Group pursuit in Pontryagin’s nonstationary example

On the interval [t ₀, ∞), we consider the following group pursuit problem with one evader: 1 $$ z_i^{(l)} + a_1 (t)z_i^{(l - 1)} + a_2 (t)z_i^{(l - 2)} + \cdots + a_l (t)z_i = u_i - v, u_i ,v \in V, z_i^{(q)} (t_0 ) = z_i^q , $$ where z _i , u _i , v ∈ R ^v , (v ≥ 2), V is a strictly convex compact set in R ^v , the functions a ₁(t), a ₂(t), …, a _l (t) are continuous, i = 1, 2, …, n and q = 0, 1, …, l ? 1. Let ? _q (t, s) be the solution of the Cauchy problem $$ \begin{gathered} \omega ^{(l)} + a_1 (t)\omega ^{(l - 1)} + a_2 (t)\omega ^{(l - 2)} + \cdots + a_l (t)\omega = 0, \omega ^{(q)} (s) = 1, \hfill \\ \omega ^{(r)} (s) = 0, r = 0, \ldots q - 1,q + 1, \ldots ,l - 1, \hfill \\ \end{gathered} $$ and let $$ \xi _\iota (t) = \varphi _0 (t,t_0 )Z_i^0 + \varphi _1 (t,t_0 )Z_i^1 + \cdots + \varphi _{l - 1} (t,t_0 )Z_i^{l - 1} . $$ We prove that if there exist continuous functions α _i (t) and ξ _i ¹ (t) such that the ξ _i ¹ (t) are Bohr almost periodic on [t ₀, ∞), α _i (t) > 0 for all t ≥ t ₀, lim _t→∞(ξ _i ¹ (t) ? α _i (t)ξ _i (t)) = 0, lim _t→∞(min _i α _i (t) ∝ _t0 ^t |? _l?1(t, s)| ds) = ∞, and there exist points h _i ⁰ ∈ H _i ¹ = {ξ _i ¹ (t), t ∈ [0, ∞)} such that 0 ∈ Int co{h _i ⁰ }, then the pursuit problem with evader discrimination is solvable.     

The integral representation of the exponential product of stochastic semigroups

An integral representation is obtained for the exponential product of stochastic semigroups $$X_s^t \otimes Z_s^t = X_s^t + \mathop \smallint \limits_{s< u< t} X_u^t dV_u X_s^u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} X_{u_2 }^t dV_{u_2 } X_{u_1 }^{u_2 } dV_{u_1 } X_s^{u_1 } + \cdots ,$$ whereV _t is the generating process of the semigroupZ _s ^t and the integrals are understood in the sense of mean-square limits of the Riemann-Stieltjes sums. This representation is different from the traditional representation $$X_s^t \otimes Z_s^t = E + \mathop \smallint \limits_{s< u< t} dW_u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} dW_{u_2 } dW_{u_1 } + \cdots ,$$ in which the integration extends over the processW _t=Y_t+V_t that is the generating process of the exponential productX _s ^t ?Z _s ^t andY _t is the generator of the semigroupX _s ^t .     

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

Boundary noncrossings of additive Wiener fields∗

Let {W _i (t), t ∈ ?₊}, i = 1, 2, be two Wiener processes, and let W ₃ = {W ₃(t), t ∈ ? ₊ ² } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P _f = P{W ₁(t ₁) + W ₂(t ₂) + W ₃(t) + f(t) ≤ u(t), t ∈ ? ₊ ² }, where f, u : ? ₊ ² → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P _γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W ₁(t ₁) + W ₂(t ₂) + W ₃(t). It turns out that our approach is also applicable for the additive Brownian pillow.     

Modification of a Finsler space by a normalized semi-parallel vector field

LetF _n be a Finsler space with metric functionF(x, y). M. Matsumoto [6] has defined a modified Finsler spaceF _n ^* whose metric functionF ^*(x, y) is given byF ^*2 = = F² + (X_i(x)yⁱ)², whereX _i are the components of a covariant vector which is a function of coordintae only. Since a concurrent vector is a function of coordinate only, Matsumoto and Eguchi [9] have studied various properties of the modified Finsler spaceF _n ^* under the assumption thatX _i are the components of a concurrent vector field inF _n. In this paper we shall introduce the concept of semi-parallel vector field inF _n and study the properties of modified Finsler spaceF _n ^* .     

