On computing the dispersion function

A two-dimensional analogue of the well-known bisection method for root finding is presented in order to solve the following problem, related to the dispersion function of a set of random variables: given distribution functionsF ₁,...,F _n and a probabilityp, find an interval [a,b] of minimum width such thatF _i(b)–F _i(a ^–)p, fori=1,...,n.The author wishes to thank Dr. I. D. Coope, for helpful advice offered during the preparation of this paper, and the referee, whose comments contributed to a clearer presentation.     

Spectral properties of operators in linear transport theory

Diagonalizability of operators occurring in linear transport theory is discussed from a general point of view. In particular, the operator A^–1T, where T is the multiplication operator in L² (–1, 1) and A is given by a formula of the type A f = f – _o ⁿ a_j < f, p_j > p_j , is investigated. Diagonalization of this operator which is connected with one-group neutron transport is carried out in the general case that the coefficients a_j are arbitrary complex numbers. Also, a peculiarity of multi-group theory, where the operator involved has a multiple continuous spectrum, is pointed out. A correct interpretation of the main result of that theory is provided.     

Decomposing the complete graph into cycles of many lengths

For all oddv 3 the complete graph onvK _v vertices can be decomposed intov – 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,...,v – 1. Also, for all oddv 7,K _v can be decomposed intov – 3 edge disjoint cycles whose lengths are 3, 4,...,v – 4,v – 2,v – 1,v. Research supported by Australian Research Council grant A49130102     

Some properties of solutions of linear homogeneous differential equations with continuous coefficients

An order-reducing formula is obtained for the Wronskian W[y₁,..., y_n] of a system of (n–l)-times continuously differentiable functions. This formula is used to prove that the condition W[y₁,..., y_n] 0 (x 1) is necessary and sufficient for linear independence on I of the solutions y₁,..., y_m of the equation ..... of order n m. A formula isderived which, given a system of n – 1 linearly independent solutions Y_1,..., Y_n–1 of Eq. (1) constructs in explicit forma solution Y₂ linearly independent with Y_1,..., Y_n–1 (a generalization of Abel's formula).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 51–56, 1985.     

The view-obstruction problem for 3-dimensional spheres

LetC be ann-dimensional sphere with diameter 1 and center at the origin inE ⁿ . The view-obstruction problem forn-dimensional spheres is to determine a constant ν(n) to be the lower bound of those α for which any half-lineL, given byx _i =a _i t (i=1,2,...,n) where parametert≥0 anda _i (i=1,2,...,n) are positive real numbers, intersects

Regular sequences in Rees and symmetric algebras,II

In continuing [7] we study necessary and sufficient conditions for a system of elements b₁,...,b_s,a₁,...,a_t of a local Noetherian ring A such that the sequence b₁,...,b_s,a₁T,a₁-a₂T,...,a_t–1-a_tT,a_t in the Rees algebra A[a₁ T,...,a_t T], T is an indeterminate, constitutes a regular sequence.     

Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

For the equationL ₀ x(t)+L ₁x(t)+...+L _n x ⁽ⁿ⁾(t)=O, whereL _k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x ^(1–1) (a)=0, 1=1, ..., p, andx ^(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx ^(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.     

On the increments of the Wiener process

Summary Let {W(t), t0} be a standard Wiener process and 0<a _t t a nondecreasing function of t. The asymptotic behaviour of several increment processes, obtained from W and a _t , is investigated in terms of their upper classes. In some cases we characterize these classes by means of an integral test. Two such processes are (W(t+a _t) – W(t))a _t ^–1/2 and      

On the existence of one-sided designs

A collection of subsets (called blocks) of a fixed vertex set (possibly with repetition) is called a (t _n , t _n –1, ..., t ₁; a _m , a _m –1, ..., a ₁)-design if it satisfies certain regularity conditions on the number of blocks which contain subsets of the vertex set of certain size, and other regularity conditions on the size of the intersections of certain numbers of the blocks. (For example, a BIBD (or (b, v, r, k, )-configuration) is a (1, 2; 1)-design, and a t-design is a (t, t–1, ..., 1; 1)-design.) A design has design-type (t _n , ..., t ₁; a _m , ..., a ₁) if it satisfies only those conditions. A one-sided design is a design with design-type (t _n , ..., t ₁;) or (;a _m , ..., a ₁). In this paper we show, by construction, that any one-sided design-type is possible.     

