Showing papers on "Remainder published in 2009"

The sharp Sobolev inequality in quantitative form

Abstract: A quantitative version of the sharp Sobolev inequality in W (R), 1 < p < n, is established with a remainder term involving the distance from extremals.

Two-loop polygon Wilson loops in N=4 SYM

Abstract: We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in = 4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n = 6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n = 7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of ``master integrals, which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the = 4 theory.

A Robust Chinese Remainder Theorem With Its Applications in Frequency Estimation From Undersampled Waveforms

Abstract: The Chinese remainder theorem (CRT) allows to reconstruct a large integer from its remainders modulo several moduli. In this paper, we propose a robust reconstruction algorithm called robust CRT when the remainders have errors. We show that, using the proposed robust CRT, the reconstruction error is upper bounded by the maximal remainder error range named remainder error bound, if the remainder error bound is less than one quarter of the greatest common divisor (gcd) of all the moduli. We then apply the robust CRT to estimate frequencies when the signal waveforms are undersampled multiple times. It shows that with the robust CRT, the sampling frequencies can be significantly reduced.

Two-Loop Polygon Wilson Loops in N=4 SYM

Abstract: We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N=4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n=6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n=7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of "master integrals", which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the N=4 theory.

SPECTRAL ASYMPTOTICS FOR ARITHMETIC QUOTIENTS OF SL(n,R)/SO(n)

Abstract: In this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space SL(n,R)/SO(n). In particular, we obtain Weyl's law with an estimation on the remainder term. This extends some of the main results of Duistermaat, Kolk, and Varadarajan ([DKV1]) to this setting

Studies in perturbation theory. XIV. Treatment of constants of motion, degeneracies and symmetry properties by means of multidimensional partitioning

Abstract: The multidimensional partitioning technique is reviewed, with particular emphasis on its formal properties. The treatment of constants of motion by this method coincides with a treatment given previously, involving a modified resolvent in the ordinary approach. On the other hand the present method, upon expansion, coincides with Van Vleck's perturbation theory for quasidegenerate states. Bounds and estimates of the remainder related to Padee approximants obtained via inner projections are discussed. The construction of effective-hamiltonians is considered, in particular the energy-independent non self-adjoint ones.

Ketoreductase polypeptides for the production of a 3-aryl-3-hydroxypropanamine from a 3-aryl-3-ketopropanamine

Abstract: Manipulated ketoreductase polypeptide capable of converting the substrate N, N-dimethyl-3-keto-3- (2-thienyl) -1-propanamine into the product (S) -N, N-dimethyl-3-hydroxy-3- (2 -thienyl) -1-propanamine at a rate that is improved compared to that of a reference polypeptide having the amino acid sequence of SEQ ID NO: 2, wherein the polypeptide comprises an amino acid sequence that is identical in at least 85% to a reference sequence having the amino acid sequence of SEQ ID NO: 2, in which the polypeptide has the following characteristics: (a) the remainder corresponding to the X94 moiety is a glycine, (b) the residue corresponding to the X145 residue is an aromatic amino acid or leucine; and (c) the remainder corresponding to the remainder X190 is proline; and (d) the remainder at position X153 is threonine or valine, the remainder at position X195 is methionine, the remainder at position X206 is phenylalanine, tryptophan or tyrosine, and / or the remainder at position X233 is glycine.

A Class of Completely Monotonic Functions Related to the Remainder of Binet's Formula with Applications

Abstract: In the note, the complete monotonicity of dierence between remainders of Binet’s formula and the star-shaped and subadditive properties of the remainder of Binet’s formula are proved.

Trees and asymptotic expansions for fractional stochastic differential equations

Abstract: In this article, we consider an n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameterH >1/3. We derive an expansion for E[f (Xt )] in terms of t, where X denotes the solution to the SDE and f :Rn →R is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl. 117 (2007) 550-574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case H >1/2.

Uniform Asymptotics of Green's Kernels for Mixed and Neumann Problems in Domains with Small Holes and Inclusions

Abstract: Uniform asymptotic approximations of Green's kernels for the harmonic mixed and Neumann boundary value problems in domains with singularly perturbed boundaries are obtained. We consider domains with small holes (in particular, cracks) or inclusions. Formal asymptotic algorithms are supplied with rigorous estimates of the remainder terms.

Best remainder norms in Sobolev-Hardy inequalities

Abstract: We exhibit the optimal norm for a remainder term in the sharp Sobolev inequality involving a Lorentz norm, and in the equivalent classical Hardy inequality. Limiting cases of the relevant inequalities are also considered.