Finite projective spaces and intersecting hypergraphs

Let ? be a family ofk-subsets on ann-setX andc be a real number 0 <c<1. Suppose that anyt members of ? have a common element (t ≧ 2) and every element ofX is contained in at mostc|?| members of ?. One of the results in this paper is (Theorem 2.9): If $$c = {{(q^{t - 1} + ... + q + 1)} \mathord{\left/ {\vphantom {{(q^{t - 1} + ... + q + 1)} {(q^t + ... + q + 1)}}} \right. \kern-\nulldelimiterspace} {(q^t + ... + q + 1)}}$$ . whereq is a prime power andn is sufficiently large, (n >n (k, c)) then The corresponding lower bound is given by the following construction. LetY be a (q ^t + ... +q + 1)-subset ofX andH ₁,H ₂, ...,H _|Y| the hyperplanes of thet-dimensional projective space of orderq onY. Let ? consist of thosek-subsets which intersectY in a hyperplane, i.e., ?={F∈( _k ^X ): there exists ani, 1≦i≦|Y|, such thatY∩F=H _i }.     

The combinatorics and extreme value statistics of protein threading

In protein threading, one is given a protein sequence, together with a database of protein core structures that may contain the natural structure of the sequence. The object of protein threading is to correctly identify the structure(s) corresponding to the sequence. Since the core structures are already associated with specific biological functions, threading has the potential to provide biologists with useful insights about the function of a newly discovered protein sequence. Statistical tests for threading results based on the theory of extreme values suggest several combinatorial problems. For example, what is the number of waysm′=# _t {L _i >x _i } _i ⁼⁰ⁿ of choosing a sequence {X _i } _i ⁼¹ⁿ from the set {1, 2, ...,t}, subject to the difference constraints {L _i =X _i+1?X _i >x _i } _i ⁼⁰ⁿ , whereX ₀=0,X _n+1=t+1, and {x _i } _i ⁼⁰ⁿ is an arbitrary sequence of integers? The quantitym′ has many attractive combinatorial interpretations and reduces in special continuous limits to a probabilistic formula discovered by the Finetti. Just as many important probabilities can be derived from de Finetti's formula, many interesting combinatorial quantities can be derived fromm′. Empirical results presented here show that the combinatorial approach to threading statistics appears promising, but that structural periodicities in proteins and energetically unimportant structure elements probably introduce statistical correlations that must be better understood.     

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

Approximation for counts of 2-runs in a two state Markov chain

We consider a Markov chain X = {Xi,i = 1,2,...} with the state space {0,1},and define W =∑i=1n XiXi+1,which is the number of 2-runs in X before time n + 1.In this paper,we prove that the negative binomial distribution is an appropriate approximation for LW when VarW is greater than EW.The error estimate obtained herein improves the corresponding result in previous literatures.     

The semigroup approach to first order quasilinear equations in several space variables

The Cauchy problem for u _t + Σ _{i = 1} ⁿ (φ _i (u)) _xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σ _{i = 1} ⁿ (φ _i (u)) _xi = h for h∈L ¹(Rⁿ), which in turn is studied via the regularized equation $$ u + \sum\nolimits_{i = 1}^n {\left( {\phi _i \left( u \right)} \right)} _{xi} - \varepsilon \Delta u = h $$ .     

Self-normalized moderate deviations for independent random variables

Let X1,X2,...be a sequence of independent random variables(r.v.s) belonging to the domain of attraction of a normal or stable law.In this paper,we study moderate deviations for the self-normalized sum ∑ni=1 Xi/Vn,p,where Vn,p =(∑ni=1 |Xi|p)1/p(p>1).Applications to the self-normalized law of the iterated logarithm,Studentized increments of partial sums,t-statistic,and weighted sum of independent and identically distributed(i.i.d.) r.v.s are considered.     

A stability theorem for entropy solutions of initial value problems for first order quasilinear hyperbolic systems in several space variables

In this paper we prove an uniqueness and stability theorem for the solutions of Cauchy problem for the systems $$\frac{\partial }{{\partial t}}u + \sum\limits_{i = 1}^n { \frac{\partial }{{\partial x_i }} } f^i (x,t,u) = g(x,t,u),$$ whereu is a vector function (u ₁ (x, t),..., u _r (x, t)),f ⁱ =(a ₁ ⁱ (x, t, u),..., a _r ⁱ (x, t, u)), i=1,...,n, g=(g ₁ (x, t, u),...,g _r (x, t, u),i G ? ⁿ and t≥0. We use the concept of entropy solution introduced by Kruskov and improved by Lax, Dafermos and others autors. We assume that the Jacobian matricesf _u ⁱ are symmetric and the Hessian(a _j ⁱ ) _uu (i=1,...,n; j=1,...,r) are positive. We obtain uniqueness and stability inL _loc ² within the class of those entropy solutions which satisfy $$\frac{{u_j (---,x_i ,---,t)---u_j (---,y_i ,---,t)}}{{x_i - y_i }} \geqslant - K(t),$$ (i=1,...,n; j=1,...,r) for (?,x _i ,?,t), (?,y _i ,?,t) on a compact setD ? ? ⁿ x (0, ∞) and a functionK(t)∈L _loc ¹ ([0, ∞)) depending onD. Here we denote by (?,x _i ,?,t) and (?,y _i ,?,t) two points whose coordinates only differ in thei-th space variable. At the end we relax the hypotheses of symmetry and convexity on the system and give a theorem of uniqueness and stability for entropy solutions which are locally Lipschitz continuous on a strip ? ⁿ x [0,T].     

