Persistent convergence on randomly deleted sets

At time t_k, a unit with magnitude X_k and lifetime L_k enters a system. Let λ be a real valued function on the finite real sequences. One such sequence, B*_t, consists of the X_k's for which t_k t < t_k + L_k. When λ(X₁,…, X_n) converges (in some sense) to φ, we find conditions under which λ(B*_t) converges or fails to converge to φ in the same sense.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

Gowers norms and pseudorandom measures of subsets

Let $A \subset {{\Bbb Z}_N}$, and ${f_A}(s) = \left\{ {\begin{array}{*{20}{l}}{1 - \frac{{|A|}}{N},}&{{\rm{for}}\;s \in A,}\\{ - \frac{{|A|}}{N},}&{{\rm{for}}\;s \notin A.}\end{array}} \right.$ We define the pseudorandom measure of order k of the subset A as follows, P_k(A, N) = $\begin{array}{*{20}{c}}{\max }\\D\end{array}$|$\mathop \Sigma \limits_{n \in {\mathbb{Z}_N}}$f_A(n + c₁)f_A(n + c₂) … f_A(n + c_k)|, where the maximum is taken over all D = (c₁, c₂, . . . , c_k) ∈ ${\mathbb{Z}^k}$ with 0 ≤ c₁ < c₂ < … < c_k ≤ N - 1. The subset A ⊂ ${{\mathbb{Z}_N}}$ is considered as a pseudorandom subset of degree k if P_k(A, N) is “small” in terms of N. We establish a link between the Gowers norm and our pseudorandom measure, and show that “good” pseudorandom subsets must have “small” Gowers norm. We give an example to suggest that subsets with “small” Gowers norm may have large pseudorandom measure. Finally, we prove that the pseudorandom subset of degree L(k) contains an arithmetic progression of length k, where L(k) = 2·lcm(2, 4, . . . , 2|$\frac{k}{2}$|), for k ≥ 4, and lcm(a₁, a₂, . . . , a_l) denotes the least common multiple of a₁, a₂, . . . , a_l.     

Multicolored forests in complete bipartite graphs

If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=(a₁,a₂,…,a_k), in which a_i denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph K_n,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of K_n,n into multicolored trees.     

Solution to an extremal problem on bigraphic pairs with a <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connected realization

Let S=(a₁,...,a_m; b₁,...,b_n), where a₁,...,a_m and b₁,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S=(a₁,...,a_m; b₁,...,b_n) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a₁,...,a_m and b₁,...,b_n are the degrees of the vertices in X and Y, respectively. Let Z₃ be the cyclic group of order 3. Define σ(Z₃, m, n) to be the minimum integer k such that every bigraphic pair S=(a₁,...,a_m; b₁,...,b_n) with a_m, b_n ≥ 2 and σ(S)=a₁ +... + a_m ≥ k has a Z₃-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z₃, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z₃, m, n) for m ≥ n ≥ 4.     

Instabilities of robot motion

Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let X denote the space of all admissible configurations of a mechanical system. A motion planner is given by a splitting X×X=F₁F₂F_k (where F₁,…,F_k are pairwise disjoint ENRs, see below) and by continuous maps s_j :F_j→PX, such that Es_j=1_Fj. Here PX denotes the space of all continuous paths in X (admissible motions of the system) and E :PX→X×X denotes the map which assigns to a path the pair of its initial–end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets F_j in any motion planner in X. We also introduce a new notion of order of instability of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space X. We study a number of specific problems: motion of a rigid body in R³, a robot arm, motion in R³ in the presence of obstacles, and others.     

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.     

Invariant record processes and applications to best choice modelling

Let X₁, X₂,…be identically distributed random variables from an unknown continuous distribution. Further let I_r(1), I_r(2),…be a sequence of indicator functions defined on X₁, X₂,…by I_r(k) = 0 if k < r, I_r(k) = 1 if X_k is a r-record AND = 0 otherwise. Suppose that we observe X₁, X₂,… at times T₁ < T₂ <… where the T_k's are realisations of some regular counting process (N(τ)) defined on the positive half-line. Having observed [0, τ], say, the problem is to predict the future behaviour of the counting processes (R_r(τ, s))_sτ = # r-records in [τ, s]. More specifically the objective of this paper is to show that these processes can be (inhomogeneous) Poisson processes even if (N(τ))_τ0 has dependent increments.

The strong link between optimal selection and optimal stopping of record sequences or record processes, perhaps not fully recognized so far, is pointed out in this paper. It is shown to lead to a unification of the treatment of problems which, at first sight, are rather different. Moreover the stopping of record processes in continuous time can lead to rigorous and elegant solutions in cases where dynamic programming is bound to fail. Several examples will be given to facilitate a comparison with other methods.     

An asymptotic representation for products of random matrices

Let {X_k} be a stationary ergodic sequence of nonnegative matrices. It is shown in this paper that, under mild additional conditions, the logarithm of the i, jth element of X_t···X₁ is well approximated by a sum of t random variables from a stationary ergodic sequence. This representation is very useful for the study of limit behaviour of products of random matrices. An iterated logarithm result and an estimation result of use in the theory of demographic population projections are derived as corollaries.     

A characterization of multivariate normal distribution and its application

Let X₁, X₂, …, X_n be i.i.d. d-dimensional random vectors with a continuous density. Let and . In this paper we find that the distribution of Z_k (or Y_k) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.     

Compatibilité entre structures d''intervalles et relations d''orde

Compatibility between interval structures and partial orderings.