Large deviations for a simple closed queueing model

We consider a simple queueing model with one service station. The arrival and service processes have intensitiesa(N–Q _t) andNf(N ^–1 Q _t), where Q_t is the queue length,N is a large integer,a>0 andf(x) is a positive continuous function. We establish the large deviation principle for the sequence of the normalized queue length processq _N ^t =N ^–1Q_t,N1 for both light (a<f(0)) and heavy (af(0)) traffic and use this result for an investigation of ergodic properties ofq _N ^t ,N 1.     

Une famille remarquable de suites recurrentes lineaires

Letu(n) be a recurrent sequence of rational integers, i.e.,u(n+s)+a _s–1 u(n+s–1)+...+a ₀ u(n)=0,n0,a _i,i=0,...,s–1. The polynomialP(x)=x ^s +a _s–1x^s +...+a ₀ is the companion or the characteristic polynomial of the recurrence. It is known that if none of the ratios of the roots ofP is a root of unity, then the setA={n,u(n)=0} is finite. A recent result of F. Beukers shows that ifs=3, then the setA has at most 6 elements and there exists, up to trivial transformations, only one recurrence of order 3 with 6 zeros, found by J. Berstel. In this paper, we construct for eachs, s2 a recurrent sequence of orders, with at leasts ²/2+s/2–1 zeroes, which generalize Berstel's sequence.

     

Law of the iterated logarithm for the quadratic variation of a wiener process

LetP _a be the class of those partitions of intervals [,T], such that |t _i –t _i–1 |>a, wherea is a constant, . It is proved that for anya lim V(T,P _a )/2Tln ₂T=1 a. s., whereln _i x=ln ln x, ifln x e,ln ₂ x=1, ifln x <e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 72–80, 1987.     

A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

Decomposition of bipartite graphs into closed trails

Let Lct(G) denote the set of all lengths of closed trails that exist in an even graph G. A sequence (t ₁,..., t _p ) of elements of Lct(G) adding up to |E(G)| is G-realisable provided there is a sequence (T ₁,..., t _p ) of pairwise edge-disjoint closed trails in G such that T _i is of length T _i for i = 1,..., p. The graph G is arbitrarily decomposable into closed trails if all possible sequences are G-realisable. In the paper it is proved that if a ⩾ 1 is an odd integer and M _a,a is a perfect matching in K _a,a , then the graph K _a,a -M _a,a is arbitrarily decomposable into closed trails.      

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.

Bayesian testing of nonparametric hypotheses and its application to global optimization

Random distribution functions are the basic tool for solving nonparametric decision-theoretic problems. In 1974, Doksum introduced the family of distributions neutral to the right, that is, distributions such thatF(t ₁),[F(t ₂)–F(t ₁)]/[1 –F(t ₁)],...,[F(t _k)–F(t _{k – 1})]/[1 –F(t _{k – 1})] are independent whenevert ₁ < ... <t _kIn practice, application of distributions neutral to the right has been prevented by the lack of a manageable analytical expression for probabilities of the typeP(F(t)<q) for fixedt andq. A subclass of such distributions can be provided which allows for a close expression of the characteristic function of log[1–F(t)], given the sample. Then, thea posteriori distribution ofF(t) is obtained by numerical evaluation of a Fourier integral. As an application, the global optimization problem is formulated as a problem of inference about the quantiles of the distributionF(y) of the random variableY=f(X), wheref is the objective function andX is a random point in the search domain.The author thanks J. Koronacki and R. Zielinski of the Polish Academy of Sciences for their valuable criticism during the final draft of the paper.     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains in a circle

Let a_k, k=1,2,3, be distinct points of the circle U={z:¦z¦<1}, a_3+k=1/¯a_k, k=1,2,3. Let D₁,...,D₆ be a system of nonoverlapping simply connected domains D₁,...,D₆ on,a_k D_k, k=1,...,6. Let R(D_k,a_k) be the conformal radius of the domain D_k with respect to the point a_k. One formulates the following theorem. For any points a_k U, k=1,2,3, and any system of the indicated domains one has the sharp inequality One points out all the cases when equality prevails in (1). One indicates the main steps of the proof of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 99–113, 1983.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

The topological entropy of the transformationx ↦ax (1−x)

A well-known conjecture about the transformationT _a :x ax (1–x) on [0, 1], where 2a4, says that the mapa h _top (T _a ) is monotone. In this paper we show that this is connected with a property of the polynomialsP _k (t) (4t8) given byP ₀ (t)=0 andP _k+1 (t)=(t–P _k t)²)/2, namely that they have in some sense a minimal number of zeros. Furthermore we show for a countable subset of [2, 4], whose limit points form a sequence converging to 4, to be in {a[2,4]:h _top (T _c ), ifc<a andd>a}.With 2 Figures