On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ? X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω ₁-monolithic compact space X, if C _p (X)is star countable then it is Lindelöf.     

Extrapolation in Grand Lebesgue Spaces with <Emphasis Type="Italic">A</Emphasis>∞ Weights

Results on extrapolation withA∞ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the productmeasure μ₁ ×· · ·×μ_n onX₁ ×· · ·×X_n, where (X_i, d_i, μ_i), i = 1,..., n, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with A_∞ weights for operators of harmonic analysis are derived.     

Existence of linear Pfaffian systems whose lower characteristic set has positive measure in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaffian system ?x/?t i = A _i (t)x, x ∈ R ⁿ , t = (t ₁, t ₂, t ₃) ∈ R ₊ ³ , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure.     

Minimal and maximal unconditional bases with respect to framings

This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if （xi, fi） is a framing of a Banach space X, and （eimin） and （eimax） are the minimal unconditional basis and the maximal unconditional basis with respect to （xi, fi）, respectively, then for any unconditional basis （ei） associated with （xi, fi）, there are A,B 〉 0 such that A｜｜i=1∑∞aieimin｜｜≤｜｜i=1∑∞aiei｜｜≤B｜｜i=1∑∞aieimax｜｜ for all （ai） ∈ c00.
It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds.     

Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

Constructive Characteristics of Full Module of Smoothness in Mixed Metric

We give constructive characteristics of full module of smoothness of natural order in mixed metrics of the spaces L_p1p2, 1 ≤ p _i ≤ ∞, i = 1, 2.     

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite-dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if f: V → V is a nonexpansive map on a finite-dimensional normed space V, then the fixed point set of f is nonempty and bounded if and only if there exist w₁,..., w _m in V such that {f(w _i ) ? w _i : i = 1,..., m} illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite-dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle

Given a three-dimensional dynamical system on the interval t ₀ < t < +∞, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighbor-hood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue λ = ? + iβ, with β ? ? > 0 and a real eigenvalue with δ < 0 with |δ| ? ?. On the arbitrary interval [t ₀, +∞), an approximate solution is sought as a polynomial P _N (?) in powers of the small parameter with coefficients from Hölder function spaces. It is proved that there exist ? _N and C _N depending on the initial data such that, for 0 < ? < ? _N , the difference between the exact and approximate solutions does not exceed C _N ? ^N+1.     

Minimal logarithmic signatures for classical groups

The minimal logarithmic signature conjecture states that in any finite simple group there are subsets A _i , 1 ≤ i ≤ k such that the size |A _i | of each A _i is a prime or 4 and each element of the group has a unique expression as a product \({\prod_{i=1}^k x_i}\) of elements \({x_i \in A_i}\). The conjecture is known to be true for several families of simple groups. In this paper the conjecture is shown to be true for the groups \({\Omega^-_{2m}(q), \Omega^+_{2m}(q)}\), when q is even, by studying the action on suitable spreads in the corresponding projective spaces. It is also shown that the method can be used for the finite symplectic groups. The construction in fact gives cyclic minimal logarithmic signatures in which each A _i is of the form \({\{y_i^j \ |\ 0 \leq j < |A_i|\}}\) for some element y _i of order ≥ |A _i |.     

On iterative methods for solving equations with covering mappings

In this paper we propose an iterative method for solving the equation Υ(x, x) = y, where the mapping Υ acts in metric spaces and is covering in the first argument and Lipschitzian in the second one. Each subsequent element x _i+1 of the sequence of iterations is defined by the previous one as a solution to the equation Υ(x, x _i) = y _i, where y _i can be an arbitrary point sufficiently close to y. Conditions for convergence and error estimates are obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine x _i+1 in practical implementation of the method in linear normed spaces, it is proposed to perform one step by using the Newton–Kantorovich method. The thus-obtained method of solving the equation of the form Υ(x, u) = ψ(x) ? φ(u) coincides with the iterative method proposed by A.I. Zinchenko,M.A. Krasnosel’skii, and I.A. Kusakin.     

Functional estimation of activity criticality indices and sensitivity analysis of expected project completion time

For a PERT network, a new method is developed for estimating the criticality index of activity i (ACI _i ) as a function of the expected duration of activity i (μ _i ) and for the sensitivity analysis of the expected project completion time (μ _T ) with respect to μ _i . The proposed method evaluates the frequency of activity i being on the critical path, and thereby its ACI _i using Monte Carlo simulation or a Taguchi orthogonal array experiment at several values of μ _i , fits a logistic regression model for estimating ACI _i as a function of μ _i , and then, using the estimated ACI _i function, evaluates the amount of change in μ _T when μ _i is changed by a given amount. Unlike the previous works, the proposed method models ACI _i as a nonlinear (ie, logistic) function of μ _i , which can be used to estimate the amount of change in μ _T for a variety of changes in μ _i . Computational results indicate that the performance of the proposed method is comparable to that of direct Monte Carlo simulation.     