If H=(X,E) is a hypergraph, n the cardinality of X,I_n the ordered set {1..n} and < an order relation on X, we call F(X,<) the set of the one-to-one functions from X to I_n which are compatible with <. If AI_n we denote by (A) the length of the smallest interval of I_n which contains A.

We first deal with the following problem: Find ƒF(X,<) which minimise . The a_e, eR are positive coefficients.

This problem can be understood as a scheduling problem and is checked to be NP-complete. We learn how to recognize in polynomial time those hypergraphs H=(X,E) which induce an optimal value of z min equal to .

Next we work on a dual question which arises about interval graphs, when some partial orderings on the vertex set of these graphs intend to represent inclusion, overlapping or anteriority relations between closed intervals of the real line.     

Convex 4-valent polytopes

Let {p_k}_k≥3 be a sequence of nonnegative integers which satisfies 8 + Σ_k≥3 (k-4) p_k = 0 and p₄ ≥ p₃. Then there is a convex 4-valent polytope P in E³ such that P has exactly p_k k-gons as faces. The inequality p₄ ≥ p₃ is the best possible in the sense that for c < 1 there exist sequences that are not 4-realizable that satisfy both 8 + Σ_{k ≥3} (k - 4) p_k = 0 and p₄ > cp₃. When Σ_{k ≥ 5} p_k ≠ 1, one can make the stronger statement that the sequence {p_k} is 4-reliazable if it satisfies 8 + Σ_{k ≥ 3} (k - 4) p_k = 0 and p₄ ≥ 2Σ_{k ≥ 5} p_k + max{k ¦ p_k ≠ 0}.     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ₁. We consider the power method, i.e. that of choosing a vector v₀ and setting v_k = A^kv₀; then the Rayleigh quotients R_k = (Av_k, v_k)/(v_k, v_k) usually converge to λ₁ as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close R_k is to λ₁. They are both based on a bound on λ₁ − R_k involving the difference of two consecutive Rayleigh quotients and a quantity ω_k. While we do not know how to directly calculate ω_k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ₁ − R_k which is proportional to (λ₂/λ₁)^2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ).     

Coincidence of the upper Kuratowski topology with the co-compact topology on compact sets, and the Prohorov property

For a Hausdorff space X, let F be the hyperspace of all closed subsets of X and H a sublattice of F. Following Nogura and Shakhmatov, X is said to be H-trivial if the upper Kuratowski topology and the co-compact topology coincide on H. F-trivial spaces are the consonant spaces first introduced and studied by Dolecki, Greco and Lechicki. In this paper, we deal with K-trivial spaces and Fin-trivial space, where K and Fin are respectively the lattices of compact and of finite subsets of X. It is proved that if C_k(X) is a Baire space or more generally if X has ‘the moving off property’ of Gruenhage and Ma, then X is K-trivial. If X is countable, then C_p(X) is Baire if and only if X is Fin-trivial and all compact subsets of X are finite. As for consonant spaces, it turns out that every regular K-trivial space is a Prohorov space. This result remains true for any regular Fin-trivial space in which all compact subsets are scattered. It follows that every regular first countable space without isolated points, all compact subsets of which are countable, is Fin-nontrivial. Examples of K-trivial non-consonant spaces, of Fin-trivial K-nontrivial spaces and of countably compact Prohorov Fin-nontrivial spaces, are given. In particular, we show that all (generalized) Fréchet–Urysohn fans are K-trivial, answering a question by Nogura and Shakhmatov. Finally, we describe an example of a continuous open compact-covering mapping f :X→Y, where X is Prohorov and Y is not Prohorov, answering a long-standing question by Topsøe.     

Quasi-shadowing Property on Random Partially Hyperbolic Sets

In this paper we investigate the quasi-shadowing property for C~1 random dynamical systems on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}_(-∞)~(+∞)on a random partially hyperbolic set there exists a "center" pseudo orbit {yk}_(-∞)~(+∞)shadowing it in the sense that yk+1 is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above "center" pseudo orbit {yk}_(-∞)~(+∞)can be chosen such that yk+1 and the image of yk lie in their common center leaf.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

Graph spectra

The k-spectrum s_k(G) of a graph G is the set of all positive integers that occur as the size of an induced k-vertex subgraph of G. In this paper we determine the minimum order and size of a graph G with s_k (G) = {0, 1, …,(₂^k)} and consider the more general question of describing those sets S {0,1, … ,(₂^k)} such that S = s_k(G) for some graph G.     

Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U₁, …, U_m) of vector subspaces of W be uniformly distributed; and consider the statistics X_m,1 dim_F(∑_i=1^m U_i) and X_m,2 dim_F (∩_i=1^m U_i). If F is finite of cardinality q, we determine lim E(X_m,1^k), and lim E(X_m,2^k), and hence, lim var(X_m,1) and lim var(X_m,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.     

Strong approximation for cross-covariances of linear variables with long-range dependence

Suppose {_k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {c_j}^∞_j=0, {d_j}^∞_j=0 are sequences of real numbers such that Σ_jc²_j < ∞, Σ_jd²_j < ∞. Then, under appropriate moment conditions on {_k, −∞ < k < ∞}, y_k Σ^∞_j=0c_jk-j, z_k Σ^∞_j=0d_jk-j exist almost surely and in ⁴ and the question of Gaussian approximation to S_[t]Σ^[t]_k=1 (y_k z_k − E{y_k z_k}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S_[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on _k for “time” k ≤ 0, weaken the stationarity assumptions on {_k, −∞ < k < ∞}, and improve the summability conditions on {c_j}^∞_j=0, {d_j}^∞_j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let d_j = c_j-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.