Dimension theory and superpositions of continuous functions

The main result of this paper is the following: IfX is a compact two dimensional metric space, and {φ _i} _i ^{= 1/4} are four functions inC(X), then there exists a functionf inC(X) which cannot be represented in the form: $$f(x) = \sum\limits_{i = 1}^4 {g_\iota (\varphi _i (x))} $$ , with $$g_\iota \in C(R)$$ .     

Combinational properties of sets of residues modulo a prime and the Erdős—Graham problem

Consider an arbitrary ε > 0 and a sufficiently large prime p > 2. It is proved that, for any integer a, there exist pairwise distinct integers x ₁, x ₂, ..., x _N , where N = 8([1/ε + 1/2] + 1)² such that 1 ≤ x _i ≤ p ^ε, i = 1, ..., N, and $$a \equiv x_1^{ - 1} + \cdots + x_N^{ - 1} (\bmod p)$$ , where x _i ^?1 is the least positive integer satisfying x _i ^?1 x _i ≡ 1 (modp). This improves a result of Sparlinski.     

On normal control processes

Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB _{p′, r′} of the control spaceL ^p′([τ, T]),l _m ^r ′ are characterized in terms ofH(t)=X(T)X ^?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.     

Positive solutions for multi-point boundary value problems for nonlinear fractional differential equations

The paper studies the problem of existence of positive solution to the following boundary value problem: $D_{0^ + }^\sigma u''(t) - g(t)f(u(t)) = 0$ , t ∈ (0, 1), u″(0) = u″(1) = 0, au(0) ? bu′(0) = Σ _i=1 ^m?2 a _i u(ξ _i ), cu(1) + du′(1) = Σ _i=1 ^m?2 b _i u(ξ _i ), where $D_{0^ + }^\sigma$ is the Riemann-Liouville fractional derivative of order 1 < σ ≤ 2 and f is a lower semi-continuous function. Using Krasnoselskii’s fixed point theorems in a cone, the existence of one positive solution and multiple positive solutions for nonlinear singular boundary value problems is established.     

New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings

Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T ₁, T ₂: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k _n ⁽ⁱ⁾ } and {δ _n ⁽ⁱ⁾ } ? [1, ∞),, i = 1, 2, respectively such that Σ _n=1 ^∞ (k _n ⁽ⁱ⁾ ? 1) < ∞, Σ _n=1 ⁽ⁱ⁾ δ _n ⁽ⁱ⁾ < ∞, and F = F(T ₁) ∩ F(T ₂) ≠ ?. For an arbitrary x ₁ ∈ C, let {x _n } be the sequence in C defined by $$ \begin{gathered} y_n = P\left( {\left( {1 - \beta _n - \gamma _n } \right)x_n + \beta _n T_2 \left( {PT_2 } \right)^{n - 1} x_n + \gamma _n v_n } \right), \hfill \\ x_{n + 1} = P\left( {\left( {1 - \alpha _n - \lambda _n } \right)y_n + \alpha _n T_1 \left( {PT_1 } \right)^{n - 1} x_n + \lambda _n u_n } \right), n \geqslant 1, \hfill \\ \end{gathered} $$ where {α _n }, {β _n }, {γ _n } and {λ _n } are appropriate real sequences in [0, 1) such that Σ _n=1 ^∞ ] γ _n < ∞, Σ _n=1 ^∞ λ _n < ∞, and {u _n }, }v _n } are bounded sequences in C. Then {x _n } and {y _n } converge strongly to a common fixed point of T ₁ and T ₂ under suitable conditions.     

Difference based estimation for partially linear regression models with measurement errors

This paper is concerned with the estimating problem of the partially linear regression models where the linear covariates are measured with additive errors. A difference based estimation is proposed to estimate the parametric component. We show that the resulting estimator is asymptotically unbiased and achieves the semiparametric efficiency bound if the order of the difference tends to infinity. The asymptotic normality of the resulting estimator is established as well. Compared with the corrected profile least squares estimation, the proposed procedure avoids the bandwidth selection. In addition, the difference based estimation of the error variance is also considered. For the nonparametric component, the local polynomial technique is implemented. The finite sample properties of the developed methodology is investigated through simulation studies. An example of application is also illustrated.