Asymptotic Inequalities Research Articles

High order asymptotic inequalities for some dissipative systems

We obtain some important fundamental inequalities concerning the long time behavior of high order derivatives for solutions of some dissipative systems in terms of their algebraic L2 decay. Some of these inequalities have not been observed in the literature even for the fundamental Navier–Stokes equations. To illustrate this new approach, we derive bounds for the asymmetric incompressible fluids equations, where improved inequalities were established for the micro-rotational field w. Finally, we show how this technique can be applied to other dissipative systems.

Double asymptotic inequalities for the generalized Wallis ratio

Asymptotic estimates for the generalized Wallis ratio \(W^*(x):=\frac{1 {\sqrt{\pi}}\cdot\frac{\Gamma(x+\frac{1}{2})}{\Gamma(x+1)}\) are presented for \(x\in\mathbb{R}^+\) on the basis of Stirling's approximation formula for the \(\Gamma\) function. For example, for an integer \(p\ge2\) and a real \(x>-\tfrac{1}{2}\) we have the following double asymptotic inequality \[A(p,x)\,<\,W^*(x)\,<\,B(p,x),\] where \begin{align*} A(p,x):=& W_p(x)\left(1-\tfrac{1}{8(x+p)}+\tfrac{1}{128(x+p)^2}+\tfrac{1}{379(x+p)^3}\right), \\ B(p,x):= & W_p(x)\left(1-\tfrac{1}{8(x+p)}+\tfrac{1}{128(x+p)^2}+\tfrac{1}{191(x+p)^3}\right),\\ W_p(x):=& \frac{1}{\sqrt{\pi\,(x+p)}}\cdot\frac{(x+1)^{(p)}}{(x+\frac{1}{2})^{(p)}}, \end{align*} with \(y^{(p)}\equiv y(y+1)\cdots(y+p-1)\), the Pochhammer rising (upper) factorial of order \(p\).

Asymptotic estimates for deviations of Fejér means on Poisson integrals

The work is devoted to the investigation of problem of approximation of continuous periodic functions of a real variable by trigonometric polynomials generated by linear methods of summation of Fourier series. The simplest example of the process of linear approximation of periodic functions is approximation of functions by partial sums of their Fourier series. However, sequences of partial Fourier sums are not uniformly convergent on the whole class of continuous periodic functions. Therefore, numerous studies are devoted to the study of approximation properties of approximating methods, which are generated by different transformations of sequences of partial sums of the Fourier series and allow obtaining sequences of trigonometric polynomials that are uniformly convergent for the entire class of continuous functions. In particular, Fejér means have been intensively studied in recent decades. One of the important tasks in this direction is study of the asymptotic behavior of the upper bounds of the deviations of trigonometric polynomials for a fixed class of periodic functions. The aim of the work is to systematize the known results concerning the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums, and to present new facts obtained for a more general case. In paper we investigate the asymptotic behavior of upper bounds on classes of Poisson integrals of periodic functions of real variable of deviations of linear means of Fourier series, which are constructed using the Fejér summation method. We study the classes consist of analytic functions of a real variable, which can be regularly extended into the corresponding strip of the complex plane. In the work, asymptotic inequalities for the upper bounds of the deviations of the Fejér means on the class of Poisson integrals were found. In certain case of parameters defining the class, the obtained formulas coincide with previously known asymptotic equalities.

Some new results related to the constant e and function (1+1/x)x

It is known that the constant e has the following series representations: e = ?? k=0 1/k! and e = ?? k=0 9k2+1/(3k)!. The second series is extremely rapidly convergent. In this paper, we present asymptotic expansions and two-sided inequalities for the remainders Rn and Rn, where Rn = e ? ?n k=0 1/k! and Rn=e??n k=0 9k2+1/(3k)!. Also, we present some inequalities and completely monotonic functions involving (1 + 1/x)x. We also consider a number of related developments on the subject of this paper.

Oracle inequalities for weighted group Lasso in high-dimensional Poisson regression model

This article considers the problem of estimating the high-dimensional Poisson regression model with group sparsity in the parameter vector using the weighted group Lasso method. Non asymptotic oracle inequalities measuring the convergence rate of the estimation and prediction errors are given, provided that the true regression coefficients and the covariates satisfy suitable conditions. The non asymptotic oracle inequalities imply that the weighted group lasso estimator is consistent. A comparison of the performance of our proposed weights with other adaptive weights in simulated data shows that the weighted group Lasso estimator provides a good sparse approximation to the true parameters.

Asymptotic expansions and inequalities relating to the gamma function

We present some asymptotic expansions and inequalities for ?(x+1)1/x ?(x+2)1/(x+1) and ?(x+1)1/x.

