Analytic Functions of a Quaternionic Variable

Abstract: Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show how many of those obstacles can be overcome. After a tiny bit of linear algebra we look at the beginnings of differential calculus. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*).


Stability in $H^{1/2}$ of the sum of $K$ solitons for the Benjamin-Ono equation

Abstract: This note proves the orbital stability in the energy space $H^{1/2}$ of the sum of widely-spaced 1-solitons for the Benjamin-Ono equation, with speeds arranged so as to avoid collisions.


New Proof of the Equation $\sum_{k=1}^\infty \frac{\mu(k)}{k}=0.$

Abstract: This is an English translation of Edmund Landau's Doctoral Dissertation.


Hilbert modular forms of weight 1/2 and theta functions

Abstract: Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result - under certain mild restrictions on the level and character - to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.


On metrics of positive Ricci curvature conformal to MxR^m

Abstract: Let (M, g) be a closed Riemannian manifold and gE the Euclidean metric. We show that for m > 1, (M x R^m, (g + gE)) is not conformal to a positive Einstein manifold. Moreover, (M x R^m, (g + gE)) is not conformal to a Riemannian manifold of positive Ricci curvature, through a smooth, radial, positive, integrable function of R^m, for m > 1. These results are motivated by some recent questions on Yamabe constants.


Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories

Abstract: We prove that the Morava-$K$-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a $p$-local finite Postnikov system with vanishing $(n+1)$st homotopy group.


Largeness of LERF and 1-relator groups

Abstract: We consider largeness of groups given by a presentation of deficiency 1, where the group is respectively free-by-cyclic, LERF or 1-relator. We give the first examples of (finitely generated free)-by-(infinite cyclic) word hyperbolic groups which are large, show that a LERF deficiency 1 group with first Betti number at least 2 is large or the integers times the integers, and show that 2-generator 1-relator groups where the relator has height 1 obey the dichotomy that either the group is large or all its finite images are metacyclic.


The elliptic GL(n) dynamical quantum group as an h-Hopf algebroid

Abstract: Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, F_ell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sl_n. We apply the generalized FRST construction and obtain an h-bialgebroid F_ell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the h-Hopf algebroid F_ell(GL(n)).


Spanning sets for Moebius vertex algebras satisfying arbitrary difference conditions

Abstract: Spanning sets for vertex operator algebras satisfying difference-zero and difference-one conditions have been extensively studied in the recent years. In this paper, we extend these results. More specifically, we show that for a suitably chosen generating set, any N-graded Moebius vertex algebra is spanned by monomials satisfying a difference-N ordering condition.


A conjectured lower bound for the cohomological dimension of elliptic spaces. Some results in some simple cases

Abstract: Here we prove some special cases of the following conjecture: that the sum of the Betti numbers of a 1-connected elliptic space is greater than the total rank of its homotopy groups. Our main tool is Sullivan's minimal model.


Convergence rates for adaptive finite elements

Abstract: In this article we prove that it is possible to construct, using newest-vertex bisection, meshes that equidistribute the error in $H^1$-norm, whenever the function to approximate can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of Partial Differential Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite element methods (AFEM) using Lagrange finite elements of any polynomial degree are obtained.


The homology of the stable non-orientable mapping class group

Abstract: Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group $N_\infty$ (after plus-construction). At odd primes p, the F_p-homology coincides with that of $Q_0(HP^\infty_+)$, but at the prime 2 the result is less clear. We identify the F_2-homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of $N_\infty$ in degrees up to six. As in the oriented case, not all of these cohomology classes have a geometric interpretation. We determine a polynomial subalgebra of $H^*(N_\infty ; F_2)$ consisting of geometrically-defined characteristic classes.


Set families with a forbidden subposet

Abstract: We asymptotically determine the size of the largest family F of subsets of {1,...,n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's theorem.


The congruence subgroup property for the hyperelliptic modular group: the open surface case

Abstract: Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $\Gamma_{g,n}$ and $H_{g,n}$, the so called Teichm{\"u}ller modular group and hyperelliptic modular group. A choice of base point on ${\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrow\Gamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichm\"uller group $\Gamma_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $\Gamma_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookrightarrow\operatorname{Out}(\pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\lambda$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $\pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\to\operatorname{Out}(\pi_1(S_{g,n})/K)$ is contained in $H^\lambda$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.


Combinatorics of binomial primary decomposition

Abstract: An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.


Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties

Abstract: A 2-category was introduced in arXiv:0803.3652 [math.QA] that categorifies Lusztig's integral version of quantum sl(2). Here we construct for each positive integer N arepresentation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).


Large intersection properties in Diophantine approximation and dynamical systems

Abstract: We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems.


Simultaneous inference: When should hypothesis testing problems be combined?

Abstract: Modern statisticians are often presented with hundreds or thousands of hypothesis testing problems to evaluate at the same time, generated from new scientific technologies such as microarrays, medical and satellite imaging devices, or flow cytometry counters. The relevant statistical literature tends to begin with the tacit assumption that a single combined analysis, for instance, a False Discovery Rate assessment, should be applied to the entire set of problems at hand. This can be a dangerous assumption, as the examples in the paper show, leading to overly conservative or overly liberal conclusions within any particular subclass of the cases. A simple Bayesian theory yields a succinct description of the effects of separation or combination on false discovery rate analyses. The theory allows efficient testing within small subclasses, and has applications to ``enrichment,'' the detection of multi-case effects.


Irreducible Representations of C*-crossed products by Finite Groups

Abstract: We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this description to derive a characterization of irreducible representations of crossed-products by finite cyclic groups in terms of representations of the C*-algebra and its fixed point subalgebra. These results are applied to crossed-products by the permutation group on three elements and illustrated by various examples.


Existence criterion for Hall subgroups of finite groups

Abstract: In the paper we obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series.


Sparse estimation of large covariance matrices via a nested Lasso penalty

Abstract: The paper proposes a new covariance estimator for large covariance matrices when the variables have a natural ordering. Using the Cholesky decomposition of the inverse, we impose a banded structure on the Cholesky factor, and select the bandwidth adaptively for each row of the Cholesky factor, using a novel penalty we call nested Lasso. This structure has more flexibility than regular banding, but, unlike regular Lasso applied to the entries of the Cholesky factor, results in a sparse estimator for the inverse of the covariance matrix. An iterative algorithm for solving the optimization problem is developed. The estimator is compared to a number of other covariance estimators and is shown to do best, both in simulations and on a real data example. Simulations show that the margin by which the estimator outperforms its competitors tends to increase with dimension.


