Showing papers on "Remainder published in 2012"

Analytic two-loop form factors in N = 4 SYM

Abstract: We derive a compact expression for the three-point MHV form factors of half- BPS operators in $ \mathcal{N} = 4 $ super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symme- tries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical poly- logarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of the two- loop Higgs plus three-gluon scattering amplitudes in QCD.

BFKL approach and 2 → 5 maximally helicity violating amplitude in N = 4 super-Yang-Mills theory

Abstract: We study maximally helicity violating amplitude for the $2\ensuremath{\rightarrow}5$ scattering in the multi-Regge kinematics. The Mandelstam cut correction to the Bern-Dixon-Smirnov amplitude is calculated in the leading logarithmic approximation and the corresponding remainder function is given to any loop order in a closed integral form. We show that the leading logarithmic approximation remainder function at two loops for $2\ensuremath{\rightarrow}5$ amplitude can be written as a sum of two $2\ensuremath{\rightarrow}4$ remainder functions due to recursive properties of the leading order impact factors. We also make some generalizations for the maximally helicity violating amplitudes with more external particles. The results of the present study are in agreement with the all leg two-loop symbol derived by Caron-Huot as shown in a parallel paper of one of the authors with collaborators.

The Operator Product Expansion Converges in Perturbative Field Theory

Abstract: We show, within the framework of the massive Euclidean \({\varphi^4}\) -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.

The two-loop six-point amplitude in ABJM theory

Abstract: In this paper we present the first analytic computation of the six-point two-loop amplitude of ABJM theory. We show that the two-loop amplitude consist of corrections proportional to two distinct local Yangian invariants which can be identified as the tree- and the one-loop amplitude respectively. The two-loop correction proportional to the tree-amplitude is identical to the one-loop BDS result of N=4 SYM plus an additional remainder function, while the correction proportional to the one-loop amplitude is finite. Both the remainder and the finite correction are dual conformal invariant, which implies that the two-loop dual conformal anomaly equation for ABJM is again identical to that of one-loop N=4 SYM, as was first observed at four-point. We discuss the theory on the Higgs branch, showing that its amplitudes are infrared finite, but equal, in the small mass limit, to those obtained in dimensional regularization.

Remainder terms in the fractional Sobolev inequality

Abstract: We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak $L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where $s$ is an even integer.

Inference of polynomial invariants for imperative programs: A farewell to Gröbner bases

Abstract: The article presents a static analysis for computing polynomial invariants for imperative programs. The analysis is derived from an abstract interpretation of a backwards semantics, and computes preconditions for equalities of the form g=0 to hold at the end of execution. A distinguishing feature of the technique is that it computes polynomial loop invariants without resorting to Grobner base computations. The analysis uses remainder computations over parameterized polynomials in order to handle conditionals and loops efficiently. The algorithm can analyze and find a large majority of loop invariants reported previously in the literature, and executes significantly faster than implementations using Grobner bases.

A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies.

Abstract: A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. Th...

Table-Based division by small integer constants

Abstract: Computing cores to be implemented on FPGAs may involve divisions by small integer constants in fixed or floating point This article presents a family of architectures addressing this need They are derived from a simple recurrence whose body can be implemented very efficiently as a look-up table that matches the hardware resources of the target FPGA For instance, division of a 32-bit integer by the constant 3 may be implemented by a combinatorial circuit of 48 LUT6 on a Virtex-5 Other options are studied, including iterative implementations, and architectures based on embedded memory blocks This technique also computes the remainder An efficient implementation of the correctly rounded division of a floating-point constant by such a small integer is also presented

T-functions and multi-gluon scattering amplitudes

Abstract: We study gluon scattering amplitudes/Wilson loops in $ \mathcal{N} = 4 $ super Yang-Mills theory at strong coupling which correspond to minimal surfaces with a light-like polygonal boundary in AdS3. We find a concise expression of the remainder function in terms of the T-function of the associated thermodynamic Bethe ansatz (TBA) system. Continuing our previous work on the analytic expansion around the CFT/regular-polygonal limit, we derive a formula of the leading-order expansion for the general 2n-point remainder function. The T-system allows us to encode its momentum dependence in only one function of the TBA mass parameters, which is obtained by conformal perturbation theory. We compute its explicit form in the single mass cases. We also find that the rescaled remainder functions at strong coupling and at two loops are close to each other, and their ratio at the leading order approaches a constant near 0.9 for large n.

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.

The Semigroups of Order 10

Abstract: The number of finite semigroups increases rapidly with the number of elements. Since existing counting formulae do not give the complete number of semigroups of given order up to equivalence, the remainder can only be found by careful search. We describe the use of mathematical results combined with distributed Constraint Satisfaction to show that the number of non-equivalent semigroups of order 10 is 12,418,001,077,381,302,684. This solves a previously open problem in Mathematics, and has directly led to improvements in Constraint Satisfaction technology.