ASYMPTOTICS OF A MULTIZONAL INTERNAL LAYER SOLUTION TO A PIECEWISE-SMOOTH SINGULARLY PERTURBED EQUATION WITH A TRIPLE ROOT OF THE DEGENERATE EQUATION

A singularly perturbed boundary value problem for a stationary equation of reaction-diffusion type in the case when reactive term undergoes discontinuity along some curve that is independent of the small parameter is studied. This is a new class of problems with triple roots of the degenerate equation, which leads to the formation of complex multizonal internal layers in the neighborhood of the discontinuity curve. By the method of asymptotic differential inequalities and matching asymptotic expansion, the existence of a contrast structure solution is proved. Using a different modified boundary layer function method, the asymptotic representation of point itself and this solution are constructed.

Asymptotic formulae and inequalities for point spectrum in max algebra

We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of nonnegative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.

Adaptive efficient analysis for big data ergodic diffusion models

We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions.

Ramanujan’s Theorem and Highest Abundant Numbers

In 1915, Ramanujan proved asymptotic inequalities for the sum of divisors function, assuming the Riemann hypothesis (RH). We consider a strong version of Ramanujan's theorem and define highest abundant numbers that are extreme with respect to the Ramanujan and Robin inequalities. Properties of these numbers are very different depending on whether the RH is true or false.

Improved asymptotic formula and inequalities for the ratio of Pochhammer k-symbol

In this paper, we study some ratios of the Pochhammer k-symbol. We propose some asymptotic formula for the ratio of Pochhammer k-symbol based on continued fraction by Multiple-correction method and prove some sharp inequalities of its bounds.

On two new inequalities for Leray solutions of the Navier-Stokes equations in [formula omitted

We derive two asymptotic inequalities, not previously observed, that are valid for spatial derivatives (of arbitrary order) of Leray solutions to the incompressible Navier-Stokes equations in Rn, n≤4. They are important to the decay investigation of solution derivatives.

Asymptotic inequalities for alternating harmonics

For [Formula: see text] the [Formula: see text]th alternating harmonic number [Formula: see text] is given in the form [Formula: see text] where [Formula: see text] is a parameter controlling the magnitude of the error term [Formula: see text] estimated as [Formula: see text]

Inequalities and asymptotic expansions related to the generalized Somos quadratic recurrence constant

In this paper, we give asymptotic expansions and inequalities related to the generalized Somos quadratic recurrence constant, using its relation with the generalized Euler constant.

Compositional Restricted Multiple Access Channel

We introduce the notion of a q-ary s-compositional code and prove that the rate, R, of the best such code satisfies for large s the asymptotic inequalities $$(q - 1)\frac{{{{\log }_q}s}}{{4s}} \lesssim 2(q - 1)\frac{{{{\log }_q}s}}{{4s}}$$ .

Non Asymptotic Sharp Oracle Inequalities for the Improved Model Selection Procedures for the Adaptive Nonparametric Signal Estimation Problem

In this paper, we consider the robust adaptive non parametric estimation problem for the periodic function observed with the Levy noises in continuous time. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Sharp oracle inequalities for the robust risks have been obtained.

New Asymptotic Formulas and Inequalities for the Gamma Function Based on Continued Fractions

In this paper, based on continued fractions, we provide two faster asymptotic formulas of the gamma function compared with former achievements. Then, we present some relevant inequalities. Finally, in order to demonstrate that our new asymptotic formulas have superiority, some numerical computations are also given.

On the stability and the concentration of extended Kalman-Bucy filters

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estimates in terms of the filter forgetting properties with respect to erroneous initial conditions. For uniformly stable and fully observable signals, we also provide explicit non-asymptotic estimates for the exponential forgetting rate of the filters and the associated stochastic Riccati equations w.r.t. Frobenius norms. These non asymptotic exponential concentration and quantitative stability estimates seem to be the first results of this type for this class of nonlinear filters. Our techniques combine $\chi $-square concentration inequalities and Laplace estimates with spectral and random matrices theory, and the non asymptotic stability theory of quadratic type stochastic processes.

Asymptotic inequalities for alternating harmonics

For n ∈ ℕ the nth alternating harmonic number Hn∗ :=∑ k=1n(−1)k−1 1 k is given in the form Hn∗ =ln 2 + (−1)n+1 4⌊n+1 2 ⌋ +∑i=1q−1 (4i − 1)B 2i (2i)(2⌊n+1 2 ⌋)2i + rq(n), where q ∈ ℕ is a parameter ...

Some Asymptotic Formulas and Inequalities About Gamma and Psi Functions Based on Continued Fractions

In this paper, based on continued fractions, some faster asymptotic formulas of gamma, psi and psi’s derivative functions are given. Then, we provide some relevant inequalities. Finally, in order to demonstrate that our new asymptotic formulas have superiority, some numerical computations are also given.