Showing papers on "Remainder published in 2010"

A Closed-Form Robust Chinese Remainder Theorem and Its Performance Analysis

Abstract: Chinese remainder theorem (CRT) reconstructs an integer from its multiple remainders that is well-known not robust in the sense that a small error in a remainder may cause a large error in the reconstruction. A robust CRT has been recently proposed when all the moduli have a common factor and the robust CRT is a searching based algorithm and no closed-from is given. In this paper, a closed-form robust CRT is proposed and a necessary and sufficient condition on the remainder errors for the closed-form robust CRT to hold is obtained. Furthermore, its performance analysis is given. It is shown that the reason for the robustness is from the remainder differential process in both searching based and our proposed closed-form robust CRT algorithms, which does no exist in the traditional CRT. We also propose an improved version of the closed-form robust CRT. Finally, we compare the performances of the traditional CRT, the searching based robust CRT and our proposed closed-form robust CRT (and its improved version) algorithms in terms of both theoretical analysis and numerical simulations. The results demonstrate that the proposed closed-form robust CRT (its improved version has the best performance) has the same performance but much simpler form than the searching based robust CRT.

Asymptotic treatment of perforated domains without homogenization

Abstract: As a main result of the paper, we construct and justify an asymptotic approximation of Greens function in a domain with many small inclusions. Periodicity of the array of inclusions is not required. We start with an analysis of the Dirichlet problem for the Laplacian in such a domain to illustrate a method of mesoscale asymptotic approximations for solutions of boundary value problems in multiply perforated domains. The asymptotic formula obtained involves a linear combination of solutions to certain model problems whose coefficients satisfy a linear algebraic system. The solvability of this system is proved under weak geometrical assumptions, and both uniform and energy estimates for the remainder term are derived. In the second part of the paper, the method is applied to derive an asymptotic representation of the Greens function in the same perforated domain. The important feature is the uniformity of the remainder estimate with respect to the independent variables.

A Survey of Gcd-Sum Functions

Abstract: We survey properties of the gcd-sum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcd-sum function and for the function defined by the harmonic mean of the gcd’s.

Energy solution to Schr\"odinger-Poisson system in the two-dimensional whole space

Abstract: We consider the Cauchy problem of the two-dimensional Schr\"odinger-Poisson system in the energy class. Though the Newtonian potential diverges at the spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to apply the perturbation method. Our argument can be adapted to the one-dimensional problem.

Some properties of extended remainder of binet's first formula for logarithm of gamma function

Abstract: In the paper, we extend Binet’s first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet’s first formula for the logarithm of the gamma function and related functions.

Hardy–Rellich inequalities with boundary remainder terms and applications

Abstract: We prove a family of Hardy–Rellich inequalities with optimal constants and additional boundary terms. These inequalities are used to study the behavior of extremal solutions to biharmonic Gelfand-type equations under Steklov boundary conditions.

Convergence study of the 1/Z expansion for the energy levels of two-electron atoms

Abstract: We perform numerical analysis of the first 20 and 14 coefficients for 1 {sup 1}S and 2 {sup 3}S states of the 1/Z expansion of the energy of two-electron atoms, respectively. The radius of convergence and large-order behavior of the coefficients are determined. The results obtained are in disagreement with those given so far in the literature. We sum the terms of the series with known coefficients and the remainder of the series where we replace the actual coefficients by their large-order values. We show that inclusion of the remainder improves agreement with variational results by more than three orders of magnitude. We argue that the energy is at least three times and most likely infinitely degenerate at the singularity. Numerical result for the effective characteristic polynomial supports this conclusion.

On lattice points in large convex bodies

Abstract: We consider a compact convex body $\mathcal{B}$ in $\mathbb{R}^d$ $(d\geqslant 3)$ with smooth boundary and nonzero Gaussian curvature and prove a new estimate of $P_{\mathcal{B}}(t)$, the remainder in the lattice point problem, which improves previously known best result.

Rigorous and Accurate Enclosure of Invariant Manifolds on Surfaces

Abstract: Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.

