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101
REDLOG Computer Algebra Meets Computer Logic
 ACM SIGSAM Bulletin
, 1996
"... . redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know a ..."
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Cited by 105 (30 self)
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. redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know about the underlying algebraic theory and make use of it. Though the focus is on simplification, parametric linear optimization, and quantifier elimination, redlog is designed as a generalpurpose system. We describe the functionality of redlog as it appears to the user, and explain the design issues and implementation techniques. ? The second author was supported by the dfg (Schwerpunktprogramm: Algorithmische Zahlentheorie und Algebra) 1 Introduction redlog stands for reduce logic system. It provides an extension of the computer algebra system (cas) reduce to a computer logic system (cls) implementing symbolic algorithms on firstorder formulas w.r.t. temporarily fixed firstorder languag...
On the spheredecoding algorithm I. Expected complexity
 IEEE Trans. Sig. Proc
, 2005
"... Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The ..."
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Cited by 76 (5 self)
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Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NPhard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worstcase complexity of the integer leastsquares problem, we study its expected complexity, averaged over the noise and over the lattice. For the “sphere decoding” algorithm of Fincke and Pohst, we find a closedform expression for the expected complexity, both for the infinite and finite lattice.
Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints
 THE JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
, 2004
"... Regularization of illposed linear inverse problems via ℓ1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ1 penalized functional is via an iterative softthresholding algorithm. We propose an alternative implem ..."
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Cited by 58 (10 self)
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Regularization of illposed linear inverse problems via ℓ1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ1 penalized functional is via an iterative softthresholding algorithm. We propose an alternative implementation to ℓ1constraints, using a gradient method, with projection on ℓ1balls. The corresponding algorithm uses again iterative softthresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations
, 2006
"... An efficient, highorder, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve highcomputational efficiency and geometric flexibility; it ..."
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Cited by 35 (24 self)
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An efficient, highorder, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve highcomputational efficiency and geometric flexibility; it utilizes the concept of discontinuous and highorder local representations to achieve conservation and high accuracy; and it is based on the finitedifference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids. KEY WORDS: Highorder; conservation laws; unstructured grids; spectral difference; spectral collocation method; Euler equations.
User interface design with matrix algebra
 ACM Transactions on CHI
, 2004
"... It is usually very hard, both for designers and users, to reason reliably about user interfaces. This article shows that ‘push button ’ and ‘point and click ’ user interfaces are algebraic structures. Users effectively do algebra when they interact, and therefore we can be precise about some importa ..."
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Cited by 22 (11 self)
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It is usually very hard, both for designers and users, to reason reliably about user interfaces. This article shows that ‘push button ’ and ‘point and click ’ user interfaces are algebraic structures. Users effectively do algebra when they interact, and therefore we can be precise about some important design issues and issues of usability. Matrix algebra, in particular, is useful for explicit calculation and for proof of various user interface properties. With matrix algebra, we are able to undertake with ease unusally thorough reviews of real user interfaces: this article examines a mobile phone, a handheld calculator and a digital multimeter as case studies, and draws general conclusions about the approach and its relevance to design.
Existence and Stability of Standing Pulses in Neural Networks
 I. Existence. SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse soluti ..."
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Cited by 15 (1 self)
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Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse solutions coexist, the small pulse is unstable, and the large pulse is stable. The large single pulse is bistable with the “alloff ” state. This bistable localized activity may have strong implications for the mechanism underlying working memory. We show that dimple pulses have similar stability properties to large pulses but double pulses are unstable.
Two likelihoodbased semiparametric estimation methods for panel count data with covariates
, 2005
"... We consider estimation in a particular semiparametric regression model for the mean of a counting process with “panel count ” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{N(t)Z} = exp(β T 0 Z)Λ0(t) where Z is a vector of covariates ..."
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Cited by 15 (7 self)
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We consider estimation in a particular semiparametric regression model for the mean of a counting process with “panel count ” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{N(t)Z} = exp(β T 0 Z)Λ0(t) where Z is a vector of covariates and Λ0 is the baseline mean function. The “panel count ” observation scheme involves observation of the counting process N for an individual at a random number K of random time points; both the number and the locations of these time points may differ across individuals. We study semiparametric maximum pseudolikelihood and maximum likelihood estimators of the unknown parameters (β0,Λ0) derived on the basis of a nonhomogeneous Poisson process assumption. The pseudolikelihood estimator is fairly easy to compute, while the maximum likelihood estimator poses more challenges from the computational perspective. We study asymptotic properties of both estimators assuming that the proportional mean model holds, but dropping the Poisson process assumption used to derive the estimators. In particular we establish asymptotic normality for the estimators of the regression parameter β0 under appropriate hypotheses. The results show that our estimation procedures are robust in the sense that the estimators converge to the truth regardless of the underlying counting process.
WBCSim: A Prototype Problem Solving Environment for WoodBased Composites Simulations
, 1999
"... This paper describes a computing environment named WBCSim that is intended to increase the productivity of wood scientists conducting research on woodbased composite materials. WBCSim integrates Fortran 77based simulation codes with a graphical front end, an optimization tool, and a visualization ..."
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Cited by 13 (8 self)
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This paper describes a computing environment named WBCSim that is intended to increase the productivity of wood scientists conducting research on woodbased composite materials. WBCSim integrates Fortran 77based simulation codes with a graphical front end, an optimization tool, and a visualization tool. WBCSim serves as a prototype for the design, construction, and evaluation of larger scale problem solving (computing) environments. Several different woodbased composite material simulations are supported. A detailed description of the prototype's software architecture and a typical scenario of use are presented. The system converts output from the simulations to the Virtual Reality Modeling Language (VRML) for visualizing simulation results.