On the Dirichlet's box principle

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong.     

Quantum approximation theorem

In this article a problem of relation between p-adic and ordinary quantum mechanics is discussed. The text was submitted by the author in English.     

On a joint strong approximation theorem for record and inter-record times

Summary We present a simple joint strong approximation for the logarithms of record and inter-record times from an exchangeable sequence, including an exact estimation for the rate of convergence in terms of upper and lower class functions of a Wiener process. The approach chosen here allows for simple proofs of exact and asymptotic (joint) results for record and inter-record times, such as the Law of Large Numbers (LLN), Central Limit Theorem (CLT) and Law of the Iterated Logarithm (LIL), and others.Research supported by the Air Force Office of Scientific Research Contract No. F49620 85 C0144     

A general poisson approximation theorem

We study the rate of convergence in a limit theorem due to Kabanov-Liptser-Shiryayev. We show how the probabilities P(N _t= k) can be computed from the compensator, when it is deterministic.     

On the real accuracy of approximation in the central limit theorem

We consider various approximations in the central limit theorem for distributions of sums of independent random variables. We study how many summands in the normalized sums guarantee the precision 10⁻³ for these approximations. It turns out that for the same distribution but different approximations this number varies from hundreds of thousands to a few tens.     

An approximation theorem for likelihood ratios

A semigroup approximation theorem is used to prove the convergence of approximate likelihood ratios.     

An approximation theorem in shape theory

In this paper it is shown that if X is a compactum in the interior of a PL manifold M and if U is a neighborhood of X in M, then there is a compactum X′ in U such that X and X′ have the same relative shape in U and the embedding dimension of X′ equals the fundamental dimension of X. Whenever the dimension of M is not equal to three, the relative shape equivalence from X′ to X can be realized by an infinite isotopy of M.     

On the real accuracy of approximation in the central limit theorem. II

In the present article, we obtain new explicit estimates for accuracy of approximation in the central limit theorem (CLT). We construct these approximations with the use of asymptotic expansions. We compare the estimates with the real accuracy of approximation for a specific distribution. We also discuss the following question: Why the estimate from the Berry–Esseen theorem cannot catch even the order of proximity of distributions in the CLT?     

A theorem concerning approximation on the sphere

Рассматривается тео рема типа Джексона и е е обращение на единичной сфереσ _n ?1≡σ пространстваR ⁿ . Сфери ческий сдвигS _y f(μ) фун кцииf(μ)∈L _p (σ), 1≦p≦∞, определяе тся как среднее значениеf(μ′) на «окружности»μ μ′=cosγ, μ,μ′∈σ. Для того, чтобы при 2k>г-2?>0 выполнялось нер авенство $$\left\| {(S_\gamma - E)^k D^\varrho f} \right\|L_p \left( \sigma \right) \leqq M\gamma ^{r - 2\varrho } $$ (гдеE — единичный опер атор, D - оператор Лапласа—Бельтрами) н еобходимо и достаточ но, чтобы наилучшее приб лижение функциf в мет рикеL _p (σ) полиномами порядкаv по сферическим гармоникам подчинял ось оценке $$E_v \left( f \right)_p \leqq \frac{K}{{v^r }} \left( {v = 1, 2, \ldots } \right).$$ Полное доказательст во дано приp=2.     

An approximation theorem for second-order evolution equations

Summary A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos for 0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.Research supported by ONR grant N00014-57-A-0298-0015Research supported by USARO grant DAAG 29-278-C-0024     

An inverse approximation theorem in various metrics

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 57–65, December, 1991.