Topological entropy of sets of generic points for actions of amenable groups

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).     

Stability of basic sequences via nonlinear <Emphasis Type="Italic">ε</Emphasis>-isometries

In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: X → Y is said to be a standard ε-isometry provided f(0) = 0 and

$$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$

(1)

for all x, y ∈ X. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(x_n)}n≥1 is a basic sequence of Y equivalent to {x_n}n≥1 whenever {x_n}n≥1 is a basic sequence in X and f: X → Y is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.

     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

A self-normalized law of the iterated logarithm for the geometrically weighted random series

Let {X, X_n; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX~2I(|X| ≤ x) is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞_(n=0)β~nX_n(0 β 1) is obtained, under some minimal conditions.     

Tightness games with bounded finite selections

For a topological space X and a point x ∈ X, consider the following game—related to the property of X being countably tight at x. In each inning n ∈ ω, the first player chooses a set A _n that clusters at x, and then the second player picks a point a _n ∈ A _n ; the second player is the winner if and only if \(x \in \overline {\left\{ {{a_n}:n \in \omega } \right\}} \).In this work, we study variations of this game in which the second player is allowed to choose finitely many points per inning rather than one, but in which the number of points they are allowed to choose in each inning has been fixed in advance. Surprisingly, if the number of points allowed per inning is the same throughout the play, then all of the games obtained in this fashion are distinct. We also show that a new game is obtained if the number of points the second player is allowed to pick increases at each inning.     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

On Lipschitz extension from finite subsets

We prove that for every n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: S → Z such that the Lipschitz constant of every function F: X → Z that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X and a function f: S → ?₂ that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ?₂ that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Real elements in groups of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

Let G be a group. An element x ∈ G is called real if x is conjugate to x ^?1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F ₄ and algebraic groups of type F ₄ defined over an arbitrary field.     

Hyperbolas over two-dimensional Fibonacci quasilattices

For the number n _s (α, β; X) of points (x ₁ , x ₂) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x ₁ ² ? ??αx ₂ ² ?=?β and such that 0?≤?x ₁? ≤?X, x ₂? ≥?0, the asymptotic formula

$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $

is established, and the coefficient c _s (α, β) is calculated exactly. Using this, we obtain the following result. Let F _m be the Fibonacci numbers, A _i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A _i in the Fibonacci numeral system. Then the number n _s (X) of all solutions (A ₁ , A ₂) of the Diophantine system

$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $

0?≤?A ₁? ≤?X, A ₂? ≥?0, satisfies the asymptotic formula

$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $

Here τ?=?(?1?+?√5)/2 is the golden ratio, and c _s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.

     

Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

On the Problem of the Optimal Choice of Record Values

Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X₁ < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X₁ > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x₁, x₂, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X _r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= lim_n→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.     

On criteria of estimation of the errors of cubature formulas

The paper considers cubature formulas for calculating integrals of functions f(X), X = (x ₁, …, x _n ) which are defined on the n-dimensional unit hypercube K ⁿ = [0, 1] ⁿ and have integrable mixed derivatives of the kind \(\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)\), 0 ≤ α _j ≤ 2. We estimate the errors R[f] = \(\smallint _{K^n } \) f(X)dX ? Σ _{k = 1} ^N c _k f(X(k)) of cubature formulas (c _k > 0) as functions of the weights c _k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K ⁿ : |R(f)| ≤ \(\sum _{\alpha _j } \) G(α _j )\(\int_{K^r } {\left| {\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)} \right|} \) dX _r , where coefficients G(α _j ) are criteria which depend only on parameters c _k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c _k and X(k).     

On judicious bipartitions of graphs

For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G)=V ₁?V ₂ such that G has at most s edges with both incident vertices in V _i . Alon proved that the limsup of \(f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)\) tends to infinity as m tends to infinity, establishing a conjecture of Erd?s. Bollobás and Scott proposed the following judicious version of Erd?s' conjecture: the limsup of \(m/4 + \left( {\sqrt {m/32} - g(m)} \right)\) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon's result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to k-partitions for odd integers k.     

An improvement of the five halves theorem of J. Boardman

Let (M ^m , T) be a smooth involution on a closed smooth m-dimensional manifold and F = ∪ _j=0 ⁿ F ^j (n ≤ m) its fixed point set, where F ^j denotes the union of those components of F having dimension j. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if F is nonbounding, then m ≤ 5/2n. In this paper we obtain an improvement of the Five Halves Theorem when the top dimensional component of F, F ⁿ , is nonbounding. Specifically, let ω = (i ₁, i ₂, …, i _r ) be a non-dyadic partition of n and s _ω (x ₁, x ₂, …, x _n ) the smallest symmetric polynomial over Z ₂ on degree one variables x ₁, x ₂, …, x _n containing the monomial \(x_1^{i_1 } x_2^{i_2 } \cdots x_r^{i_r }\). Write s _ω (F ⁿ ) ∈ H ⁿ (F ⁿ , Z ₂) for the usual cohomology class corresponding to s _ω (x ₁, x ₂, …, x _n ), and denote by ?(F ⁿ ) the minimum length of a nondyadic partition ω with s _ω (F ⁿ ) ≠ 0 (here, the length of ω = (i ₁, i ₂, …, i _r ) is r). We will prove that, if (M ^m , T) is an involution for which the top dimensional component of the fixed point set, F ⁿ , is nonbounding, then m ≤ 2n + ?(F ⁿ ); roughly speaking, the bound for m depends on the degree of decomposability of the top dimensional component of the fixed point set. Further, we will give examples to show that this bound is best possible.     

Rectangular <Emphasis Type="Italic">R</Emphasis>-Transform as the Limit of Rectangular Spherical Integrals

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maïda proved for Hermitian matrices in (J. Funct. Anal. 222(2):435–490, 2005). More specifically, we study the limit, as n and m tend to infinity, of \(\frac{1}{n}\log\mathbb{E}\{\exp[\sqrt{nm}\theta X_{n}]\}\), where θ∈?, X _n is the real part of an entry of U _n M _n V _m and M _n   is a certain n×m deterministic matrix and U _n and V _m are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n and m×m. We prove that when the singular law of M _n converges to a probability measure μ, for θ small enough, this limit actually exists and can be expressed with the rectangular R-transform of μ. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.     

A limit theorem for continuous selectors

Any (measurable) function K from Rⁿ to R defines an operator K acting on random variables X by K(X) = K(X₁,..., X_n), where the X_j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x₁, x₂,..., x_n) ∈ {x₁, x₂,..., x_n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ω_H in the interval (0, 1) so that, for any random variable X, the iterates H^(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

Stabilization by output of continuous and pulse-modulated uncertain systems

The system ? _i = ? _i (?) + x _i+2, \(i \in \overline {1,n - 2} \), ? _n?1 = ? _n?1(?) + u ₁, ? _n = ? _n (?) + u ₂,where ? _i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u ₁ and u ₂ are the controls is considered. It is assumed that only the outputs x ₁ and x ₂ are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method.