Lower bounds for the blow-up time in a non-local reaction–diffusion problem

For a non-local reaction–diffusion problem with either homogeneous Dirichlet or homogeneous Neumann boundary conditions, the questions of blow-up are investigated. Specifically, if the solutions blow up, lower bounds for the time of blow-up are derived.     

On waiting time in the Markov-Pólya scheme

We consider the Markov-Pólya urn scheme. The urn contains a given number of balls of each of N different colors. The balls are sequentially drawn from the urn one at a time, independently of each other. and with equal probabilities to be drawn. After each draw, the ball drawn is returned into the urn together with c balls of the same color, c∈{−1, 0, 1, 2, ...}. The drawing process halts when, for the first time, the frequencies of k unspecified colors attain or exceed the corresponding (random) levels settled before the beginning of the trials. Limit distributions of the stopping time υ_c(N, K) are considered as N→∞, k=k(N); in particular, the dependence of the asymptotic properties of the waiting time on the parameter c is studied. Results concerning υ_c(N, K) are derived as consequences of general limit theorems for decomposable statistics

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

Approximation by the Bézier variant of the MKZ-Kantorovich operators in the case α < 1

The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators for the case 0 < α < 1. In the end we propose the q-analogue of MKZK operators.      

Further results on the relationship between μ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains

This note considers continuous-time Markov chains whose state space consists of an irreducible class, C, and an absorbing state which is accessible from C. The purpose is to provide results on μ-invariant and μ-subinvariant measures where absorption occurs with probability less than one. In particular, the well-known premise that the μ-invariant measure, m, for the transition rates be finite is replaced by the more natural premise that m be finite with respect to the absorption probabilities. The relationship between μ-invariant measures and quasi-stationary distributions is discussed.     

Inequalities for upper bounds of functionals on the classes w r H ω and their applications

We show that the well-known results on estimates of upper bounds of functionals on the classes W ^r H ^ω of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes W ^r H ^ω , establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class H ^ω on the derivatives of trigonometric polynomials or polynomial splines in terms of the L ^ϱ -norms of these polynomials and splines.     

$$L^p$$ Christoffel–Minkowski problem: the case $$1< p<k+1$$1<p<k+1

We consider a fully nonlinear partial differential equation associated to the intermediate \(L^p\) Christoffel–Minkowski problem in the case \(1<p<k+1\). We establish the existence of convex body with prescribed k-th even p-area measure on \(\mathbb S^n\), under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain \(C^{1,1}\) regularity estimates for admissible solutions of the equation when \( p\ge \frac{k+1}{2}\).     

Optimal models for the first arrival time distribution function in continuous time — With a special case

Optimal Models for first arrival time ( _H ) and first arrival target total return (W _H ) distribution functions on MDP in continuous time are presented. Asymptotic expansions of _H andW _H are derived and expressed in simple, explicit forms, and some of their properties are discussed. Two methods to find an optimal policy for distribution function of _H are given. Several necessary and sufficient conditions for the existence of the optimal policy are obtained. This result leads to that the scope of finding the optimal policy is greatly reduced. A special case is also discussed and some deep results are given.     

Busy period,virtual waiting time and number of customers in G<Superscript><Emphasis Type="Italic">δ</Emphasis></Superscript>|M<Superscript><Emphasis Type="Italic">ϰ</Emphasis></Superscript>|1|B system

In this article, we determine the integral transforms of several two-boundary functionals for a difference of a compound Poisson process and a compound renewal process. Another part of the article is devoted to studying the above-mentioned process reflected at its infimum. We use the results obtained to study a G ^δ |M ^ϰ |1|B system with batch arrivals and finite buffer in the case when δ∼ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes. The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process.     

On the Nörlund logarithmic means with respect to Vilenkin system in the martingale Hardy space <Emphasis Type="Italic">H</Emphasis><Subscript>1</Subscript>

We prove and discuss a new divergence result of Nörlund logarithmic means with respect to Vilenkin system in Hardy space H ₁.     

Testing for the maximum cell probabilities in multinomial distributions

This paper investigates one-sided hypotheses testing for p, the largest cell probability of multinomial distribution. A small sample test of Ethier (1982) is extended to the general cases. Based on an estimator of p, a kind of large sample tests is proposed. The asymptotic power of the above tests under local alternatives is derived. An example is presented at the end of this paper.     

A waiting time phenomenon for modulations of pattern in reaction–diffusion systems

In order to describe slow modulations in time and space of stable or slightly unstable spatially periodic stationary solutions of pattern forming reaction–diffusion systems, so-called phase diffusion equations and Cahn–Hilliard equations can be derived via multiple scaling analysis as formal approximation equations. In the case that these equations degenerate, waiting time phenomena are well known to occur. In this paper, we prove that such waiting time phenomena can also occur approximately in the original reaction–diffusion systems by proving estimates between the formal approximations and the exact solutions of the original systems.     

Grüss-type bounds for covariances and the notion of quadrant dependence in expectation

We show that Grüss-type probabilistic inequalities for covariances can be considerably sharpened when the underlying random variables are quadrant dependent in expectation (QDE). The herein established covariance bounds not only sharpen the classical Grüss inequality but also improve upon recently derived Grüss-type bounds under the assumption of quadrant dependency (QD), which is stronger than QDE. We illustrate our general results with examples based on specially devised bivariate distributions that are QDE but not QD. Such results play important roles in decision making under uncertainty, and particularly in areas such as economics, finance, and insurance.     

Some test length bounds for nonrepeating functions in the {&;, ∨} basis

Testing relative to a nonrepeating alternative in a conjunction-disjunction basis is considered. A lower bound on the test length is established for all nonrepeating functions in this basis. A subsequence of easily testable functions is constructed and the corresponding tests are described. Individual lower test length bounds are proved for functions of a special form; minimality of the tests is established for the functions of the constructed subsequence.     

A smoothing Newton-type method for solving the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> spectral estimation problem with lower and upper bounds

This paper discusses the L ₂ spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.     

Interdeparture time distributions in ΣiMi/Gi/1 priority queues

This paper reviews existing results for the stationary interdeparture time distribution in the M/G/1 nonpreemptive and preemptive resume queues, and introduces a unified approach which exploits for the first time the common structure for the interdeparture time process that is present in all classical preemptive priority service disciplines. This approach confirms previously known results for the preemptive resume discipline, and presents new results for several variants of the preemptive repeat model. Exact expressions for the squared coefficient of variation of the interdeparture time distribution are also provided. Several numerical examples are given and discussed. This revised version was published online in June 2006 with corrections to the Cover Date.