Skip sequencing: A decision problem in questionnaire design

Abstract: This paper studies questionnaire design as a formal decision problem, focusing on one element of the design process: skip sequencing. We propose that a survey planner use an explicit loss function to quantify the trade-off between cost and informativeness of the survey and aim to make a design choice that minimizes loss. We pose a choice between three options: ask all respondents about an item of interest, use skip sequencing, thereby asking the item only of respondents who give a certain answer to an opening question, or do not ask the item at all. The first option is most informative but also most costly. The use of skip sequencing reduces respondent burden and the cost of interviewing, but may spread data quality problems across survey items, thereby reducing informativeness. The last option has no cost but is completely uninformative about the item of interest. We show how the planner may choose among these three options in the presence of two inferential problems, item nonresponse and response error.


Coordinate descent algorithms for lasso penalized regression

Abstract: Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known $\ell_2$ algorithm is based on cyclic coordinate descent. Our new $\ell_1$ algorithm is based on greedy coordinate descent and Edgeworth's algorithm for ordinary $\ell_1$ regression. Each algorithm relies on a tuning constant that can be chosen by cross-validation. In some regression problems it is natural to group parameters and penalize parameters group by group rather than separately. If the group penalty is proportional to the Euclidean norm of the parameters of the group, then it is possible to majorize the norm and reduce parameter estimation to $\ell_2$ regression with a lasso penalty. Thus, the existing algorithm can be extended to novel settings. Each of the algorithms discussed is tested via either simulated or real data or both. The Appendix proves that a greedy form of the $\ell_2$ algorithm converges to the minimum value of the objective function.


A statistical framework for testing functional categories in microarray data

Abstract: Ready access to emerging databases of gene annotation and functional pathways has shifted assessments of differential expression in DNA microarray studies from single genes to groups of genes with shared biological function. This paper takes a critical look at existing methods for assessing the differential expression of a group of genes (functional category), and provides some suggestions for improved performance. We begin by presenting a general framework, in which the set of genes in a functional category is compared to the complementary set of genes on the array. The framework includes tests for overrepresentation of a category within a list of significant genes, and methods that consider continuous measures of differential expression. Existing tests are divided into two classes. Class 1 tests assume gene-specific measures of differential expression are independent, despite overwhelming evidence of positive correlation. Analytic and simulated results are presented that demonstrate Class 1 tests are strongly anti-conservative in practice. Class 2 tests account for gene correlation, typically through array permutation that by construction has proper Type I error control for the induced null. However, both Class 1 and Class 2 tests use a null hypothesis that all genes have the same degree of differential expression. We introduce a more sensible and general (Class 3) null under which the profile of differential expression is the same within the category and complement. Under this broader null, Class 2 tests are shown to be conservative. We propose standard bootstrap methods for testing against the Class 3 null and demonstrate they provide valid Type I error control and more power than array permutation in simulated datasets and real microarray experiments.


Integral geometry under $G_2$ and $Spin(7)$

Abstract: A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.


Julia and John revisited

Abstract: We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet-Eckmann and Topological Collet-Eckmann maps and maps verifying a summability condition (as considered by Graczyk and Smirnov).


Accounting for self-protective responses in randomized response data from a social security survey using the zero-inflated Poisson model

Abstract: In 2004 the Dutch Department of Social Affairs conducted a survey to assess the extent of noncompliance with social security regulations. The survey was conducted among 870 recipients of social security benefits and included a series of sensitive questions about regulatory noncompliance. Due to the sensitive nature of the questions the randomized response design was used. Although randomized response protects the privacy of the respondent, it is unlikely that all respondents followed the design. In this paper we introduce a model that allows for respondents displaying self-protective response behavior by consistently giving the nonincriminating response, irrespective of the outcome of the randomizing device. The dependent variable denoting the total number of incriminating responses is assumed to be generated by the application of randomized response to a latent Poisson variable denoting the true number of rule violations. Since self-protective responses result in an excess of observed zeros in relation to the Poisson randomized response distribution, these are modeled as observed zero-inflation. The model includes predictors of the Poisson parameters, as well as predictors of the probability of self-protective response behavior.


More Discriminants with the Brezing-Weng Method

Abstract: The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.


Genus and braid index associated to sequences of renormalizable Lorenz maps

Abstract: We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant $(X,Y)*(S,W)^{*n}$, concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.


Lightlike hypersurfaces in indefinite $\mathcal{S}$-manifolds

Abstract: In a metric $g.f.f$-manifold we study lightlike hypersurfaces $M$ tangent to the characteristic vector fields, and owing to the presence of the $f$-structure, we determine some decompositions of $TM$ and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the existence of a $g.f.f$-structure on a lightlike hypersurface and, under suitable hypotheses, we obtain an indefinite $\mathcal{S}$-structure on the leaves of an integrable distribution. The existence of totally umbilical lightlike hypersurfaces of an indefinite $\mathcal{S}$-space form is also discussed. Finally, we explicitely describe a lightlike hypersurface of an indefinite $\mathcal{S}$-manifold.


Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type

Abstract: We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.


Transcription factor binding site prediction with multivariate gene expression data

Abstract: Multi-sample microarray experiments have become a standard experimental method for studying biological systems. A frequent goal in such studies is to unravel the regulatory relationships between genes. During the last few years, regression models have been proposed for the de novo discovery of cis-acting regulatory sequences using gene expression data. However, when applied to multi-sample experiments, existing regression based methods model each individual sample separately. To better capture the dynamic relationships in multi-sample microarray experiments, we propose a flexible method for the joint modeling of promoter sequence and multivariate expression data. In higher order eukaryotic genomes expression regulation usually involves combinatorial interaction between several transcription factors. Experiments have shown that spacing between transcription factor binding sites can significantly affect their strength in activating gene expression. We propose an adaptive model building procedure to capture such spacing dependent cis-acting regulatory modules. We apply our methods to the analysis of microarray time-course experiments in yeast and in Arabidopsis. These experiments exhibit very different dynamic temporal relationships. For both data sets, we have found all of the well-known cis-acting regulatory elements in the related context, as well as being able to predict novel elements.


Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories

Abstract: We compute the Grothendieck group of certain 2-Calabi--Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin--Zelevinsky's cluster algebras. In this setup, we also prove a generalization of Fomin--Zelevinsky's mutation rule.


Optimal factorial designs for cDNA microarray experiments

Abstract: We consider cDNA microarray experiments when the cell populations have a factorial structure, and investigate the problem of their optimal designing under a baseline parametrization where the objects of interest differ from those under the more common orthogonal parametrization. First, analytical results are given for the $2\times 2$ factorial. Since practical applications often involve a more complex factorial structure, we next explore general factorials and obtain a collection of optimal designs in the saturated, that is, most economic, case. This, in turn, is seen to yield an approach for finding optimal or efficient designs in the practically more important nearly saturated cases. Thereafter, the findings are extended to the more intricate situation where the underlying model incorporates dye-coloring effects, and the role of dye-swapping is critically examined.


The Reverse of The Law of Large Numbers

Abstract: The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists. In this paper, we investigate the opposite effect: keeping the sample size fixed while increasing the number of outcomes (M) available to a discrete random variable. We establish sufficient conditions for the variance of the sample mean to increase monotonically with the number of outcomes, such that the sample mean ``diverges'' from the population mean, acting like an ``reverse'' to the law of large numbers. These results, we believe, are relevant to many situations which require sampling of statistics of certain finite discrete random variables.


Assessing surrogate endpoints in vaccine trials with case-cohort sampling and the Cox model

Abstract: Assessing immune responses to study vaccines as surrogates of protection plays a central role in vaccine clinical trials. Motivated by three ongoing or pending HIV vaccine efficacy trials, we consider such surrogate endpoint assessment in a randomized placebo-controlled trial with case-cohort sampling of immune responses and a time to event endpoint. Based on the principal surrogate definition under the principal stratification framework proposed by Frangakis and Rubin [Biometrics 58 (2002) 21--29] and adapted by Gilbert and Hudgens (2006), we introduce estimands that measure the value of an immune response as a surrogate of protection in the context of the Cox proportional hazards model. The estimands are not identified because the immune response to vaccine is not measured in placebo recipients. We formulate the problem as a Cox model with missing covariates, and employ novel trial designs for predicting the missing immune responses and thereby identifying the estimands. The first design utilizes information from baseline predictors of the immune response, and bridges their relationship in the vaccine recipients to the placebo recipients. The second design provides a validation set for the unmeasured immune responses of uninfected placebo recipients by immunizing them with the study vaccine after trial closeout. A maximum estimated likelihood approach is proposed for estimation of the parameters. Simulated data examples are given to evaluate the proposed designs and study their properties.


Dress induction and the Burnside quotient Green ring

Abstract: We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups.


Hyperfocused arcs in PG(2,32)

Abstract: In PG(2,32) the following two results are proven by a computer aided search. (i) Uniqueness of hyperfocused 12-arcs, up to projectivities; (ii) Non-existence of hyperfocused 14-arcs. The existence problem for hyperfocused 16-arcs remains open.


Convex Hypersurfaces in Hadamard Manifolds

Abstract: We prove a theorem about an extremal property of Lobachevsky space among simply connected Riemannian manifolds of nonpositive curvature


A hidden spatial-temporal Markov random field model for network-based analysis of time course gene expression data

Abstract: Microarray time course (MTC) gene expression data are commonly collected to study the dynamic nature of biological processes. One important problem is to identify genes that show different expression profiles over time and pathways that are perturbed during a given biological process. While methods are available to identify the genes with differential expression levels over time, there is a lack of methods that can incorporate the pathway information in identifying the pathways being modified/activated during a biological process. In this paper we develop a hidden spatial-temporal Markov random field (hstMRF)-based method for identifying genes and subnetworks that are related to biological processes, where the dependency of the differential expression patterns of genes on the networks are modeled over time and over the network of pathways. Simulation studies indicated that the method is quite effective in identifying genes and modified subnetworks and has higher sensitivity than the commonly used procedures that do not use the pathway structure or time dependency information, with similar false discovery rates. Application to a microarray gene expression study of systemic inflammation in humans identified a core set of genes on the KEGG pathways that show clear differential expression patterns over time. In addition, the method confirmed that the TOLL-like signaling pathway plays an important role in immune response to endotoxins.


On the Global Structure of Hopf Hypersurfaces in Complex Space Form

Abstract: It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed C^(2n-1) regular Hopf hypersurface is a geodesic hypersphere


An approximation algorithm for counting contingency tables

Abstract: We present a randomized approximation algorithm for counting contingency tables, mxn non-negative integer matrices with given row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial N^{O(ln N)} complexity, where N=r_1+...+r_m=c_1+...+c_n. Various classes of margins are smooth, e.g., when m=O(n), n=O(m) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1+sqrt{5})/2 = 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for log-concave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables.


Dynamics of rational symplectic mappings and difference Galois theory

Abstract: In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.


A solution to a problem and the Diophantine equation X^2+bX+c=Y^2

Abstract: We prove that for given integers b and c, the diophantine equation x^2+bx+c=y^2, has finitely many integer solutions(i.e. pairs in ZxZ),in fact an even number of such solutions(including the zero or no solutions case).We also offer an explicit description of the solution set. Such a description depends on the form of the integer b^2-4c. Some Corollaries do follow. Furthermore, we show that the said equation has exactly two integer solutions, precisely when b^2-4c= 1,4,16,-4,or-16. In each case we list the two solutions in terms of the coefficients b and c.


A sharp stability estimate in tensor tomography

Abstract: We prove a sharp stability estimate for the problem of reconstructing a symmetric 2-tensor from its integrals along all maximal geodesics on a simple manifold.


Invariant Krein subspaces, regular irreducibility and integral representations

Abstract: We study unitary representations of groups in Krein spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Krein subspaces and their (reproducing) kernels and a variant of Choquet's theorem.


The KP hierarchy, branched covers, and triangulations

Abstract: The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plucker relations from geometry. We give a solution to the Plucker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the m-hypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond.


{Spaces of Infinite Measure and Pointwise Convergence of the Bilinear Hilbert and Ergodic Averages Defined by $L^{p}$-Isometries

Abstract: We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sigma-finite measure spaces and take the form of a.e. convergence of such discrete averages, as well as of their continuous variable counterparts, when these averages are defined by Lebesgue space isometries and act on $L^{p_{1}}\times L^{p_{2}}$ ($ 1<p_{1},p_{2}<\infty $, $p_{1}^{-1}+p_{2}^{-1}<3/2$). In the setting of an arbitrary measure space, this yields the a.e. convergence of these discrete bilinear averages when they act on $L^{p_{1}}\times L^{p_{2}}$ and are defined by an invertible measure-preserving point transformation.


