Asymptotic Analysis and Stability of Inviscid Liquid Sheets

Asymptotic methods are employed to determine the leading-order equations that govern the fluid dynamics of slender, and thin and slender, inviscid, irrotational, planar liquid sheets subject to pressure differences and gravity. Two flow regimes have been identified depending on the Weber number, and analytical solutions to the steady state equations are provided. Linear stability studies indicate that the sinuous mode corresponds to Weber numbers on the order of unity, while the varicose mode is associated with small Weber numbers. For small Weber numbers, the nonlinear stability of liquid sheets is determined analytically in terms of elliptic integrals of the first and second kinds. It is also shown that the sinuous mode of thin and slender liquid sheets is identical to the same mode for slender sheets.     

The Symmetric Level-Index System

The purpose of this paper is to present the details of an arithmeticsystem which virtually abolishes the phenomena of computer overflowand underflow in a logically symmetric manner. A generalizedexponential function is used in such a way as to enable verylarge numbers to be represented with a uniform precision, andvery small numbers by reciprocation.     

Ramsey numbers for bipartite graphs with small bandwidth

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.     

Betti numbers in multidimensional persistent homology are stable functions

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of scalar‐valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, that is, the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector‐valued filtering functions, we can consider the multidimensional analog of persistent Betti numbers. Varying the lower level sets, we obtain that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non‐negative integers. In this paper, we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector‐valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector‐valued filtering functions and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalar‐valued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers, we obtain a lower bound for the natural pseudo‐distance. Copyright © 2013 John Wiley & Sons, Ltd.     

Expansion of a Rarefied Gas Cloud in a Vacuum: Asymptotic Treatment

The unsteady expansion of a rarefied gas of finite mass in an unlimited space is studied. The long-time asymptotic behavior of the solution is examined at Knudsen numbers tending to zero. An asymptotic analysis shows that, in the limit of small Knudsen numbers, the behavior of the macroscopic parameters of the expanding gas cloud at long times (i.e., for small density values) has nothing to do with the free-molecular or continuum flow regimes. This conclusion is unexpected and not obvious, but follows from a uniformly suitable solution constructed by applying the method of outer and inner asymptotic expansions. In particular, the unusual temperature behavior is of interest as applied to remote sensing of rocket exhaust plumes.     

Probabilistic algorithm for testing primality

We present a practical probabilistic algorithm for testing large numbers of arbitrary form for primality. The algorithm has the feature that when it determines a number composite then the result is always true, but when it asserts that a number is prime there is a provably small probability of error. The algorithm was used to generate large numbers asserted to be primes of arbitrary and special forms, including very large numbers asserted to be twin primes. Theoretical foundations as well as details of implementation and experimental results are given.     

Application of the Finite Pointset Method (FPM) to the kinetic BGK model

The BGK model of rarefied gas dynamics [1] is solved numerically by using the Finite Pointset Method (FPM) [2], which is a particle method developed at the ITWM Kaiserslautern. For the implementation a semilagrangian scheme [3] is used. Numerical results are shown on the example of a Shock tube problem for different Knudsen numbers. The solutions are compared to the solutions of exact Euler in the case of small Knudsen numbers and to DSMC solutions for higher Knudsen numbers. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the non-orthogonal Stagnation flow of the Oldroyd-B fluid

This paper is concerned with a non-orthogonal stagnation flow of an Oldroyd-B fluid between two parallel plates. We reduce the problem to a set of ordinary differential equations (ODE's), which is then solved with finite differences using a parameter continuation method. Perturbation analyses are also carried out for small Reynolds numbers and small Weissenberg numbers respectively. The solution of the set of ODE's is discussed. It is known that for a Newtonian fluid, the stagnation point shifts from the potential flow case in the opposite direction of the tangential velocity. The effect of the fluid elasticity is to reduce this shift. It is also shown that the Oldroyd-B model has a limiting Weissenbeg number, depending on the angle of the injected flow.     

