The k-record processes are i.i.d.

Let X ₁, X ₂, ... be i.i.d. positive random variables, and let _n be the initial rank of X _n (that is, the rank of X _n among X ₁, ..., X _n). Those observations whose initial rank is k are collected into a point process N ^k on ⁺, called the k-record process. The fact that {itN^k; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N ¹, ..., N ^k, the lifetimes in rank k of all observations of initial rank at most k are independent geometric random variables.These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a time-space Poisson process. Initially, we think of this Poisson process as having values in ⁺, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes.     

Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

Let us consider the boundary‐value problem where g: ? → ? is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ?² and ∈ C [0, π ] with ∫^π ₀ (x) sin x dx = 0. If λ ₁ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ₁, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new model updating method for damped structural systems

In this paper, the following two are considered: Problem IQEP Given M_a∈ SR ^n×n, Λ=diag{λ₁, …, λ_p}∈ C ^p×p, X=[x₁, …, x_p]∈ C ^n×p, and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in {\mathbf{C}}In this paper, the following two are considered: Problem IQEP Given M_a∈ SR ^n×n, Λ=diag{λ₁, …, λ_p}∈ C ^p×p, X=[x₁, …, x_p]∈ C ^n×p, and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in {\mathbf{C}}$, x_2j=x?_2j?1∈ C ⁿ for j = 1,…,l, and λ_k∈ R , x_k∈ R ⁿ for k=2l+1,…,p, find real‐valued symmetric (2r+1)‐diagonal matrices D and K such that ∥M_aXΛ²+DXΛ+KX∥=min. Problem II Given real‐valued symmetric (2r+1)‐diagonal matrices D_a, K_a∈ R ^n×n, find $(\hat{D},\hat{K}) \in {\mathscr{S}}_{DK}$ such that $\|\hat{D}-D_a \|^2+ \| \hat{K}-K_a \|^2=\rm{inf}_{(D,K) \in {\mathscr{S}}_{DK}}(\|D-D_a\|^2+\|K-K_a\|^2)$, where ??_DK is the solution set of IQEP. By applying the Kronecker product and the stretching function of matrices, the general form of the solution of Problem IQEP is presented. The expression of the unique solution of Problem II is derived. A numerical algorithm for solving Problem II is provided. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On Abel—Poisson type and Riesz means

В работе исследуются ядра методов суммиро вания типа Абеля—Пуассона и Рис са, применяемых к кратны м интегралам Фурье. Вы ясняются условия на параметры, определяющие эти методы, при которы х их ядра неотрицател ьны. Полученные результа ты можно сформулировать в тер минах положительной определенности неко торых функций. Наприм ер, функция exp(? ¦x¦^α) при 0<а ≦2 является, а при α>2 не является положитель но определенной в евклидовом простра нствеE _N размерностиN (N=1, 2, ...). Далее, еслиt ₊=max (t, 0), то при любом натураль номN на интервале 0<λ<2 существует неубываю щая непрерывная функцияk _N (λ) такая, что функция (1 ? ¦х¦^λ) ₊ ^k приk≧k _N (λ) является, а приk<k _N (λ) не является положительно опреде ленной в пространств еE _N . При этом $$k_N (1) = \frac{{N + 1}}{2}, k_N (2 - 0) = + \infty , k_N (\lambda ) \geqq \lambda + \frac{{N - 1}}{2}.$$ Если же λ≧2, то функция (1?¦x¦^λ) ₊ ^k ни при каком значении параметраk не является положите льно определенной в прост ранствеE _N ,N=1, 2, .... Кроме того, исследует ся порядок приближен ия функцийN переменных класса Н икольскогоk _P ^α , 1≦р<∞, 0<а<2, операторам и типа Абеля—Пуассон а в метрикеL _p (E _N ).     

