ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 06 Feb 2023 23:02:39 GMT2023-02-06T23:02:39Z50231- Minimal sequences and the Kadison-Singer problemhttps://scholarbank.nus.edu.sg/handle/10635/103550Title: Minimal sequences and the Kadison-Singer problem
Authors: Lawton, W.
Abstract: The Kadison-Singer problem asks: does every pure state on the C*-algebra ℓ∞ (Z) admit a unique extension to the C*-algebra β (ℓ2 (Z))? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We prove that for measurable S ⊂ T, {χS e2πikt}k∈Z is a finite union of Riesz sequences in L2 (T) if and only if there exists a nonempty A ⊂ Z such that χA is a minimal sequence and {χS e2πikt} k∈A is a Riesz sequence. We also suggest some directions for future research.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035502010-01-01T00:00:00Z
- Matrix completion problems in multidimensional systemshttps://scholarbank.nus.edu.sg/handle/10635/53023Title: Matrix completion problems in multidimensional systems
Authors: Lawton, Wayne M.; Lin, Zhiping
Abstract: Hermite rings and matrix completions are closely related to factorizations of polynomial matrices. A ring with identity is Hermite if every unimodular row vector over the ring can be completed to form a unimodular square matrix over the ring. Examples and counterexamples of Hermite rings are constructed and several open problems which have potential applications in multidimensional systems are formulated.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/530231999-01-01T00:00:00Z
- Matrix completion problems in multidimensional systemshttps://scholarbank.nus.edu.sg/handle/10635/53024Title: Matrix completion problems in multidimensional systems
Authors: Lawton, Wayne M.; Lin, Zhiping
Abstract: Hermite rings and matrix completions are closely related to factorizations of polynomial matrices. A ring is Hermite if every unimodular vector over the ring can be completed to form a unimodular matrix over the ring. Examples and counterexamples of Hermite rings are constructed and several open problems which have potential applications in multidimensional systems are formulated.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/530241999-01-01T00:00:00Z
- Proteinmorphosis: a mechanical model for protein conformational changes.https://scholarbank.nus.edu.sg/handle/10635/113234Title: Proteinmorphosis: a mechanical model for protein conformational changes.
Authors: Meiyappan, S.; Raghavan, R.; Viswanathan, R.; Yu, Y.; Lawton, W.
Abstract: Proteinmorphosis is a physically-based interactive modeling system for simulating large or small conformational changes of proteins and protein complexes. It takes advantage of the cross-linked one-dimensional nature of protein chains. The user can, based on her chemical knowledge, pull pairs of points (lying either on a single protein or on different molecules) together by specifying geometric distance constraints. The resulting conformation(s) of the molecule(s) of interest is computed by an efficient finite element formalism taking into account elasticity of the protein backbone, van der Waals repulsions, hydrogen bonds, salt bridges and the imposed distance constraints. The conformational change is computed incrementally and the result can be visualized as an animation; complete interactivity is provided to position and view the proteins as desired by the user. Physical properties of regions on the protein can also be chosen interactively. The conformational change of calmodulin upon peptide binding is examined as a first experiment. It is found that the result is satisfactory in reproducing the conformational change that follows on peptide binding. We use Proteinmorphosis to study the cooperative hemoglobin oxygen binding mechanism in a second, more sophisticated, experiment. Different modeling strategies are designed to understand the allosteric (cooperative) binding process in this system and the results are found to be consistent with existing hypotheses.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1132341999-01-01T00:00:00Z
- Tubes in tubes: Catheter navigation in blood vessels and its applicationshttps://scholarbank.nus.edu.sg/handle/10635/113236Title: Tubes in tubes: Catheter navigation in blood vessels and its applications
Authors: Lawton, W.; Raghavan, R.; Ranjan, S.R.; Viswanathan, R.R.
Abstract: The construction of realistic simulators for medical procedures is increasingly important. We describe here a physical model and a numerical algorithm to simulate the insertion and navigation of a catheter into an arterial system. A novel formulation of the elasticity of thin rods was developed for modeling the catheter. The catheter bends and twists within the blood vessels, not simply tracking a central curve. The inner artery walls are modeled as rigid surfaces; this constraint of catheter containment within rigid walls is implemented through the use of a wall potential. The model has been integrated into an interactive system, with visualization and a direct catheter input interface (previously described), called da Vinci (Visual navigation of catheter insertion). © 2000 Elsevier Science Ltd. All rights reserved.
