On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

The isometry group of the Urysohn space as a Lévy group

We prove that the isometry group Iso(U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that Iso(U) has a dense locally finite subgroup.     

Countable connected Hausdorff and Urysohn bunches of arcs in the plane

In this paper, we answer a question by Krasinkiewicz, Reńska and Sobolewski by constructing countable connected Hausdorff and Urysohn spaces as quotient spaces of bunches of arcs in the plane. We also consider a generalization of graphs by allowing vertices to be continua and replacing edges by not necessarily connected sets. We require only that two “vertices” be in the same quasi-component of the “edge” that contains them. We observe that if a graph G cannot be embedded in the plane, then any generalized graph modeled on G is not embeddable in the plane. As a corollary we obtain not planar bunches of arcs with their natural quotients Hausdorff or Urysohn. This answers another question by Krasinkiewicz, Reńska and Sobolewski.     

A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of U (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that U admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.     

Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J₃ of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that

(a)

the set {0}∪{±³−(an+1)|n∈N} is quasi-convex in T if and only if a₀>0 and gn>1 for every n∈N;

(b)

the set {0}∪{±an³|n∈N} is quasi-convex in the group J₃ of 3-adic integers if and only if gn>1 for every n∈N.

Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347-1356] providing a complete characterization of the sequences such that {0}∪{±²−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±²−(an+1)|n∈N} is quasi-convex in R.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Countably infinite quasi-convex sets in some locally compact abelian groups

We produce a class of countably infinite quasi-convex sets (sequences converging to zero) in the circle group T and in the group J₂ of 2-adic integers determined by sequences of integers satisfying a mild lacunarity condition. We also extend our results to the group R of real numbers. All these quasi-convex sets have a stronger property: Every infinite (necessarily) symmetric subset containing 0 is still quasi-convex.     

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A^∗), the differential associated to the Lie algebroid structure on A^∗ has the form d_∗=A[Λ,⋅]+Ω, where Λ is a section of ^∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form , where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.     

On commutators of equivariant homeomorphisms

The long-standing problem of the perfectness of the compactly supported equivariant homeomorphism group on a G-manifold (with one orbit type) is solved in the affirmative. The proof is based on an argument different than that for the case of diffeomorphisms. The theorem is a starting point for computing H₁(HG(M)) for more complicated G-manifolds.     

Minimum Hellinger distance estimation in a two-sample semiparametric model

We investigate the estimation problem of parameters in a two-sample semiparametric model. Specifically, let X₁,…,X_n be a sample from a population with distribution function G and density function g. Independent of the X_i’s, let Z₁,…,Z_m be another random sample with distribution function H and density function h(x)=exp[α+r(x)β]g(x), where α and β are unknown parameters of interest and g is an unknown density. This model has wide applications in logistic discriminant analysis, case-control studies, and analysis of receiver operating characteristic curves. Furthermore, it can be considered as a biased sampling model with weight function depending on unknown parameters. In this paper, we construct minimum Hellinger distance estimators of α and β. The proposed estimators are chosen to minimize the Hellinger distance between a semiparametric model and a nonparametric density estimator. Theoretical properties such as the existence, strong consistency and asymptotic normality are investigated. Robustness of proposed estimators is also examined using a Monte Carlo study.     

Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations

Stationary and isotropic iteration stable random tessellations are considered, which are constructed by a random process of iterative cell division. The collection of maximal polytopes at a fixed time t within a convex window W⊂R^d is regarded and formulas for mean values, variances and a characterization of certain covariance measures are proved. The focus is on the case d≥3, which is different from the planar one, treated separately in Schreiber and Thäle (2010) [12]. Moreover, a limit theorem for suitably rescaled intrinsic volumes is established, leading — in sharp contrast to the situation in the plane — to a non-Gaussian limit.     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

Adaptive Monte Carlo methods for matrix equations with applications

This paper discusses empirical studies with both the adaptive correlated sequential sampling method and the adaptive importance sampling method which can be used in solving matrix and integral equations. Both methods achieve geometric convergence (provided the number of random walks per stage is large enough) in the sense: e_ν≤cλ^ν, where e_ν is the error at stage ν, λ∈(0,1) is a constant, c>0 is also a constant. Thus, both methods converge much faster than the conventional Monte Carlo method. Our extensive numerical test results show that the adaptive importance sampling method converges faster than the adaptive correlated sequential sampling method, even with many fewer random walks per stage for the same problem. The methods can be applied to problems involving large scale matrix equations with non-sparse coefficient matrices. We also provide an application of the adaptive importance sampling method to the numerical solution of integral equations, where the integral equations are converted into matrix equations (with order up to 8192×8192) after discretization. By using Niederreiter’s sequence, instead of a pseudo-random sequence when generating the nodal point set used in discretizing the phase space Γ, we find that the average absolute errors or relative errors at nodal points can be reduced by a factor of more than one hundred.     

H-spectral spaces

Let X be a T₀-space, we say that X is H-spectral if its T₀-compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T₁-spaces the T₀-compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T₁-space X in order to get its Wallman compactification spectral.     

Inverse problem of groundwater modeling by iteratively regularized Gauss-Newton method with a nonlinear regularization term

A nonlinear minimization problem ‖F(d)−u‖?min, ‖u−u_δ‖≤δ, is a typical mathematical model of various applied inverse problems. In order to solve this problem numerically in the lack of regularity, we introduce iteratively regularized Gauss-Newton procedure with a nonlinear regularization term (IRGN-NRT). The new algorithm combines two very powerful features: iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process. The convergence analysis is carried out in the presence of noise in the data and in the modified source condition. Numerical simulations for a parameter identification ill-posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method.     

Counting subsets of the random partition and the ‘Brownian Bridge’ process

Let Ω_m be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ω_m, and define X_ms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n^(m)(t), t?[0, 1], and centering constants b_ms, 0?s? m, such that

$\text{Z}{}^{\mathrm{(m)}}\text{(t)=}\text{∑}\text{s=0}\text{n(m)(t)}\text{(X}{}_{\mathrm{ms}}\text{?b}{}_{\mathrm{ms}}\text{)}\text{√}\text{∑}\text{s=0}\text{m}\text{b}{}_{\mathrm{ms}}$

converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions.     

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

It is now known that the intersection of two Magnus subgroups M_i=〈Y_i〉 (1≤i≤2) in a one-relator group is either the free group F on Y₁∩Y₂ or the free product of F together with an infinite cyclic group (so-called exceptional intersection). Using this, we give conditions under which two embedding theorems for cyclically presented groups can be obtained. This provides a new method for proving such groups infinite. We also give a combinatorial method for checking the presence of exceptional intersections.     

On topologizable and non-topologizable groups

A group G   is called hereditarily non-topologizable if, for every H?G$H?G$, no quotient of H admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked k-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.