On highly transcendental quantities which cannot be expressed by integral formulas

Abstract: E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series $\sum x^{t_n}$ where $t_n$ is the $n$th triangular number, this could lead to a proof of Fermat's theorem that every positive integer is the sum of three triangular numbers. This doesn't end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (\S 9) is not clear to me. My best reading is that there are infinitely many "levels" of transcendental numbers and that this is unexpected or remarkable.


On the Ramsey multiplicity of complete graphs

Abstract: We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \approx 2.18$ is an explicitly defined constant. The old lower bound, due to Erd\H{o}s \cite{E62}, and based upon the standard bounds for Ramsey's theorem, is \[\frac{n^t}{4^{(1+o(1))t^2}}.\]


Complex hyperbolic hyperplane complements

Abstract: We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n>1 the fundamental group of M-S satisfies Mostow-type Rigidity, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of M-S is biautomatic and satisfies Strong Tits Alternative.


A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals

Abstract: A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after each application of the Poisson summation. In other words, our method relies on modular properties of the theta function. Applications to the numerical computation of the Riemann zeta function and to finding the number of solutions of Waring type Diophantine equations are discussed.


Contact equations, Lipschitz extensions and isoperimetric inequalities

Abstract: We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.


A note on lower bounds for hypergraph Ramsey numbers

Abstract: We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erd\H{o}s and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]


Fast methods to compute the Riemann zeta function

Abstract: The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Sch\"onhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author's algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method employs an algorithm, developed in this paper, to compute cubic exponential sums. Its complexity has exponent 4/13 (approximately, 0.307).


A bound on the exponent of the cohomology of BC-bundles

Abstract: We give a lower bound for the exponent of certain elements in the integral cohomology of the total spaces of principal BC-bundles for C a finite cyclic group. As applications we give a proof of the theorem of A. Adem and H.-W. Henn that a p-group is elementary abelian if and only if its integral cohomology has exponent p, and we exhibit some infinite groups of finite virtual cohomological dimension whose Tate-Farrell cohomology contains torsion of order greater than the l.c.m. of the orders of their finite subgroups. We also give an upper bound for the exponent of all but finitely many of the integral cohomology groups of a finite group, in terms of the permutation representations of the group.


Smoothness criteria for Navier-Stokes equations in terms of regularity along the steam lines

Abstract: This article is devoted to a regularity criteria for solutions of the Navier-Stokes equations in terms of regularity along the stream lines. More precisely, we prove that a suitable weak solution for the Navier-Stokes equations is regular under some constraint on the second derivative of |u| along the stream lines.


The Yagita invariant of general linear groups

Abstract: We give a definition of the Yagita invariant at a prime p of an arbitrary group G, and compute the invariant for each prime for the general linear groups over any integrally closed subring of the complex numbers. We also compute the invariants for special linear groups over the same rings, except in some cases when both the degree of the linear group and the ring are `small' compared to p.


On subgroups of Coxeter groups

Abstract: A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are related to homological properties of the simplicial complex whose simplices are the sets of commuting generators. Using these techniques, we construct torsion-free groups which are Poincare duality groups over some rings but not over others, and a group whose integral cohomological dimension is finite but strictly greater than its cohomological dimension over any field. We determine which Coxeter groups have finite virtual cohomological dimension (it is classical that all finitely generated Coxeter groups have finite vcd, but there are others). We also give minimal presentations for certain torsion-free finite-index subgroups of right-angled Coxter groups. Finally we give a `bare-hands' construction (using free products with amalgamation and HNN extensions) of a torsion-free group whose integral cohomological dimension is strictly greater than its rational cohomological dimension.


Chern classes and extraspecial groups

Abstract: The mod-p cohomology ring of the extraspecial p-group of exponent p is studied for odd p. We investigate the subquotient ch(G) generated by Chern classes modulo the nilradical. The subring of ch(G) generated by Chern classes of one-dimensional representations was studied by Tezuka and Yagita. The subring generated by the Chern classes of the faithful irreducible representations is a polynomial algebra. We study the interplay between these two families of generators, and obtain some relations between them.


On universally stable elements

Abstract: We show that certain subrings of the cohomology of a finite p-group P may be realised as the images of restriction from suitable virtually free groups. We deduce that the cohomology of P is a finite module for any such subring. Examples include the ring of `universally stable elements' defined by Evens and Priddy, and rings of invariants such as the mod-2 Dickson algebras.


On the GL(V)-module structure of K(n)^*(BV)

Abstract: We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses) for the automorphism group of V. We use Brauer characters and computer calculations. We construct and implement an algorithm for finding permutation submodules of maximal dimension inside modules for p-groups in characteristic p.


On the integral cohomology of wreath products

Abstract: Under mild conditions on the space X, we describe the additive structure of the integral cohomology of the space $X^p \times_{C_p}EC_p$ in terms of the cohomology of X. We give weaker results for other similar spaces, and deduce various corollaries concerning the cohomology of finite groups.


The cohomology of Bestvina-Brady groups

Abstract: For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals.


The cohomology of certain groups

Abstract: Computations in the cohomology of finite groups.


The p-adic closure of a subgroup of rational points on a commutative algebraic group

Abstract: Let G be a commutative algebraic group over Q. Let Gamma be a subgroup of G(Q) contained in the union of the compact subgroups of G(Q_p). We formulate a guess for the dimension of the closure of Gamma in G(Q_p), and show that its correctness for certain tori is equivalent to Leopoldt's conjecture.


The Computation of the Logarithmic Cohomology for Plane Curves

Abstract: We give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables. (Fixed typos of v1)


New classification techniques for ordinary differential equations

Abstract: The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of \'Elie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODE, that are regarded as well-known, or at least well-studied. The dictionary considered in this article are ODE in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.


The Structure of a Bernoulli Process Variation of the Fibonacci Sequence

Abstract: We consider the structure of a variation of the Fibonacci sequence which is determined by a Bernoulli process. The associated structure of all Bernoulli variations of the Fibonacci sequence can be represented by a directed binary tree, which we denote X, with vertex labels representing the specific state of the recurrence variation. Since X is a binary tree, we can consider the term of a sequence variation given by a finite traversal of X represented by a binary code t. We then prove that the traversal of X that is the reflection of the digits of t gives exactly the integer term corresponding to t. We consider how to further this result with the statement of an additional conjecture. Finally, we give connections to Fibonacci expansions, the Stern-Brocot tree, and we apply our methods to the Three Hat Problem as seen in ``Puzzle Corner'' of the ``Technology Review'' magazine.


Equilibrium points for Optimal Investment with Vintage Capital

Abstract: The paper concerns the study of equilibrium points, namely the stationary solutions to the closed loop equation, of an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. Sufficient conditions for existence of equilibrium points in the general case are given and later applied to the economic problem of optimal investment with vintage capital. Explicit computation of equilibria for the economic problem in some relevant examples is also provided. Indeed the challenging issue here is showing that a theoretical machinery, such as optimal control in infinite dimension, may be effectively used to compute solutions explicitly and easily, and that the same computation may be straightforwardly repeated in examples yielding the same abstract structure. No stability result is instead provided: the work here contained has to be considered as a first step in the direction of studying the behavior of optimal controls and trajectories in the long run.


A linear equation for Minkowski sums of polytopes relatively in general position

Abstract: The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is satisfied by f-vectors of the sum and the summands. We study some of the implications of this equation.


Pluripolarity of manifolds

Abstract: By the classical result of E. Bedford, a real-analytic non-generic manifold is pluripolar. We extend this result for manifolds of the Gevrey class. This also gives a generalization of the recent result of D. Coman, N. Levenberg and E. Poletsky on pluripolarity of curves of the Gevrey class.


Slider-pinning Rigidity: a Maxwell-Laman-type Theorem

Abstract: We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks.


Fillings method in number theory

Abstract: Number of results in number theory have been developed using a new method. The Goldbach binary conjecture in strengthened formulation have been among them.


Periodic orbits of period 3 in the disc

Abstract: Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.


Elliptic solutions of the Toda chain and a generalization of the Stieltjes-Carlitz polynomials

Abstract: We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials which can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials. Relations between characteristic (i.e. positive definite) functions, Toda chain and orthogonal polynomials are developed in order to derive main properties of these polynomials. The recurrence coefficients and the weight function of these polynomials are expressed explicitly. In the degenerated cases of the elliptic functions the modified Meixner polynomials and the Krall-Laguerre polynomials appear.


Janet's Algorithm

Abstract: We have introduced the Janet's algorithm for the Stanley decomposition of a monomial ideal I in a polynomial ring S = K[x_1,...,x_n] and prove that Janet's algorithm gives the squarefree Stanley decomposition of S/I for a squarefree monomial ideal I. We have also shown that the Janet's algorithm gives a partition of a simplicial complex.


Generalized Bochner theorem: characterization of the Askey-Wilson polynomials

Abstract: Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$ A(s) P_n(z(s+1)) + B(s) P_n(z(s)) + C(s) P_n(z(s-1)) = \lambda_n P_n(z(s)), n=0,1,2,..., N$$ where $z(s), A(s), B(s), C(s)$ are some functions of the discrete argument $s$ and $N$ may be either finite or infinite. The irreducibility condition $A(s-1)C(s) \ne 0$ is assumed for all admissible values of $s$. In the finite case we assume that there are $N+1$ distinct grid points $z(s), \: s=0,1,..., N$ such that $z(i) \ne z(j), \: i \ne j$. If $N=\infty$ we assume that the grid $z(s)$ has infinitely many different values for different values of $s$. In both finite and infinite cases we assume also that the problem is non-degenerate, i.e. $\lambda_n \ne \lambda_m, n \ne m$. Then we show that necessarily: (i) the grid $z(s)$ is at most quadratic or q-quadratic in $s$; (ii) corresponding polynomials $P_n(z)$ are at most the Askey-Wilson polynomials corresponding to the grid $z(s)$. This result can be considered as generalizing of the Bochner theorem (characterizing the ordinary classical polynomials) to generic case of arbitrary difference operator on arbitrary grids.


Birational geometry of Fano double covers

Abstract: We prove divisorial canonicity of Fano double hypersurfaces of general position.


The hyperbolic mean curvature flow

Abstract: We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.


Symplectic Jacobi diagrams and the Lie algebra of homology cylinders

Abstract: Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial extension of the Le-Murakami-Ohtsuki invariant, we show that the graded Lie algebra associated to the Y-filtration is isomorphic to the Lie algebra of ``symplectic Jacobi diagrams.'' This Lie algebra consists of the primitive elements of a certain Hopf algebra whose multiplication is a diagrammatic analogue of the Moyal-Weyl product. The mapping cylinder construction embeds the Torelli group into the monoid of homology cylinders, sending the lower central series to the Y-filtration. We give a combinatorial description of the graded Lie algebra map induced by this embedding, by connecting Hain's infinitesimal presentation of the Torelli group to the Lie algebra of symplectic Jacobi diagrams. This Lie algebra map is shown to be injective in degree two, and the question of the injectivity in higher degrees is discussed.


Zero diffusion-dispersion limits for scalar conservation laws

Abstract: We consider solutions of hyperbolic conservation laws regularized with vanishing diffusion and dispersion terms. Following a pioneering work by Schonbek, we establish the convergence of the regularized solutions toward discontinuous solutions of the hyperbolic conservation law. The proof relies on the method of compensated compactness in the $L^2$ setting. Our result improves upon Schonbek's earlier results and provides an optimal condition on the balance between the relative sizes of the diffusion and the dispersion parameters. A convergence result is also established for multi-dimensional conservation laws by relying on DiPerna's uniqueness theorem for entropy measure-valued solutions.


Auxiliary Information and A Priori Values in Construction of Improved Estimators

Abstract: This volume is a collection of six papers on the use of auxiliary information and 'a priori' values in construction of improved estimators. The work included here will be of immense application for researchers and students who emply auxiliary information in any form.


Global wellposedness in the energy space for the Maxwell-Schr\"odinger system

Abstract: We prove that the Maxwell-Schr\"odinger system in $\R^{3+1}$ is globally well-posed in the energy space. The key element of the proof is to obtain a short time wave packet parametrix for the magnetic Schr\"odinger equation, which leads to linear, bilinear and trilinear estimates. These, in turn, are extended to larger time scales via a bootstrap argument.


Univoque numbers and an avatar of Thue-Morse

Abstract: Univoque numbers are real numbers $\lambda > 1$ such that the number 1 admits a unique expansion in base $\lambda$, i.e., a unique expansion $1 = \sum_{j \geq 0} a_j \lambda^{-(j+1)}$, with $a_j \in \{0, 1, ..., \lceil \lambda \rceil -1\}$ for every $j \geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set $\{0, 1, >..., b\}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b, b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3 \to 31$, $2 \to 30$, $1 \to 03$, $0 \to 02$, occurs in a "universal" manner.


Symmetric Systems and their Applications to Root Systems Extended by Abelian Groups

Abstract: We investigate the class of root systems R obtained by extending an irreducible root system by a torsion-free group G. In this context there is a Weyl group W and a group U with the presentation by conjugation. We show under additional hypotheses that the kernel of the natural homomorphism from U to W is isomorphic to the kernel of the homomorphism from the abelianization of U to that of W. For this we introduce the concept of a symmetric system, a discrete version of the concept of a symmetric space. Mathematics Subject Classification 2000: 20F55, 17B65, 17B67, 22E65, 22E40. Key Words and Phrases: Weyl group, root system, presentation by conjugation, extended affine Weyl group (EAWeG), extended affine root system (EARS), irreducible root system extended by an abelian group.


Hessian estimates for the sigma-2 equation in dimension three

Abstract: We derive a priori interior Hessian estimates for the special Lagrangian equation $\sigma_{2}=1$ in dimension three.


Morse-Novikov cohomology of locally conformally K\"ahler manifolds

Abstract: A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same role as the Kahler class in Kahler geometry. If these classes for two LCK-structures coincide, the difference between these structures can be expressed by a smooth potential, similar to the Kahler case. We show that the Morse-Novikov class and the Bott-Chern class of a Vaisman manifold vanishes. Moreover, for any LCK-structure on a Vaisman manifold, we prove that its Morse-Novikov class vanishes. We show that a compact LCK-manifold $M$ with vanishing Bott-Chern class admits a holomorphic embedding to a Hopf manifold, if $\dim_\C M \geq 3$, a result which parallels the Kodaira embedding theorem.


On the moduli of constant mean curvature cylinders of finite type in the 3-sphere

Abstract: We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.


Random sampling of plane partitions

Abstract: This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is $O(n (\ln n)^3)$ in approximate-size sampling and $O(n^{4/3})$ in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for $(a\times b)$-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.


Similarity of operators and geometry of eigenvector bundles

Abstract: We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators.


Generalized Cohn's Theorem

Abstract: We introduce the notion of a free associative $\mathcal{Z}_2$-algebra on the union of two disjoint sets and prove a generalization of Cohn's Theorem on Jordan algebras.


On the partition of numbers into parts of a given type and number

Abstract: E394 in the Enestrom index. Translated from the Latin original, "De partitione numerorum in partes tam numero quam specie datas" (1768). Euler finds a lot of recurrence formulas for the number of partitions of $N$ into $n$ parts from some set like 1 to 6 (numbers on the sides of a die). He starts the paper talking about how many ways a number $N$ can be formed by throwing $n$ dice. There do not seem to be any new results or ideas here that weren't in "Observationes analyticae variae de combinationibus", E158 and "De partitione numerorum", E191. In this paper Euler just does a lot of special cases. My impression is that Euler is trying to make his theory of partitions more approachable,. Also, maybe for his own benefit he wants to say it all again in different words, to make it clear.


Gorenstein Global Dimensions and Cotorsion Dimension of Rings

Abstract: In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.


Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media

Abstract: We consider a space-homogeneous gas of {\it inelastic hard spheres}, with a {\it diffusive term} representing a random background forcing (in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_* \in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.


Global Gorenstein dimensions of polynomial rings and of direct product of rings

Abstract: In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct product of rings, which gives examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.


Rings over which all modules are strongly Gorenstein projective

Abstract: One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective.


Pr\"ufer-Like Conditions in Subring Retracts and Applications

Abstract: In this paper, we consider five possible extensions of the Pr\"ufer domain notion to the case of commutative rings with zero divisors. We investigate the transfer of these Pr\"ufer-like properties between a commutative ring and its subring retract. Our results generate new families of examples of rings subject to a given Pr\"ufer-like conditions.


Symmetry p-adic invariant integral on Z_p for Bernoulli and euler polynomials

Abstract: The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integral on Z_p.


The Algebra of Graph Invariants - Lower and Upper Bounds for Minimal Generators

Abstract: In this paper we study the algebra of graph invariants, focusing mainly on the invariants of simple graphs. All other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this algebra. In fact, every graph invariant is a linear combination of the basic graph invariants which we study in this paper. To prove that two graphs are isomorphic, a number of basic invariants are required, which are called separator invariants. The minimal set of separator invariants is also the minimal basic generator set for the algebra of graph invariants. We find lower and upper bounds for the minimal number of generator/separator invariants needed for proving graph isomorphism. Finally we find a sufficient condition for Ulam's conjecture to be true based on Redfield's enumeration formula.


Graded modules for Virasoro-like algebra

Abstract: In this paper, we consider the classification of irreducible ${\bf Z}$- and ${\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonzero center. Finally, we prove that there does not exist any nontrivial ${\bf Z}$-graded modules of intermediate series.


The Ring of Graph Invariants - Graphic Values

Abstract: The ring of graph invariants is spanned by the basic graph invariants which calculate the number of subgraphs isomorphic to a given graph in other graphs. These subgraphs counting invariants are not algebraically independent. In our view the most important problem in graph theory of unlabeled graphs is the problem of determining graphic values of arbitrary sets of graph invariants. This corresponds to explaining the syzygy of the graph invariants when the number of vertices is unbounded. We introduce two methods to explore this complicated structure. Sets of graphs with a small number of vertices impose constraints on larger sets. We describe families of inequalities of graph invariants. These inequalities allow to loop over all values of graph invariants which look like graphic from the small sets point of view. We also develop strong notion of graphic values where the existence of the corresponding graphs is guaranteed once the constraints are satisfied by the basic graph invariants. These constraints are necessary and sufficient for graphs whose local neighborhoods are generated by a finite set of locally connected graphs. The reconstruction of the graph from the basic graph invariants is shown to be NP-complete in this restricted case. Finally we apply these results to formulate the problem of Ramsey numbers as an integer polyhedron problem of moderate and adjustable dimension.


Generating functions of Cauchy-Stieltjes type for orthogonal polynomials

Abstract: We characterize by the use of free probability the family of measures for which the mulitiplicative renormalization method applies with $h(x) = (1-x)^_{-1}$. This provides a representation formula for their Voiculescu Transforms.


Branching of Hitchin's Prym cover for SL(2)

Abstract: It is shown that the map from the Jacobian of the spectral curve to the moduli of stable bundles of rank 2 is generically simply branched along an irreducible divisor. This observation falsifies the key step in the "abelianization of the SU(2) WZW connection" presented in a recent paper [Yoshida, Annals 2006]


The zero-product problem for Toeplitz operators with radial symbols

Abstract: For any bounded measurable function $f$ on the unit ball $B_n$, let $T_f$ be the Toeplitz operator with symbol $f$ acting on the Bergman space $A^2(B_n)$. The Zero-Product Problem asks: if $f_1,..., f_N$ are bounded measurable functions such that $T_{f_1}... T_{f_N}=0$, does it follow that one of the functions must be zero almost everywhere? This paper give the affirmative answer to this question when all except possibly one of the symbols are radial functions. The answer in the general case remains unknown.


A novel numerical technique used in the solution of ordinary differential equations with a mixture of integer and fractional derivatives

Abstract: Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some algorithms where four discrete forms of the Caputo derivative and three different numerical techniques of solving ordinary differential equations are proposed. We then illustrate how to introduce classical initial conditions into equations where the Riemann-Liouville derivative is included.


Rates of convergence for minimal distances in the central limit theorem under projective criteria

Abstract: In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.


Summarization and Classification of Non-Poisson Point Processes

Abstract: Fitting models for non-Poisson point processes is complicated by the lack of tractable models for much of the data. By using large samples of independent and identically distributed realizations and statistical learning, it is possible to identify absence of fit through finding a classification rule that can efficiently identify single realizations of each type. The method requires a much wider range of descriptive statistics than are currently in use, and a new concept of model fitting which is derive from how physical laws are judged to fit data.


The Non-Backtracking Spectrum of the Universal Cover of a Graph

Abstract: A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, $B$, indexed by $H$'s directed edges and related to Ihara's Zeta function. We show how to determine $B$'s spectrum in the case where $H$ is a tree covering a finite graph. We show that when $H$ is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of $B$'s spectrum, the corresponding Green function has ``periodic decay ratios.'' The existence of such a ``ratio system'' can be effectively checked, and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly $\sqrt\gr$, where $\gr$ is the growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras \cite{ST}. Finally, we give experimental evidence that for a fixed, finite graph, $H$, a random lift of large degree has non-backtracking new spectrum near that of $H$'s universal cover. This suggests a new generalization of Alon's second eigenvalue conjecture.


Some finiteness results for Fourier-Mukai partners

Abstract: We develop some methods for studying the Fourier-Mukai partners of an algebraic variety. As applications we prove that abelian varieties have finitely many Fourier-Mukai partners and that they are uniquely determined by their derived category of coherent $D$-modules. We also generalize a famous theorem due to A. Bondal and D. Orlov.


Vector bundles with Theta divisors I: Bundles on Castelnuovo curves

Abstract: In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.


The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions

Abstract: It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on $R^d$, which are the Jurek class, the Goldie-Steutel-Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type $G$ distributions, are identical with the closure of the class of stable distributions. More general results are also given.


The String Topology Loop Coproduct and Cohomology Operations

Abstract: This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory.


Upper bound for isometric embeddings \ell_2^m\to\ell_p^n

Abstract: The isometric embeddings $\ell_{2;K}^m\to\ell_{p;K}^n$ ($m\geq 2$, $p\in 2\N$) over a field $K\in{R, C, H}$ are considered, and an upper bound for the minimal $n$ is proved. In the commutative case ($K\neq H$) the bound was obtained by Delbaen, Jarchow and Pe{\l}czy{\'n}ski (1998) in a different way.


A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

Abstract: We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group $G$ and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in $\Theta(\sqrt{|G|})$ steps. Moreover, for the parallelized distinguished points algorithm on $J$ processors we find that $\Theta(\sqrt{|G|}/J)$ steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from $G$.


Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space

Abstract: In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula.


On completing three cyclic transversals to a latin square

Abstract: Let $P$ be a partial latin square of prime order $p>7$ consisting of three cyclically generated transversals. Specifically, let $P$ be a partial latin square of the form: \[ P=\{(i,c+i,s+i),(i,c'+i,s'+i),(i,c''+i,s''+i)\mid 0 \leq i< p\} \] for some distinct $c,c',c''$ and some distinct $s,s',s''$. In this paper we show that any such $P$ completes to a latin square which is diagonally cyclic.


Lyapunov conditions for logarithmic Sobolev and Super Poincar\'e inequality

Abstract: We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and weak Poincar\'e inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...).


The variational particle-mesh method for matching curves

Abstract: Diffeomorphic matching (only one of several names for this technique) is a technique for non-rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time-series of vector fields. One seeks the flow between two topologically-equivalent curves or surfaces which minimises some metric defined on the vector fields, \emph{i.e.} the flow closest to the identity in some sense. In this paper, we describe a new particle-mesh discretisation for the evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms are very natural for this problem because Lagrangian particles (particles moving with the flow) can represent the movement of the shape whereas the vector field is Eulerian and hence best represented on a static mesh. We explain the derivation of the method, and prove conservation properties: the discrete method has a set of conserved momenta corresponding to the particle-relabelling symmetry which converge to conserved quantities in the continuous problem. We also introduce a new discretisation for the geometric current matching condition of (Vaillant and Glaunes, 2005). We illustrate the method and the derived properties with numerical examples.


On the Structure of Equidistant Foliations of Euclidean Space

Abstract: This thesis is concerned with equidistant foliations of Euclidean space, i.e. partitions into complete, connected, properly embedded smooth submanifolds. The space of leaves is an Alexandrov space of nonnegative curvature and the canonical projection is a submetry. Generalizing a result of Gromoll and Walschap we show that an equidistant foliation always has an affine leaf and we prove homogeneneity of the foliation under certain additional assumptions. Moreover, we give several reducibility results and construct new (noncompact) inhomogeneous examples of equidistant foliations.


Pac-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning

Abstract: This monograph deals with adaptive supervised classification, using tools borrowed from statistical mechanics and information theory, stemming from the PACBayesian approach pioneered by David McAllester and applied to a conception of statistical learning theory forged by Vladimir Vapnik. Using convex analysis on the set of posterior probability measures, we show how to get local measures of the complexity of the classification model involving the relative entropy of posterior distributions with respect to Gibbs posterior measures. We then discuss relative bounds, comparing the generalization error of two classification rules, showing how the margin assumption of Mammen and Tsybakov can be replaced with some empirical measure of the covariance structure of the classification model.We show how to associate to any posterior distribution an effective temperature relating it to the Gibbs prior distribution with the same level of expected error rate, and how to estimate this effective temperature from data, resulting in an estimator whose expected error rate converges according to the best possible power of the sample size adaptively under any margin and parametric complexity assumptions. We describe and study an alternative selection scheme based on relative bounds between estimators, and present a two step localization technique which can handle the selection of a parametric model from a family of those. We show how to extend systematically all the results obtained in the inductive setting to transductive learning, and use this to improve Vapnik's generalization bounds, extending them to the case when the sample is made of independent non-identically distributed pairs of patterns and labels. Finally we review briefly the construction of Support Vector Machines and show how to derive generalization bounds for them, measuring the complexity either through the number of support vectors or through the value of the transductive or inductive margin.


Small overlap monoids: the word problem

Abstract: We develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions. In contrast to existing geometric methods, our approach facilitates a sequential left-right analysis of words which lends itself to the development of practical, efficient computational algorithms. In particular, we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition C(4), and a polynomial time solution to the uniform word problem for presentations satisfying the same condition.


Die lokale Struktur von T-Dualit\"atstripeln

Abstract: We show that the $C^*$-algebraic approach to T-duality of Mathai and Rosenberg is equivalent to the topological approach of Bunke and Schick.


Koszul complexes and fully faithful integral functors

Abstract: We characterise those objects in the derived category of a scheme which are a sheaf supported on a closed subscheme in terms of Koszul complexes. This is applied to generalize to arbitrary schemes the fully faithfullness criteria of an integral functor.


An Extension of the Classical Gauss Series-product Identity by Fermionic Construction of \hat{sl}_n

Abstract: The main result of this paper is two infinity classes of series-product identities which is based on classical Gauss identity and two different interpretations of character formula for irreducible highest weight modules of affine Lie algebras.


Deformation of Brody curves and mean dimension

Abstract: The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.


Random graphs with forbidden vertex degrees

Abstract: We study the random graph G_{n,\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \mux given as the root of a certain `characteristic equation' of S that maximises a certain function \psis(\mu). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \emph{even} graph is related to the random-cluster model on the complete graph K_n.


Heisenberg modules over real multiplication noncommutative tori and related algebraic structures

Abstract: We review some aspects of the theory of noncommutative two-tori with real multiplication focusing on the role played by Heisenberg groups in the definition of algebraic structures associated to these noncommutative spaces.


On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras

Abstract: We investigate the automorphism group of the quantised enveloping algebra U of the positive nilpotent part of certain simple complex Lie algebras g in the case where the deformation parameter q \in \mathbb{C}^* is not a root of unity. Studying its action on the set of minimal primitive ideals of U we compute this group in the cases where g=sl_3 and g=so_5 confirming a Conjecture of Andruskiewitsch and Dumas regarding the automorphism group of U. In the case where g=sl_3, we retrieve the description of the automorphism group of the quantum Heisenberg algebra that was obtained independently by Alev and Dumas, and Caldero. In the case where g=so_5, the automorphism group of U was computed in [16] by using previous results of Andruskiewitsch and Dumas. In this paper, we give a new (simpler) proof of the Conjecture of Andruskiewitsch and Dumas in the case where g=so_5 based both on the original proof and on graded arguments developed in [17] and [18].


Wavelet methods in statistics: Some recent developments and their applications

Abstract: The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of applications. After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise. Most of these methods are fitted into the general concept of regularization with appropriately chosen penalty functions. A narrow range of applications in major areas of statistics is also discussed such as partial linear regression models and functional index models. The usefulness of all these methods are illustrated by means of simulations and practical examples.


Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach

Abstract: We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.


An algorithm for evaluating the Gamma function and ramifications

Abstract: We provide a new algorithm for evaluating the gamma function at any (rational) point and a new infinite product representation free from the presence of Euler and Mascheroni constant.Formulae and inequalities seemingly new are obtained as byproducts of the method.


Liapunov's direct method for Birkhoffian systems: Applications to electrical networks

Abstract: In this paper, the concepts and the direct theorems of stability in the sense of Liapunov, within the framework of Birkhoffian dynamical systems on manifolds, are considered. The Liapunov-type functions are constructed for linear and nonlinear LC and RLC electrical networks, to prove stability under certain conditions.


Constrained BSDE and Viscosity Solutions of Variation Inequalities

Abstract: In this paper, we study the relation between the smallest $g$-supersolution of constraint backward stochastic differential equation and viscosity solution of constraint semilineare parabolic PDE, i.e. variation inequalities. And we get an existence result of variation inequalities via constraint BSDE, and prove a uniqueness result under certain condition.


Difference Problems and Differential Problems

Abstract: We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a significant contribution to the understanding of the foundations of differential and integral calculus.


Point fixe li\'e \`a une orbite p\'eriodique d'un diff\'eomorphisme de R2

Abstract: Given a diffeomorphism of the plane, which has a periodic orbit, we show how Nielsen fixed point theory can be used to establish the existence of a fixed point which is linked with this periodic orbit.


A new upper bound for the cross number of finite Abelian groups

Abstract: In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite Abelian groups. Given a finite Abelian group, this upper bound appears to depend only on the rank and on the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite Abelian groups holds asymptotically in at least two different directions.


Duality of Chordal SLE

Abstract: We derive some geometric properties of chordal SLE$(\kappa;\vec{\rho})$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(\kappa;\vec{\rho})$ process has the same distribution as the image of a chordal SLE$(\kappa';\vec{\rho'})$ trace, where $\kappa>4$, $\kappa'=16/\kappa$, and the forces $\vec{\rho}$ and $\vec{\rho'}$ are suitably chosen. We find that for $\kappa\ge 8$, the boundary of a standard chordal SLE$(\kappa)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/\kappa;\vec{\rho})$ trace started from $x$. Then we obtain a new proof of the fact that chordal SLE$(\kappa)$ trace is not reversible for $\kappa>8$. We also prove that the reversal of SLE$(4;\vec{\rho})$ trace has the same distribution as the time-change of some SLE$(4;\vec{\rho'})$ trace for certain values of $\vec{\rho}$ and $\vec{\rho'}$.


A maximum principle for relaxed stochastic control of linear SDE's with application to bond portfolio optimization

Abstract: We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence of an optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity.


The Decomposition Theorem and the topology of algebraic maps

Abstract: We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples.


A Mathematical Proof of Dodgson's Algorithm

Abstract: In this paper we give a mathematical proof of Dodgson algorithm [1]. Recently Zeilberger [2] gave a bijective proof. Our techniques are based on determinant properties and they are obtained by induction.


Coincidences in 4 dimensions

Abstract: The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions $d>3$. Here, we discuss the CSLs of the root lattice $A_4$ and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalised for 4--dimensional $\Z$--modules by discussing the icosian ring.


Functions with support in a lacunary system of intervals and cyclicity for the semi-group of left translations

Abstract: We prove in the vector-valued spaces $L^2(\mathbb{R}_+, X)$ (where $X$ is a finite dimensional Hilbert space) the cyclicity for the semi-group of left translations of some particular functions with support included in a lacunary system of intervals.


Global regular solutions for the Navier-stokes system with small initial data in $\Phi(2)$: an elementary approach

Abstract: We show existence and regularity result for the Navier Stokes system for small data in the space $\Phi(2)$, and we show relations with some classical results.


The $(L^{p},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages

Abstract: More work needs to be done to move from the tail to the averages themselves. So at this time we prefer to withdraw the paper about the averages. However a previous version of the paper which deals with the tail has been checked and we believe it to be complete and correct.


Rips Complexes of Planar Point Sets

Abstract: Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of $D$. There is a natural ``shadow'' projection map from the Rips complex to $\euc^n$ that has as its image a more accurate $n$-dimensional approximation to the homotopy type of $D$. We demonstrate that this projection map is 1-connected for the planar case $n=2$. That is, for planar domains, the Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to `quasi'-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.


Hopf algebras of dimension 16

Abstract: We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed.


Inverse Problems for Representation Functions in Additive Number Theory

Abstract: For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product, if X is nonabelian) of h not necessarily distinct elements of X. The direct problem for representation functions in additive number theory begins with a subset A of X and seeks to understand its representation functions. The inverse problem for representation functions starts with a function f:X ->N_0 U {\infty} and asks if there is a set A whose representation function is f, and, if the answer is yes, to classify all such sets. This paper is a survey of recent progress on the inverse representation problem.


Structure theorems for embedded disks with mean curvature bounded in L^P

Abstract: After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi.


About the logarithm function over the matrices

Abstract: We prove the following results: let x,y be (n,n) complex matrices such that x,y,xy have no eigenvalue in ]-infinity,0] and log(xy)=log(x)+log(y). If n=2, or if n>2 and x,y are simultaneously triangularizable, then x,y commute. In both cases we reduce the problem to a result in complex analysis.


Sequential Tracking of a Hidden Markov Chain Using Point Process Observations

Abstract: We study finite horizon optimal switching problems for hidden Markov chain models under partially observable Poisson processes. The controller possesses a finite range of strategies and attempts to track the state of the unobserved state variable using Bayesian updates over the discrete observations. Such a model has applications in economic policy making, staffing under variable demand levels and generalized Poisson disorder problems. We show regularity of the value function and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our results with several computational examples.


Closed categories, star-autonomy, and monoidal comonads

Abstract: This paper determines what structure is needed for internal homs in a monoidal category C to be liftable to the category C^G of Eilenberg-Moore coalgebras for a monoidal comonad G on C. We apply this to lift star-autonomy with the view to recasting the definition of quantum groupoid.


The number of hypergraphs and colored Hypergraphs with hereditary properties

Abstract: As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an n-vertex graph which has no subgraph in P. Kohayakawa et al. (2003) extended it from monotone to hereditary and from graphs to 3-uniform hypergraphs. We extend it to general hypergraphs. This may be a simple example illustrating how to apply a recent hypergraph regularity lemma by the author.


Solvable automorphism groups of a compact Kaehler manifold

Abstract: Let X be a compact Kaehler manifold of complex dimension n. Let G be a connected solvable subgroup of the automorphism group Aut(X), and let N(G) be the normal subgroup of G of elements of null entropy. One of the goals of this paper is to show that G/N(G) is a free abelian group of rank r(G) less than or equal to n-1 and that the rank estimate is optimal. This gives an alternative proof of the conjecture of Tits type. In addition, we show some non-obvious implications on the structure of solvable automorphism groups of compact Kaehler manifolds. Furthermore, we also show that if the rank r(G) of the quotient group G/N(G) is equal to n-1 and the identity component Aut_0(X) of Aut(X) is trivial, then N(G) is a finite set. The main strategy of this paper is to combine the method of Dinh and Sibony and the theorem of Birkhoff-Perron-Frobenius (or Lie-Kolchin type), and one argument of D.-Q. Zhang originated from the paper of Dinh and Sibony plays an important role.


Projective normality of nonsingular toric varieties of dimension three

Abstract: We show that if an ample line bundle L on a nonsingular toric 3-fold satisfies h^0(L+2K)=0, then L is normally generated. As an application, we show that the anti-canonical divisor on a nonsingular toric Fano 4-fold is normally generated.


Large Deviations for Heavy-Tailed Factor Models

Abstract: We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms contribute to the behaviour of the tail-probability of the sum. A simple conditional Monte Carlo algorithm is also provided together with a comparison between the simulations and the large deviation approximation. We also study large deviation probabilities for stochastic processes with factor structure. The processes involved are assumed to be Levy processes with regularly varying jump measures. Based on the results of the first part of the paper, we show that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process.


Multiple equilibria of nonhomogeneous Markov chains and self-validating web rankings

Abstract: PageRank is a ranking of the web pages that measures how often a given web page is visited by a random surfer on the web graph, for a simple model of web surfing. It seems realistic that PageRank may also have an influence on the behavior of web surfers. We propose here a simple model taking into account the mutual influence between web ranking and web surfing. Our ranking, the T-PageRank, is a nonlinear generalization of the PageRank. It is defined as the limit, if it exists, of some nonlinear iterates. A positive parameter T, the temperature, measures the confidence of the web surfer in the web ranking. We prove that, when the temperature is large enough, the T-PageRank is unique and the iterates converge globally on the domain. But when the temperature is small, there may be several T-PageRanks, that may strongly depend on the initial ranking. Our analysis uses results of nonlinear Perron-Frobenius theory, Hilbert projective metric and Birkhoff's coefficient of ergodicity.


An Exact Exprsssion of Pi(x)

Abstract: The author states an exact expression of the distribution of primes.


Integral means and boundary limits of Dirichlet series

Abstract: We study the boundary behavior of functions in the Hardy spaces HD^p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in HD^\infty, i.e., for ordinary Dirichlet series in H^\infty of the right half-plane. We discuss an important embedding problem for HD^p, the solution of which is only known when p is an even integer. Viewing HD^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.


Sharp Spectral Asymptotics for Dirac Energy. II. Magnetic Schroedinger operator

Abstract: I derive sharp semiclassical asymptotics of \int |e_h(x,y,0)|^2\omega (x,y)dxdy where e_h(x,y,\tau) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and \omega (x,y) is singular as x=y. I also consider asymptotics of more general expressions.


Rational Extensions of C(X) via Hausdorff Continuous Functions

Abstract: The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.


Newhouse phenomenon and homoclinic classes

Abstract: We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources.


Lyapunov stable chain recurrent classes

Abstract: We show that for a $C^1$ residual subset of diffeomorphisms far away from homoclinic tangency, the stable manifolds of periodic points cover a dense subset of the ambient manifold. This gives a partial proof to a conjecture of C. Bonatti.


From Quantum Universal Enveloping Algebras to Quantum Algebras

Abstract: The ``local'' structure of a quantum group G_q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping algebra of a n-dimensional Lie algebra g=Lie(G). However, we show how, by starting from the generators of the underlying Lie bialgebra (g,\delta), the analyticity in the deformation parameter(s) allows us to determine in a unique way a set of n ``almost primitive'' basic objects in U_q(g), that could be properly called the ``quantum algebra generators''. So, the analytical prolongation (g_q,\Delta) of the Lie bialgebra (g,\delta) is proposed as the appropriate local structure of G_q. Besides, as in this way (g,\delta) and U_q(g) are shown to be in one-to-one correspondence, the classification of quantum groups is reduced to the classification of Lie bialgebras. The su_q(2) and su_q(3) cases are explicitly elaborated.


Bernoulli coding map and almost sure invariance principle for endomorphisms of $\mathbb{P}^k$

Abstract: Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $\mu$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,\mu)$. Our class $\cal{U}$ of observables includes the H\"older functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $\omega: (\Sigma, s, \nu) \to (\mathbb{P}^k,f,\mu)$. We obtain the invariance principle for an observable $\psi$ on $(\mathbb{P}^k,f,\mu)$ by applying Philipp-Stout's theorem for $\chi = \psi \circ \omega$ on $(\Sigma, s, \nu)$. The invariance principle implies the Central Limit Theorem as well as several statistical properties for the class $\cal{U}$. As an application, we give a \emph{direct} proof of the absolute continuity of the measure $\mu$ when it satisfies Pesin's formula. This approach relies on the Central Limit Theorem for the unbounded observable $\log \textsf{Jac} f \in \cal{U}$.


K-spectral sets and intersections of disks of the Riemann sphere

Abstract: We prove that if two closed disks X_1 and X_2 of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then the intersection X_1\cap X_2 is a complete (2+2/\sqrt{3})-spectral set for A. When the intersection of X_1 and X_2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).


Comparing Gr\"obner bases and word reversing

Abstract: Gr\"obner bases, in their noncommutative version, and word reversing are methods for solving the word problem of a presented monoid, and both rely on iteratively completing the initial list of relations. Simple examples may suggest to conjecture that both completion procedures are closely related. Here we disprove this conjecture by exhibiting families of presentations for which they radically differ.


Large deviations for return times in non-rectangle sets for axiom A diffeomorphisms

Abstract: For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.


${\cal T}$-class algorithms for pseudocontractions and $\kappa$-strict pseudocontractions in Hilbert spaces

Abstract: In this paper we study iterative algorithms for finding a common element of the set of fixed points of $\kappa$-strict pseudocontractions or finding a solution of a variational inequality problem for a monotone, Lipschitz continuous mapping. The last problem being related to finding fixed points of pseudocontractions. These algorithms were already studied in [G.L. Acedo, H.-K. Xu] and [N. Nadezhkina, W. Takahashi] but our aim here is to provide the links between these know algorithms and the general framework of ${\cal T}$-class algorithms studied in [H.H. Bauschke, P.L. Combettes].


A simplified multidimensional integral

Abstract: We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.


Fragmenting random permutations

Abstract: Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each n a fragmentation process (\Pi_{n,k}, 1 \leq k \leq n) taking values in the space of partitions of {1,2,...,n} such that \Pi_{n,k} is distributed like the partition generated by cycles of a uniform random permutation of {1,2,...,n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.


Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems

Abstract: We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze's and Helmke's compactification arises naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze's compactification as a Quot scheme is given, whereas Helmke's compactification is shown to be an algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric description of both compactifications. As an application, we determine the cohomology ring of Helmke's compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive.


Differential Equations in Metric Spaces with Applications

Abstract: This paper proves the local well posedness of differential equations in metric spaces under assumptions that allow to comprise several different applications. We consider below a system of balance laws with a dissipative non local source, the Hille-Yosida Theorem, a generalization of a recent result on nonlinear operator splitting, an extension of Trotter formula for linear semigroups and the heat equation.


Commensurability and QI classification of free products of finitely generated abelian groups

Abstract: Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\Z$ with a $\Z^n$ for each $n$ occurring in $S'$.


Large deviations for local time fractional Brownian motion and applications

Abstract: Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0\}$ be the local time process at zero of a strictly stable L\'evy process $X=\{X_t, t \ge 0\}$ of index $1<\alpha\leq 2$ independent of $W^H$. The $\a$-stable local time fractional Brownian motion $Z^H=\{Z^H(t), t \ge 0\}$ is defined by $Z^H(t) = W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1 - \frac 1 \alpha)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coupleCTRW,limitCTRW}). However, $Z^H$ does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process $Z^H$. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for $Z^H$.


Inverse problems for regular variation of linear filters, a cancellation property for $\sigma$-finite measures and identification of stable laws

Abstract: In this paper, we consider certain $\sigma$-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have regularly varying tails as well. This turns out to be related to the presence of a particular cancellation property in $\sigma$-finite measures, which in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of i.i.d. random variables, to products of independent random variables, and to stochastic integrals with respect to L\'evy motions.


Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

Abstract: We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive L\'evy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.


Enhanced delay to Bifurcation

Abstract: This article provides an example of fast-slow system such that most orbits remain as close as possible to the unstable manifold of the fast dynamics for an arbitrarily long time.


Cardinal sequences of LCS spaces under GCH

Abstract: We give full characterization of the sequences of regular cardinals that may arise as cardinal sequences of locally compact scattered spaces under GCH. The proofs are based on constructions of universal locally compact scattered spaces.


Non group-theoretical semisimple Hopf algebras from group actions on fusion categories

Abstract: Given an action of a finite group G on a fusion category C we give a criterion for the category of G-equivariant objects in C to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in math/0203060. Namely, we show that certain Z/2Z-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami are equivalent to representation categories of non group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.


Maximal small extensions of o-minimal structures

Abstract: A proper elementary extension of a model is called small if it realizes no new types over any finite set in the base model. We answer a question of Marker, and show that it is possible to have an o-minimal structure with a maximal small extension. Our construction yields such a structure for any cardinality. We show that in some cases, notably when the base structure is countable, the maximal small extension has maximal possible cardinality.


Some families of increasing planar maps

Abstract: Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations.


Irreducible Representations of Groupoid $C^*$-algebras

Abstract: If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.


A folk model structure on omega-cat

Abstract: We establish a model structure on the category of strict omega-categories. The constructions leading to the model structure in question are expressed entirely within the scope of omega-categories, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while cofibrant objects are exactly the free ones. Our model structure transfers to n-categories along right-adjoints, for each n, thus recovering the known cases n = 1 and n = 2.


Filtrations

Abstract: In this article, we define the notion of a filtration and then give the basic theorems on initial and progressive enlargements of filtrations.


Modular unit and cuspidal divisor class groups of X_1(N)

Abstract: In this article, we consider the group $F_1^\infty(N)$ of modular units on $X_1(N)$ that have divisors supported on the cusps lying over $\infty$ of $X_0(N)$, called the $\infty$-cusps. For each positive integer $N$, we will give an explicit basis for the group $F_1^\infty(N)$. This enables us to compute the group structure of the rational torsion subgroup $C_1^\infty(N)$ of the Jacobian $J_1(N)$ of $X_1(N)$ generated by the differences of the $\infty$-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the $p$-primary part of $C_1^\infty(p^n)$ for a regular prime $p$.


Overpartitions and class numbers of binary quadratic forms

Abstract: We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as $q$-series identities of the mock theta type.


Topologically unique maximal elementary Abelian group actions on compact oriented surfaces

Abstract: We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $\sigma\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.


Controlled Synchronization of One Class of Nonlinear Systems under Information Constraints

Abstract: Output feedback controlled synchronization problems for a class of nonlinear unstable systems under information constraints imposed by limited capacity of the communication channel are analyzed. A binary time-varying coder-decoder scheme is described and a theoretical analysis for multi-dimensional master-slave systems represented in Lurie form (linear part plus nonlinearity depending only on measurable outputs) is provided. An output feedback control law is proposed based on the Passification Theorem. It is shown that the synchronization error exponentially tends to zero for sufficiantly high transmission rate (channel capacity). The results obtained for synchronization problem can be extended to tracking problems in a straightforward manner, if the reference signal is described by an {external} ({exogenious}) state space model. The results are applied to controlled synchronization of two chaotic Chua systems via a communication channel with limited capacity.


