Some properties of Ramsey numbers

In this paper, some properties of Ramsey numbers are studied, and the following results are presented.

1. (1) For any positive integers k₁, k₂, …, k_m l₁, l₂, …, l_m (m> 1), we have

.

2. (2) For any positive integers k₁, k₂, …, k_m, l₁, l₂, …, l_n , we have

. Based on the known results of Ramsey numbers, some results of upper bounds and lower bounds of Ramsey numbers can be directly derived by those properties.

     

Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

With the notation ,

we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies

||p||_L2(K)An^1/2 and ||p′||_L2(K)Bn^3/2.

Then

M₄(p)−M₂(p)M₂(p)

with

We also prove that

and

M₂(p)−M₁(p)10⁻³¹M₂(p)

for every , where denotes the collection of all trigonometric polynomials of the form

     

A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function ₂F₁(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that

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is the solution of the Gauss's hypergeometric differential equation,

x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0

. where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given.     

Entropy numbers of Sobolev embeddings of radial Besov spaces

Let be the radial subspace of the Besov space . We prove the independence of the asymptotic behavior of the entropy numbers

from the difference s₀−s₁ as long as the embedding itself is compact. In fact, we shall show that

This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces B_p,q^s(Ω) on bounded domains Ω.     

Curves of positive solutions of boundary value problems on time-scales

Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem

(2)

(4)

u(a)=u(b)=0,

where λR₊:=[0,∞), and satisfies the conditions

We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ,u) of (1)–(2), u is positive on T a,b . In addition, we show that there exists λ_max>0 (possibly λ_max=∞), such that, if 0λ<λ_max then (1)–(2) has a unique solution u(λ), while if λλ_max then (1)–(2) has no solution. The value of λ_max is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).     

On the recursive sequence

Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation

where B, C and α, β, γ are positive, k {1, 2, 3, … }, and the initial conditions x_−2k+1, … , x₋₁, x₀ are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples.     

Singular boundary value problems for the one-dimension p-Laplacian

The singular boundary value problem

where φ(s)=|s|^p−2s, p>1, is studied in this paper. The singularity may appear at u=0, t=0 and t=1, and the function g may change sign. The existence of solutions is obtained via an upper and lower solution method.     

Estimates for n-widths of the Hardy-type operators (Addendum to “Improved estimates for the approximation numbers of the Hardy-type operators”)

Consider the Hardy-type operator T : L^p(a,b)→L^p(a,b),-∞a<b∞, which is defined by

It is shown that

where ρ_n(T) stands for any of the following: the Kolmogorov n-width, the Gel’fand n-width, the Bernstein n-width or the nth approximation number of T.     

The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations

In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form

Δ²(y_n+py_n−k)+f(n,y_n−ℓ)=0,n

, n ε, and

Δ²(y_n+py_n−k)+f(n,y_n−ℓ,Δy_n−ℓ)=0,n

,n ε using some difference inequalities. We establish conditions under which all nonoscillatory solutions are asymptotic to an + b as n → ∞ with a and b ε .     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

On recurrence coefficients for rapidly decreasing exponential weights

Let, for example,

where α>0, k1, and exp_k=exp(exp(…exp())) denotes the kth iterated exponential. Let {A_n} denote the recurrence coefficients in the recurrence relation

xp_n(x)=A_np_n+1(x)+A_n-1p_n-1(x)

for the orthonormal polynomials {p_n} associated with W². We prove that as n→∞,

where log_k=log(log(…log())) denotes the kth iterated logarithm. This illustrates the relationship between the rate of convergence to of the recurrence coefficients, and the rate of decay of the exponential weight at ±1. More general non-even exponential weights on a non-symmetric interval (a,b) are also considered.     

BV solutions to a degenerate parabolic equation for image denoising

In this article, the authors consider equation ut = div(ψ(Гu)A(|Du|2)Du) -(u- I), where ψ is strictly positive and Г is a known vector-valued mapping, A: R → R is decreasing and A(s) ～ 1/√a as s → ∞. This kind of equation arises naturally from image denoising. For an initial datum I ∈ BVloc ∩ L∞, the existence of BV solutions to the initial value problem of the equation is obtained.     

Peakons and periodic cusp wave solutions in a generalized Camassa–Holm equation

By using the bifurcation theory of planar dynamical systems to a generalized Camassa–Holm equation

m_t+c₀u_x+um_x+2mu_x=-γu_xxx

with m = u − α²u_xx, α ≠ 0, c₀, γ are constant, which is called CH-r equation, the existence of peakons and periodic cusp wave solutions is obtained. The analytic expressions of the peakons and periodic cusp wave solutions are given and numerical simulation results show the consistence with the theoretical analysis at the same time.     

On the Krein and Friedrichs extensions of a positive Jacobi operator

We show that for a positive linear operator acting in ℓ² and defined from

a_nx_n+1+b_nx_n+a_n-1x_n-1

its so-called Friedrichs and Krein extensions may be explicitly characterized by boundary conditions as n→∞.     

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,+∞)].     

Existence of multiple positive solutions for nonlinear m-point boundary-value problems

In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian

(φ_p(u^′))^′+f(t,u)=0, t(0,1),

     

Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:

u_tt−u_xx+u−|u|²u=0.