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Wavelet electrodynamics II: Atomic Composition of Electromagnetic Waves, Applied and Computational Harmonic Analysis 1:246–260
, 1994
"... The representation of solutions of Maxwell’s equations as superpositions of scalar wavelets with vector coefficients developed earlier is generalized to wavelets with polarization, which are matrixvalued. The construction proceeds in four stages: (1) A Hilbert space H of solutions is considered, ba ..."
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The representation of solutions of Maxwell’s equations as superpositions of scalar wavelets with vector coefficients developed earlier is generalized to wavelets with polarization, which are matrixvalued. The construction proceeds in four stages: (1) A Hilbert space H of solutions is considered, based on a conformally invariant inner product. (2) The analyticsignal transform is used to extend solutions from real spacetime to a complex spacetime domain T. The evaluation maps Ez, which send any solution F = B +iE to the values ˜ F(z) of its extension at points z ∈ T, are bounded linear maps on H. Their adjoints Ψz ≡ E ∗ z are the electromagnetic wavelets. (3) The eight real parameters z = x + iy ∈ T are given a complete physical interpretation: x = (x, t) ∈ R 4 is interpreted as a spacetime point about which Ψz is focussed. The imaginary spacetime vector y = (y, s) is timelike, i.e., y  < s. The sign of s is interpreted as the helicity of the wavelet, while s  is its scale. The 3vector v ≡ y/s is the velocity of its center. Thus wavelets parameterized by the set of Euclidean points E = {(x, is)} (real space and imaginary time) have stationary centers, and wavelets with y = 0 are Dopplershifted versions of ones with stationary centers. All the wavelets can be obtained from a single “mother ” by conformal transformations. (4) A resolution of unity is established in H, giving a representation of solutions as “atomic compositions ” of wavelets parameterized by z ∈ E. This yields a constructive method for generating solutions with initial data specified locally in space and by scale. Other representations, employing wavelets with moving centers, are obtained by applying conformal transformations to the stationary representation. This could be useful in the analysis of electromagnetic waves reflected or emitted by moving objects, such as radar signals. 1
Understanding the illposed twofluid model,” The
 10th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH10), Seoul, Korea
"... Multifield models are central to the modeling and simulation of transport processes in multiphase, homogenized systems. The approach is based on an interpenetrating continua description, in which conservation laws are applied to each phase as a separate continuum (field) and constitutive laws are p ..."
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Cited by 4 (2 self)
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Multifield models are central to the modeling and simulation of transport processes in multiphase, homogenized systems. The approach is based on an interpenetrating continua description, in which conservation laws are applied to each phase as a separate continuum (field) and constitutive laws are provided to represent interfield interactions. The resulting model is illposed and mathematically complex, in the sense that the equation system is nonhyperbolic, nonlinear and nonconservative. These features are thought to be at the root of difficulties in the numerical solution of the multifield model equation system. Consequently, the fidelity of the numerical solution is confounded by the interplay between uncertainty in physical closure relationships (constitutive laws) and numerical errors due to numerical diffusion and unphysical oscillations. In this paper, we provide a synthesis toward understanding the illposed twofluid model. Issues related to nonhyperbolicity are revisited broadly and brought under the perspective of their physical and mathematical origins. We emphasize the connection between the model’s mathematical properties and approaches of mathematical and/or numerical regularization. Lessons learned from past experiences are highlighted. The synthesis leads to principal questions of mathematical, physical and numerical nature that need to be addressed if further progress is to be made. A new approach to mathematical regularization through Virtual Spacetime Relaxation (VSR) is described, and numerical examples that clarify the roles of hyperbolicity and conservatism are offered. 1.
Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U (2, 2) and Born’s Reciprocity
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GENERALIZED FRAMES AND THEIR REDUNDANCY
"... Abstract. Let h be a generalized frame in a separable Hilbert space H indexed by a measure space (M,S,µ), and assume its analysing operator is surjective. It is shown that h is essentially discrete; that is, the corresponding index measure space (M,S,µ) can be decomposed into atoms E1,E2, ··· such t ..."
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Abstract. Let h be a generalized frame in a separable Hilbert space H indexed by a measure space (M,S,µ), and assume its analysing operator is surjective. It is shown that h is essentially discrete; that is, the corresponding index measure space (M,S,µ) can be decomposed into atoms E1,E2, ··· such that L2 (µ) is isometrically isomorphic to the weighted space ℓ2 w of all sequences {ci} of complex numbers with {ci}  2 = � ci  2wi < ∞, where wi = µ(Ei), i=1, 2, ·· ·. This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in L2 (R). 1.
