Showing papers on "Remainder published in 2019"

Uniformity and the delta method

Abstract: When are asymptotic approximations using the delta-method uniformly valid? We provide sufficient conditions as well as closely related necessary conditions for uniform negligibility of the remainder of such approximations. These conditions are easily verified by empirical practitioners and permit to identify settings and parameter regions where pointwise asymptotic approximations perform poorly. Our framework allows for a unified and transparent discussion of uniformity issues in various sub-fields of statistics and econometrics. Our conditions involve uniform bounds on the remainder of a first-order approximation for the function of interest.

Effective Algorithms for Finding the Remainder of Multi-Digit Numbers

Abstract: In this paper, the theoretical foundations of the developed methods for finding the remains of multi-digit numbers and the Mersenne number in a binary number system are given, the use of which allows reducing the time complexity. The numerical experiments of the time characteristics of program implementations of the proposed approaches and the classical method are given. The scheme of the algorithm for finding the remainder of the multi-digit number and the multi-digit Mersenne number is given.

On differences of linear positive operators

Abstract: In this paper we consider two different general linear positive operators defined on unbounded interval and obtain estimates for the differences of these operators in quantitative form. Our estimates involve an appropriate K-functional and a weighted modulus of smoothness. Similar estimates are obtained for Chebyshev functional of these operators as well. All considerations are based on rearrangement of the remainder in Taylor’s formula. The obtained results are applied for some well known linear positive operators.

All two-loop MHV remainder functions in multi-Regge kinematics

Abstract: We introduce a method to extract the symbol of the coefficient of (2πi)2 of MHV remainder functions in planar $$ \mathcal{N} $$ = 4 Super Yang-Mills in multi-Regge kinematics region directly from the symbol in full kinematics. At two loops this symbol can be uplifted to the full function in a unique way, without any beyond-the-symbol ambiguities. We can therefore determine all two-loop MHV amplitudes at function level in all kinematic regions with different energy signs in multi-Regge kinematics. We analyse our results and we observe that they are consistent with the hypothesis of a contribution from the exchange of a three-Reggeon composite state starting from two loops and eight points in certain kinematic regions.

Lace Expansion and Mean-Field Behavior for the Random Connection Model

Abstract: We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider three versions of the connection function $\varphi$: a finite-variance version (including the Boolean model), a spread-out version, and a long-range version. For sufficiently large dimension (resp., spread-out parameter and $d>6$), we then prove the convergence of the lace expansion, derive the triangle condition, and establish an infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent $\gamma$ takes its mean-field value \gamma=1$ and that the percolation function is continuous.

Edge-localized states on quantum graphs in the limit of large mass

Abstract: In this work, we construct and quantify asymptotically in the limit of large mass a variety of edge-localized stationary states of the focusing nonlinear Schrodinger equation on a quantum graph. The method is applicable to general bounded and unbounded graphs. The solutions are constructed by matching a localized large amplitude elliptic function on a single edge with an exponentially smaller remainder on the rest of the graph. This is done by studying the intersections of Dirichlet-to-Neumann manifolds (nonlinear analogues of Dirichlet-to-Neumann maps) corresponding to the two parts of the graph. For the quantum graph with a given set of pendant, looping, and internal edges, we find the edge on which the state of smallest energy at fixed mass is localized. Numerical studies of several examples are used to illustrate the analytical results.

Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

Abstract: This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies.In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants.In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures.In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.

Algorithms and Data Structures for Sparse Polynomial Arithmetic

Abstract: We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, respectively, where the divisor set is assumed to form a triangular set. Our operations make use of two data structures for sparse distributed polynomials and sparse recursively viewed polynomials, with a keen focus on locality and memory usage for optimized performance on modern memory hierarchies. Experimentation shows that these new implementations compare favorably against competing implementations, performing between a factor of 3 better (for multiplication over the integers) to more than 4 orders of magnitude better (for pseudo-division with respect to a triangular set).

