A New Integrable Case for the Kirchhoff Equation

A new integrable case is found for the Kirchhoff equation. The additional integral of motion is a fourth-degree polynomial, the principal metric is diagonal with the eigenvalues a ₁ = a ₂ = 1 and a ₃ = 2, and the other two metrics are nondiagonal.     

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

This paper is concerned with the two‐species chemotaxis‐competition system where Ω is a bounded domain in with smooth boundary ∂Ω, n≥2; χ_i and μ_i are constants satisfying some conditions. The above system was studied in the cases that a₁,a₂∈(0,1) and a₁>1>a₂, and it was proved that global existence and asymptotic stability hold when are small. However, the conditions in the above 2 cases strongly depend on a₁,a₂, and have not been obtained in the case that a₁,a₂≥1. Moreover, convergence rates in the cases that a₁,a₂∈(0,1) and a₁>1>a₂ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a₁,a₂>0 which covers the case that a₁,a₂≥1, and lead to convergence rates for solutions of the above system in the cases that a₁,a₂∈(0,1) and a₁≥1>a₂.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, T_A1T_{A_1}, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, T_A1T_{A_1} , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.      

The arithmetic‐geometric mean inequality: a short proof

This article gives a simple inductive proof of the following classical inequality connecting the arithmetic and geometric means of the n positive quantities a ₁, a ₂,......a_n :

by using the fact that the geometric mean of the n — 1 numbers

is the same as that of a ₁, a ₂,......a_n : An inductive procedure is then applicable to proving the inequality from the case n — 1 to the case n by repeated applications of a simple inequality involving two pairs of positive numbers with the same product.     

Contact problem for a layer with two stamps : PMM vol. 42, no. 6, 1978, pp. 1068–1073

A problem of impressing coaxial stamps of circular cross section into the upper and lower surface of a homogeneous elastic layer is studied. The bases of the stamps have axial symmetry. The parts of the layer surfaces lying oustide the contact zone are stress-free, there is no friction or coupling between the layer and the stamps. A system of two integral equations with two unknown functions is obtained, and provides a solution of the problem. The method of separating the singularities provides the way of reducing this system to the Fredholm equations of second kind. An approximate solution of the equations is obtained for the case of flat stamps under the assumptions that the two parameters entering the system are sufficiently small.

Problems of a layer with various boundary conditions were formulated and solved in many papers and books, e.g. [1, 2]. However, to the best of the author's knowledge, in all these problems the conditions at the boundary were assumed different only on one side of the layer; in the present problem the boundary conditions are mixed at both sides of the layer, and this results in a system of two integral equations.     

Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a₀ and on the two first off‐diagonals the constants a₁(lower) and a₋₁(upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is (a₀,a₁,a₋₁)=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a₀,a_ω,a_−ω), but where the two off‐diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a₀, a_ω=a_−ω, ω=2,3,…, and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.     

Inequalities for probability contents of convex sets via geometric average

It is shown that: If (X₁, X₂) is a permutation invariant central convex unimodal random vector and if A is a symmetric (about 0) permutation invariant convex set then P{(aX₁, X₂/a) A} is nondecreasing as a varies from )+ to 1 and is non-increasing as a varies from 1 to ∞ (that is, P{(a₁X₁, a₂X₂) ε A} is a Schur-concave function of (log a₁, log a₂). Some extensions of this result for the n-dimensional case are discussed. Applications are given for elliptically contoured distributions and scale parameter families.     

Rado Numbers fora(x+y)=bz

In the case of existence the smallest numberN=Ra_kis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Ra_k(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {x_n} generated bya(x_n+x_n+1)=bx_n+2 is discussed.     

