One-step optical trinary signed-digit arithmetic using redundant bit representations

A simple one-step fully parallel trinary signed-digit arithmetic is proposed for parallel optical computing. This technique performs multidigit carry-free addition and borrow-free subtraction in constant time. The trinary signed-digit arithmetic operations are based on redundant bit representation of the digits. Optical implementation of the proposed arithmetic can be carried out using correlation or matrix multiplication based schemes. An efficient matrix multiplication based optical implementation that employs a fixed number of minterms for any operand length is developed. It is shown that only 30 minterms (less than recently reported techniques) are enough for implementing the one-step trinary addition and subtraction.     

A simple proof of the Uncertainty Principle for compact groups

We give a simple proof of the Uncertainty Principle for finite nonabelian groups, which generalizes directly to compact groups.     

HOW TO CONSTRUCT LAX REPRESENTATION FOR CONSTRAINED FLOWS OF THEBOUSSINESQ HIERARCHY VIA ADJOINT REPRESENTATIONS

1IntroductionTheBoussinesqequationarisesinseveralphysicalapplicationandhasbeenstudiedquiteextensivelyinthepast[1--3].Inarecentpaper[4],itwasfoundthattheBoussinesqhierarchycanbeobtainedfromthezero-curvatureconditionassociatedwiththegroupSL(3,R).ThisshowsadirectrelationshipbetweenthegroupSL(3,R)andtheW3algebraofZamolodchikov.Recentlytherehasbeenconsiderableinterestinthedecompositionofsolitonequationsviaconstraintsrelatingpotentialandeigenfunctions,becausethedecompositionprovidesaneffectiveme…     

A MODEL FOR WHITE NOISE ANALYSIS IN P-ADIC NUMBER FIELDS

ＡＭＯＤＥＬＦＯＲＷＨＩＴＥＮＯＩＳＥＡＮＡＬＹＳＩＳＩＮＰ－ＡＤＩＣＮＵＭＢＥＲＦＩＥＬＤＳ（黄志远）￥ＡｎｄｒｅｗＫｈｒｅｎｎｉｋｏｖ（ＭｏｓｃｏｗＩｎｓｔ．ｏｆＥｌｅｃｔｒｏｎｉｃＥｎｇｉｎｅｅｒｉｎｇ，１０３４９８，Ｚｅｌｅｎｏｇｒａｄ，Ｋ－...     

On Fourier transform of generalized Brownian functionals

Let $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ and $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ into $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ is defined by ??(?) = ∫$\text{S}$¹: exp[?i ∫_$\text{R}$?(t) ?(t) dt]: ${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}\text{?(}\text{B}\text{?}\text{) dμ(}\text{B}\text{?}$), where : :${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}$ denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space $\text{S}$¹ of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.     

An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is total-variation-diminishing (TVD). Much attention has been paid to these representations in the literature.

In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them.

Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible Shu-Osher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods.

In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).

     

Nonlinear Commutation Relations: Representations by Point-Supported Operators

We present a class of non-Lie commutation relations admitting representations by point-supported operators (i.e., by operators whose integral kernels are generalized point-supported functions). For such relations we construct all operator-irreducible representations (up to equivalence). Each representation is realized by point-supported operators in the Hilbert space of antiholomorphic functions. We show that the reproducing kernels of these spaces can be represented via hypergeometric series and the theta function, as well as via their modifications. We construct coherent states that intertwine abstract representations with irreducible representations.     

The Magnus representation of the Torelli group

In this paper we consider the Magnus representation of the Torelli group. We prove that it is not faithful by showing a non-trivial element in the kernel of this representaion.

     

Efficient parallel exponentiation in using normal basis representations

Von zur Gathen proposed an efficient parallel exponentiation algorithm in finite fields using normal basis representations. In this paper we present a processor-efficient parallel exponentiation algorithm in GF(qⁿ) which improves upon von zur Gathen's algorithm. We also show that exponentiation in GF(qⁿ) can be done in O((log₂n)²/log_qn) time using n/(log₂n)² processors. Hence we get a processor-time bound of O(n/log_qn), which matches the best known sequential algorithm. Finally, we present an efficient on-line processor assignment scheme which was missing in von zur Gathen's algorithm.