Mon, 01 May 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1132362000-05-01T00:00:00Z
- Analytic signals and radar processinghttps://scholarbank.nus.edu.sg/handle/10635/102861Title: Analytic signals and radar processing
Authors: Lawton, Wayne M.
Abstract: The analytic signal representation introduced by Ville has found extensive use in matched filter processing and in time/frequency analysis. We discuss the role of the analytic signal representation in radar signal processing. In particular we analyze errors associated with this representation using quantized calculus and wavelets. We also discuss the classical problem of ambiguity synthesis, both narrowband and wideband, in the context of the analytic signal representation and highlight the importance of the upper half plane and its group of conformal transformations for wideband synthesis.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028611999-01-01T00:00:00Z
- Infinite convolution products and refinable distributions on lie groupshttps://scholarbank.nus.edu.sg/handle/10635/103420Title: Infinite convolution products and refinable distributions on lie groups
Authors: Lawton, W.
Abstract: Sufficient conditions for the convergence in distribution of an infinite convolution product μ1 * μ2 * ... of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a refinement operator constructed from a measure n and a dilation automorphism A. The existence of A implies G is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore', there exists a unique minimal compact subset K. ⊂ G such that for any open set U containing K., and for any distribution f on Q with compact support, there exists an integer n(U, f) such that n ≥ n(U, f) implies supp(Tnf) C U. If μ is supported on an A-invariant uniform subgroup Γ, then T is related, by an intertwining operator, to a transition operator W on C(Γ). Necessary and sufficient conditions for Tn f to converge to φ ∈ L2, and for the Γ-translates of φ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of W to functions supported on Ω := KK-1 ⊂ Γ. ©2000 American Mathematical Society.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034202000-01-01T00:00:00Z
- Construction of conjugate quadrature filters with specified zeroshttps://scholarbank.nus.edu.sg/handle/10635/115043Title: Construction of conjugate quadrature filters with specified zeros
Authors: Lawton, W.; Micchelli, C.A.
Abstract: Let ℂ denote the complex numbers and ℒ denote the ring of complex-valued Laurent polynomial functions on ℂ \ {0}. Furthermore, we denote by ℒR, ℒN the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials P1, P2 ∈ ℒN, which have no common zeros in ℂ \ {0} there exists a pair of Laurent polynomials Q1, Q2 ∈ ℒN satisfying the equation Q1P1 + Q2P2 = 1. We provide some information about the minimal length Laurent polynomials Q1 and Q2 with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ ⊂ ℂ \ {0} with the property that for every λ, μ ∈ Λ, λ ≠ μ implies λ ≠ -μ and λ ≠ -1/μ̄.
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1150431997-01-01T00:00:00Z
- A Fast Algorithm to Map Functions Forwardhttps://scholarbank.nus.edu.sg/handle/10635/111126Title: A Fast Algorithm to Map Functions Forward
Authors: Lawton, W.
Abstract: Mapping functions forward is required in image warping and other signal processing applications. The problem is described as follows: specify an integer d ≥ 1, a compact domain D ⊂ Rd, lattices L1, L2 ⊂ Rd, and a deformation function F : D → Rd that is continuously differentiable and maps D one-to-one onto F(D). Corresponding to a function J : F(D) → R, define the function I = J ○ F. The forward mapping problem consists of estimating values of J on L2 ∩ F(D), from the values of I and F on L1 ∩ D. Forward mapping is difficult, because it involves approximation from scattered data (values of I ○ F-1 on the set F(L1 ∩ D)), whereas backward mapping (computing I from J) is much easier because it involves approximation from regular data (values of J on L2 ∩ D). We develop a fast algorithm that approximates J by an orthonormal expansion, using scaling functions related to Daubechies wavelet bases. Two techniques for approximating the expansion coefficients are described and numerical results for a one dimensional problem are used to illustrate the second technique. In contrast to conventional scattered data interpolation algorithms, the complexity of our algorithm is linear in the number of samples.
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1111261997-01-01T00:00:00Z
- Characterization of compactly supported refinable splineshttps://scholarbank.nus.edu.sg/handle/10635/111150Title: Characterization of compactly supported refinable splines
Authors: Lawton, W.; Lee, S.L.; Shen, Z.
Abstract: We prove that a compactly supported spline function φ of degree k satisfies the scaling equation {Mathematical expression} for some integer m ≥ 2, if and only if {Mathematical expression} where p(n) are the coefficients of a polynomial P(z) such that the roots of P(z)(z - 1)k+1 TM are mapped into themselves by the mapping z →zm, and Bk is the uniform B-spline of degree k. Furthermore, the shifts of φ form a Riesz basis if and only if P is a monomial. © 1995 J.C. Baltzer AG, Science Publishers.
Sun, 01 Jan 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1111501995-01-01T00:00:00Z
- Information theory, wavelets, and image compressionhttps://scholarbank.nus.edu.sg/handle/10635/111184Title: Information theory, wavelets, and image compression
Authors: Lawton, W.
Abstract: We examine practical, theoretical, and speculative aspects of wavelet transform-based image compression. Section I summarizes objectives and compares experimental results using a JPEG-standard cosine-based algorithm with a wavelet based algorithm developed at ISS. Section II analyzes image compression requirements using information theory to explain why wavelet transform-based image compression works well. The wavelet transform is shown to be a simple transform that effectively exploits second-order image statistics. Section III speculates about next-generation image compression and pattern recognition. It outlines a research plan to develop a probabilistic image model that incorporates higher-order image statistics by using wavelet expansions to provide a convergent series of finite dimensional marginal image probability densities. Physicists have successfully used similar cell cluster expansions to analyze lattice fields, Ising models, and Euclidean quantum fields. © 1996 John Wiley & Sons, Inc.
Mon, 01 Jan 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1111841996-01-01T00:00:00Z
- A civil engineering model of protein conformational changehttps://scholarbank.nus.edu.sg/handle/10635/114975Title: A civil engineering model of protein conformational change
Authors: Lawton, W.; Meiyappan, S.; Raghavan, R.; Viswanathan, R.; Yu, Y.
Abstract: We present a variational approach for the simulation of large conformational changes of proteins (including multiple protein chains/ligands) which takes advantage of their cross-linked one-dimensional nature, a structure which often occurs in civil engineering. Conformational changes are computed by incremental energy minimisation. We use an efficient finite element method for finding equilibria of complexes composed of inter-linked chains; this method is based on recent advances in the description of one-dimensional elasticity. Protein backbone elasticity, van der Waals repulsions, hydrogen bonds and salt bridges are taken into account, together with user-defined geometric distance constraints that may be imposed for purposes of simulating various binding processes based on chemical knowledge. These computational methods have been integrated into a system, Proteinmorphosis, which includes interactive visualisation. The conformational change of calmodulin upon peptide binding is examined as a first experiment. Allostery in hemoglobin, which consists of a cooperative oxygen binding mechanism, is a second, more sophisticated, numerical experiment. Different modelling strategies are designed to understand the allostery. The results for both molecules are consistent with existing hypotheses, and reproduce the known atomic positions after binding to within the experimental error. The modelling system is part of an on-going program to model structural biology, from protein structure to cell and tissue properties. © Springer-Verlag 1999.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1149751999-01-01T00:00:00Z
- Stability and orthonormality of multivariate refinable functionshttps://scholarbank.nus.edu.sg/handle/10635/111211Title: Stability and orthonormality of multivariate refinable functions
Authors: Lawton, W.; Lee, S.L.; Shen, Z.
Abstract: This paper characterizes the stability and orthonormality of the shifts of a multidimensional (M, c) refinable function φ in terms of the eigenvalues and eigenvectors of the transition operator Wcau defined by the autocorrelation cau of its refinement mask c, where M is an arbitrary dilation matrix. Another consequence is that if the shifts of φ form a Riesz basis, then Wcau has a unique eigenvector of eigenvalue 1, and all of its other eigenvalues lie inside the unit circle. The general theory is applied to two-dimensional nonseparable (M, c) refinable functions whose masks are constructed from Daubechies' conjugate quadrature filters.
Tue, 01 Jul 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1112111997-07-01T00:00:00Z
- Design of conjugate quadrature filters having specified zeroshttps://scholarbank.nus.edu.sg/handle/10635/111245Title: Design of conjugate quadrature filters having specified zeros
Authors: Lawton, Wayne; Micchelli, Charles A.
Abstract: Conjugate quadrature filters with multiple zeros at 1 have classical applications to unitary subband coding of signals using exact reconstruction filter banks. Recent work shows how to construct, given a set of n negative numbers, a CQF whose degree does not exceed 2n-1 and whose zeros contain the specified negative numbers, and applies such filters to interpolatory subdivision and to wavelet construction in Sobelov spaces. This paper describes a recent result of the authors which extends this construction for an arbitrary set of n nonzero complex numbers that contains no negative or negative reciprocal conjugate pairs. Detailed derivations are to be given elsewhere. We design several filters using an exchange algorithm to illustrate a conjecture concerning the minimal degree and we discuss an application to coding transient acoustic signals.
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1112451997-01-01T00:00:00Z
- Computing the inertia operator of a rigid bodyhttps://scholarbank.nus.edu.sg/handle/10635/103032Title: Computing the inertia operator of a rigid body
Authors: Lawton, W.; Noakes, L.
Abstract: We prove that the inertia operator A of a rigid body is genetically determined, up to a scalar multiple, by the curve Ω in R3 that describes its angular velocity in the body. The precise condition is that Ω not be contained in a two-dimensional subspace of R3. We derive two indirect methods to compute A from the values of O over an arbitrary interval, and a direct method to compute A from the second-and fourth-order moments of Ω. The direct method utilizes moment identities derived from symmetries in Euler's equation. © 2001 American Institute of Physics.
Sun, 01 Apr 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030322001-04-01T00:00:00Z
- A continuum mechanics-based model for cortical growthhttps://scholarbank.nus.edu.sg/handle/10635/111125Title: A continuum mechanics-based model for cortical growth
Authors: Raghavan, R.; Lawton, W.; Ranjan, S.R.; Viswanathan, R.R.
Abstract: One method for the synthesis of object shapes is by using physical laws. A continuum mechanics-based model of growth is proposed here. An energy functional, a function of the shape of an elastic object, is defined. At every instant of the growth process, the shape of the object corresponds to a minimum of this energy functional; growth is taken to be a quasistatic process. The model is used to simulate the growth of a one-dimensional 'brain cortex'. Starting from almost smooth initial configurations, growth leads to the formation of complex folds or convolutions. It is demonstrated that apart from the constraint of fitting in the skull, two other constraints are both necessary and sufficient to robustly generate patterns actually seen in cortical contours. These are: a minimum thickness for cortical folds due to white matter, and a shear constraint on the white matter tracts. Finally, an interesting difference between periodic and non-periodic initial conditions is pointed out.
Mon, 21 Jul 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1111251997-07-21T00:00:00Z
- Spectral relationships between kicked Harper and on-resonance double kicked rotor operatorshttps://scholarbank.nus.edu.sg/handle/10635/53186Title: Spectral relationships between kicked Harper and on-resonance double kicked rotor operators
Authors: Lawton, W.; Mouritzen, A.S.; Wang, J.; Gong, J.
Abstract: Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectra of these operators. In this paper we apply C -algebra methods to explain this resemblance. We show that each pair of corresponding operators belongs to a common rotation C -algebra Bα, prove that their spectra are equal if α is irrational, and prove that the Hausdorff distance between their spectra converges to zero as q increases if α=p/q with p and q coprime integers. Moreover, we show that corresponding operators in Bα are homomorphic images of mother operators in the universal rotation C -algebra Aα that are unitarily equivalent and hence have identical spectra. These results extend analogous results for almost Mathieu operators. We also utilize the C -algebraic framework to develop efficient algorithms to compute the spectra of these mother operators for rational α and present preliminary numerical results that support the conjecture that their spectra are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the ten Martini problem, was recently proven after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators. © 2009 American Institute of Physics.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/531862009-01-01T00:00:00Z
- Proof of the hyperplane zeros conjecture of Lagarias and Wanghttps://scholarbank.nus.edu.sg/handle/10635/103976Title: Proof of the hyperplane zeros conjecture of Lagarias and Wang
Authors: Lawton, W.