Free subexponentiality

Abstract: In this article, we introduce the notion of free subexponentiality, which extends the notion of subexponentiality in the classical probability setup to the noncommutative probability spaces under freeness We show that distributions with regularly varying tails belong to the class of free subexponential distributions This also shows that the partial sums of free random elements having distributions with regularly varying tails are tail equivalent to their maximum in the sense of Ben Arous and Voiculescu [Ann Probab 34 (2006) 2037-2059] The analysis is based on the asymptotic relationship between the tail of the distribution and the real and the imaginary parts of the remainder terms in Laurent series expansion of Cauchy transform, as well as the relationship between the remainder terms in Laurent series expansions of Cauchy and Voiculescu transforms, when the distribution has regularly varying tails

From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-box Identity Test for Depth-3 Circuits

Abstract: We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d^{k^k}-time black-box identity test over rationals (Kayal-Saraf, FOCS 2009) to one that takes d^{k^2}-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir-Shpilka (STOC 2005) and Kayal-Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for black-box identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional Sylvester-Gallai theorems and the rank of depth-3 identities in a very transparent manner. The existence of this was hinted at by Dvir-Shpilka (STOC 2005), but first proven, for reals, by Kayal-Saraf (FOCS 2009). We introduce the concept of Sylvester-Gallai rank bounds for any field, and show the intimate connection between this and depth-3 identity rank bounds. We also prove the first ever theorem about high dimensional Sylvester-Gallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth-3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.

Voronoi summation formulae and multiplicative functions on permutations

Abstract: We prove a Tauberian theorem for the Voronoi summation method of divergent series with an estimate of the remainder term. The results on the Voronoi summability are then applied to analyze the mean values of multiplicative functions on random permutations.

Singular equivariant asymptotics and Weyl's law

Abstract: We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed G-manifold M, where G is a compact, connected Lie group acting effectively and isometrically on M. Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it with the reduction of the corresponding Hamiltonian flow, proving that the equivariant spectral counting function satisfies Weyl's law, together with an estimate for the remainder.

Positive solutions to a linearly perturbed critical growth biharmonic problem

Abstract: Existence and nonexistence results for positive solutions to a linearly perturbed critical growth biharmonic problem under Steklov boundary conditions, are determined. Furthermore, by investigating the critical dimensions for this problem, a Sobolev inequality with remainder terms, of both interior and boundary type, is deduced.

On fuzzy taylor formulae

Abstract: We present Fuzzy Taylor formulae with integral remainder in the univariate and multivariate cases, analogs of the real setting. This chapter is based on [19].

Unified Nearly Optimal Algorithms for Structured Integer Matrices

Abstract: Our subject is the solution of a structured linear system of equations, which is closely linked to computing a shortest displacement generator for the inverse of its structured coefficient matrix We consider integer matrices with the displacement structure of Toeplitz, Hankel, Vandermonde, and Cauchy types and combine the unified divide-and-conquer MBA algorithm (due to Morf 1974, 1980 and Bitmead and Anderson 1980) with the Chinese remainder algorithm to solve both computational problems within nearly optimal randomized Boolean and word time bounds The bounds cover the cost of both solution and its correctness verification The algorithms and nearly optimal time bounds are extended to the computation of the determinant of a structured integer matrix, its rank and a basis for its null space and further to some fundamental computations with univariate polynomials that have integer coefficients

Remainder terms in a higher order Sobolev inequality

Abstract: For higher order Hilbertian Sobolev spaces, we improve the embedding inequality for the critical L p -space by adding a remainder term with a suitable weak norm.

Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

Abstract: Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "N-point" functions for the simple case of zero-dimensional 4 field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymp- totic level. The Mellin-Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A nume- rical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.

