Asymptotic uniformity of the quantization error of self-similar measures

Let??? be a self-similar measure on ${\mathbb{R}^d}$ associated with a family of contractive similitudes {S ₁, . . . , S _N } and a probability vector {p ₁, . . . , p _N }. Let ${(\alpha_n)_{n=1}^\infty}$ be a sequence of n-optimal sets for??? of order r. For each n, we denote by ${\{P_a(\alpha_n) : a \in \alpha_n\}}$ a Voronoi partition of ${\mathbb{R}^d}$ with respect to ?? _n . Under the strong separation condition for {S ₁, . . . , S _N }, we show that the nth quantization error of??? of order ${r \in [1, \infty)}$ satisfies the following asymptotic uniformity property: $$\int \limits _{P_a(\alpha_n)}{\rm d}(x, a)^rd\mu(x) \asymp \frac{1}{n}V_{n,r}(\mu),\quad {\rm for\,all}\,a \in \alpha_n.$$      

Ballisticity conditions for random walk in random environment

Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d ?? 2. In 2002, Sznitman introduced for each ${\gamma\in (0, 1)}$ the ballisticity conditions (T) _?? and (T??), the latter being defined as the fulfillment of (T) _?? for all ${\gamma\in (0, 1)}$ . He proved that (T??) implies ballisticity and that for each ${\gamma\in (0.5, 1)}$ , (T) _?? is equivalent to (T??). It is conjectured that this equivalence holds for all ${\gamma\in (0, 1)}$ . Here we prove that for ${\gamma\in (\gamma_d, 1)}$ , where ?? _d is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) _?? is equivalent to (T??). This is achieved by a detour along the effective criterion, the fulfillment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.     

Teichmüller spaces for pointed Fuchsian groups

Let T(G) be the Teichmüller space of a Fuchsian group G and T(G) be the pointed Teichmüller space of a corresponding pointed Fuchsian group G.We will discuss the existence of holomorphic sections of the projection from the space M(G) of Beltrami coefficients for G to T(G) and of that from T(G) to T(G) as well.We will also study the biholomorphic isomorphisms between two pointed Teichmüller spaces.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Convergence of representation averages and of convolution powers

LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:

1. if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU ⁿ (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
2. If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U ⁿ converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
3. If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .

     

The cardinality of β A (S J )

Let J be an infinite set and let $I=\mathcal{P}_{f}( J)$ . For i??I, define $\mathcal{B}_{J}( i) =\{ f\mid f:\mathcal{P}( i) \rightarrow \mathcal{P}( i) \} $ and let $$S_{J}=\{ ( i,f) \mid i\in I\text{ and } f\in \mathcal{B}_{J}( i) \}.$$ For (i,f), (k,g)??S _J , define $f\ast g:\mathcal{P}( i\cup k) \rightarrow \mathcal{P}( i\cup k) $ as follows. For $x\in \mathcal{P}( i\cup k) $ , let $$( f\ast g) ( x) =\left\{\begin{array}{l@{\quad }l}g( x) , & \text{if\ }x=\emptyset, \\g( x\cap k) , & \text{if\ }x\cap k\neq \emptyset, \\f( x) , & \text{if\ }x\in \mathcal{P}( i\backslash k)\text{ and }x\neq \emptyset.\end{array}\right.$$ Define (i,f)?(k,g)=(i??k,f?g). It is shown that (S _J ,?) is a semigroup. Let ??S _J denote the collection of all ultrafilters on the set S _J . We consider (??S _J ,?), the compact (Hausdorff) right topological semigroup that is the Stone?C?ech Compactification of the semigroup (S _J ,?) equipped with the discrete topology. Similar to the construction in Grainger (Semigroup Forum 73:234?C242, 2006), it is shown that there is an injective map A???? _A (S _J ) of $\mathcal{P}( J) $ into $\mathcal{P}( \beta S_{J}) $ such that each ?? _A (S _J ) is a closed subsemigroup of (??S _J ,?), the set ?? _J (S _J ) is the smallest ideal of (??S _J ,?) and the collection $\{ \beta_{A}( S_{J}) \mid A\in \mathcal{P}( J) \} $ is a partition of???S _J . The main result is establishing that the cardinality of??? _A (S _J ) is $2^{2^{\vert J\vert }}$ for any?A?J.     

A Spanning Tree with High Degree Vertices

Let G be a connected graph, let ${X \subset V(G)}$ and let f be a mapping from X to {2, 3, . . .}. Kaneko and Yoshimoto (Inf Process Lett 73:163–165, 2000) conjectured that if |N _G (S) ? X| ≥ f (S) ? 2|S| + ω _G (S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ . In this paper, we show a result with a stronger assumption than this conjecture; if |N _G (S) ? X| ≥ f (S) ? 2|S| + α(S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ .     

