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).     

A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ ∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.     

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.     

On linearly related sequences of derivatives of orthogonal polynomials

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (P_n)_n and (Q_n)_n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as

for all n=0,1,2,…, where M and N are fixed nonnegative integer numbers, and r_i,n and s_i,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (P_n)_n and (Q_n)_n (resp.). Assuming 0mk, we prove the existence of four polynomials Φ_M+m+i and Ψ_N+k+i, of degrees M+m+i and N+k+i (resp.), such that

the (k−m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k=m, then u and v are connected by a rational modification. If k=m+1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k>m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k−m with polynomial coefficients.     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

Positive solutions for one-dimensional -Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian

subject to the boundary value conditions:

where _p(s)=|s|^p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.     

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

     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

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.     

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.     

Hardy–Sobolev critical elliptic equations with boundary singularities

Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in (n3), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω,

when 0<s<2, , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:

where f is a lower order perturbative term at infinity and f(x,0)=0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.     

Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum–Delsarte–Janssen–Vries problem

One establishes inequalities for the coefficients of orthogonal polynomials

which are orthogonal with respect to a constant weight on the arc of the unit circle S={e^iθ,απ<θ<2π-απ}, with 0<α<1. Recurrence relations (Freud equations), and differential relations are used. Among other results, it is shown that Φ_n(0)>0,n=1,2,…     

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 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

     

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.

     

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),

     

On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A₁,…,A_n be finite nonempty subsets of F, and let

with k{1,2,3,…}, a₁,…,a_nF{0} and degg<k. We show that

When kn and |A_i|i for i=1,…,n, we also have

consequently, if nk then for any finite subset A of F we have

In the case n>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.     

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.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1