adjustment procedures, an interface between system alignment sensors and the system controller which provides efficient handling of system generated data and system inputs and outputs, and live measurements and display of all alignment quantities so that interdependent changes between adjustable alignment quantities are observable.

Recursive flexible multibody system dynamics using spatial operators

Abstract: This paper uses spatial operators to develop new spatially recursive dynamics algorithms for flexible multibody systems. The operator description of the dynamics is identical to that for rigid multibody systems. Assumed-mode models are used for the deformation of each individual body. The algorithms are based on two spatial operator factorizations of the system mass matrix. The first (Newton-Euler) factorization of the mass matrix leads to recursive algorithms for the inverse dynamics, mass matrix evaluation, and composite-body forward dynamics for the systems. The second (innovations) factorization of the mass matrix, leads to an operator expression for the mass matrix inverse and to a recursive articulated-body forward dynamics algorithm. The primary focus is on serial chains, but extensions to general topologies are also described. A comparison of computational costs shows that the articulated-body, forward dynamics algorithm is much more efficient than the composite-body algorithm for most flexible multibody systems.

The reconstruction of Sturm-Liouville operators

Abstract: The authors give a constructive algorithm for the inverse Sturm-Liouville operator in non-potential form. They emphasize the unknown impedance case, i.e. the recovery of the coefficient a(x) in (a(x)u'(x))'+ lambda a(x)u(x)=0 from spectral data. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. An iterative procedure for solving this latter problem is developed and numerical examples are presented. In particular, it will be shown that this iteration scheme will converge if log a is Lipschitz continuous, and numerical experiments indicate that the approach can be used to reconstruct impedances in a much wider class.

The interior Radon transform

Abstract: The interior Radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured. This problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the X-ray dose. The corresponding operator $\mathcal{R}_I $ is investigated as a mapping between weighted $L_2 $-spaces. The main result is a singular value decomposition (SVD) for this operator for functions of unbounded support in $\mathbb{R}^2 $. The proof is based on the construction of intertwining differential operators. The techniques used are unified in the sense that SVDs for other Radon transforms with rotational symmetry can easily be derived in the same way. Consistency conditions are also obtained for $\mathcal{R}_I $ for functions of bounded and unbounded support. Many of the results generalize to higher dimensions; they are stated whenever they follow directly from the two-dimensional case.

The algebraic structures of isospectral Lax operators and applications to integrable equations

Abstract: For a general spectral operator, the author establishes types of algebraic structures of the spaces of the corresponding isospectral Lax operators, which essentially form the theoretical basis of the Lax operator method. Furthermore the author introduces the concepts of tau -algebras and master algebras to describe time-polynomial-dependent symmetries of nonlinear integrable equations. Finally the author applies the theory of Lax operators to the KP hierarchy of integrable equations as an illustrative example, and thus obtain the master symmetry algebra of the KP hierarchy.

Common theoretical framework for quantum chemical solvent effect theories

Abstract: Quantum chemical solvent effect theories deal with the description of the electronic structure of a molecular subsystem embedded in a solvent or other molecular environment. The average reaction field theories, which describe electrostatic and polarization interactions between solute and solvent, can be formulated in terms of a nonlinear reaction potential operator. This operator depends on the one hand on the reaction potential function of the solvent, and on the other hand on the charge density operators, which appear in the solute-solvent interaction. The former quantity is determined by the physical model of the solvent (e.g. dielectric continuum, discrete model, crystal lattice, etc.). The charge density operator can be approximated at different levels, like exact, one-centered and multicentered multipolar forms. These two ingredients of the theory, the reaction potential response function and the specific charge density operator, define unequivocally different solvent effect models. Various versions of average reaction field models are critically reviewed on the basis of this common theoretical framework.

An approach for constructing nonisospectral hierarchies of evolution equations

Abstract: For a given isospectral ( lambda t=0) hierarchy of evolution equations, the author proposes a simple method of constructing its corresponding nonisospectral ( lambda t= lambda n,n>or=0) hierarchy of evolution equations closely related to tau -symmetries. It is crucial to find an initial Lax operator W0 and an initial vector field g0 satisfying the key equation (W0,L)=L'(g0)-I, in which L,I are spectral and identity operators, respectively. As examples, the author presents the corresponding nonisospectral hierarchies of equations and displays the fundamental relations generating symmetry algebras for the KdV hierarchy, the AKNS hierarchy and a new integrable hierarchy.

Generative modeling: a symbolic system for geometric modeling

Abstract: This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higher-dimensioned shapes such as surfaces deforming in time, and volumes with a spatially varying mass density. The system also supports powerful operations on shapes such as “reparameterize this curve by arclength”, “compute the volume, center of mass, and moments of inertia of the solid bounded by these surfaces”, or “solve this constraint or ODE system”. The system has been used for a wide variety of applications, including creating surfaces for computer graphics animations, modeling the fur and body shape of a teddy bear, constructing 3D solid models of elastic bodies, and extracting surfaces from magnetic resonance (MR) data. Shapes in the system are specified using a language which builds multidimensional parametric functions. The language is baaed on a set of symbolic operators on continuous, piecewise differentiable parametric functions. We present several shape examples to show bow conveniently shapes can be specified in the system. We also discuss the kinds of operators useful in a geometric modeling system, including arithmetic operators, vector and matrix operators, integration, differentiation, constraint solution, and constrained minimisation. Associated with each operator are several methods, which compute properties about the parametric functions represented with the operators. We show how many powerful rendering and analytical operations can be supported with only three methods: evaluation of the parametric function at a point, symbolic dlfferentiation of the parametric function, and evacuation of an inclusion function for the parametric function. Like CSG, and unlike most other geometric modeling approaches, 3Ms modeling approach is closed, meaning that further modeling operations cart be applied to any results of modeling operations, yielding valid models. Because of this closure property, the symbolic operators can be composed very flexibly, allowing the construction of higher-level operators without changing the underlying implementation of the system. Because the modeling operations are described symbolically, specified models can capture the designer’s intent without approximation error.

Hardy Space Estimates for Multilinear Operators, II

Abstract: We continue the study of multilinear operators given by products of finite vectors of Calderon-Zygmund operators. We determine the set of all r = 1 for which these operators map products of Lebesgue spaces Lp(Rn) into the Hardy spaces Hr(Rn). At the endpoint case r = n/(n + m + 1), where m is the highest vanishing moment of the multilinear operator, we prove a weak type result.

Fuzzy operator for sharpening of noisy images

Abstract: An operator is presented which is able to sharpen the details of an image, by applying fuzzy reasoning rules to the input luminance values. The particular processing rules used allow the operator to be insensitive to noise.

The Multicritical Kontsevich-Penner Model

Abstract: We consider the Hermitian matrix model with an external field entering the quadratic term tr(ΛXΛX) and Penner-like interaction term αN(log(1+X)-X). An explicit solution in the leading order in N is presented. The critical behavior is given by the second derivative of the free energy in α which appears to be a pure logarithm, that is a feature of c=1 theories. Various critical regimes are possible, some of them corresponds to critical points of the usual Penner model, but there exists an infinite set of multicritical points which differ by values of scaling dimensions of proper conformal operators. Their correlators with the puncture operator are given in genus zero by Legendre polynomials whose argument is determined by an analog of the string equation.