Production management system

Abstract: A monitoring system includes a control circuit configured to determine scrap values, yield values, and remainder values for at least a first operation and a second operation. The control circuit is configured to transmit one or more display signals. The one or more display signals include instructions to display a first operation status bar and a second operation status bar. The first operation status bar includes a first operation yield value, a first operation scrap value, and a first operation remainder value. The second operation status bar includes a second operation yield value, a second operation scrap value, and a second operation remainder value.

Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

Abstract: We study the asymptotic behavior of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude e tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of e −2 , the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as e goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale.

The fractional Hardy inequality with a remainder term

Abstract: We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]

Diffraction from arbitrarily shaped bodies of revolution: analytical regularization

Abstract: A mathematically rigorous and numerically efficient approach, based on analytical regularization, for solving the scalar wave diffraction problem with a Dirichlet boundary condition imposed on an arbitrarily shaped body of revolution is described. Seeking the solution in an integral-equation formulation, the singular features of its kernel are determined, and the initial equation transformed so that its kernel can be decomposed into a singular canonical part and a regular remainder. An analytical transformation technique is used to reduce the problem equivalently to an infinite system of linear algebraic equations of the second kind. Such system can be effectively solved with any prescribed accuracy by standard numerical methods. The matrix elements of this algebraic system are expressible in the terms of the Fourier coefficients of the remainder. Due to the smoothness of the remainder a robust and efficient technique is obtained to calculate the matrix elements. Numerical investigations of structures, such as the prolate spheroid and bodies obtained by rotation of “Pascal’s Limacon” and of the “Cassini Oval”, exhibit the high accuracy and wide possibilities of the approach.

Arithmetic complexity

Abstract: We obtain lower bounds on the cost of computing various arithmetic functions and deciding various arithmetic relations from specified primitives. This includes lower bounds for computing the greatest common divisor and deciding coprimeness of two integers, from primitives like addition, subtraction, division with remainder and multiplication. Some of our results are in terms of recursive programs, but they generalize directly to most (plausibly all) algorithms from the specified primitives. Our methods involve some elementary number theory as well as the development of some basic notions and facts about recursive algorithms.

Obtaining More Karatsuba-Like Formulae over The Binary Field.

Abstract: The aim of this study is to find more Karatsuba-like formulae for a fixed set of moduli polynomials in GF(2)[x]. To this end, a theoretical framework is established. The authors first generalise the division algorithm, and then present a generalised definition of the remainder of integer division. Finally, a generalised Chinese remainder theorem is used to achieve their initial goal. As a by-product of the generalised remainder of integer division, the authors rediscover Montgomery's N-residue and present a systematic interpretation of definitions of Montgomery's multiplication and addition operations.

Bernstein-type operators on tetrahedrons

Abstract: The aim of the paper is to construct Bernstein-type operators on tetrahedron with all straight edges and on tetrahedron with three curved edges defined by some given functions. We study the interpolation prop- erties, the approximation accuracy (degree of exactness, precision set) and the remainder of the corresponding approximation formulas. The accuracy is also illustrated by numerical examples.

The Baire property in remainders of topological groups and other results

Abstract: It is established that a remainder of a non-locally compact topological group G has the Baire property if and only if the space G is not y Cech-complete. We also show that if G is a non-locally compact topological group of countable tightness, then either G is submetrizable, or G is the y Cech-Stone remainder of an arbitrary remainder Y of G. It follows that if G and H are non-submetrizable topological groups of countable tightness such that some remainders of G and H are homeomorphic, then the spaces G and H are homeomorphic. Some other corollaries and related results are presented.

Sparse Matrix Padding

Abstract: Zero elements are added to respective lines (e.g., rows/columns) of a sparse matrix. The added zero elements increase the number of elements in the respective lines to be a multiple of a predetermined even number “n” (e.g., 2, 4, 8, etc.), based upon an n-fold unrolling loop, where n=2, 4, 8, etc. By forming a sparse matrix having lines (e.g., rows or columns) that are multiples of the predetermined number “n”, the n-fold unrolling loop thereby acts upon a predetermined number of elements in respective iterations, avoiding unnecessarily costly operations (e.g., additional loop unrolling code) on remainder non-zero elements (e.g. remainder row/column elements not within an n-fold unrolling loop) left in a row or column after unrolling. This improves the efficiency of sparse matrix linear algebra solvers and key sparse linear algebra kernels (e.g., SPMV) thereby improving the overall performance of a computer (e.g., running an application).

Numerical integration of bivariate functions over a non rectangular area by using Fluctuationlessness theorem

Abstract: "Fluctuation free matrix representation approximation Method" developed by M. Demiralp can be used in approximating the multiple remainder terms of the integral of the Multivariate Taylor expansion. This provides us with a new numerical integration method for multivariate functions. However in this paper instead of dealing with a single formula which takes care of the multiple remainder terms, a new approach is undertaken. At every step of a multivariate integration only one variable is taken care of. Thus an iterative procedure which speeds up the computation rate is obtained.