The first eigenvalue of the p‐Laplace operator under powers of mean curvature flow

In this paper, we show the following main results. Let (Mⁿ,g(t)), t ∈ [0,T), be a solution of the unnormalized H^k ? flow on a closed manifold, and λ_1,p(t) be the first eigenvalue of the p‐Laplace operator. If there exists a nonnegative constant ε such that in M × [0,T) and in M × [0,T),then λ_1,p(t) is increasing and the differentiable almost everywhere along the unnormalized H^k ? flow on [0,T). At last, we discuss some useful monotonic quantities. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ?Ω, 1 < p < N,0 ≤ a < (N ? p) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (N ? pd),d = a + 1 ? b,a ≤ b < a + 1. By the variational methods, we prove that problem 0.1 admits a sequence of solutions u_k under the appropriate assumptions on the weight functions H(x) and H(x). For the critical case, s = q,h(x) = | x | ^{? bq}, we obtain that problem 0.1 has at least a nonnegative solution with p < r < q and a sequence of solutions u_k with 1 < r < p < q and J(u_k) → 0 as k → ∞ , where J(u) is the energy functional associated to problem 0.1 . Copyright © 2013 John Wiley & Sons, Ltd.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Some limit results for lag sums of independent,non-i.i.d., random variables

Summary Let X _k be an independent, but not necessarily i.i.d., sequence of random variables. Suppose EX _k 0 and that _Nk ² = E(S _N –S _N–k )² where S _N =X ₁+...+X_N. We obtain conditions under which and obtain other related results. We assume either that P{¦X _k ¦t}Mt^–r for all k and all t>0, or that E(expX _k )b< for all k and all &#x2266;t₀. The results are like earlier results for the i.i.d. case, but the method of proof is entirely different.The research of both authors was supported by National Science Foundation Grant No. MCS 81-00740     

Finitely Presented Functors and Degenerations#

Given two d-dimensional Λ-modules M and N, then M degenerates to N if and only if there exists an exact sequence of the form 0→ U→ U ⊕ M→ N→ 0 for some U ? mod Λ (Zwara, 1998 Zwara , G. ( 1998 ). A degeneration-like order for modules . Arch. Math. 71 : 437 – 444 . [CSA] [CROSSREF]  [Google Scholar]). Having this as a starting point, in this article we give a characterization of degenerations by the existence of a certain finitely presented functor. This gives new information about U in the sequence above. We show how this new information can be used to prove that M   _deg N even when M ⊕ X ≤  _deg N ⊕ X for some X for modules over Λ _q  = k〈 x,y〉/〈 x ²,y ²,xy + qyx〉, q ≠ 0 in k, where k is an algebraically closed field.     

A Fujita-type global existence—global non-existence theorem for a system of reaction diffusion equations with differing diffusivities

In this paper we condiser non-negative solutions of the initial value problem in ?^N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.

• (a) If δ = 0 then u=v=0 is the only non-negative global solution of the system.
• (b) If δ>0, non-negative non-globle solutions always exist for suitable initial values.
• (c) If 0<?1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.
• (d) If N > 2, 0 < δ ? 1 and max (α β) < (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L^∞(?^N×[0, ∞]) and satisfy 0 < u(X, t) ? c∣x∣^?2α and 0 < v(X, t) ? c ∣x∣^?2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data.
• (e) Suppose 0 > δ 1 and max (α, β) < N/2. If N> = 1,2 or N > 2 and max (p, q)? N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H¹ (K) where K(x) ? exp(¼∣x∣²). They decay like e^{[max(α,β)-(N/2)+ε]t} for every ε > 0. These solutions are classical solutions for t > 0.
• (f) If max (α, β) < N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u₀, v₀) < (N/2)?;max(α, β). Suppose min(p, q) ? 1.
• (g) If pq ≥ 1, all non-negative solutions are global. Suppose min(p, q) < 1.
• (h) If pg > 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.
• (i) If 0 < δ ? 1, pq > 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.
• (j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N/2, there are both global and non-negative solutions.

We also indicate some extensions of these results to moe general systems and to othere geometries.     

Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group

Denoting Δ_? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ? ^N , we prove some nonexistence results for solutions of inequalities of the three types

Conditional limit theorems for exponential families and finite versions of de Finetti's theorem

Consider an exponential familyP _λ which is maximal, smooth, and has uniformly bounded standardized fourth moments. Consider a sequenceX ₁,X ₂,... of i.i.d. random variables with parameter λ. LetQ _nsk be the law ofX ₁,...,X _k given thatS _n=X ₁+...+X _n=s. Choose λ so thatE _λ(X ₁)=s/n. Ifk andn→∞ butk/n→0, then $$\parallel Q_{nsk} - P_\lambda ^k \parallel = \gamma \frac{k}{n} + o\left( {\frac{k}{n}} \right)$$ where γ=1/2E{|1?Z ²|} andZ isN(0,1). The error term is uniform ins, the value ofS _n. Similar results are given fork/n→θ and for mixtures of theP _λ ^k . Versions of de Finetti's theorem follow.     

On Modules and Complexes Without Self-Extensions

Let Λ be an Artin algebra over a commutative Artinian ring, k. If M is a finitely generated left Λ -module, we denote by Ω (M) the kernel of η _M : P _M  → M a minimal projective cover. We prove that if M and N are finitely generated left Λ -modules and Ext _Λ ¹ (M, M) = 0, Ext _Λ ¹ (N, N) = 0, then M? N if and only if M/rad M? N/rad N and Ω (M)? Ω (N).

Now if k is an algebraically closed field and (d _i ) _i?? is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X ?  ^b (Λ), the bounded derived category of Λ, with Hom ^b (Λ)(X,X[1]) = 0 and dim _k H ⁱ (X) = d _i for all i ? ?, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001 Huisgen-Zimmermann , B. , Saorín , M. ( 2001 ). Geometry of chain complexes and outer automorphisms under derived equivalence . Trans. Amer. Math. Soc. 353 : 4757 – 4777 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]).     

On a class of coupled Schrödinger systems with critical Sobolev exponent growth

Consider the following two critical nonlinear Schrödinger systems: (0.1) (0.2) where is a smooth bounded domain, N≥3,?λ(Ω) < λ₁,λ₂<0,μ₁,μ₂>0,α,β≥1 with α + β = 2^?,γ ≠ 0,λ(Ω) is the first eigenvalue of ?Δ with the Dirichlet boundary condition and For N = 3,λ₁=λ₂,γ > 0 small, we obtain the existence of positive least energy solution of 0.1 and 0.2 . For N≥5,γ > 0, the existence of positive least energy solution of 0.2 is established. For N≥5,γ ≠ 0, we prove that 0.1 possesses a positive least energy solution. The limit behavior of the positive least energy solutions when γ→?∞ and phase separation for 0.1 are also considered. Copyright © 2015 John Wiley & Sons, Ltd.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

A variational characterisation of spherical designs

In this paper we first establish a new variational characterisation of spherical designs: it is shown that a set , where , is a spherical L-design if and only if a certain non-negative quantity A_L,N(X_N) vanishes. By combining this result with a known “sampling theorem” for the sphere, we obtain the main result, which is that if is a stationary point set of A_L,N whose “mesh norm” satisfies h_XN<1/(L+1), then X_N is a spherical L-design. The latter result seems to open a pathway to the elusive problem of proving (for fixed d) the existence of a spherical L-design with a number of points N of order (L+1)^d. A numerical example with d=2 and L=19 suggests that computational minimisation of A_L,N can be a valuable tool for the discovery of new spherical designs for moderate and large values of L.     

On the number of interior peak solutions for a singularly perturbed Neumann problem

We consider the following singularly perturbed Neumann problem: where Δ = Σ ?²/?x is the Laplace operator, ? > 0 is a constant, Ω is a bounded, smooth domain in ?^N with its unit outward normal ν, and f is superlinear and subcritical. A typical f is f(u) = u^p where 1 < p < +∞ when N = 2 and 1 < p < (N + 2)/(N ? 2) when N ≥ 3. We show that there exists an ?₀ > 0 such that for 0 < ? < ?₀ and for each integer K bounded by where α_{N, Ω, f} is a constant depending on N, Ω, and f only, there exists a solution with K interior peaks. (An explicit formula for α_{N, Ω, f} is also given.) As a consequence, we obtain that for ? sufficiently small, there exists at least [α_{N, Ωf}/?^N (|ln ?|)^N] number of solutions. Moreover, for each m ∈ (0, N) there exist solutions with energies in the order of ?^N?m. © 2006 Wiley Periodicals, Inc.