On the Fundamental Properties of Linear Parameter-Varying Dynamic Systems Under Parametrical Multi-Perturbations. Applications to Time-Delay Systems. Preliminary Results

Abstract: This paper deals with a unifying approach to the problems of computing the admissible sets of parametrical multi perturbations in appropriate bounded sets such that some fundamental properties of parameter-varying linear dynamic systems are maintained provided that the so-called (i.e. perturbation-free) nominal system possesses such properties. The sets of parametrical multi perturbations include any combinations of parametrical multi perturbations in the matrix of dynamics as well as in the control, output and input-output interconnection matrices which belong to some prescribed bounded domain in the complex space. The various properties which are investigated are controllability, observability, output controllability and existence of minimal state-space realizations together with the associate existence or not of associate decoupling, transmission and invariant zeros. All the matrices of parameters including the nominal and the disturbed ones which parameterize the dynamic system may be real or complex. The radii of the multi- parametrical perturbations are calculated in a simple way. The obtained results are then applied to systems subject to a finite number of point internal delays and parametrical multi perturbations by comparing the state- space descriptions of such systems with the general descriptions previously investigated. In particular, the contributions of the delays to the spectral descriptions are assimilated to the contributions of a set of varying parameters in a domain for the general description.


How many distribution functions are there? Bracketing entropy bounds for high dimensional distribution functions

Abstract: This paper has been withdrawn by the authors due to a crucial error in a bound on page 19 and some other errors earlier in the paper.


On the Asymptotic Expansions for Time-Varying Scalar Differential Equations Possesssing Limiting Differential Equations by Application of the Residue Theorem to their Discretized Counterparts. Preliminary Results

Abstract: In this paper, it is proved that, in a dual context, asymptotic expansions of ordinary linear time-differential equations which possess limiting equations to their limiting equations might be obtained by first discretizing them and then using residues calculation for such discrete- time counterparts


Local approximation of the solutions of algebraic equations

Abstract: A method of local approximation of holomorphic solutions of algebraic equations is discussed


Fuzzy Statistical Limits

Abstract: Statistical limits are defined relaxing conditions on conventional convergence. The main idea of the statistical convergence of a sequence l is that the majority of elements from l converge and we do not care what is going on with other elements. At the same time, it is known that sequences that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they converge or statistically converge in the strict mathematical sense. To overcome these limitations, fuzzy convergence was introduced earlier in the context of neoclassical analysis and fuzzy statistical convergence is introduced and studied in this paper. We find relations between fuzzy statistical convergence of a sequence and fuzzy statistical convergence of its subsequences (Theorem 2.1), as well as between fuzzy statistical convergence of a sequence and conventional convergence of its subsequences (Theorem 2.2). It is demonstrated what operations with fuzzy statistical limits are induced by operations on sequences (Theorem 2.3) and how fuzzy statistical limits of different sequences influence one another (Theorem 2.4). In Section 3, relations between fuzzy statistical convergence and fuzzy convergence of statistical characteristics, such as the mean (average) and standard deviation, are studied (Theorems 3.1 and 3.2).


Description of the inelastic collision of two solitary waves for the BBM equation

Abstract: We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves).


The tropical $j$-invariant

Abstract: If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C to get a tropical curve Trop(C). If the Newton subdivision induced by f is a triangulation, then Trop(C) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of the j-invariant of C.


Dynamical systems method for solving linear finite-rank operator equations

Abstract: A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.


Super-Mathematics Functions

Abstract: In this paper we talk about the so-called SuperMathematics Functions (SMF), which often constiture the base for generating technical neo-geometrical objects.


Free algebras and the Freiheitssatz

Abstract: There have been gaps found in the proofs. The paper is withdrawn until further notice.


On semistable principal bundles over a complex projective manifold

Abstract: Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.


Ergodic Properties of a Class of Discrete Abelian Group Extensions of Rank-One Transformations

Abstract: We define a class of discrete abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.


The invertible double of elliptic operators

Abstract: First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderon projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderon projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderon projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last re- maining gaps between the geometric Dirac operator type situation and the general linear elliptic case.


Involutive Yang-Baxter Groups

Abstract: In 1992 Drinfeld posed the question of finding the set theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of $I$-type. A group $\mathcal{G}$ of $I$-type is a group isomorphic to a subgroup of the natural semidirect product of $Fa_n$, the free abelian group of rank $n$, by $Sym_n$, the symmetric group on $n$ letters, so that the projection onto $Fa_n$ is a bijective map. The projection of $\mathcal{G}$ onto $Sym_n$ we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld's problem for involutive non-degenerate set theoretic solutions. First classify the IYB groups and second, for a given IYB group $G$, classify the groups of $I$-type with $G$ as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of $I$-type (and hence infinitely many involutive non-degenerate set theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.


Letter to the editor. NAA and JFK: Can revisionism take us home?

Abstract: Occasionally during the course of the human learning experience we are faced with an anomaly. An aberration of sorts, which try as we might, defies appropriate classification. The recent paper by Spiegelman et al.--Chemical and forensic analysis of JFK assassination bullet lots: Is a second shooter possible?--is one such aberration. It is riddled with both misconceptions and errors of fact. Purporting to cast doubt on the NAA (neutron activation analysis) work conducted by Dr. Vincent Guinn in the investigation of the assassination of President John F. Kennedy, it fails miserably. The paper offers two central conclusions, one which is demonstrably false, and the other which is specious. The authors opine; ``If the assassination fragments are derived from three or more separate bullets, then a second assassin is likely, as the additional bullet would not be attributable to the main suspect, Mr. Oswald.'' This statement relating to the likelihood of a second assassin based on the premise of three or more separate bullets is demonstrably false. The available evidence indicates that Oswald fired three shots, one of which is believed to have missed. However, on the off chance that all three shots hit (even though there is absolutely no other supporting forensic evidence for such a notion) those three shots alone in no way would indicate then that ``a second assassin is likely.'' The authors' erroneous conclusion was achieved because they have either been misled (which I personally believe is the case) or they simply aren't familiar with the evidence.


Response to the Letter to the Editor

Abstract: This paper has attracted interest around the world from the media (both TV and newspapers). In addition, we have received letters, emails and telephone calls. One of our favorites was a voicemail message asking us to return a call to Australia at which point we would learn who really killed JFK. We welcome the opportunity to respond to the letter to the editor from Mr. Fiorentino. Mr. Fiorentino claims that our ``statement relating to the likelihood of a second assassin based on the premise of three or more separate bullets is demonstrably false.'' In response we would like to simply quote from page 327 of Gerald Posner's book Case Closed, one of the most well known works supporting the single assassin theory: ``If Connally was hit by another bullet, it had to be fired from a second shooter, since the Warren Commission's own reconstructions showed that Oswald could not have operated the bolt and refired in 1.4 seconds.'' Mr. Fiorentino also claims that the ``second fatal flaw is the use of a rather uncomplicated formula based on Bayes Theorem.'' Let $E$ denote the evidence and $T$ denote the theory that there were just two bullets (and hence a single shooter). We used Bayes Theorem to hypothetically calculate $P(T|E)$ from $P(E|T)$ and the prior probability $P(T)$. In order to make $P(T|E)$ ten times more likely than $P(\bar{T}|E)$, the ratio of the prior probabilities [i.e., $P(T) / P(\bar{T})$] would have to be greater than 15. Thus, we again conclude that this casts serious doubt on Dr. Guinn's conclusion that the evidence supported just two bullets. Sadly, this is far from the first time that probability has been misunderstood and/or misapplied in a case of public interest. A notable British example is the Clark case. See Nobles and Schiff (2005) for details. Finally, we welcome and, in fact, encourage members of the scientific community to provide alternative analyses of the data.


