Remainder Term Research Articles

Asymptotic linear expansion of regularized M-estimators

Parametric high-dimensional regression requires regularization terms to get interpretable models. The respective estimators correspond to regularized M-functionals which are naturally highly nonlinear. Their Gâteaux derivative, i.e., their influence curve linearizes the asymptotic bias of the estimator, but only up to a remainder term which is not guaranteed to tend (sufficiently fast) to zero uniformly on suitable tangent sets without profound arguments. We fill this gap by studying, in a unified framework, under which conditions the M-functionals corresponding to convex penalties as regularization are compactly differentiable, so that the estimators admit an asymptotically linear expansion. This key ingredient allows influence curves to reasonably enter model diagnosis and enable a fast, valid update formula, just requiring an evaluation of the corresponding influence curve at new data points. Moreover, this paves the way for optimally-robust estimators, bounding the influence curves in a suitable way.

On higher order Poincaré inequalities with radial derivatives and Hardy improvements on the hyperbolic space

In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, $$\begin{aligned} \int _{\mathbb {H}^{N}} |\nabla _{r,\mathbb {H}^{N}}^{k} u|^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ge \bigg (\frac{N-1}{2}\bigg )^{2(k-l)} \int _{\mathbb {H}^{N}} |\nabla _{r,\mathbb {H}^{N}}^{l} u|^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ \ \text { for all } u\in H^k(\mathbb {H}^{N}), \end{aligned}$$ where underlying space is N-dimensional hyperbolic space $$\mathbb {H}^{N}$$ , $$0\le l<k$$ are integers and the constant $$\big (\frac{N-1}{2}\big )^{2(k-l)}$$ is sharp. Furthermore we improve the above inequalities by adding Hardy-type remainder terms and the sharpness of some constants is also discussed.

Точные равенства наближения функций класса Соболева их обобщенными интегралами Пуассона

Решение задач о движении системы взаимодействующих материальных точек в большинстве случаев сводится как к обычным дифференциальным уравнениям, так и к уравнениям в частных производных. Одним из решений такого типа уравнений являются так называемые обобщенные интегралы Пуассона, которые в отдельных случаях превращаются в хорошо известные интегралы Абеля-Пуассона или бигармонические интегралы Пуассона. Существует ряд результатов по приближению различных классов дифференцируемых периодических и непериодических функций вышеупомянутыми интегралами (так называемая задача Колмогорова-Никольского в терминологии А.И. Степанця). Но практически во всех решенных задачах Колмогорова-Никольского как для интегралов Абеля-Пуассона, так и для бигармонических интегралов Пуассона с точки зрения математического моделирования (вычислительного эксперимента) существенный недостаток. Суть этого недостатка состоит в том, что в большинстве решенных ранее задач Колмо-горова-Никольского для интегралов Абеля-Пуассона и бигармонических интегралов Пуассона (в конечном итоге) был получен только главный и остаточный члены приближения, что существенно может влиять на точность вычислительного опыта. Данная работа посвящена получению чётких равенств приближения функций классов Соболева их обобщенными интегралами Пуассона. Итак, доказанная в работе теорема является обобщением и уточнением ранее известных результатов, характеризующих апроксимативные свойства интегралов Абеля-Пуассона и бигармонических интегралов Пуассона на классах дифференцируемых периодических функций. Особенностью решенной в работе задачи приближения для обобщенного интеграла Пуас-сона на классах дифференцирующих функций является то, что полученный результат удалось записать с помощью известных констант Ахиезера-Крейна-Фавара. Указанный факт значительно повышает точность результата математического моделирования (вычислительного эксперимента) любого реального процесса, описываемого с помощью обобщенного интеграла Пуассона. Эти результаты в дальнейшем смогут значительно расширить рамки применения задач Колмогорова-Никольского к математическому моделированию.

A second order numerical method for singularly perturbed problem with non-local boundary condition

The aim of this paper is to present a monotone numerical method on uniform mesh for solving singularly perturbed three-point reaction–diffusion boundary value problems. Firstly, properties of the exact solution are analyzed. Difference schemes are established by the method of the integral identities with the appropriate quadrature rules with remainder terms in integral form. It is then proved that the method is second-order uniformly convergent with respect to singular perturbation parameter, in discrete maximum norm. Finally, one numerical example is presented to demonstrate the efficiency of the proposed method.

On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers

We obtain asymptotic formulas with remainder terms for the hyperbolic summations sum _{mnle x} f((m,n)) and sum _{mnle x} f([m,n]), where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions f(n)=tau (n), log n, omega (n) and Omega (n). We also define a common generalization of the latter three functions, and prove a corresponding result.

Asymptotics with remainder term for moments of the total cycle number of random A-permutation

Abstract Dedicated to the memory of Alexander Ivanovich Pavlov. We consider the set of n-permutations with cycle lengths belonging to some fixed set A of natural numbers (so-called A-permutations). Let random permutation τ n be uniformly distributed on this set. For some class of sets A we find the asymptotics with remainder term for moments of total cycle number of τ n .

