The range of the p-Laplacian

We prove that for λ ≥ 0, p ≥ 3, there exists an open ball B L²(0,1) such that the problem

− (|u′|^p−2 u′)′ − λ|u|^p−2u = f, in (0,1)

, subject to certain separated boundary conditions on (0,1), has a solution for f B.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Two theorems on Euclidean distance matrices and Gale transform

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ(E−C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones

where is the elliptope (set of correlation matrices) and is the (closed convex) cone of EDMs.

The characterization is given using the Gale transform of the points generating D. We also show that given points , for any scalars λ₁,λ₂,…,λ_n such that

∑_j=1ⁿλ_jp^j=0, ∑_j=1ⁿλ_j=0,

we have

∑_j=1ⁿλ_jpⁱ−p^j2= forall i=1,…,n,

for some scalar independent of i.     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.     

Poisson evolution in word selection

This paper presents the finding that the invocation of new words in human language samples is governed by a slowly changing Poisson process. The time dependent rate constant for this process has the form

λ(t) = λ₁(1−λ₂t)e^-λ₂t+λ₃(1−λ₄t)e^-λ₄t+λ₅

, where

λ_i > 0, I=1,…,5

.

This form implies that there are opening, middle and final phases to the introduction of new words, each distinguished by a dominant rate constant, or equivalently, rate of decay. With the occasional exception of the phase transition from beginning to middle, the rate λ(t) decays monotonically. Thus, λ(t) quantifies how the penchant of humans to introduce new words declines with the progression of their narratives, written or spoken.     

Strong continuity of the relative rearrangement maps and application to a Galerkin approach for nonlocal problems

We study the strong continuity of the map u   (b_*u, b_*u(| > u(·)|)). Here,

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for σ]0 means Ω[, u_* (respectively, (b|_{u=u*(σ)})*) denotes the decreasing rearrangement of u (respectively b restricted to the set {u = u_*(σ)}) and |E| denotes the Lebesgue measure of a set E included in a domain Ω. The results are useful for solving plasmas physics equations or any nonlocal problems involving the monotone rearrangement, its inverse or its derivatives.     

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes

in the space , where fL²([0,T]ⁿ) is a given function, and {η(t)}_>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t₁,…,t_n)=1_{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t₁,…,t_n)=f₁(t₁)f_n(t_n)1_{t1<t2<<tn}, with f_iL²([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

Oscillation criteria for first-order neutral nonlinear difference equations with variable coefficients

Consider the first-order neutral nonlinear difference equation of the form

, where τ > 0, σ_i ≥ 0 (i = 1, 2,…, m) are integers, {p_n} and {q_n} are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σ_n=0^∞ q_n = ∞ or Σ_n=0^∞ nq_n Σ_j=n^∞ q_j = ∞ commonly used in the literature.     

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type

(|u′|^m−2u′)′ + f(t,u,u′)=0, m 2

are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

An extension theorem for t-designs

In 1994, van Trung (Discrete Math. 128 (1994) 337–348) [9] proved that if, for some positive integers d and h, there exists an S_λ(t,k,v) such that

then there exists an S_λ(v−t+1)(t,k,v+1) having v+1 pairwise disjoint subdesigns S_λ(t,k,v). Moreover, if B_i and B_j are any two blocks belonging to two distinct such subdesigns, then d|B_i∩B_j|<k−h. In 1999, Baudelet and Sebille (J. Combin. Des. 7 (1999) 107–112) proved that if, for some positive integers, there exists an S_λ(t,k,v) such that

where m=min{s,v−k} and n=min{i,t}, then there exists an

having pairwise disjoint subdesigns S_λ(t,k,v). The purpose of this paper is to generalize these two constructions in order to produce a new recursive construction of t-designs and a new extension theorem of t-designs.     

Coupling and harmonic functions in the case of continuous time Markov processes

Consider two transient Markov processes (X^v_t)_tεR·, (X^μ_t)_tεR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process.

We show that there exist (randomized) stopping times S for (X^v_t), T for (X^μ_t) with common final distribution, L(X^v_S|S < ∞) = L(X^μ_T|T < ∞), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown

where denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.     

Multiplicity results on a nonlinear biharmonic equation

Let ω be a bounded open set in Rn with smooth boundary ω We are concerned with a fourth order semilinear elliptic boundary value problem Δ²u + cΔu = bu⁺ + s inω under Dirichlet boundary condition. We investigate the existence of solutions of the fourth order nonlinear equation (0.1) when the nonlinearity bu⁺ crosses eigenvalues of Δ² + cΔ under Dirichlet boundary condition.     

Stability properties of non-negative solutions to a non-autonomous p-Laplacian equation

We study the stability of non-negative stationary solutions of

where Δ_p denotes the p-Laplacian operator defined by Δ_pz = div(z^p−2z); p > 2, Ω is a bounded domain in R^N(N  1) with smooth boundary where [0,1],h:∂Ω→R⁺ with h = 1 when  = 1, λ > 0, and g:Ω×[0,∞)→R is a continuous function. If g(x, u)/u^p−1 be strictly increasing (decreasing), we provide a simple proof to establish that every non-trivial non-negative solution is unstable (stable).     

Cubic theta functions

Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show that

a′(q,z)³=b(q,z)³+c(q)²c(q,z).

This is a counterpart of the identity

a(q,z)³=b(q)²b(q,z³)+c(q,z)³,

which was found by Hirschhorn et al.

The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained.