The limit sets of Schottky quasiconformal groups are uniformly perfect
Xiaosheng
Li
2119-2132
Abstract: In this paper we study Schottky quasiconformal groups. We show that the limit sets of Schottky quasiconformal groups are uniformly perfect, and that the limit set of a given discrete non-elementary quasiconformal group has positive Hausdorff dimension.
Analysis of a coupled system of kinetic equations and conservation laws: Rigorous derivation and existence theory via defect measures
M.
Tidriri
2133-2164
Abstract: In this paper we introduce a coupled system of kinetic equations of B.G.K. type and then study its hydrodynamic limit. We obtain as a consequence the rigorous derivation and existence theory for a coupled system of kinetic equations and their hydrodynamic (conservation laws) limit. The latter is a particular case of the coupled system of Boltzmann and Euler equations. A fundamental element in this study is the rigorous derivation and justification of the interface conditions between the kinetic model and its hydrodynamic conservation laws limit, which is obtained using a new regularity theory introduced herein.
Stable rank and real rank for some classes of group $C^\ast$-algebras
Robert
J.
Archbold;
Eberhard
Kaniuth
2165-2186
Abstract: We investigate the real and stable rank of the $C^\ast$-algebras of locally compact groups with relatively compact conjugacy classes or finite-dimensional irreducible representations. Estimates and formulae are given in terms of the group-theoretic rank.
Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations
Rafal
Goebel
2187-2203
Abstract: Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.
Orbifolds and analytic torsions
Xiaonan
Ma
2205-2233
Abstract: In this paper, we calculate the behavior of the Quillen metric by orbifold immersions. We thus extend a formula of Bismut-Lebeau to the orbifold case. R´ESUMÉ. Dans cet article, on calcule le comportement de métrique de Quillen par immersions d'orbifold. On étend ainsi une formule de Bismut-Lebeau au cas d'orbifold.
Anosov automorphisms on compact nilmanifolds associated with graphs
S.
G.
Dani;
Meera
G.
Mainkar
2235-2251
Abstract: We associate with each graph $(S,E)$ a $2$-step simply connected nilpotent Lie group $N$ and a lattice $\Gamma$ in $N$. We determine the group of Lie automorphisms of $N$ and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold $N/\Gamma$ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every $n\geq 17$ there exist a $n$-dimensional $2$-step simply connected nilpotent Lie group $N$ which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice $\Gamma$ in $N$ such that $N/\Gamma$ admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups $N$ of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.
Homological algebra for the representation Green functor for abelian groups
Joana
Ventura
2253-2289
Abstract: In this paper we compute some derived functors ${Ext}$ of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product. When the group is a cyclic $2$-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor ${Ext}$. When the group is $G=\mathbb{Z} /2\times\mathbb{Z} /2$, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of $G$ by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired ${Ext}$ functors.
Positivity preserving transformations for $q$-binomial coefficients
Alexander
Berkovich;
S.
Ole
Warnaar
2291-2351
Abstract: Several new transformations for $q$-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new $q$-binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.
Symmetric functions and the phase problem in crystallography
J.
Buhler;
Z.
Reichstein
2353-2377
Abstract: The calculation of crystal structure from X-ray diffraction data requires that the phases of the ``structure factors'' (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with $n$ equal atoms in the unit cell, with $n$ small. We rephrase the problem as a question about multiplicative invariants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases.
On degrees of irreducible Brauer characters
W.
Willems
2379-2387
Abstract: Based on a large amount of examples, which we have checked so far, we conjecture that
Characteristic subsurfaces and Dehn filling
Steve
Boyer;
Marc
Culler;
Peter
B.
Shalen;
Xingru
Zhang
2389-2444
Abstract: Let $M$ be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in $M$ which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of $M(\alpha)$ has no non-abelian free subgroup, and if $M(\beta)$ is a reducible manifold which is not homeomorphic to $S^1 \times S^2$ or $P^3 \char93 P^3$, then $\Delta(\alpha, \beta)\le 5$. Under the same condition on $M(\beta)$, it is shown that if $M(\alpha)$ is Seifert fibered, then $\Delta(\alpha, \beta)\le 6$. Moreover, in the latter situation, character variety techniques are used to characterize the topological types of $M(\alpha)$ and $M(\beta)$ in case the bound of $6$ is attained.
Weighted rearrangement inequalities for local sharp maximal functions
Andrei
K.
Lerner
2445-2465
Abstract: Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.
Smoothness of equisingular families of curves
Thomas
Keilen
2467-2481
Abstract: Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{\vert D\vert}^{irr}\big(\mathcal{S}_1,\ldots,\mathcal{S}_r\big)$ of irreducible curves in the linear system $\vert D\vert _l$ with precisely $r$ singular points of types $\mathcal{S}_1,\ldots,\mathcal{S}_r$ is T-smooth. Considering different surfaces including the projective plane, general surfaces in $\mathbb{P} _{\mathbb{C} }^3$, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type \begin{displaymath}\sum\limits_{i=1}^r\gamma_\alpha(\mathcal{S}_i) < \gamma\cdot (D- K_\Sigma)^2, \end{displaymath} where $\gamma_\alpha$ is some invariant of the singularity type and $\gamma$ is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the $\gamma_\alpha$-invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.
Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws
Gregory
A.
Freiman;
Boris
L.
Granovsky
2483-2507
Abstract: We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that $n^{\frac{1}{l+1}}$ is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as $n\to \infty.$ This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.
On some constants in the supercuspidal characters of $\operatorname{GL}_l$, $l$ a prime $\neq p$
Tetsuya
Takahashi
2509-2526
Abstract: The article gives explicit values of some constants which appear in the character formula for the irreducible supercuspidal representation of $\operatorname{GL}_l(F)$ for $F$ a local field of the residual characteristic $p\neq l$.
Upper bounds for the number of solutions of a Diophantine equation
M.
Z.
Garaev
2527-2534
Abstract: We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the $L_1$-norm of certain exponential sums.