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

Descent of Line Bundles to Git Quotients of Flag Varieties by Maximal Torus

Let G be a connected, simply connected semisimple complex algebraic group with a maximal torus T and let P be a parabolic subgroup containing T. Let $ \mathcal{L}_{P} {\left( \lambda \right)} $ be a homogeneous ample line bundle on the ag variety Y?=?G?=?P. We give a necessary and sufficient condition for $ \mathcal{L}_{P} {\left( \lambda \right)} $ to descend to a line bundle on the GIT quotient Y(λ)//T. As a consequence of this result, we get the precise list of P-regular weights λ for which the line bundle $ \mathcal{L}_{P} {\left( \lambda \right)} $ descends to the GIT quotient Y(λ)//T.     

Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

We analyze the perturbations T?+?B of a selfadjoint operator T in a Hilbert space H with discrete spectrum ${\{ t_k\}, T \phi_k = t_k \phi_k}$ . In particular, if t _k+1 ? t _k ?? ck ^{?? - 1}, ?? > 1/2 and ${\| B \phi_k \| = o(k^{\alpha - 1})}$ then the system of root vectors of T?+?B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions ${t_{k+p} - t_k \geq d > 0, \forall k}$ (with d and p fixed) and ${\| B \phi_k \| \rightarrow 0}$ a Riesz system {P _k } of projections on invariant subspaces of T?+?B, Rank P _k ?? p, is constructed (Theorem 3).     

Absolute Nil-Ideals of Abelian Groups

It is known that in an Abelian group G that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on G is $\bigcap\limits_{p} pT(G)$ , where T(G) is the torsion part of G. In this work, we define a pure fully invariant subgroup G*???T(G) of an arbitrary Abelian mixed group G and prove that if G contains no nonzero torsion-free subgroups, then the subgroup $\bigcap\limits_{p} pG^{*}$ is a nil-ideal in any ring on G, and the first Ulm subgroup G1 is its nilpotent ideal.     

Estimating asymptotic dependence functionals in multivariate regularly varying models

This paper deals with semiparametric estimation of the asymptotic portfolio risk factor ?? _?? introduced in [G. Mainik and L. Rüschendorf, On optimal portfolio diversification with respect to extreme risks, Finance Stoch., 14:593?C623, 2010] for multivariate regularly varying random vectors in $ \mathbb{R}_{+}^d $ . The functional ?? _?? depends on the spectral measure ??, the tail index ??, and the vector ?? of portfolio weights. The representation of ?? _?? is extended to characterize the portfolio loss asymptotics for random vectors in ? ^d . The earlier results on uniform strong consistency and uniform asymptotic normality of the estimates of ?? _?? are extended to the general setting, and the regularity assumptions are significantly weakened. Uniform consistency and asymptotic normality are also proved for the estimators of the functional $ \gamma_\xi^{{{1} \left/ {\alpha } \right.}} $ that characterizes the asymptotic behavior of the portfolio loss quantiles. The techniques developed here can also be applied to other dependence functionals.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Lines Avoiding Balls in Three Dimensions Revisited

Let $\mathcal{B}$ be a collection of n arbitrary balls in ?³. We establish an almost-tight upper bound of O(n ^3+?? ), for any ??>0, on the complexity of the space $\mathcal{F}(\mathcal{B})$ of all the lines that avoid all the members of $\mathcal{B}$ . In particular, we prove that the balls of $\mathcal{B}$ admit O(n ^3+?? ) free isolated tangents, for any ??>0. This generalizes the result of Agarwal et al.?(Discrete Comput. Geom. 34:231?C250, 2005), who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of ??(n ³) of Glisse and Lazard (Proc. 26th Annu. Symp. Comput. Geom., pp. 48?C57, 2010). Our approach is constructive and yields an algorithm that computes the discrete representation of the boundary of $\mathcal{F}(B)$ in O(n ^3+?? ) time, for any ??>0.     

Concentration of measures via size-biased couplings

Let Y be a nonnegative random variable with mean??? and finite positive variance ?? ², and let Y ^s , defined on the same space as Y, have the Y size-biased distribution, characterized by $$ E[Yf(Y)]=\mu E f(Y^s) \quad {\rm for\,all\,functions}\,f\,{\rm for\,which\,these\,expectations\,exist}. $$ Under a variety of conditions on Y and the coupling of Y and Y ^s , including combinations of boundedness and monotonicity, one sided concentration of measure inequalities such as $$ P\left(\frac{Y-\mu}{\sigma} \ge t\right)\le {\rm exp}\left(-\frac{t^2}{2(A+Bt)} \right) \quad {\rm for\,all}\,t\, > 0 $$ hold for some explicit A and B. The theorem is applied to the number of bulbs switched on at the terminal time in the so called lightbulb process of Rao et?al. (Sankhy?? 69:137?C161, 2007).     

