On the cumulative quantile regression process

Let (X, Y) be a bivariate random vector and F(x) the marginal distribution function of X. The quantile regression (QR) function of Y on X is defined as r(u) = E[Y | F(X) = u] and the cumulative QR function (CQR) M(u) as its integral over [0, u]. The empirical counterpart based on a sample of size n is M _n (u). In this paper, we construct strong Gaussian approximations of the associated CQR process under appropriate assumptions. The construction provides a firm basis for the study of functional statistics based on M _in (u). A law of the iterated logarithm for the CQR process follows from our result.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Vanishing exponential integrability for functions whose gradients belong to L(log(e+L))

If the gradient of u(x) is nth power locally integrable on Euclidean n-space, then the integral average over a ball B of the exponential of a constant multiple of |u(x)−u_B|^n/(n−1), u_B=average of u over B, tends to 1 as the radius of B shrinks to zero—for quasi almost all center points. This refines a result of N. Trudinger (1967). We prove here a similar result for the class of gradients in Lⁿ(log(e+L))^α, 0?α?n−1. The results depend on a capacitary strong-type inequality for these spaces.     

Trees with second minimum general Randić index for α>0

The general Randi? index R _α (G) of a graph G is the sum of the weights (d(u)d(v)) ^α of all edges uv of G, where α is a real number(α≠0) and d(u) denotes the degree of the vertex u. We have known that P _n has minimum general Randi? index for α>0 among trees when n≥5. In this paper, we prove that P _n,3 has second minimum general Randi? index for α>0 among trees when n≥7.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Fractional BVPs with strong time singularities and the limit properties of their solutions

In the first part, we investigate the singular BVP \(\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u\) , u(0) = A, u(1) = B, ^c D ^α u(t)| _t=0 = 0, where \(\mathcal{H}\) is a continuous operator, α ∈ (0, 1) and a < 0. Here, ^c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems \(\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)\) , u(0) = A, u(1) = B, \(\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0\) where a < 0, 0 < β _n ≤ α _n < 1, lim _n→∞ β _n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.     

Existence of a unique solution to a quasilinear elliptic equation

The purpose of this paper is to prove the existence of a unique classical solution u(x) to the quasilinear elliptic equation −∇⋅(a(u)∇u)+v⋅∇u=f, where u(x₀)=u₀ at x₀∈Ω and where n⋅∇u=g on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution u(x) exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location x₀∈Ω, and where n⋅∇u is known on the boundary.     

Functional Chung laws for small increments of the empirical process and a lower bound in the strong invariance principle

Consider the set Θ _n of all a _n -sized increment processes of the uniform empirical process α _n on [0, 1]. We assume that a _n ↓ 0, na _n ↑ ∞, d _n = na _n (log _n)^?1 → ∞ and na _n (log n)^?7/3 = O(1). In Berthet (1996, 2005) the fourth assumption was shown to be critical with respect to the pointwise rates of convergence in the functional law of Deheuvels and Mason (1992) for Θ _n because strong approximation methods become ineffective at such a small scale a _n . We are now able to study directly these small empirical increments and compute the exact rate of clustering of Θ _n to any Strassen function having Lebesgue derivative of bounded variation by making use of a sharp small deviation estimate for a Poisson process of high intensity due to Shmileva (2003a). It turns out that the best rates are of order d _n ^1/4 (log n)^?1 and are faster than in the Brownian case whereas the slowest rates are of order d _n ^?1/2 and correspond to the apparently crude ones obtained in Berthet (2005) by means of Gaussian small ball probabilities. These different sharp properties of the empirical and Brownian paths imply an almost sure lower bound in the strong invariance principle and provide a new insight into the famous KMT approximation of α _n .     

The asymptotic distribution of weighted empirical distribution functions

Let G_n denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between G_n(u) and u of the forms 6w_v(u)D_n(u)6 and 6w_v(G_n(u))D_n(u)6, where D_n(u) = G_n(u)?u, w_v(u) = (u(1?u))^?1+v and 0 ? v < $\text{1}\text{2}$ is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function w_v with $\text{1}\text{2}$ ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function w_v with 0 ? v < $\text{1}\text{2}$ are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.     

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

We consider random self-adjoint Jacobi matrices of the form

(Jωu)(n)=an(ω)u(n+1)+bn(ω)u(n)+an−1(ω)u(n−1)     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

Central and local limit theorems for the coefficients of polynomials of binomial type

We introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials P_n(x) of binomial type; that is, a sequence P_n(x) satisfying $\text{exp}\text{(xg(u)) = ∑}{}_{\mathrm{n=0}}{}^{\mathrm{\infty }}\text{P}{}_{\mathrm{n}}\text{(x)(}\text{u}{}^{\mathrm{n}}\text{n!}\text{)}$ for some (formal) power series g(u) lacking constant term. We give a complete answer in the case when g(u) is a polynomial, and point out the widest known class of nonpolynomial power series g(u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of P_n(x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g(u) which assures these conditions on the coefficients of P_n(x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of P_n(x) are developed.     

Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

We consider a convolution-type integral equation u = k ? g(u) on the half line (???; a), a ?? ?, with kernel k(x) = x ^???1, 0 < ??, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if ?? ?? (1, 4), then for any two nontrivial solutions u ₁, u ₂ there exists a constant c ?? ? such that u ₂(x) = u ₁(x +c), ??? < x. The results are obtained by applying Hilbert projective metrics.     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Trees of extremal connectivity index

The connectivity index w_α(G) of a graph G is the sum of the weights (d(u)d(v))^α of all edges uv of G, where α is a real number (α≠0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w₁(T). Also, for -1?α<0, we give a sharp lower bound and a upper bound for w_α(T).     

Separation of variables of a generalized porous medium equation with nonlinear source

This paper considers a general form of the porous medium equation with nonlinear source term: u_t=(D(u)u_xⁿ)_x+F(u), n≠1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions to the resulting equations are constructed, and their behavior are also investigated.     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

Loi fonctionnelle du logarithme itéré pour les incréments du processus des espacements

Let a_n(l): 0 ≤ /denote the reseated empirical process based upon uniform spacings. Let {h_n, n ≥ 1 } be a sequence decreasing to 0. Under appropriate conditions on h_n. We give a functional lnw of the iterated logarithm for the set of increment functions {a_n (l + h_n.) — a_n(l): 0 ≤ / ≤ 1 -h_n}.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.