Frames: a Maximum Entropy Statistical Estimate of the Inverse Problem
 J. of Mathematical Physics
, 1997
"... A Maximum Entropy statistical treatment of an inverse problem concerning frame theory is presented. The problem arises from the fact that a frame is an overcomplete set of vectors that defines a mapping with no unique inverse. Although any vector in the concomitant space can be expressed as linea ..."
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Cited by 3 (1 self)
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A Maximum Entropy statistical treatment of an inverse problem concerning frame theory is presented. The problem arises from the fact that a frame is an overcomplete set of vectors that defines a mapping with no unique inverse. Although any vector in the concomitant space can be expressed as linear combination of frame elements, the coefficients of the expansion are not unique. Frame theory guarantees the existence of a set of coefficients which is "optimal" in a Minimum Norm sense. Weshow here that these coefficients are also "optimal" from a Maximum Entropy viewpoint.
Recursive biorthogonalisation approach and orthogonal projectors, mathph/0209026 (2002); L. RebolloNeira, New Topics
 in Mathematical Physics Research, Nova Science Publishers
, 2006
"... An approach is proposed which, given a family of linearly independent functions, constructs the appropriate biorthogonal set so as to represent the orthogonal projector operator onto the corresponding subspace. The procedure evolves iteratively and it is endowed with the following properties: i) it ..."
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An approach is proposed which, given a family of linearly independent functions, constructs the appropriate biorthogonal set so as to represent the orthogonal projector operator onto the corresponding subspace. The procedure evolves iteratively and it is endowed with the following properties: i) it yields the desired biorthogonal functions avoiding the need of inverse operations ii) it allows to quickly update a whole family of biorthogonal functions each time that a new member is introduced in the given set. The approach is of particular relevance to the approximation problem arising when a function is to be represented as a finite linear superposition of non orthogonal waveforms. I.
Twostate dynamics for replicating twostrand systems
 Open Systems and Information Dynamics
, 2007
"... Abstract. We propose a formalism for describing twostrand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann ..."
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Abstract. We propose a formalism for describing twostrand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann equations and entangled states of systems consisting of two subsystems evolving in time in opposite directions. Such a time evolution has analogies in realistic DNA where the polymerazes move on leading and lagging strands in opposite directions. 1. TwoStrand Systems and Mutually TimeReflected Turing Machines According to Adleman [1] the process of DNA replication may be analyzed in terms of Turing machines: One strand plays a role of an instruction tape, a polymeraze is the read/write head, and the second strand contains the results of instructions. At a molecular level each strand is a sequence of molecules. In simple models one can represent sequences of molecules in a strand as chains of twolevel systems (bits) in a state ψ(t) 〉 =∑B1...Bn ψ(t)B1...Bn B1... Bn〉, Bj = 0, 1. Thinking of the motion of the head in terms of a dynamics, one can write ψ(t) 〉 = U(t, 0)ψ(0)〉, where 0 ≤ t ≤ T. The final time T is the time of arrival of the head at the end of the strand, and U(t1, t2) is a unitary operator which, in principle, may be different for different initial states of the system (this type of evolution occurs for nonlinear Schrödinger equations). A twostrand system can be represented by an entangled state Ψ(t) 〉 =
On the truncation of the harmonic oscillator wavepacket
, 2005
"... We present an interesting result regarding the implication of truncating the wavepacket of the harmonic oscillator. We show that disregarding the nonsignificant tails of a function which is the superposition of eigenfunctions of the harmonic oscillator has a remarkable consequence. Namely, there ex ..."
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We present an interesting result regarding the implication of truncating the wavepacket of the harmonic oscillator. We show that disregarding the nonsignificant tails of a function which is the superposition of eigenfunctions of the harmonic oscillator has a remarkable consequence. Namely, there exit infinitely many different superpositions giving rise to the same function on the interval. Uniqueness, in the case of a wavepacket, is restored by a postulate of quantum mechanics. PACS: 03.65.w, 03.65.Ca 1
A Gelfand Triple Approach to Wigner and Husimi Representations
, 2002
"... The notion of Gelfand triples is applied to interpret mathematically a family of phasespace representations of quantum mechanics interpolating between the Wigner and Husimi representations. Gelfand triples of operators on Hilbert space, and Gelfand triples of functions on phasespace are introduced ..."
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The notion of Gelfand triples is applied to interpret mathematically a family of phasespace representations of quantum mechanics interpolating between the Wigner and Husimi representations. Gelfand triples of operators on Hilbert space, and Gelfand triples of functions on phasespace are introduced in order to get isomorphic correspondences between operators and their phasespace representations. The phasespace Gelfand triples are characterized by means of growth conditions on the analytic continuation of the functions. We give integral expressions for the sesquilinear forms belonging to the phasespace Gelfand triples. This provides mathematically rigorous phasespace analogues for quantum mechanical expectation values of bounded operators. 1.