On embeddings of certain spherical homogeneous spaces in prime characteristic

Let $ \mathcal{G} $ be a reductive group over an algebraically closed field of characteristic p?>?0. We study embeddings of homogeneous $ \mathcal{G} $ -spaces that are induced from the G?×?G-space G, G a suitable reductive group, along a parabolic subgroup of $ \mathcal{G} $ . We give explicit formulas for the canonical divisors and for the divisors of B-semi-invariant functions. Furthermore, we show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen?CMacaulay. Examples are the G?×?G-orbits in normal reductive monoids with unit group G. Further examples are the open $ \mathcal{G} $ -orbits of the well known determinantal varieties and the varieties of (circular) complexes. Finally, we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.     

Index-modules and applications

Let K be a commutative field, ${A\subseteq K}$ be a Dedekind ring and V be a K-vector space. For any pair of A-lattices R????0 and S of V, we define an A-submodule ${\left[R : S\right]^{\prime}_{A}}$ of K, their A-index-module. Once the basic properties of these modules are stated, we show that this notion can be used to recover more usual ones: the group-index, the relative invariant, the Fitting ideal of R/S when ${S\subseteq R}$ , and the generalized index of Sinnott. As an example, we consider the following situation. Let F/k be a finite abelian extension of global function fields, with Galois group G, and degree g. Let ?? be a place of k which splits completely in F/k. Let ${{\mathcal O}_{F}}$ be the ring of functions of F, which are regular outside the places of F sitting over ??. Then one may use Stark units to define a subgroup ${\mathcal E_F}$ of ${{\mathcal O}_{F}^{\times}}$ , the group of units of ${\mathcal O_F}$ . We use the notion of index-module to prove that for every nontrivial irreducible rational character ?? of G, the ??-part of ${\mathbb Z\left[g^{-1}\right]\otimes_\mathbb Z\left(\mathcal O_F^\times/\mathcal E_F\right)}$ and the ??-part of ${\mathbb Z\left[g^{-1}\right]\otimes_\mathbb ZCl(\mathcal O_F)}$ have the same order.     

The reverse H?lder inequality for power means

For a function ?? non-negative on the interval [0, 1], the power mean of order ??????0 is defined by the equality $ \mathcal{M}_{\alpha \varphi} (t) = {\left( {\frac{1}{t}\int_0^t {{\varphi^\alpha }(u)du} } \right)^{1/\alpha }},\,0 < t \leqslant 1 $ . We consider the class $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ of functions ?? satisfying the reverse H?lder inequality $$ {\mathcal{M}_\beta }_\varphi \leqslant B \cdot {\mathcal{M}_\alpha }_\varphi $$ at some ???<???,??·??????0,???>?1. The sharp estimates for the summability exponents of the compositions of power means are established. As a result, we determine the properties of self-improvement of the summability exponents of functions from $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ .     

Ascending chain conditions on principal left and right ideals for semidirect products of semigroups

In this paper we investigate the ascending chain conditions on principal left and right ideals for semidirect products of semigroups and show how this is connected to the corresponding problem for rings of skew generalized power series. Let S be a left cancellative semigroup with a unique idempotent e, T a right cancellative semigroup with an idempotent f and $\omega: T \to \operatorname {End}(S)$ a semigroup homomorphism such that ??(f)=id _S . We show that in this case the semidirect product S? _?? T satisfies the ascending chain condition for principal left ideals (resp. right ideals) if and only if S and T satisfy the ascending chain condition for principal left ideals (resp. right ideals and $\operatorname {Im}\omega(t)$ is closed for complete inverses for all t??T). We also give several examples to show that for more general semigroups these implications may not hold.     

Sturmian maximizing measures for the piecewise-linear cosine family

Let T be the angle-doubling map on the circle $\mathbb{T}$ , and consider the 1-parameter family of piecewise-linear cosine functions $f_\theta :\mathbb{T} \to \mathbb{R}$ , defined by $f_\theta (x) = 1 - 4d_\mathbb{T} (x,\theta )$ . We identify the maximizing T-invariant measures for this family: for each ?? the f _?? -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with ?? of the maximum ergodic average for f _?? .