Remainder

About: Remainder is a research topic. Over the lifetime, 3443 publications have been published within this topic receiving 35379 citations. The topic is also known as: dividend mod divisor.

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.

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.

Efficient Implementations of the Chinese Remainder Theorem for Sign Detection and Residue Decoding

Abstract: Two conversion techniques based on the Chinese remainder theorem are developed for use in residue number systems. The new implementations are fast and simple mainly because adders modulo a large and arbitrary integer M are effectively replaced by binary adders and possibly a lookup table of small address space. Although different in form, both techniques share the same principle that an appropriate representation of the summands must be employed in order to evaluate a sum modulo M efficiently. The first technique reduces the sum modulo M in the conversion formula to a sum modulo 2 through the use of fractional representation, which also exposes the sign bit of numbers. Thus, this technique is particularly useful for sign detection and for any operation requiring a comparison with a binary fraction of M. The other technique is preferable for the full conversion from residues to unsigned or 2's complement integers. By expressing the summands in terms of quotients and remainders with respect to a properly chosen divisor, the second technique systematically replaces the sum modulo M by two binary sums, one accumulating the quotients modulo a power of 2 and the other accumulating the remainders the ordinary way. A final recombination step is required but is easily implemented with a small lookup table and binary adders.

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.