Abstract: We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics, and Fourier analysis. © 2008 Birkhäuser Boston.
Fri, 01 Aug 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039762008-08-01T00:00:00Z
- Fixed point properties of C*-algebrashttps://scholarbank.nus.edu.sg/handle/10635/103276Title: Fixed point properties of C*-algebras
Authors: Dhompongsa, S.; Fupinwong, W.; Lawton, W.
Abstract: This paper derives relations between the following properties of a C*-algebra: (i) it has the fpp, (ii) the spectrum of every self-adjoint element is finite, (iii) it is finite dimensional, (iv) it is generated by two projections p and q and the spectrum of p+q is homeomorphic to a compact ordinal α
Tue, 01 Feb 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032762011-02-01T00:00:00Z
- Hermite interpolation in loop groups and conjugate quadrature filter approximationhttps://scholarbank.nus.edu.sg/handle/10635/103367Title: Hermite interpolation in loop groups and conjugate quadrature filter approximation
Authors: Lawton, W.M.
Abstract: A classical result of Weierstrass ensures that any continuous finite length trajectory in a vector space can be uniformly approximated by one whose coordinates are trigonometric functions. We derive an analogous result for trajectories in spheres and apply it to show that a continuous frequency response of a conjugate quadrature filter can be uniformly approximated by the frequency response of a finitely supported conjugate quadrature filter. We also extend this result so as to preserve specified roots of the frequency response and derive an approximation result for refinable functions whose integer translates are orthonormal. Our methods utilize properties of loop groups jets and the Brouwer topological degree. © Kluwer Academic Publishers 2004.
Wed, 01 Dec 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033672004-12-01T00:00:00Z
- An algorithm for matrix extension and wavelet constructionhttps://scholarbank.nus.edu.sg/handle/10635/52779Title: An algorithm for matrix extension and wavelet construction
Authors: Lawton, W.; Lee, S.L.; Shen, Z.
Abstract: This paper gives a practical method of extending an n x r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.
Mon, 01 Apr 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/527791996-04-01T00:00:00Z
- Ribbons and groups: A thin rod theory for catheters and filamentshttps://scholarbank.nus.edu.sg/handle/10635/113235Title: Ribbons and groups: A thin rod theory for catheters and filaments
Authors: Lawton, W.; Raghavan, R.; Ranjan, S.R.; Viswanathan, R.
Abstract: We use the rotation group and its algebra to provide a novel description of deformations of special Cosserat rods or thin rods that have negligible shear. Our treatment was motivated by the problem of the simulation cf catheter navigation in a network of blood vessels, where this description is directly useful. In this context, we derive the Euler differential equations that characterize equilibrium configurations of stretch-free thin rods. We apply perturbation methods, used in time-dependent quantum theory, to the thin rod equations to describe incremental deformations of partially constrained rods. Further, our formalism leads naturally to a new and efficient finite element method valid for arbitrary deformations of thin rods with negligible stretch. Associated computational algorithms are developed and applied to the simulation of catheter motion inside an artery network.
Fri, 05 Mar 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1132351999-03-05T00:00:00Z
- Applications of complex valued wavelet transforms to subband decompositionhttps://scholarbank.nus.edu.sg/handle/10635/111140Title: Applications of complex valued wavelet transforms to subband decomposition
Authors: Lawton, Wayne
Abstract: A method for constructing complex valued linear phase FIR conjugate quadrature filters and associated wavelet bases is described. Each filter is derived by replacing certain zeros of a real valued FIR conjugate quadrature filter by their reciprocal conjugates. The derived filters have the same frequency response magnitudes as the original filters and their liner phase property permits the use of symmetrization in subband decomposition to avoid border discontinuities that result from signal periodization. Subband decomposition and reconstruction using both a length 6 filter associated with a Daubechies wavelet bases and a related length 6 complex valued linear phase filter are compared to illustrate the reduced border effects.
Wed, 01 Dec 1993 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1111401993-12-01T00:00:00Z