Extremes of a class of non-stationary Gaussian processes and maximal deviation of projection density estimates

In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. We focus on the case when the processes have finite number of points attaining their maximal variance, but, unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maximal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.

A uniform numerical method for solving singularly perturbed Fredholm integro-differential problem

In this paper, we deal with a class of boundary-value problems for the singularly perturbed Fredholm integro-differential equation. To solve the problem, we construct a new difference scheme by the method of integral identities using interpolating quadrature rules with remainder terms in integral form. We prove that the method is convergent in the discrete maximum norm, uniformly with respect to the perturbation parameter. We present numerical experiments which support the theoretical results.

Equidistribution of random walks on compact groups

Let X1,X2,… be independent, identically distributed random variables taking values from a compact metrizable group G. We prove that the random walk Sk=X1X2⋯Xk, k=1,2,… equidistributes in any given Borel subset of G with probability 1 if and only if X1 is not supported on any proper closed subgroup of G, and Sk has an absolutely continuous component for some k≥1. More generally, the sum ∑k=1Nf(Sk), where f:G→R is Borel measurable, is shown to satisfy the strong law of large numbers and the law of the iterated logarithm. We also prove the central limit theorem with remainder term for the same sum, and construct an almost sure approximation of the process ∑k≤tf(Sk) by a Wiener process provided Sk converges to the Haar measure in the total variation metric.

Method of moments with a choice of special basic functions for fourth-order partial differential equations

This paper presents an approximation method to solve the boundary value problem (BVP) for partial differential equations (PDEs) of a kind of the fourth order with the idea of discretization of a spatial variable by using the method of moments with a choice of special basic functions. A system of ordinary differential equations (ODEs) is obtained by multiplying the original equation by some auxiliary functions, followed by interpolation and integration over the spatial variable. Newton-Stirling, Hermite-Birkhoff and Hermite interpolations are flexibly applied to internal and pre-boundary nodes. In addition, boundary conditions are automatically satisfied without being approximated separately as in classical numerical methods (for example, method of grids, method of lines). Thus, the proposed schemes have a higher order of approximation. The main goal of the work is the construction of basic functions for a fourth-order differential operator and a possible increase in the order of the remainder term when passing to difference equations.

Some weighted Hardy and Rellich inequalities on the Heisenberg group

In this paper, we establish some weighted Hardy and Rellich inequalities and discuss its best constants on the Heisenberg group. Moreover, we also present a class of higher-order weighted Hardy–Rellich inequalities with the remainder term.

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

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

Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale hbar = N^{-1/3} in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

HDO-CO Complex: D-Bonded and H-Bonded Isomers and Intra- and Intermolecular Rovibrational States from Full-Dimensional and Fully Coupled Quantum Calculations.

We report full-dimensional and fully coupled quantum bound-state calculations of the J = 0, 1 intra- and intermolecular rovibrational states of the isotopically asymmetric HDO-CO complex. They are performed on the ab initio nine-dimensional (9D) potential energy surface (PES) [Liu, Y.; Li, J. Phys. Chem. Chem. Phys. 2019, 21, 24101]. The present study complements our earlier theoretical investigation of the 9D rovibrational level structure of the H2O-CO and D2O-CO complexes [Felker, P. M.; Bačić, Z. J. Chem. Phys. 2020, 153, 074107]. What distinguishes HDO-CO is that, unlike the two isotopically symmetric isotopologues, it does not display hydrogen-interchange tunneling but has two distinct isomers, the lower-energy D-bonded HOD-CO and the higher-energy H-bonded DOH-CO. The highly efficient methodology employed in the present calculations derives from our earlier study referenced above, taking into account the lower symmetry of HDO-CO. The full 9D rovibrational Hamiltonian is partitioned into three reduced-dimension Hamiltonians: the 5D rigid-monomer intermolecular vibrational Hamiltonian and two intramolecular vibrational Hamiltonians, one for the HDO monomer (3D) and another for the CO monomer (1D), and a 9D remainder term. The reduced-dimension Hamiltonians are diagonalized separately, and small portions of their low-energy eigenstates are incorporated in the compact final 9D product contracted basis covering all internal, intra- and intermolecular degrees of freedom of the complex. The 9D rovibrational Hamiltonian is diagonalized in this fully contracted basis. The calculations show that the eigenstates belonging to the D-bonded and H-bonded isomers, designated as D and H, respectively, are easy to identify, owing to the near-complete localization of their wave functions in either of the two minima on the PES. The computed intramolecular vibrational frequencies of the two monomers are either blue- or red-shifted, depending on the mode. The excitations of the intramolecular vibrational modes affect the energies of the low-lying D and H intermolecular vibrational states in the respective intramolecular manifolds. Comparison is made with the experimental data available in the literature.