On Excess-time Correlated Cumulative Processes

In this paper a class of correlated cumulative processes, B _s (t) = ∑^N(t)_i=1 H _s (X _i )X _i , is studied with excess level increments X _i ?s, where {N(t), t ?0} is the counting process generated by the renewal sequence T _n , T _n and X _n are correlated for given n, H _s (t) is the Heaviside function and s?0 is a given constant. Several useful results, for the distributions of B _s (t), and that of the number of excess (non-excess) increments on (0, t) and the corresponding means, are derived. First passage time problems are also discussed and various asymptotic properties of the processes are obtained. Transform results, by applying a flexible form for the joint distribution of correlated pairs (T _n , X _n ) are derived and inverted. The case of non-excess level increments, X _i < s, is also considered. Finally, applications to known stochastic shock and pro-rata warranty models are given.     

A Minimax Approach to Errors-in-Variables Linear Models

The paper considers a simple Errors-in-Variables (EiV) model Y_i = a + bX_i + εξ_i; Z_i= X_i + σζ_i, where ξ_i, ζ_i are i.i.d. standard Gaussian random variables, X_i ∈ ? are unknown non-random regressors, and ε, σ are known noise levels. The goal is to estimates unknown parameters a, b ∈ ? based on the observations {Y_i, Z_i, i = 1, …, n}. It is well known [3] that the maximum likelihood estimates of these parameters have unbounded moments. In order to construct estimates with good statistical properties, we study EiV model in the large noise regime assuming that n → ∞, but \({\epsilon ^2} = \sqrt n \epsilon _ \circ ^2,{\sigma ^2} = \sqrt n \sigma _ \circ ^2\) with some \(\epsilon_\circ^2, \sigma_\circ^2>0\). Under these assumptions, a minimax approach to estimating a, b is developed. It is shown that minimax estimates are solutions to a convex optimization problem and a fast algorithm for solving it is proposed.     

Isotopes of the alternative monster and the Skosyrsky algebra

We prove that the isotopes of the alternative monster and the Skosyrsky algebra satisfy the identity П_i=1⁴ [x_i, y_i] = 0. Hence, the algebras themselves satisfy the identity П_i=1⁴ (c, x_i, y_i) = 0. We also show that none of the identities П_i=1ⁿ(c, x_i, y_i) = 0 holds in all commutative alternative nil-algebras of index 3. Thus, we refute the Grishkov–Shestakov hypothesis about the structure of the free finitely generated commutative alternative nil-algebras of index 3.     

Theory of (<Emphasis Type="Italic">q</Emphasis> <Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis> <Subscript>2</Subscript>)-quasimetric spaces and coincidence points

We introduce (q₁, q₂)-quasimetric spaces and examine their properties. Covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Sufficient conditions for the existence of a coincidence point of two mappings acting between (q₁, q₂)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition are obtained.     

Coloring number and on-line Ramsey theory for graphs and hypergraphs

Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G ₀=(V,E ₀), where E ₀=Ø and V is determined by Builder. On the ith round Builder constructs a new edge e _i (distinct from previous edges) and sets G _i =(V,E _i ), where E _i =E _i?1∪{e _i }. Painter responds by coloring e _i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K _s ^t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges.We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K _s ^t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H ₀ = (V,Ø) with an arbitrary finite number of vertices and no edges. Let H _i?1=(V _i?1,E _i?1) be the hypergraph constructed in the first i ? 1 rounds. On the i-th round Presenter plays by presenting a p-subset P _i ?V _i?1 and Chooser responds by choosing an s-subset X _i ?P _i . The vertices in P _i ? X _i are discarded and the edge X _i added to E _i?1 to form E _i . Presenter wins the survival game if H _i contains a copy of K _s ^t for some i. We show that for positive integers p,s,t with s≤p, Presenter has a winning strategy.     

Ergodic theorems for coset spaces

We study in this paper the validity of the Mean Ergodic Theorem along left Følner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Følner sequence in G, we provide examples where the Mean Ergodic Theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g., if G is virtually nilpotent, then the Mean Ergodic Theorem holds along any left Følner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a “sufficiently thin” subgroup, we show that the Mean Ergodic Theorem holds for the induced representation along any left Følner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Følner sequence (F_n) in L, there exists a sequence (s_n) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (F_ns_n) in G. Finally, we provide examples of countable (not necessarily amenable) groups G with proper, infinite-index subgroups H, so that the Pointwise Ergodic Theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset G/H.     

Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Let {X _i = (X _1,i ,...,X _m,i )^?, i ≥ 1} be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X ₁ are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence {X _i , i ≥ 1}. Under several mild assumptions, precise large deviations for S _n = Σ _i=1 ⁿ X _i and S _N(t) = Σ _i=1 ^N(t) X _i are investigated. Meanwhile, some simulation examples are also given to illustrate the results.