On the Berg--Chen--Ismail theorem and the Nevanlinna-Pick problem

Abstract: In 2002 C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinancy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna--Pick interpolation problem.


Statistical advances and challenges for analyzing correlated high dimensional SNP data in genomic study for complex diseases

Abstract: Recent advances of information technology in biomedical sciences and other applied areas have created numerous large diverse data sets with a high dimensional feature space, which provide us a tremendous amount of information and new opportunities for improving the quality of human life. Meanwhile, great challenges are also created driven by the continuous arrival of new data that requires researchers to convert these raw data into scientific knowledge in order to benefit from it. Association studies of complex diseases using SNP data have become more and more popular in biomedical research in recent years. In this paper, we present a review of recent statistical advances and challenges for analyzing correlated high dimensional SNP data in genomic association studies for complex diseases. The review includes both general feature reduction approaches for high dimensional correlated data and more specific approaches for SNPs data, which include unsupervised haplotype mapping, tag SNP selection, and supervised SNPs selection using statistical testing/scoring, statistical modeling and machine learning methods with an emphasis on how to identify interacting loci.


A Note on Ternary Sequences of Strings of 0 and 1

Abstract: B. D. Acharya has conjectured that if $\bigl(A_i: i=1, 2, ..., 2^{|X|}-1\bigr)$ is a permutation of all nonempty subsets of a set $X$ with at least two elements such that for each even positive integer $j<2^{|X|}-1$, $A_{j-1}\triangle A_j\triangle A_{j+1}=\emptyset$, then $|X|=2$. In this article, we show that if the cardinality of a set $X$ is more than four, then a permutation as described above indeed exists.


On Central Automorphisms Fixing the Center Element-wise

Abstract: Let $G$ be a finite $p$-group of nilpotency class 2. We find necessary and sufficient conditions on $G$ such that each central automorphism of $G$ fixes the center of $G$ element-wise.


Profinite homotopy theory

Abstract: We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the \'etale homotopy theory of schemes in which higher profinite \'etale homotopy groups fit well with the \'etale fundamental group which is always profinite. We show that the profinite \'etale topological realization functor is a good object in several respects.


Analysis of boosting algorithms using the smooth margin function

Abstract: We introduce a useful tool for analyzing boosting algorithms called the ``smooth margin function,'' a differentiable approximation of the usual margin for boosting algorithms. We present two boosting algorithms based on this smooth margin, ``coordinate ascent boosting'' and ``approximate coordinate ascent boosting,'' which are similar to Freund and Schapire's AdaBoost algorithm and Breiman's arc-gv algorithm. We give convergence rates to the maximum margin solution for both of our algorithms and for arc-gv. We then study AdaBoost's convergence properties using the smooth margin function. We precisely bound the margin attained by AdaBoost when the edges of the weak classifiers fall within a specified range. This shows that a previous bound proved by R\"{a}tsch and Warmuth is exactly tight. Furthermore, we use the smooth margin to capture explicit properties of AdaBoost in cases where cyclic behavior occurs.


K-stability of constant scalar curvature K\"ahler manifolds

Abstract: We show that a polarised manifold with a constant scalar curvature K\"ahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.


Brill-Noether theory of curves on Enriques surfaces I: the positive cone and gonality

Abstract: We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality.


Measuring and testing dependence by correlation of distances

Abstract: Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented.


Estimation of the covariance matrix of random effects in longitudinal studies

Abstract: Longitudinal studies are often conducted to explore the cohort and age effects in many scientific areas. The within cluster correlation structure plays a very important role in longitudinal data analysis. This is because not only can an estimator be improved by incorporating the within cluster correlation structure into the estimation procedure, but also the within cluster correlation structure can sometimes provide valuable insights in practical problems. For example, it can reveal the correlation strengths among the impacts of various factors. Motivated by data typified by a set from Bangladesh pertinent to the use of contraceptives, we propose a random effect varying-coefficient model, and an estimation procedure for the within cluster correlation structure of the proposed model. The estimation procedure is optimization-free and the proposed estimators enjoy asymptotic normality under mild conditions. Simulations suggest that the proposed estimation is practicable for finite samples and resistent against mild forms of model misspecification. Finally, we analyze the data mentioned above with the new random effect varying-coefficient model together with the proposed estimation procedure, which reveals some interesting sociological dynamics.


Classifying Hilbert functions of fat point subschemes in $\mathbb P^2$

Abstract: A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the projective plane. We study this problem for $r$ points in the plane over an algebraically closed field $k$ of arbitrary characteristic in case either $r \le 8$ or the points lie on a (possibly reducible) conic. In either case, it follows from work of the second author that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say $p_1,...,p_r$ and $p'_1,...,p'_r$ have the same {\it configuration type} if for all choices of nonnegative integers $m_i$, $Z=m_1p_1+...+m_rp_r$ and $Z'=m_1p'_1+...+m_rp'_r$ have the same Hilbert function.) Assuming either that $7 \le r\le 8$ (see recent work of Guardo and the second author for the cases $r\le 6$) or that the points $p_i$ lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients $m_i$ determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of $Z=m_1p_1+...+m_rp_r$. We demonstrate our results by explicitly listing all Hilbert functions for schemes of $r\le 8$ double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.


Examples of cohomology manifolds which are not homologically locally connected

Abstract: Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions $n \ge 3$.


Fr\'{e}chet-Urysohn fans in free topological groups

Abstract: In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fr\'echet-Urysohn fan $S_\w$ in a topological group $G$ admitting a functorial embedding $[0,1]\subset G$. The latter means that each autohomeomorphism of $[0,1]$ extends to a continuous homomorphism of $G$. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space $X$ containing a topological copy of the space $\IQ$ of rationals give topological groups containing $S_\w$.


Correction. Strong invariance principles for sequential Bahadur--Kiefer and Vervaat error processes of long-range dependent sequences

Abstract: Correction to The Annals of Statistics (2006) 34, 1013--1044 [URL: http://projecteuclid.org/euclid.aos/1151418250]


A diagrammatic approach to categorification of quantum groups I

Abstract: To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.


Preserving $Z$-sets by Dranishnikov's resolution

Abstract: We prove that Dranishnikov's $k$-dimensional resolution $d_k\colon \mu^k\to Q$ is a UV$^{n-1}$-divider of Chigogidze's $k$-dimensional resolution $c_k$. This fact implies that $d_k^{-1}$ preserves $Z$-sets. A further development of the concept of UV$^{n-1}$-dividers permits us to find sufficient conditions for $d_k^{-1}(A)$ to be homeomorphic to the N\"{o}beling space $\nu^k$ or the universal pseudoboundary $\sigma^k$. We also obtain some other applications.


Regularity of conjugacies of algebraic actions of Zariski dense groups

Abstract: Let \alpha_0 be an affine action of a discrete group \Gamma on a compact homogeneous space X and \alpha_1 a smooth action of \Gamma on X which is C^1-close to \alpha_0. We show that under some conditions, every topological conjugacy between \alpha_0 and \alpha_1 is smooth. In particular, our results apply to Zariski dense subgroups of SL_d(Z) acting on the torus T^d and Zariski dense subgroups of a simple noncompact Lie group G acting on a compact homogeneous space X of G with an invariant measure.


On extending actions of groups

Abstract: Problems of dense and closed extension of actions of compact transformation groups are solved. The method developed in the paper is applied to problems of extension of equivariant maps and of construction of equivariant compactifications.


Homological stability for certain classical groups

Abstract: We prove homological stability for standard unitary groups over R, C and H and for general linear groups over skew-fields with infinite centre. We focus on the similarities and differences of these proofs. Both proofs are due to Chih-Han Sah (Homology of classical Lie groups made discrete I: Stability theorems and Schur multipliers. Comment. Math. Helv. 61(2), 1986).


Surface subgroups from homology

Abstract: Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.


Principalization of ideals in abelian extensions of number fields

Abstract: We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$


The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices

Abstract: This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663--714] which finds the Tracy--Widom limit for the largest eigenvalue of a nonsingular $p$-dimensional complex Wishart matrix $W_{\mathbb{C}}(\Omega_p,n)$ to the case of several of the largest eigenvalues of the possibly singular $(n<p)$ matrix $W_{\mathbb{C}}(\Omega_p,n).$ As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643--1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.


The Calderon Projection: New Definition and Applications

Abstract: We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze various ways of writing A|U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well--posed selfadjoint Fredholm extensions under continuous variation of the data.


Strong approximation methods in group theory, an LMS/EPSRC Short course lecture notes

Abstract: These are the lecture notes for the LMS/EPSRC short course on strong approximation methods in linear groups organized by Dan Segal in Oxford in September 2007.


Relative differential K-characters

Abstract: We define a group of relative differential K-characters associated with a smooth map between two smooth compact manifolds. We show that this group fits into a short exact sequence as in the non-relative case. Some secondary geometric invariants are expressed in this theory.


Average growth of the spectral function on a Riemannian manifold

Abstract: We study average growth of the spectral function of the Laplacian on a Riemannian manifold. Two types of averaging are considered: with respect to the spectral parameter and with respect to a point on a manifold. We obtain as well related estimates of the growth of the pointwise zeta-function along vertical lines in the complex plane. Some examples and open problems regarding almost periodic properties of the spectral function are also discussed.


Parity patterns associated with lifts of Hecke groups

Abstract: Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the parity behaviour of the generalized subgroup numbers of $\Ga_m(q)$ which were defined in [T. W. M\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\Ga_m(q)$ into a given finite group $H$, in the case when $\gcd(m,| H|)=1$. This computation depends upon the solution of three counting problems in the Hecke group $\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_2$ and of $C_\infty$; (iii) determination of the parity of the number of index $n$ subgroups in $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. M\"uller, in: {\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. M\"uller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of Problems (ii) and (iii).


An approximation formula for holomorphic functions by interpolation on the ball

Abstract: We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is constructed from the knowledge of $f$ and its derivatives on these lines. We prove that this formula approximates any function when the number of lines increases. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.


Two-sided Grassmann-Rayleigh quotient iteration

Abstract: The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left-right eigenvectors of a matrix $C$. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of $p$-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left-right $p$-dimensional invariant subspaces of $C$. Moreover, Grassmannian versions of the Rayleigh quotient iteration are given for the generalized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian eigenproblem.


A normalization formula for the Jack polynomials in superspace and an identity on partitions

Abstract: We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.


When is sinx+cosx+tanx+cotx+secx+cscx an integer ?

Abstract: In this paper, we investigate the one-variable equation, sinx+cosx+tanx+cotx+secx+cscx=n, where n is an integer. We prove that if n lies between(inclusively) -1 and 6; then the above equation has no real number solutions. While if n is greater than or equal to 7; or less than or equal to -2, then the said equation has a nonempty solution set which we describe.


Morphisms between spaces of leaves viewed as fractions

Abstract: Reprint of a 1989 paper including minor corrections of misprints. Added comments (11 pages) about later related papers in the literature concerning comparison of Gabriel-Zisman calculus of (right) fractions and the use of generalized morphims in the sense of Haefliger-Skandalis-Hilsum for inverting differentiable equivalences between Lie groupoids.


Toroidalization of Locally Toroidal Morphisms from N-folds to Surfaces

Abstract: The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of \emph{locally toroidal} morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.


Local moduli of holomorphic bundles

Abstract: We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.


A New two-dimensional Second Order Non-oscillatory Central Scheme Applied to multiphase flows in heterogeneous porous media

Abstract: We compare the Kurganov-Tadmor (KT) two-dimensional second order semi-discrete central scheme in dimension by dimension formulation with a new two-dimensional approach introduced here and applied in numerical simulations for two-phase, two-dimensional flows in heterogeneous formations. This semi-discrete central scheme is based on the ideas of Rusanov's method using a more precise information about the local speeds of wave propagation computed at each Riemann Problem in two-space dimensions. We find the KT dimension by dimension has a much simpler mathematical description than the genuinely two-dimensional one with a little more numerical diffusion, particularly in the presence of viscous fingers. Unfortunately, as one can see, the KT with the dimension by dimension approach might produce incorrect boundary behavior in a typical geometry used in the study of porous media flows: the quarter of a five spot. This problem has been corrected by the authors with the new semi-discrete scheme proposed here. We conclude with numerical examples of two-dimensional, two-phase flow associated with two distinct flooding problems: a two-dimensional flow in a rectangular heterogeneous reservoir (called slab geometry) and a two-dimensional flow in a 5-spot geometry homogeneous reservoir.