On Gelfond-type problem for generalized Zeckendorf representations

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.

Generalized Rauzy tilings and linear recurrence sequences

Rauzy introduced a fractal set associated with the two-dimensional toric shift by the vector (𝛽−1, 𝛽−2), where 𝛽 is the real root of the equation 𝛽3 = 𝛽2 + 𝛽 + 1 and showed that this fractal is divided into three fractals that are bounded remainder sets with respect to a given toric shift. The introduced set was named as Rauzy fractal. It obtains many applications in the combinatorics of words, geometry, theory of dynamical systems and number theory. Later, an infinite sequence of tilings of 𝑑 − 1-dimensional Rauzy fractals associated with algebraic Pisot units of the degree 𝑑 into fractal sets of 𝑑 types were introduced. Each subsequent tiling is a subdivision of the previous one. These tilings are closely related to some irrational toric shifts and allowed to obtain new examples of bounded remainder sets for these shifts, and also to get some results on self-similarity of shift orbits. In this paper, we continue the study of generalized Rauzy tilings related to Pisot numbers. A new approach to definition of Rauzy fractals and Rauzy tilings based on expansions of natural numbers on linear recurrence sequences is proposed. This allows to improve the results on the connection of Rauzy tilings and bounded remainder sets and to show that the corresponding estimate of the remainder term is independent on the tiling order. The geometrization theorem for linear recurrence sequences is proved. It states that the natural number has a given endpoint of the greedy expansion on the linear recurrence sequence if and only if the corresponding point of the orbit of toric shift belongs to some set, which is the union of the tiles of the Rauzy tiling. Some number-theoretic applications of this result is obtained. In conclusion, some open problems related to generalized Rauzy tilings are formulated.

Spectral Splitting Method for Nonlinear Schrodinger Equations with Quadratic Potential

In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a nonlinear Schroedinger equation by solving the linear problem and treating the nonlinear term separately, with a rigorous estimate of the remainder term. Furthermore, we show by means of numerical experiments that such a modified approximation is more efficient than the standard one.

АСИМПТОТИКА РЕШЕНИЯ ДВУХЗОННОЙ ДВУХТОЧЕЧНОЙ КРАЕВОЙ ЗАДАЧИ

The article investigates the asymptotic behavior of the solution of a two-point boundary value problem on an interval for a linear inhomogeneous ordinary differential equation of the second order with a small parameter at the highest derivative. The essential features of the problem are the presence of a small parameter in front of the second-order derivative of the desired function, the existence of a twodimensional boundary layer at the left end of the segment at x = 0, and the non-smoothness of the solution to the corresponding unperturbed boundary value problem. It is required to construct a uniform asymptotic expansion of the solution to a two-zone two-point boundary value problem on a unit interval, with any degree of accuracy, as the small parameter tends to zero. Due to the second and third features of the problem, it is not easy to construct an asymptotic solution expansion with respect to the small parameter using the known asymptotic methods. When solving the problem, the following methods are used: methods of integration of ordinary differential equations; the method of a small parameter; the classical method of boundary functions; and the generalized method of boundary functions and the maximum principle. The problem is solved in two stages: in the first stage, a formal expansion of the solution to the two-point boundary value problem is constructed, and in the second stage, the justification of this expansion is given, i.e. the remainder term of the expansion is estimated. In the first stage, a formal asymptotic solution is sought in the form of the sum of three solutions: a smooth outer solution on the entire segment; classical boundary layer solution in the vicinity of x = 0, which exponentially decreases outside the boundary layer; and an intermediate boundary layer solution at x = 0, which decreases in power mode outside the boundary layer. The constructed asymptotic expansion of the solution to the two-point boundary value problem is asymptotic in the sense of Erdei.

On some additive problems of Goldbach’s type

In this paper, we find an asymptotic formula with power-saving remainder term for the number of solutions of one non-linear ternary problem with primes. The proof is based on the precise formulafor Chebyshev’s function involving the zeros of Riemann zeta function. In fact, a ternary problem each zerois solved. I. M. Vinogradov’s solution of the ternary Goldbach problem (1937, see [1], [2]) opened the way of solving a wide class of problems of the above type. In 1938, he found a power-saving estimate (with respect to the length of the summation interval) for the mean value of the modulus of the exponential sum with primes (see [2], theorem 3, p.82; theorems 6 and 7, p.86). Starting at 1996, G.I.Arkhipov, K.Buriev and the author have obtained several results concerning the exceptional sets in some binary problems of Goldbach’s type. These results used both the tools of the theory of Diophantine approximations and the precise formulasfrom Riemann’s zeta function theory. At the same time, the method of estimating of linear sums with primes based on the measure theory was derived in the papers of G. L. Watson, D. Bruedern, R. D. Cook and A. Perelli.