An infinite color analogue of Rado's theorem

Let R be a subring of the complex numbers and a be a cardinal. A system L of linear homogeneous equations with coefficients in R is called a-regular over R if, for every a-coloring of the nonzero elements of R, there is a monochromatic solution to L in distinct variables. In 1943, Rado classified those finite systems of linear homogeneous equations that are a-regular over R for all positive integers a. For every infinite cardinal a, we classify those finite systems of linear homogeneous equations that are a-regular over R. As a corollary, for every positive integer s, we have ℵ₀²>ℵs if and only if the equation x₀+sx₁=x₂+?+xs+2 is ℵ₀-regular over R. This generalizes the case s=1 due to Erd?s.     

Representations by quaternary quadratic forms whose coefficients are 1, 4, 9 and 36

Explicit formulae are determined for the number of representations of a positive integer by the quadratic forms ax²+by²+cz²+dt² with a,b,c,d∈{1,4,9,36}, gcd(a,b,c,d)=1 and a?b?c?d.     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On Hilbert''s Integral Inequality

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L²[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π²is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫^∞₀(c(t²x)/(1 + t²)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].     

Admissibility of the natural estimator of the mean of a Gaussian process

Let {X(t): t [a, b]} be a Gaussian process with mean μ L₂[a, b] and continuous covariance K(s, t). When estimating μ under the loss ∫_a^b ( (t)−μ(t))² dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫_a^b K(t, t) dt and admissible if and only if the three by three matrix whose entries are K(t_i, t_j) has a determinant which vanishes identically in t_i [a, b], i = 1, 2, 3.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

On the signless Laplacian spectral characterization of the line graphs of T-shape trees

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic graph has the same signless Laplacian spectrum (simply G is DQS). Let T (a, b, c) denote the T-shape tree obtained by identifying the end vertices of three paths P _a+2, P _b+2 and P _c+2. We prove that its all line graphs L(T(a, b, c)) except L(T(t, t, 2t+1)) (t ? 1) are DQS, and determine the graphs which have the same signless Laplacian spectrum as L(T(t, t, 2t + 1)). Let µ₁(G) be the maximum signless Laplacian eigenvalue of the graph G. We give the limit of µ₁(L(T(a, b, c))), too.     

An Excursion into Nonlinear Ramsey Theory

Formulas for two-color Rado numbers have been established for many families of linear equations. However, there are no explicit formulas for two-color Rado numbers for any nonlinear equations. In this paper, we will establish formulas for the two-color Rado numbers for three families of equations: x + y ⁿ = z, x + y ² + c = z, and x +  y ² = az , where c and a are positive integers.     

Parameter estimation in linear filtering

Suppose on a probability space (Ω, F, P), a partially observable random process (x_t, y_t), t ≥ 0; is given where only the second component (y_t) is observed. Furthermore assume that (x_t, y_t) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W₁(t)) and (W₂(t)): dx_t=−βx_tdt+dW₁(t), x₀=0, dy_t=αx_tdt+dW₂(t), y₀=0; α, β∞(a,b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter θ = (α, β). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and θ is a k-dimensional vector.     

Strongly Singular Sturm – Liouville Problems

We consider a Sturm – Liouville operator Lu = —(r(t)u′)′ + p (t)u , where r is a (strictly) positive continuous function on ]a, b [ and p is locally integrable on ]a, b[. Let r₁(t) = (1/r) ds andchoose any c ∈]a, b[. We are interested in the eigenvalue problem Lu = λm(t)u, u (a) = u (b) = 0,and the corresponding maximal and anti .maximal principles, in the situation when 1/r ∈ L¹ (a, c),1 /r ∉ L¹ (c, b), pr₁ ∉ L¹ (a, c) and pr₁ ∉ L¹(c, b).     

A new reduction method in integer programming

The system of two equations in nonnegative integer unknowns x_j and expressed in the form ∑ⁿ_j=1 a_1jx_j=b₁, ∑ⁿ_j=1 a_2jx_j=b₂ has equivalent solutions to the single equation ∑ⁿ_j=1 (t₁a_ij+t₂a_2j)a_j= t₁b₁+t₂b₂, provided t₁ and t₂ are suitably chosen. Glover [2] has n inequalities for determining t₁ and t₂. Elimam and Elmaghraby [1] try to achieve a single inequality for t₁ and t₂. We show, however, that the inequality of [1] may give t₁ and t₂ values that do not produce equivalence. We present a new theorem which leads to a single inequality constraint giving t₁ and t₂ values that consistently produce equivalence.     

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.     

Triangle-free distance-regular graphs

Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a ₁=0 and a ₂≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)²(b+2)/c ₂, (b−2)(b−1)b(b+1)/(2+2b−c ₂) is an integer and that c ₂≤b(b+1). This upper bound for c ₂ is optimal, since the Hermitian forms graph Her₂(d) is a triangle-free distance-regular graph that satisfies c ₂=b(b+1). Work partially supported by the National Science Council of Taiwan, R.O.C.     

Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case

Under general growth assumptions, that include some cases of linear growth, we prove existence of Lipschitzian solutions to the problem of minimizing ∫_a^bL(x(s),x′(s)) ds with the boundary conditions x(a)=A, x(b)=B.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

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 Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.