On the Trace of Algebraic Integers of Small Height

We give an upper bound for the modulus of the first non–zero trace among natural powers of an algebraic integer of small house. An upper bound for this power is obtained for the Pisot and Salem numbers. Although the house of these numbers is not at all small, similar bounds for the first non–zero trace are also established. Finally, we give an upper bound for the trace of an algebraic number with the Mahler measure bounded above by the square root of the degree.     

Are the hyperharmonics integral? A partial answer via the small intervals containing primes

In a recent work, the authors have used Bertrand's postulate to give a partial answer to the conjecture of Mez? which says that the hyperharmonic numbers – iterations of partial sums of harmonic numbers – are not integers. In this Note, using small intervals containing prime numbers, we prove that a great class of hyperharmonic numbers are not integers.     

On some polynomial values of repdigit numbers

We study the equal values of repdigit numbers and the k dimensional polygonal numbers. We state some effective finiteness theorems, and for small parameter values we completely solve the corresponding equations.     

Quadratic Spline Wavelets with Short Support for Fourth-Order Problems

In the paper, we propose constructions of new quadratic spline-wavelet bases on the interval and the unit square satisfying homogeneous Dirichlet boundary conditions of the second order. The basis functions have small supports and wavelets have one vanishing moment. We show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small.     

Recurrence Formulas for Kostka and Inverse Kostka Numbers via Quantum Cohomology of Grassmannians

A Gröbner basis for the small quantum cohomology of Grassmannian G _k,n is constructed and used to obtain new recurrence relations for Kostka numbers and inverse Kostka numbers. Using these relations it is shown how to determine inverse Kostka numbers which are related to the mod-p Wu formula.     

On zero sum Ramsey numbers: Multiple copies of a graph

As a consequence of our main result, a theorem of Schrijver and Seymour that determines the zero sum Ramsey numbers for the family of all r-hypertrees on m edges and a theorem of Bialostocki and Dierker that determines the zero sum Ramsey numbers for r-hypermatchings are combined into a single theorem. Another consequence is the determination of zero sum Ramsey numbers of multiple copies of some small graphs.     

Shortened recurrence relations for Bernoulli numbers

Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.     

Constructing heterogeneous hash families by puncturing linear transversal designs

Constructions that use hash families to select columns from small covering arrays in order to construct larger ones can exploit heterogeneity in the numbers of symbols in the rows of the hash family. For specific distributions of numbers of symbols, the efficacy of the construction is improved by accommodating more columns in the hash family. Known constructions of such heterogeneous hash families employ finite geometries and their associated transversal designs. Using thwarts in transversal designs, specific constructions of heterogeneous hash families are developed, and some open questions are posed.     

有限个任意小同胚连接区域的性质

本文给出了一个条件,证明了在此条件下任意小同胚连接区域是道路连通的．     

Nonstationary mass transfer of a reacting spherical particle in laminar stream of a viscous fluid : PMM vol. 38, n 6, 1974, pp. 1041–1047

The process of the formation of a stationary mass transfer mode for a moving reacting particle is examined. An analytic expression valid for a nonstationary distribution of the concentration of matter in a steady stream of viscous fluid, flowing past a spherical particle, was obtained for the case when at a certain instant a chemical reaction of the first order begins at the surface of the sphere. The problem is solved for small finite Reynolds and Péclet numbers. The solution of the corresponding stationary problem has been obtained in [1]. Paper [2] examined a nonstationary heat transfer of a fluid spherical drop in an inviscid flow with spasmodic change of initial temperature at high Péclet numbers. Paper [3] contains an analysis of the problem of a nonstationary heat transfer of a rigid spherical particle for small Reynolds and Péclet numbers at spasmodic change of temperature of the particle surface. The results obtained in [3] can be used to describe the mass transfer for a moving reacting particle only in the case of a diffusion mode of the chemical reaction.     

On some polynomial values of repdigit numbers

We study the equal values of repdigit numbers and the k-dimensional polygonal numbers. We state some effective finiteness theorems, and for small parameter values we completely solve the corresponding equations.     

The structured sensitivity of Vandermonde-like systems

Summary We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.