A robust Chinese remainder theorem with its applications in moving target Doppler estimation

Abstract: The Chinese remainder theorem (CRT) is an ancient result about simultaneous congruences in number theory, which reconstructs a large integer from its remainders modulo several moduli. It is well known that the CRT has tremendous applications in many fields, such as computing and cryptography, an important one of which could be radar signal processing and radar imaging. However, it is also well-known that CRT is not robust in the sense that a small error in any remainders may cause a larger error in the reconstruction result, which will lead to a non-robust estimation. In this paper, we introduce a robust reconstruction algorithm called robust CRT. We show that, using this robust CRT algorithm, 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. Although CRT has existed for about 2500 years, this robustness is the first time in the literature. Then, we show how this robust CRT can be used into the field of radar detection and Doppler ambiguity resolution, especially for fast moving targets, and later, simulations are given to illustrate the effectiveness and validness of this robust CRT algorithm.

The non-compact elliptic genus: mock or modular

Abstract: We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.

Device and method for detecting remainder of sealed electronic element based on random vibration

Abstract: The invention discloses a device and a method for detecting a remainder of a sealed electronic element based on random vibration, relates to the device and the method for detecting the remainder of the sealed electronic element, and solves the problem that the device and the method for detecting the remainder of the sealed electronic element cannot play the best detection capacity of detection equipment due to the adoption of fixed-frequency sinusoidal vibration. The device comprises that: a waveform generating module of the device receives random waveform data of an intelligent control module to form a random waveform drive vibration table of a specified frequency band. The method comprises the following steps: acquiring acceleration signals of the vibration table in real time by using the intelligent control module; performing power spectrum estimation; comparing the powder spectrum with a reference spectrum; performing spectrum equalization; performing Fourier transform on the equalized powder spectrum after phase randomization; transmitting a converted result after time domain randomization into a waveform generating circuit, and outputting and loading the converted result to the vibration table; and executing repeatedly for realizing closed loop control. The device and the method are suitable to be used in a process of detecting the remainder of the sealed electronic device.

Geometrically convex functions and estimation of remainder terms for taylor expansion of some functions

Abstract: zmeˇ) Abstract. In this paper, we establish two integral inequalities for geometrically convex functions. As consequences, we get the estimation for remainder terms of Taylor series for e −x ,s inx and cosx.

A digit-set-interleaved radix-8 division/square root kernel for double-precision floating point

Abstract: A common and very efficient approach to division and square root is the subtractive SRT algorithm combined with a redundant partial remainder representation like carry-save. A recently proposed modification of the SRT algorithm for division reduces the number of comparators inside the Quotient Digit Selection Function (QDSF) to the number necessary in a non-redundant implementation and derives partial remainders directly from comparison results calculated inside the QDSF. In this paper it is shown that this modified approach is also applicable to square root operations in an efficient way. A combined radix-8 division and square root kernel for double-precision floating point was synthesized using a 40-nm general-purpose cell library. The implementation comprises a critical path of only 20.8 fanout-4 inverter delays at worst case conditions which is comparable to 20.0 inverter delays published for a high-speed radix-4 SRT implementation. Furthermore, the proposed algorithm reduces the total area compared to equivalent SRT-based implementations.

Arithmetical functions involving exponential divisors: note on two papers by L. Toth

Abstract: Asymptotic estimates of L. Toth (5, 6) on the summatory functions of three arithmetical functions involving exponential divisors are improved. For two of them the improvement is on the upper bound of the size of the remainder term (O-estimate), and is reached by appealing to lattice points estimates using exponent pairs due to Kratzel (1), and by having as well a closer look at the flrst terms of the generating Dirichlet series. For the third one, a lower bound on the size of the remainder term (›-estimate) is replaced by two-sided oscillation (›§-estimate), by appealing to a method of Petermann and Wu (2).

Telemetry Coding and Surface Detection for a Mud Pulser

Abstract: A method (100) for encoding a non-negative integer, for example, representative of MWD/LWD data, includes encoding (104) at least a portion of the integer using at least a first order Fibonacci derived sequence. The remainder of the integer may be encoded (106) using conventional Fibonacci encoding. The invention tends to improve coding efficiency, downhole and surface synchronization, and surface detection. The encoding method may preferably be used for transmitting non-negative integer representative of downhole data to a surface location. There is also disclosed a related method of for receiving an encoded integer.