Exchanged toric developments and bounded remainder sets

Abstract: Using exchanged toric developments, we construct tilings of toric by bounded remainder sets. To this end, two special methods for stretching the unit cubes and a general method for multiplying toric developments are used. A multidimensional analog of Hecke’s theorem on the distribution of fractional parts is proved. Bibliography: 7 titles.

Sharp hardy inequalities in the half space with trace remainder term

Abstract: In this paper, we deal with a class of inequalities which interpolate the Kato inequality and the Hardy inequality in the half space. Starting from the classical Hardy’s inequality in the half space R + n = R n − 1 × ( 0 , ∞ ) , we show that, if we replace the optimal constant ( n − 2 ) 2 4 with a smaller one ( β − 2 ) 2 4 , 2 ≤ β n , then we can add an extra trace-term similar to that one that appears in the Kato inequality. The constant in the trace remainder term is optimal and it tends to zero when β goes to n , while it is equal to the optimal constant in Kato’s inequality when β = 2 . The approach is based on a very classical method of Calculus of Variation due to Weierstrass (and developed by Hilbert) that usually is considered to prove that the solutions of the Euler–Lagrange equation associated to a functional are, in fact, extremals. In this paper, we will show how this method is well suited also to functionals that have no extremals.

Bernstein-type Operators on a Triangle with One Curved Side

Abstract: We construct Bernstein-type operators on a triangle with one curved side. We study univariate operators, their product and Boolean sum, as well as their interpolation properties, the order of accuracy (degree of exactness, precision set) and the remainder of the corresponding approximation formulas. We also give some illustrative examples.

Some properties of log-convex function and applications for the exponential function

Abstract: In this paper, some properties of log-convex function are researched, and integral inequalities of log-convex functions are proved. As an application, an estimation formula of remainder terms in Taylor series expansion is given.

Control area network based quotient remainder compression-algorithm for automotive applications

Abstract: The present day automotives have variety of attractive features. Incorporating these features is possible with more electronics or embedded systems inside the vehicle. Controller Area Network (CAN) protocol offers low cost solution for communication between these embedded systems. These embedded systems communicate on CAN bus via message passing. With limited speed and bandwidth offered by CAN, there are limitations in communication. One of the solutions to overcome these limitations is the use of Data-Reduction (DR) algorithms or techniques. These algorithms make it possible to send fewer amount of data in the given time, thus reducing the bandwidth per message. This paper develops an alternative approach to data reduction technique called Quotient Remainder Compression (QRC) algorithm. It provides double or at least equal range to the variation in the parameter as compared to the Enhanced Data Reduction (EDR) algorithm. The compression ratio of QRC is comparable to the earlier algorithms.

Asymptotics of solutions of the heat equation in cones and dihedra

Abstract: The authors deal with the asymptotics of solutions of the first boundary value problem for the heat equation near vertices of cones or edges They obtain estimates for the remainder in weighted Lp,

Oscillations localisées sur les diviseurs

Abstract: Let f be a real arithmetic function and ∆(n, f) denote the corresponding generalization of Hooley's Delta-function. We investigate weighted moments of ∆(n; f) for oscillating functions f, typical cases being those of a non principal Dirichlet character or of the Mobius function. We obtain, in particular, sharp bounds up to factors (log x) o(1) for all weighted finite integral, even moments computed on the integers not exceeding x. This is the key step to the proof, given in a subsequent work, of Manin's conjecture, in the strong form conjectured by Peyre and with an e↵ective remainder term, for all Châtelet surfaces. The proof of the main results rest upon a genuinely new approach for Hooley-type functions.

Exact spectral asymptotics on the Sierpinski gasket

Abstract: One of the ways that analysis on fractals is more complicated than analysis on manifolds is that the asymptotic behavior of the spectral counting function $N(t)$ has a power law modulated by a nonconstant multiplicatively periodic function. Nevertheless, we show that for the Sierpinski gasket it is possible to write an exact formula, with no remainder term, valid for almost every $t$. This is a stronger result than is valid on manifolds.

Hexagon remainder function in the limit of self-crossing up to three loops

Abstract: We consider Wilson loops in planar $\mathcal{N} = 4$ SYM for null polygons in the limit of two crossing edges. The analysis is based on a renormalisation group technique. We show that the previously obtained result for the leading and next-leading divergent term of the two loop hexagon remainder is in full agreement with the appropriate continuation of the exact analytic formula for this quantity. Furthermore, we discuss the coefficients of the leading and next-leading singularity for the three loop remainder function for null n-gons with n ≥ 6.

A new generalization of some integral inequalities and their applications

Abstract: In this paper, a new identity for convex functions is derived. A consequence of the identity is that we can derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are convex. Some applications to special means of real numbers are also given.

Fast rates in learning with dependent observations

Abstract: In this paper we tackle the problem of fast rates in time series forecasting from a statistical learning perspective. In a serie of papers (e.g. Meir 2000, Modha and Masry 1998, Alquier and Wintenberger 2012) it is shown that the main tools used in learning theory with iid observations can be extended to the prediction of time series. The main message of these papers is that, given a family of predictors, we are able to build a new predictor that predicts the series as well as the best predictor in the family, up to a remainder of order $1/\sqrt{n}$. It is known that this rate cannot be improved in general. In this paper, we show that in the particular case of the least square loss, and under a strong assumption on the time series (phi-mixing) the remainder is actually of order $1/n$. Thus, the optimal rate for iid variables, see e.g. Tsybakov 2003, and individual sequences, see \cite{lugosi} is, for the first time, achieved for uniformly mixing processes. We also show that our method is optimal for aggregating sparse linear combinations of predictors.

Filtration pressure field in an inhomogeneous bed in constant drainage

Abstract: UDC 532.546 Based on the modification of the "exact-on-average" method, simple analytical formulas have been found for calculation, in the zero and first asymptotic approximations, of pressure fields in an inhomogeneous anisot- ropic bed in constant drainage for the case of one-dimensional linear flow. The results of calculations of the fields from the obtained formulas have been given. Introduction. Problems on pressure fields appearing in filtration of liquids in porous media form the basis for the theory of mass transfer in these media and are of great practical importance for oil and gas production, hydrogeol- ogy, ecology, etc. (1). The problem on filtration of a liquid occupies a special place among the indicated problems be- cause of the variety of conditions of such filtration and its practical significance. Analytical dependences for description of pressure fields appearing in the indicated bed as a result of the liquid filtration in it can be obtained by modification of asymptotic methods whose capabilities have not been completely realized. Such modification is ef- fected through selection of the formal parameter of asymptotic expansion. It becomes necessary to construct the zero and first coefficients of this expansion, boundary-layer functions, and estimating expressions for the remainder term. The resulting zero coefficient of asymptotic expansion describes the averaged value of a physical parameter. Construc- tion of the first expansion coefficient requires supplementary conditions that are based on the trivial solution of the av- eraged problem for the remainder term. From this viewpoint, expressions for the zero and first approximations are called "exact-on-average" (2). This work seeks to determine the zero and first coefficients of asymptotic expansion that describe pressure fields in an inhomogeneous anisotropic bed with constant drainage from one-dimensional linear flow. In this case the zero coefficient describes pressure values averaged over the bed's thickness, whereas the first coefficient refines de- scription of the fields in the averaging zone and determines the steady-state field at large times. 1. Formulation of the Problem for Linear Flow in Constant Drainage. Figure 1 gives the flow geometry in a plane coordinate system (x, z) whose z axis coincides with the axis of the well. The medium is represented by three regions with plane boundaries z = 1; the covering bed and the underlying bed are assumed to be low-perme- able in the horizontal direction k 1x = 0; the central region −1 < z < 1 is highly permeable in the horizontal k x and in the vertical k z directions. For the sake of simplicity it is assumed that flow is one-dimensional and linear along the x axis, the surround- ing rocks are strongly anisotropic, and the vertical permeability k 1z dominates the horizontal permeability k 1x in them. This enables us to disregard the term with the second derivative with respect to the horizontal coordinate x for the sur- rounding medium. Next, we assume that the properties of the underlying and covering beds are identical. In accordance with this, we can simplify the formulation of the problem, using the symmetry condition ∂P ⁄ ∂z = 0 and z = 0. The considered problem is substantially simplified through the use of the so-called quasistationary approxima- tion without perceptible distortions of the basic regularities under study. This approximation is widely used in electro- dynamics for studying electromagnetic fields in electric circuits; its essence is neglect of the time derivative in the

Trudinger-Moser inequality with remainder terms

Abstract: The paper gives an improvement of the Trudinger-Moser inequality, in which the constraint set is defined not by the squared gradient norm, but with the squared gradient norm minus a remainder term of the weighted L^p-type. This is a two-dimensional counterpart of the Hardy-Sobolev-Mazya inequality in higher dimensions, which is a similar refinement of the limiting Sobolev inequality. In particular, we generalize two known cases of remainder terms of potential type (i.e. weighted L^2-terms) found by Adimurthi and Druet and by Wang and Ye. In addition, we prove the inequality with a L^p-remainder, p>2, as well as give an analogous improvement for the Onofri-Beckner inequality.

Method and device for generating big prime

Abstract: The invention discloses a method for generating a large prime number and a system thereof. The method includes Step 1, generating a random number in size corresponding to number of digits input by a user, in which number of digits of the random number is identical to number of digits input by the user; Step 2, obtaining remainders by dividing all prime numbers in a predetermined little prime number table by a current value of the random number so as to form a remainder array; Step 3, determining whether there is a remainder 0 in the remainder array, if yes, going to Step 4; otherwise, going to Step 5; Step 4, updating the random number with a predetermined step, updating remainders in remainder array and going to Step 3; Step 5, checking for whether a current value of the random number is a prime number, if yes, going to Step 6; otherwise, going to Step 4; and Step 6, storing or outputting the current value of the random number. The solution provided by embodiments of the invention reduces primality test times, and thus saves time of generating a large prime number.

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.