Analytic Subordination for Free Compression

Abstract: We extend the free difference quotient coalgebra approach to analytic subordination to the case of a free compression in free probability.


Koszulity of algebras with non-pure resolutions

Abstract: We discuss certain homological properties of graded algebras whose trivial modules admit non-pure resolutions. Such algebras include both of Artin-Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module with non-pure resolution is decomposed to form an extension by two modules with pure resolutions.


Variational Principles on Triangulated Surfaces

Abstract: We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.


A Bijection Between Partially Directed Paths in the Symmetric Wedge and Matchings

Abstract: We give a bijection between partially directed paths in the symmetric wedge y= +/-x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of Prellberg et al. that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y= +/-x with k north steps is equal to the number of matchings on [2n] with k nestings.


Small Deviations of Smooth Stationary Gaussian Processes

Abstract: We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson's upper bound for small deviations and shed some light on the limits of sharpness for that estimate.


A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

Abstract: We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950's.


Some isoperimetric inequalities with application to the Stekloff problem

Abstract: In this paper we establish isoperimetric inequalities for the product of some moments of inertia. As an application, we obtain an isoperimetric inequality for the product of the $N$ first nonzero eigenvalues of the Stekloff problem in $\mathbb{R}^N$.


Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Abstract: Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{\omega_1}$ is homeomorphic to $I^{\omega_1}$. An analogous result is also proved for the cone over $I^{\omega_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.


Constructing near-embeddings of codimension one manifolds with countable dense singular sets

Abstract: The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}.


Spaces of idempotent measures of compact metric spaces

Abstract: We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.


Hereditary invertible linear surjections and splitting problems for selections

Abstract: Let $A+B$ be the pointwise (Minkowski) sum of two convex subsets $A$ and $B$ of a Banach space. Is it true that every continuous mapping $h:X \to A+B$ splits into a sum $h=f+g$ of continuous mappings $f:X \to A$ and $g:X \to B$? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.


Non-crossing Knight's Tour in 3-Dimension

Abstract: Non-crossing knight's tours in 3-dimension is a new field of research. The author has shown its possibility in small cuboids and in cubes up to 8x8x8 size. It can also be extended to larger size cubes and cuboids. The author has achieved jumps of length 15, 46, 88, 159, 258 and 395 in cubes of size 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7 and 8x8x8 respectively. This amounts to covering 59%, 73%, 71%, 74%, 76% and 77% cells in these cubes.


Appell polynomials and their relatives III. Conditionally free theory

Abstract: We extend to the multivariate non-commutative context the descriptions of a "once-stripped" probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map Phi on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bozejko, Leinert, and Speicher. This theory includes as two extreme cases the free and Boolean probability theories. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map Phi above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath-Segall polynomials, their combinatorics, and Hilbert space representations.


Free evolution on algebras with two states

Abstract: The key result in the paper concerns two transformations, Phi(rho, psi) and B_t(psi) on states on the algebra of non-commutative polynomials, or equivalently on joint distributions of d-tuples of non-commuting operators. These transformations are related to free probability: Phi intertwines the action of B_t and the free convolution with the semigroup {rho_t}. The maps {B_t} were introduced by Belinschi and Nica as a semigroup of transformations such that B_1 is the bijection between infinitely divisible distributions in the Boolean and free probability theories. They proved the intertwining property above for a single-variable version of the map Phi and the particular case of the free heat semigroup. The more general two-variable map Phi comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bozejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in arXiv:0803.4279, is closely related to the single-variable map Phi. On the other hand, more general free Meixner families behave well under all the transformations above, and provide clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Phi and descriptions of its fixed points and range.


C^1 actions of the mapping class group on the circle

Abstract: Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.


Homotopy type of the complement of an immersion and classification of embeddings of tori

Abstract: This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.


Overview of some general results in combinatorial enumeration

Abstract: This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicity of numbers of MSOL-definable structures.


Singularities of projected immersions revisited

Abstract: Sz\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic immersion is cobordant to the $\Sigma^{1_{r-1}}$-point manifold of its generic projection. Here we slightly extend this by showing that the natural mappings of these manifolds are bordant to each other. The main novelty of our approach is that we construct the bordism explicitly.


Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of the Urysohn space

Abstract: We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$ contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space $F$ there exists another finite metric space $\bar F$ and isometric imbedding $j$ of $F$ to $\bar F$ such that isometry $j$ induces the imbedding of the group monomorphism of the group of isometries of the space $F$ to the group of isometries of space $\bar F$ and each partial isometry of $F$ can be extended up to global isometry in $\bar F$. The fact that theorem $G$, is true was announced in 2005 by author without proof, and was proved by S.Solecki in \cite{Sol} (see also \cite{P,P1}) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from \cite{CV}.


Sums of Hecke eigenvalues over quadratic polynomials

Abstract: Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived.


Metabelian SL(n,C) representations of knot groups

Abstract: We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2,C) representations of knot groups. Finally we deduce the existence irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial.


A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)

Abstract: This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key role in conformal differential geometry in dimension 4 and is now known as the Paneitz operator.


Real and complex zeros of Riemannian random waves

Abstract: We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of $M$ tends to a limit current.


Products and Factors of Banach function spaces

Abstract: Given two Banach function spaces we study the pointwise product space E.F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E . M(E, F)=F, where M(E,F) denotes the space of multiplication operators from E into F.


Commutators of contactomorphisms

Abstract: The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways.


Locally constant functors

Abstract: We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. At last, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.


On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian

Abstract: Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a Riesz basis ({\it i.e.,} unconditionalbasis) for $L_2[-\pi,\pi]$. Given a function $f\in L_2(\mathbb R)$ whose Fourier transform is zero almost everywhere outside the interval $[-\pi,\pi]$, there is a unique square-summable sequence $(a_j:j\in\mathbb Z)$, depending on $\lambda$ and $f$, such that the function$$I_\lambda(f)(x):=\sum_{j\in\mathbb Z}a_je^{-\lambda(x-x_j)^2}, \qquad x\in\mathbb R, $$ is continuous and square integrable on $(-\infty,\infty)$, and satisfies the interpolatory conditions $I_\lambda (f)(x_j)=f(x_j)$, $j\in\mathbb Z$. It is shown that $I_\lambda(f)$ converges to $f$ in $L_2(\mathbb R)$, and also uniformly on $\mathbb R$, as $\lambda\to0^+$. A multidimensional version of this result is also obtained. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on $\ell_p(\mathbb Z)$ for every $p\in[1,\infty]$.