Remarks on the Hardy type inequalities with remainder terms in the framework of equalities

Abstract: We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers.

Faster remainder by direct computation: Applications to compilers and software libraries

Abstract: On common processors, integer multiplication is many times faster than integer division. Dividing a numerator n by a divisor d is mathematically equivalent to multiplication by the inverse of the divisor (n / d = n x 1/d). If the divisor is known in advance---or if repeated integer divisions will be performed with the same divisor---it can be beneficial to substitute a less costly multiplication for an expensive division. Currently, the remainder of the division by a constant is computed from the quotient by a multiplication and a subtraction. But if just the remainder is desired and the quotient is unneeded, this may be suboptimal. We present a generally applicable algorithm to compute the remainder more directly. Specifically, we use the fractional portion of the product of the numerator and the inverse of the divisor. On this basis, we also present a new, simpler divisibility algorithm to detect nonzero remainders. We also derive new tight bounds on the precision required when representing the inverse of the divisor. Furthermore, we present simple C implementations that beat the optimized code produced by state-of-art C compilers on recent x64 processors (e.g., Intel Skylake and AMD Ryzen), sometimes by more than 25%. On all tested platforms including 64-bit ARM and POWER8, our divisibility-test functions are faster than state-of-the-art Granlund-Montgomery divisibility-test functions, sometimes by more than 50%.

Uniform Approximations by Fourier Sums in Classes of Generalized Poisson Integrals

Abstract: We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums in the uniform metric in classes of 2π-periodic functions, representable in the form of convolutions of functions φ, which belong to the unit balls of the spaces Lp, with generalized Poisson kernels. For the asymptotic equalities obtained we introduce the estimates of the remainder, which are expressed in an explicit form via the parameters of the problem.

A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points

Abstract: In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, $E_\lambda(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than $\lambda^2$ as $\lambda \to\infty$. In the regime where $x,y$ are restricted to a compact neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $E_\lambda$ and its derivatives of all orders, which generalizes a result of Berard, who treated the on-diagonal case $E_\lambda(x,x)$. When $x,y$ avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for $E_\lambda$. Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the $C^\infty$ topology to a universal scaling limit at an inverse logarithmic rate.

On the remainder term of the Weyl law for congruence subgroups of Chevalley groups

Abstract: Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.

On the asymptotics of Hecke operators for reductive groups

Abstract: In this paper, we study the asymptotic behavior of the traces of Hecke operators for spherical discrete automorphic representations of fixed level on general split reductive groups over $\mathbb{Q}$. Under a condition on the analytic behavior of intertwining operators, which is known for the classical groups and the exceptional group $G_2$, we obtain the expected asymptotics in terms of the spherical Plancherel measure and an explicit estimate for the remainder.

Improved interpolation inequalities and stability

Abstract: For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carr\'e du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

An Improved Remainder Estimate in the Weyl Formula for the Planar Disk

Abstract: Y. Colin de Verdiere proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is no more than $$O(\mu ^{2/3})$$ . In this paper, we study this problem from spectral geometry by using analysis and number theory. More precisely, by combining with the method of exponential sum estimation, we give a sharper remainder term estimate $$O(\mu ^{2/3-1/495})$$ .

Refinements of Majorization Inequality Involving Convex Functions via Taylor’s Theorem with Mean Value form of the Remainder

Abstract: The aim of this paper is to establish some refined versions of majorization inequality involving twice differentiable convex functions by using Taylor theorem with mean-value form of the remainder. Our results improve several results obtained in earlier literatures. As an application, the result is used for deriving a new fractional inequality.

On the Weyl Law for Quantum Graphs

Abstract: Roughly speaking, the Weyl law describes the asymptotic distribution of eigenvalues of the Laplacian that can be attached to different objects and can be analyzed in different settings. The form of the remainder term in the Weyl law is very significant in applications, and a power-saving exponent in the remainder term is appreciated. We are dealing with Laplacian defined on compact metric graph with general self-adjoint boundary conditions. The main purpose of this paper is to present application of the special form of the Tauberian theorem for the Laplace transform to the suitably transformed trace formula in the above-mentioned quantum graphs setting. The key feature of our method is that it produces a power-saving form of the reminder term and hence represents improvement in classical methods, which may be applied in other settings as well. The obtained form of the Weyl law is with the power saving of 1 / 3 in the remainder term.

