Showing papers on "Remainder published in 2007"

Two-Level RAM Lookup Table for Block and Page Allocation and Wear-Leveling in Limited-Write Flash-Memories

Abstract: A restrictive multi-level-cell (MLC) flash memory prohibits regressive page-writes. When a regressive page-write is requested, an empty block having a low wear-level count is found, and data from the regressive page-write and data from pages stored in the old block are written to the empty block in page order. The old block is erased and recycled. A two-level look-up table is stored in volatile random-access memory (RAM). A logical page address from a host is divided by a modulo divider to generate a quotient and a remainder. The quotient is a logical block address that indexes a first-level look-up table to find a mapping entry with a physical block address that selects a row in a second-level look-up table. The remainder locates a column in the row in the second-level look-up table. If any page-valid bits above the column pointed to by the remainder are set, the write is regressive.

Phase Unwrapping and A Robust Chinese Remainder Theorem

Abstract: In the conventional Chinese remainder theorem (CRT), a small error in a remainder may cause a large error in the solution of an integer, i.e., CRT is not robust. In this letter, we first propose a robust phase unwrapping algorithm with applications in radar signal processing. Motivated from the phase unwrapping algorithm, we then derive a type of robust CRT

Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations

Abstract: It is shown that, if f is a meromorphic function of order zero and q is an element of C, then[GRAPHICS]for all r on a set of logarithmic density 1. The remainder of the paper consists of applications of identity (double dagger) to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of q-difference equations. The results obtained include q-shift analogues of the second main theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.

On large deviations of sums of independent random variables

Abstract: Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramer's condition. The large deviation x-region under consideration is broader than in the classical Cramer's theorem, and the estimate of the remainder is uniform with respect to x. The corresponding asymptotic expansion with arbitrarily many summands is also obtained.

Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes

Abstract: Problem of simulation of multivariate Levy processes is investigated. The method based on shot noise series expansions of such processes combined with Gaussian approximation of the remainder is established in full generality. Formulas that can be used for simulation of tempered stable, operator stable and other multivariate processes are obtained.

Bch code with 256 information bytes and up to 8 bytes of parity check elements

Abstract: A coding system comprises pre-multiply the message u(x) by Xn−k. Obtain the remainder b(x), i.e. the parity check digits. And combine b(x) and Xn−ku(x) to obtain the code polynomial. A decoding method comprises calculating a syndrome; finding an error-location polynomial; and computing a set of error location numbers.

A Generalization of the Hyers-Ulam-Rassias Stability of the Pexiderized Quadratic Equations, II

Abstract: In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder is defined by , , where (G, *) is a group, X is a real or complex Hausdorff topological vector space and f, g, h, k are functions from G into X.

A gmp-based implementation of schönhage-strassen's large integer multiplication algorithm

Abstract: Schonhage-Strassen's algorithm is one of the best known algorithms for multiplying large integers. Implementing it ef?ciently is of utmost importance, since many other algorithms rely on it as a subroutine. We present here an improved implementation, based on the one distributed within the GMP library. The following ideas and techniques were used or tried: faster arithmetic modulo 2n + 1, improved cache locality, Mersenne transforms, Chinese Remainder Reconstruction, the √2 trick, Harley's and Granlund's tricks, improved tuning.

System and method for processing wireless high definition video data using a shortened last codeword

Abstract: A method and system for processing high definition video data to be transmitted over a wireless medium is disclosed. In one embodiment, the method includes receiving an information packet having the length of L bytes, wherein L=(M×n×K)+A, and where: M is the depth of an interleaver, n is the number of interleavers, K is an encoding code length and A is the number of remainder bytes with respect to M×n×K bytes, wherein the remainder bytes are located at the end of the information packet. M×n×K bytes represent M×n codewords, wherein the remainder bytes sequentially form a plurality of remainder codewords, and wherein the plurality of remainder codewords comprise a last codeword which is located at the end of the remainder codewords. The method further includes i) shortening the last codeword such that the resultant shortened codeword is shorter in length than each of the remaining codewords of the information packet and ii) adding dummy bits to the outer encoded data so as to meet a predefined size requirement for an outer interleaver.

Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n)

Abstract: In this paper we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space S=SL(n,R)/SO(n) In particular, we obtain Weyl's law with an estimation on the remainder term This extends results of Duistermaat-Kolk-Varadarajan on spectral asymptotics for compact locally symmetric spaces to this non-compact setting

The area under a curve specified by measured values

Abstract: The problem is addressed of determining numerical approximations to the area under a curve specified by arbitrarily spaced data. A formulation of this problem is given in which the data are used to model the curve as a piecewise polynomial, each piece having the same degree. That piecewise function is integrated to provide an approximation to the area. A corresponding compound quadrature rule is derived. Different degrees of polynomial give rise to different orders of quadrature rule. The widely used trapezoidal rule is a special case, as is the Gill–Miller rule. The remainder of the paper is concerned with evaluating the measurement uncertainties associated with the approximations to the area obtained by the use of these rules in the case where the data ordinates correspond to measured values having stated associated uncertainties. The case in which there is correlation associated with the measured values, frequently arising when the measured values are obtained using the same measuring instrument, is also treated. A statistical test is used to select a suitable polynomial degree.

Random modulation of solitons for the stochastic Korteweg–de Vries equation

Abstract: We study the asymptotic behavior of the solution of a Korteweg–de Vries equation with an additive noise whose amplitude e tends to zero. The noise is white in time and correlated in space and the initial state of the solution is a soliton solution of the unperturbed Korteweg–de Vries equation. We prove that up to times of the order of 1/e2, the solution decomposes into the sum of a randomly modulated soliton, and a small remainder, and we derive the equations for the modulation parameters. We prove in addition that the first order part of the remainder converges, as e tends to zero, to a Gaussian process, which satisfies an additively perturbed linear equation

A radix-10 SRT divider based on alternative BCD codings

Abstract: In this paper we present the algorithm and architecture a radix-10 floating-point divider based on an SRT non-restoring digit-by-digit algorithm. The algorithm uses conventional techniques developed to speed-up radix-2k division such as signed-digit (SD) redundant quotient and digit selection by constant comparison using a carry-save estimate of the partial remainder. To optimize area and latency for decimal, we include novel features such as the use of alternative BCD codings to represent decimal operands, estimates by truncation at any binary position inside a decimal digit, a single customized fast carry propagate decimal adder for partial remainder computation, initial odd multiple generation and final normalization with rounding, and register placement to exploit advanced high fanin mux-latch circuits. The rough area-delay estimations performed show that the proposed divider has a similar latency but less hardware complexity (1.3 area ratio) than a recently published high performance digit-by-digit implementation.

Resurgence of the Euler-MacLaurin summation formula

Abstract: The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, we give a Borel summable transseries expression for the remainder term, as well as a Laplace integral formula, with an explicit integrand which is a resurgent function itself. In particular, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval. Finally, we give two applications of our results. One concerns the construction of solutions of linear difference equations with a small parameter. And another concerns the problem of proving resurgence of formal power series associated to knotted objects.

Estimates from Below for the Spectral Function and for the Remainder in Local Weyl’s Law

Abstract: We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’s law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle. Our results develop and extend the unpublished thesis of A. Karnaukh [Ka].

Explicit estimates for summatory functions linked to the Möbius µ-function

Abstract: Let $M(x)$ be the summatory function of the Mobius function and $R(x)$ be the remainder term for the number of squarefree integers up to $x$. In this paper, we prove the explicit bounds $|M(x)|

A Lower Bound for the Remainder in Weyl's Law on Negatively Curved Surfaces

Abstract: We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace asymptotics for long times, equidistribution of closed geodesics and small-scale microlocalization.

A theorem on the approximation of a trigonometric sum by a shorter one

Abstract: We prove a theorem on the approximation of a trigonometric sum by a shorter one with the constants in the remainder calculated concretely.