Birational automorphisms of nodal quartic threefolds

Abstract: It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete the proof of birational rigidity over a non-closed field and describe the situations when some of the birational involutions in question become regular (and, in particular, complete the proof of the initial theorem on birational rigidity, since some details were not established in the original paper of M.Mella).


The identity type weak factorisation system

Abstract: We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.


Spectral conditions on Lie and Jordan algebras of compact operators

Abstract: We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.


Abstract: In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D'Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss numerous connections between these spaces and various classical objects.


Transfer maps and nonexistence of joint determinant

Abstract: Transfer Maps, sometimes called norm maps, for Milnor's $K$-theory were first defined by Bass and Tate (1972) for simple extensions of fields via tame symbol and Weil's reciprocity law, but their functoriality had not been settled until Kato (1980). On the other hand, functorial transfer maps for the Goodwillie group are easily defined. We show that these natural transfer maps actually agree with the classical but difficult transfer maps by Bass and Tate. With this result, we build an isomorphism from the Goodwillie groups to Milnor's $K$-groups of fields, which in turn provides a description of joint determinants for the commuting invertible matrices. In particular, we explicitly determine certain joint determinants for the commuting invertible matrices over a finite field, the field of rational numbers, real numbers and complex numbers into the respective group of units of given field.


Regularity of monoids under Schutzenberger products

Abstract: In this paper we give a partial answer to the problem which is about the regularity of Schutzenberger products in semigroups asked by Gallagher in his thesis [Problem 6.1.6]{Gallagher} and, also, we investigate the regularity for the new version of the Schutzenberger product which was defined in [Ates].


The Kostant form of $\mathfrak{U}(sl_n^+)$ and the Borel subalgebra of the Schur algebra S(n,r)

Abstract: Let $A_n(K)$ be the Kostant form of $\mathfrak{U}(sl_n^+)$ and $\Gamma$ the monoid generated by the positive roots of $sl_n$. For each $\lambda\in \Lambda(n,r)$ we construct a functor $F_{\lambda}$ from the category of finitely generated $\Gamma$-graded $A_n(K)$-modules to the category of finite dimensional $S^+(n,r)$-modules, with the property that $F_{\lambda}$ maps (minimal) projective resolutions of the one-dimensional $A_n(K)$-module $K_{A}$ to (minimal) projective resolutions of the simple $S^+(n,r)$-module $K_{\lambda}$.


Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals

Abstract: Mayer's theory of cluster integrals allows one to write the partition function of a gas model as a generating function of weighted graphs. Recently, Labelle, Leroux and Ducharme have studied the graph weights arising from the one-dimensional hard-core gas model and noticed that the sum of the weights over all connected graphs with $n$ vertices is $(-n)^{n-1}$. This is, up to sign, the number of rooted Cayley trees on $n$ vertices and the authors asked for a combinatorial explanation. The main goal of this article is to provide such an explanation.


A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers

Abstract: In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated.


On the image of code polynomials under theta map

Abstract: The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for $g\leq 3$ and the surjectivity of the theta map follows. Instead it is known that this map is not surjective for $g\geq 5$. In this paper we discuss the possibility of an embedding between the associated projective varieties. We prove that this is not possible for $g\geq 4$ and consequently we get the non surjectivity of the graded rings for the remaining case $g=4$.


On Riemann sums and maximal functions in $\ZR^n$

Abstract: In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $\ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $f\in L^p$ with $p>1$.


Elementary proof of Rayleigh formula for graphs

Abstract: The Rayleigh monotonicity is a principle from the theory of electrical networks. Its combinatorial interpretation says for each two edges of a graph G, that the presence of one of them in a random spanning tree of G is negatively correlated with the presence of the other edge. In this paper we give a self-contained (inductive) proof of Rayleigh monotonicity for graphs.


Quelques calculs d'espaces $\R^i f_* G$ sur des courbes

Abstract: We give some properties (cancellation, representability, stratification) of the sheaf R^i f_* G for an affine relative curve f:U -> S admitting a smooth compactification and G a solvable group.


Rademacher averages on noncommutative symmetric spaces

Abstract: Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let $(\epsilon_k)_k$ be a Rademacher sequence, on some probability space $\Omega$. For finite sequences $(x_k)_k of E(M), we consider the Rademacher averages $\sum_k \epsilon_k\otimes x_k$ as elements of the noncommutative function space $E(L^\infty(\Omega)\otimes M)$ and study estimates for their norms $\Vert \sum_k \epsilon_k \otimes x_k\Vert_E$ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, the latter norm is equivalent to the infimum of $\Vert (\sum y_k^*y_k)^{{1/2}}\Vert + \Vert (\sum z_k z_k^*)^{{1/2}}\Vert$ over all $y_k,z_k$ in E(M) such that $x_k=y_k+z_k$ for any k. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. We also study Rademacher averages for doubly indexed families of E(M).


Completely 1-complemented subspaces of Schatten spaces

Abstract: We consider the Schatten spaces S^p in the framework of operator space theory and for any $1\leq p\not=2<\infty$, we characterize the completely 1-complemented subspaces of S^p. They turn out to be the direct sums of spaces of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a description of all 1-complemented subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L^p-spaces. Also we show that for any $n\geq 2$, there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is not completely isometric, and we investigate L^p-versions of such isomorphisms.


Dilations and rigid factorisations on noncommutative L^p-spaces

Abstract: We study some factorisation and dilation properties of completely positive maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for positive contractions on classical (=commutative) L^p-spaces has no reasonable analog in the noncommutative setting. Our study relies on non symmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper.


Separable states and positive maps II

Abstract: Using the natural duality between linear functionals on tensor products of C*-algebras with the trace class operators on a Hilbert space H and linear maps of the C*-algebra into B(H), we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF C*-algebras


On the number of graphs not containing $K_{3,3}$ as a minor

Abstract: We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{3,3}$-minor-free graphs, like the expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{3,3}$ plus an edge as a minor.


Quantum Isometry Group of the n tori

Abstract: We show that the Quantum Isometry Group(as introduced in \cite{goswami}) of the n tori is the classical isometry group. Moreover, using a result in \cite{bhowmick goswami}, we conclude that the Quantum Isometry group of the noncommutative n tori is a Rieffel deformation of the Quantum Isometry Group of the commutative n tori.


Yang-Mills equation for stable Higgs sheaves

Abstract: We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.