Faster Remainder by Direct Computation: Applications to Compilers and Software Libraries

Abstract: On common processors, integer multiplication is many times faster than integer division. Dividing a numerator n by a divisor d is mathematically equivalent to multiplication by the inverse of the divisor (n / d = n x 1/d). If the divisor is known in advance---or if repeated integer divisions will be performed with the same divisor---it can be beneficial to substitute a less costly multiplication for an expensive division. Currently, the remainder of the division by a constant is computed from the quotient by a multiplication and a subtraction. But if just the remainder is desired and the quotient is unneeded, this may be suboptimal. We present a generally applicable algorithm to compute the remainder more directly. Specifically, we use the fractional portion of the product of the numerator and the inverse of the divisor. On this basis, we also present a new, simpler divisibility algorithm to detect nonzero remainders. We also derive new tight bounds on the precision required when representing the inverse of the divisor. Furthermore, we present simple C implementations that beat the optimized code produced by state-of-art C compilers on recent x64 processors (e.g., Intel Skylake and AMD Ryzen), sometimes by more than 25%. On all tested platforms including 64-bit ARM and POWER8, our divisibility-test functions are faster than state-of-the-art Granlund-Montgomery divisibility-test functions, sometimes by more than 50%.

A Generalization of the Remainder Theorem and Factor Theorem.

Abstract: We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division metho...

Rellich inequalities in bounded domains

Abstract: We find necessary and sufficient conditions for the validity of weighted Rellich inequalities in Lp for functions in bounded domains vanishing at the boundary. General operators like L = Delta+ c\|x|^2x nabla-b\|x|^2 are considered. Critical cases and remainder terms are also investigated.

Boolean Minimization of Projected Sums of Products via Boolean Relations

Abstract: Projected Sums of Products (PSOPs) are a Generalized Shannon Decomposition (GSD) with remainder that restructures a logic function into three logic blocks corresponding to a logic bi-decomposition plus a reminder generated by a cofactoring function. In this paper we discuss a Boolean synthesis technique for PSOPs, which exploits the fact that the resulting logical structure induces don't care conditions that can be exploited to reduce the problem of area minimization to Boolean relation minimization, with the guarantee that all valid realizations of the circuit are considered. This technique is more general than the algebraic methods investigated so far. Moreover, we characterize the points that are in the remainder with a simple procedure that implies a fast construction of the Boolean relation for important classes of cofactoring functions like the chain of XORs or ANDs. We report experiments confirming the effectiveness in area of the proposed approach based on Boolean relations, with better run times for some cost functions.

Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems

Abstract: In our article, three iterative methods are performed to solve the nonlinear differential equations that represent the straight and radial fins affected by thermal conductivity. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. For comparison purposes, the numerical solutions were further achieved by using the fourth Runge-Kutta (RK4) method, Euler method and previous analytical methods that available in the literature. Moreover, the convergence of the proposed methods was discussed and proved. In addition, the maximum error remainder values are also evaluated which indicates that the proposed methods are efficient and reliable. Our computational works have been done by using the computer algebra system MATHEMATICA®10 to evaluate the terms in the iterative processes.

Bounded remainder sets for rotations on the adelic torus

Abstract: In this paper we give an explicit construction of bounded remainder sets of all possible volumes, for any irrational rotation on the adelic torus $\mathbb A/\mathbb Q$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument which originated in the study of deformation properties of mathematical quasicrystals.