The Natural Philosophy of Kazuo Kondo

Abstract: Kazuo Kondo (1911-2001) was Chair of the Department of Mathematical Engineering at the University of Tokyo, Japan. Over a period of 50 years, he and a few colleagues wrote and published a voluminous series of papers and monographs on the applications of analytical geometry within a diverse range of subjects in the natural sciences. Inspired by Otto Fischer's attempt at a quaternionic unified theory in the late 1950's he adopted the mathematics of the revered Akitsugu Kawaguchi to produce his own speculative unified theory. The theory appears to successfully apply Kawaguchi's mathematics to the full range of natural phenomena, from the structure of fundamental particles to the geometry of living beings. The theories are testable and falsifiable. Kondo and his theories are now almost completely unknown and this paper serves as the barest introduction to his work


Free curves and periodic points for torus homeomorphisms

Abstract: We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincar\'e-Birkhoff Theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has nonempty interior.


A practical illustration of the importance of realistic individualized treatment rules in causal inference

Abstract: The effect of vigorous physical activity on mortality in the elderly is difficult to estimate using conventional approaches to causal inference that define this effect by comparing the mortality risks corresponding to hypothetical scenarios in which all subjects in the target population engage in a given level of vigorous physical activity. A causal effect defined on the basis of such a static treatment intervention can only be identified from observed data if all subjects in the target population have a positive probability of selecting each of the candidate treatment options, an assumption that is highly unrealistic in this case since subjects with serious health problems will not be able to engage in higher levels of vigorous physical activity. This problem can be addressed by focusing instead on causal effects that are defined on the basis of realistic individualized treatment rules and intention-to-treat rules that explicitly take into account the set of treatment options that are available to each subject. We present a data analysis to illustrate that estimators of static causal effects in fact tend to overestimate the beneficial impact of high levels of vigorous physical activity while corresponding estimators based on realistic individualized treatment rules and intention-to-treat rules can yield unbiased estimates. We emphasize that the problems encountered in estimating static causal effects are not restricted to the IPTW estimator, but are also observed with the $G$-computation estimator, the DR-IPTW estimator, and the targeted MLE. Our analyses based on realistic individualized treatment rules and intention-to-treat rules suggest that high levels of vigorous physical activity may confer reductions in mortality risk on the order of 15-30%, although in most cases the evidence for such an effect does not quite reach the 0.05 level of significance.


Least Area Planes in Hyperbolic 3-Space are Properly Embedded

Abstract: We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.


Quasi-kernels and quasi-sinks in infinite graphs

Abstract: Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We investigate the following conjecture here: for any digraph G=(V,E) there is a a partition (V_0,V_1) of the vertex set such that the induced subgraph G[V_0] has a quasi-kernel and the induced subgraph G[V_1] has a quasi-sink.


Modules with Finite Cousin Cohomologies Have Uniform Local Cohomological Annihilators

Abstract: Let A be a Noetherian ring. It is shown that any finite A--module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying (S_2) which is over a local ring with Cohen--Macaulay formal fibres.


An Improved Error Bound for Multiquadric and Inverse Multiquadric Interpolation

Abstract: A new error bound which is better than the current exponential-type error bound is presented in this paper.


Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents

Abstract: For an ergodic measure preserving action on a probability space, consider the corresponding crossed product von Neumann algebra. We calculate the Fuglede-Kadison determinant for a class of operators in this von Neumann algebra in terms of the Ljapunov exponents of an associated measurable cocycle. The proof is based on recent work of Dykema and Schultz. As an application one obtains formulas for the Fuglede-Kadison determinant of noncommutative polynomials in the von Neumann algebra of the discrete Heisenberg group. These had been previously obtained by Lind and Schmidt via entropy considerations.


Integral Lattices of the SU(2)-TQFT-Modules

Abstract: We find bases for naturally defined lattices over certain rings of integers in the SU(2)-TQFT-theory modules of surfaces. We consider the TQFT where the Kauffman's A variable is a root of unity of order four times an odd prime. As an application, we show that the Frohman Kania-Bartoszynska ideal invariant for 3-manifolds with boundary using the SU(2)-TQFT-theory is equal to the product of the ideals using the 2^{'}-theory and the SO(3)-TQFT-theory under a certain change of coefficients.


Asymptotic normality of the Quasi Maximum Likelihood Estimator for multidimensional causal processes

Abstract: Strong consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) are given for a general class of multidimensional causal processes. For particular cases already studied in the literature (for instance univariate or multivariate GARCH, ARCH, ARMA-GARCH processes) the assumptions required for establishing these results are often weaker than existing conditions. The QMLE asymptotic behavior is also given for numerous new examples of univariate or multivariate processes (for instance TARCH or NLARCH processes).


Compact embeddings of model subspaces of the Hardy space

Abstract: We study embeddings of model (star-invariant) subspaces $K^p_{\Theta}$ of the Hardy space $H^p$, associated with an inner function $\Theta$. We obtain a criterion for the compactness of the embedding of $K^p_{\Theta}$ into $L^p(\mu)$ analogous to the Volberg--Treil theorem on bounded embeddings and answer a question posed by Cima and Matheson. The proof is based on Bernstein inequalities for functions in $K^p_{\Theta}$. Also we study measures $\mu$ such that the embedding operator belongs to a Schatten--von Neumann ideal.


Ergodic Theory, Abelian Groups, and Point Processes Induced by Stable Random Fields

Abstract: We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be computed using the structure theorem of finitely generated abelian groups. The corresponding point processes sequence is not tight and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques.


Bounds for Behrend's conjecture on the canonical reduction

Abstract: We prove Behrend's conjecture on the rationality of the canonical reduction of principal bundles and reductive group schemes for classical groups and give new bounds for the conjecture for exceptional groups. However we find a counterexample in the case of G_2-bundles in characteristic 2.


Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Abstract: Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in $\R^{n+1}$ with $H^F_r={constant}$ is the Wulff shape, up to translations and homotheties. In case $r=1$, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.


Pontrjagin-Thom maps and the homology of the moduli stack of stable curves

Abstract: We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.


Orbit projections of proper Lie groupoids as fibrations

Abstract: Let $\mathcal{G} \rightrightarrows M$ be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection $M \to M/\mathcal{G}$ is a fibration if and only if $\mathcal{G} \rightrightarrows M$ is regular.


System reliability and weighted lattice polynomials

Abstract: The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of "indicator" variables. A connection is studied between Y and order statistics of the set of arguments.


Transfer Principle for the Fundamental Lemma

Abstract: The purpose of this paper is to explain how the identities of various fundamental lemmas fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one collection of fields to others. In particular, once the fundamental lemma has been established for one collection of fields (for example, fields of positive characteristic), it is also valid for others (fields of characteristic zero).


Asymptotically optimal multistage tests of simple hypotheses

Abstract: A family of variable stage size multistage tests of simple hypotheses is described, based on efficient multistage sampling procedures. Using a loss function that is a linear combination of sampling costs and error probabilities, these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero. A numerical study shows significant improvement over group sequential tests in a binomial testing problem.