Dynamics of body separation—analytical procedure

Abstract: In this paper, an analytical procedure for the determination of the dynamic parameters of a remainder body after mass separation is developed. The method is based on the general principles of momentum and angular momentum of a body and system of bodies. The kinetic energy of motion of the whole body and also of the separated and remainder body is considered. The derivatives of kinetic energies with respect to the generalized velocity determine the velocity and angular velocity of the remainder body. To confirm the proposed procedure, the results are compared with those obtained using the method of momenta and angular momenta. In the paper, the theorem about increase of kinetic energies of the separated and remainder bodies for perfectly plastic separation is proved. The increase of the kinetic energies correspond to the relative velocities and angular velocities of the separated and remainder bodies. As an example, the mass separation from a pendulum is considered. The kinematic properties of the remainder pendulum are obtained using the analytic procedure. The results are in agreement with those obtained by applying the basic principles of Newton’s mechanics.

Locating method of index nodes of disk file and device thereof

Abstract: The embodiment of the invention provides a locating method of index nodes of a disk file and a device thereof. The locating method comprises the following steps: obtaining a corresponding digital file name of the disk file; dividing by a preset value with the digital file name to obtain a corresponding quotient and a remainder; and locating the index nodes of the file according to the quotient and the remainder. On the other hand, the embodiment of the invention provides a locating device of the index nodes of the disk file. The locating device comprises an obtaining unit for obtaining the corresponding digital file name of the file; a division unit for dividing by the preset value with the digital file name to obtain the corresponding quotient and the remainder; and a locating unit for locating the index nodes of the file according to the quotient and the remainder. The locating method of index nodes of the disk file and the device thereof improve search speed of the disk file, and enhance performance of a file system.

Cyclic code processing circuit, network interface card, and cyclic code processing method

Abstract: A cyclic code processing circuit, network interface card, and method for calculating a remainder from input data comprising a plurality of bits arranged in parallel. The calculation is performed by first computing a first remainder obtained by dividing an integral multiple data block by a generator polynomial, the integral multiple data block comprising a plurality of words that precede the final word of the input data. Then, a second remainder is computed by dividing the final word by the generator polynomial, the final word comprising the parallel bits located at the end of the input data. The input data remainder is calculated using the first and the second previously calculated remainders.

An Improved Algorithm for Dempster-Shafer Theory of Evidence

Abstract: The computational complexity of reasoning within the Dempster-Shafer theory of evidence is one of the major points of criticism this formalism has to face. Various approximation algorithms have been suggested that aim at overcoming this difficulty. This paper presents an improved practical algorithm through reducing the number of focal elements in the belief function involved. In this proposed algorithm, all focal elements of every piece of evidence are classified into dereliction and remainder, and the basic probability assignments of those derelictions are reassigned to the remainders when they are correlative or the dereliction is nested to the remainder. Finally, an illustrative example shows that the improved practical algorithm is effective and feasible through comparing with other approximations.

Communication apparatus, method of checking received data size, multiple determining circuit, and multiple determination method

Abstract: A dividing unit sets an actual packet length transferred from a packet receiving section to a variable U, and then sets 2α to a variable V. If a positive number determining section determines that a subtraction result of subtracting a remainder N0 from a quotient M0, both found by dividing U by V, is a positive number, the dividing unit overwrites the subtraction result to U. The dividing unit repeats such operations of dividing the subtraction result by V, until the positive number determining section determines that the subtraction result of subtracting the remainder from the quotient, both found by dividing U by V, is a non-positive number. When the subtraction result becomes a non-positive number and the quotient and the remainder match, a packet length determining section determines that received data has a normal size, and notifies it to a discard determining section.

Modified Euclidean Algorithms for Decoding Reed-Solomon Codes

Abstract: The extended Euclidean algorithm (EEA) for polynomial greatest common divisors is commonly used in solving the key equation in the decoding of Reed-Solomon (RS) codes, and more generally in BCH decoding. For this particular application, the iterations in the EEA are stopped when the degree of the remainder polynomial falls below a threshold. While determining the degree of a polynomial is a simple task for human beings, hardware implementation of this stopping rule is more complicated. This paper describes a modified version of the EEA that is specifically adapted to the RS decoding problem. This modified algorithm requires no degree computation or comparison to a threshold, and it uses a fixed number of iterations. Another advantage of this modified version is in its application to the errors-and-erasures decoding problem for RS codes where significant hardware savings can be achieved via seamless computation.