An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights

Abstract: We consider self-adjoint unbounded Jacobi matrices with diagonal q n = n and weights λ n = c n n, where c n is a 2-periodical sequence of real numbers The parameter space is decomposed into several separate regions, where the spectrum is either purely absolutely continuous or discrete This constitutes an example of the spectral phase transition of the first order We study the lines where the spectral phase transition occurs, obtaining the following main result: either the interval (−∞; 1/2) or the interval (1/2; +∞) is covered by the absolutely continuous spectrum, the remainder of the spectrum being pure point The proof is based on finding asymptotics of generalized eigenvectors via the Birkhoff-Adams Theorem We also consider the degenerate case, which constitutes yet another example of the spectral phase transition

Modular multiplication method with precomputation using one known operand

Abstract: A modular multiplication method implemented in an electronic digital processing system takes advantage of the case where one of the operands W is known in advance or used multiple times with different second operands V to speed calculation. The operands V and W and the modulus M may be integers or polynomials over a variable X. A possible choice for the type of polynomials can be polynomials of the binary finite field GF (2N). Once operand W is loaded (30; 60) into a data storage (12) location, a value P = Lw-Xn+δ/M J is pre- computed (32; 62) by the processing system (10). Then when a second operand V is loaded (34; 64), the quotient qΛ for the product V.W being reduced modulo M is quickly estimated (36; 66), qΛ = Lv-P/Xn+δJ, optionally randomized (40; 70), q' = qΛ - E, and can be used to obtain (44; 74) the remainder r' = V.W - q'-M, which is congruent to (V.W) mod M. A final reduction (46; 76) can be carried out, and the later steps repeated (52; 82) with other second operands V.

System and method for processing wireless high definition video data using remainder bytes

Abstract: A method and system for processing high definition video data using remainder bytes is disclosed. In one embodiment, the method includes receiving an information packet having the length of L bytes, wherein L=(M×n×K)+A, and where: M is the depth of an interleaver, n is the number of interleavers, K is an encoding code length and A is the number of remainder bytes with respect to M×n×K bytes, wherein the remainder bytes are located at the end of the information packet, and wherein M×n×K bytes represent M×n codewords. The method further includes converting the A remainder bytes into a plurality of shortened codewords, wherein each of the shortened codewords is shorter in length than each of the M×n codewords. At least one embodiment of the invention provides much lower padding efficiency while improving the decoding performance.

The Remainder in Ruijsenaars’ Asymptotic Expansion of Barnes Double Gamma Function

Abstract: We show that the remainder in Ruijsenaars’ asymptotic expansion of the logarithm of Barnes double gamma function gives rise to a completely monotone function. Fourier expansions of the multiple Bernoulli polynomials are also obtained.

Sommation effective d’une somme de Borel par séries de factorielles

Abstract: In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions.

A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem

Abstract: Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the -sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.

Methods and Systems for Rate-Adaptive Transmission of Video

Abstract: Embodiments of the present invention comprise systems and methods for: authorizing the transmission of audiovisual data based on an iterative changing bit budget, where the bit budget may be based on a value of a total size of the set of Network Abstraction Layer units and a value of an initial size of the group of frames; and determining a bit budget remainder and an adjusted bit budget remainder.

Asplund Decomposition of Monotone Operators

Abstract: We establish representations of a monotone mapping as the sum of a maximal subdifferential mapping and a “remainder” monotone mapping, where the remainder is “acyclic” in the sense that it contains no nontrivial subdifferential component. This is the nonlinear analogue of a skew linear operator. Examples of indecomposable and acyclic operators are given. In particular, we present an explicit nonlinear acyclic operator.

Remainder Pade Approximants for the Exponential Function

Abstract: Following our earlier research, we propose a new method for obtaining the complete Pade table of the exponential function. It is based on an explicit construction of certain Pade approximants, not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and nontrivial coincidence is proved more generally for type II simultaneous Pade approximants for a family \((\exp(a_jz))_{j=1,\ldots, r}\) with distinct complex a's and we recover Hermite's classical formulas. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesu, Coussement, and van Assche, which generalize the classical Charlier orthogonal polynomials.

Methods for transmitting buffer size information

Abstract: A method of transmitting information about a buffer size includes transmitting a bit string comprising a first bit string and a second bit string when the buffer size is greater than or equal to a first value, the first bit string indicating a quotient which is acquired by dividing the buffer size by a second value, and the second bit string indicating a value corresponding to a remainder which is acquired by dividing the buffer size by the second value.