Reduced Collatz Dynamics Data Reveals Properties for the Future Proof of Collatz Conjecture

Abstract: Collatz conjecture is also known as 3X + 1 conjecture. For verifying the conjecture, we designed an algorithm that can output reduced dynamics (occurred 3 × x + 1 or x/2 computations from a starting integer to the first integer smaller than the starting integer) and original dynamics of integers (from a starting integer to 1). Especially, the starting integer has no upper bound. That is, extremely large integers with length of about 100,000 bits, e.g., 2100000 − 1, can be verified for Collatz conjecture, which is much larger than current upper bound (about 260). We analyze the properties of those data (e.g., reduced dynamics) and discover the following laws; reduced dynamics is periodic and the period is the length of its reduced dynamics; the count of x/2 equals to minimal integer that is not less than the count of (3 × x + 1)/2 times ln(1.5)/ln(2). Besides, we observe that all integers are partitioned regularly in half and half iteratively along with the prolonging of reduced dynamics, thus given a reduced dynamics we can compute a residue class that presents this reduced dynamics by a proposed algorithm. It creates one-to-one mapping between a reduced dynamics and a residue class. These observations from data can reveal the properties of reduced dynamics, which are proved mathematically in our other papers (see references). If it can be proved that every integer has reduced dynamics, then every integer will have original dynamics (i.e., Collatz conjecture will be true). The data set includes reduced dynamics of all odd positive integers in [3, 99999999] whose remainder is 3 when dividing 4, original dynamics of some extremely large integers, and all computer source codes in C that implement our proposed algorithms for generating data (i.e., reduced or original dynamics).

On zero-remainder conditions in the Bethe ansatz

Abstract: We prove that physical solutions to the Heisenberg spin chain Bethe ansatz equations are exactly obtained by imposing two zero-remainder conditions. This bridges the gap between different criteria, yielding an alternative proof of a recently devised algorithm based on $QQ$ relations, and solving its minimality issue.

A Hybrid Method for Spectral Translation Equivalent Boolean Functions

Abstract: The equivalence of Boolean functions with respect to five invariance (aka translation) operations has been well considered with respect to the Rademacher-Walsh spectral domain. In this paper, we introduce a hybrid approach that uses both the Reed-Muller and the Rademacher-Walsh spectra. A novel hybrid algorithm that maps a Boolean function to a representative function for the equivalence class containing the original function is presented. The algorithm can be used to determine a sequence of translations that maps one function to an equivalent function. We present experimental results that show the hybrid algorithm can determine the equivalence classes for 5 variables much more efficiently than before. We also show that for 6 variables where there are 150,357 equivalence classes, 8 are very difficult, a further 58 are difficult and the remainder are straightforward in terms of the CPU time required by the hybrid algorithm.

Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder

Abstract: We reproduce the two-loop seven-point remainder function in planar, maximally supersymmetric Yang-Mills theory by direct integration of conformally-regulated chiral integrands. The remainder function is obtained as part of the two-loop logarithm of the MHV amplitude, the regularized form of which we compute directly in this scheme. We compare the scheme-dependent anomalous dimensions and related quantities in the conformal regulator with those found for the Higgs regulator.

Frequency Estimator Based on Spectrum Correction and Remainder Sifting for Undersampled Real-Valued Waveforms

Abstract: Frequency estimation of undersampled waveforms receives increasing attention in communication, radar signal processing, instrumentation and measurements, and so on. However, due to the lack of recognizing the correct remainder between two side spectra, the existing Chinese Remainder Theorem (CRT)-based frequency estimators can hardly deal with real-valued signals. To achieve this goal, this paper proposes an estimator combining spectrum correction (aiming to enhance reconstruction accuracy by incorporating the fractional parts of DFT remainders), closed-form CRT, and a remainder sifting approach. Based on the detection of an undersampled waveform’s zero crossing point, this solution can pick out the correct remainder between two side spectra, which ensures that the CRT achieves a valid reconstruction. Compared with the existing Maroosi-Bizaki estimator, the proposed method not only enlarges the upper bound of frequency recovery but also possesses higher reconstruction accuracy (the relative error is less than 0.002%) with lower consumption of computational complexity. The numerical results verify the superior performances of our estimator.