On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

This paper is devoted to the investigation of the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a product of Banach spaces. The obtained results are applied to a two‐group transport operators with general boundary conditions in the Banach space L_p ([–a, a ] × [–1, 1]) × L_p ([–a, a ] × [–1, 1]), a > 0, p ≥ 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Separation of the elasticity theory equations with radial inhomogeneity : PMM vol. 38, n 6, 1974, pp. 1139–1144

The separation of a system of three elasticity theory equations in the static case to a system of two equations and one independent equation for a space with a radial inhomogeneity is presented in a spherical coordinate system. These equations are solved by separation of variables for specific kinds of radial inhomogeneity. In particular, solutions are found for the Lamé coefficients μ = const, λ (ifr) is an arbitrary function, μ = μ_or^β, λ = λ_or^β.While methods of solving problems associated with the equilibrium of an elastic homogeneous sphere have been studied sufficiently [1], problems with spherical symmetry of the boundary conditions have mainly been solved for an inhomogeneous sphere [2, 3],For a particular kind of inhomogeneity dependent on one Cartesian coordinate, the equations have been separated completely in [4], A system of three equations with a radial inhomogeneity in a spherical coordinate system is separated below by a method analogous to [4].     

Latin bitrades,dissections of equilateral triangles,and abelian groups

Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}$. Suppose that $\bar{b}_{i}\not= \bar{c}_{i}$ for all b, c∈T^* such that $\bar{b}_{i}\not= \bar{c}_{i}$ and i∈{1, 2, 3}. We prove that then T^? can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T^* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010     

Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

We study the upper-lower class behavior of weighted sums ∑ _k=1 ⁿ a _k X _k , where X _k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X _j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a _n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a _n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.     

An improvement of the binary merge algorithm

In this paper, we are concerned with the merging of two linearly-ordered listsA andB consisting of elements:a ₁<a ₂<...<b _n . The Hwang-Lin merging algorithm was considered very efficient for merging two lists of arbitrary sizes. Recently, Manacher was able to develop methods which reduce the number of pairwise comparisons required in the Hwang-Lin algorithm by a factor 31/336m. We develop in this paper a new method which further improves this factor to 52/336m. It is possible that even larger improvements could be achieved.     

Integration with respect to fractal functions and stochastic calculus. I

The classical Lebesgue–Stieltjes integral ∫ ^b _a fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and Weyl derivatives. In the special case of H?lder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved. The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce a stochastic integral for which the classical It? formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion. Received: 14 January 1998 / Revised version: 9 April 1998     

S-forcing,I. A “black-box” theorem for morasses,with applications to super-Souslin trees

We formulate, for regular μ>ω, a “forcing principle” S_μ which we show is equivalent to the existence of morasses, thus providing a new and systematic method for obtaining applications of morasses. Various examples are given, notably that for infinitek, if 2 ^k =k ⁺ and there exists a (k ⁺, 1)-morass, then there exists ak ⁺⁺-super-Souslin tree: a normalk ⁺⁺ tree characterized by a highly absolute “positive” property, and which has ak ⁺⁺-Souslin subtree. As a consequence we show that CH+SH_{ℵ 2}⟹ℵ₂ is (inaccessible)^L. This author thanks the US-Israel Binational Science Foundation for partial support of this research.     

On product representations of powers, I

The solvability of the equation a₁a₂ … a_k = x², a₁, a₂, …, a_k ε is studied for fixed k and ‘dense’ sets of positive integers. In particular, it is shown that if k is even and k 4, and is of positive upper density, then this equation can be solved.     

Remarks on a stability problem

The paper considers a system of three differential equations of the following form:x ₁=a₁₁x₁+₂(x₂)+₃(x₃),x ₂=a₂₁x₁+a₂₂x₂+a₂₃x₃,x ₃=a₃₁x₁+a₃₂x₂+a₃₃x₃.It is established that the given system can have periodic solutions if the generalized Routh-Hurwitz conditions hold.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 533–536, November, 1971.     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010     

Distributive multiplication rings

A ringR is said to be a left (right)n-distributive multiplication ring, n>1 a positive integer, if aa₁a₂...a_n=aa₁aa₂...aa_n (a₁a₂...a_na=a₁aa₂a...a_na) for all a, a₁,...,a_n R. It will be shown that the semi-primitive left (right)n-distributive rings are precisely the generalized boolean ringsA satisfying aⁿ=a for all a A. An arbitrary left (right)n-distributive multiplication ring will be seen to be an extension of a nilpotent ringN satisfyingN ⁿ⁺¹=0 by a generalized boolean ring described above. Under certain circumstances it will be shown that this extension splits.