Simultaneous packing and covering in the two-dimensional Euclidean plane II

Abstract: This paper determines the optimal upper bound for the simultaneous packing and covering constants of the two-dimensional centrally symmetric convex domains. It solved a problem opening for more than thirty years.


A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

Abstract: A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method.


Characters of representations of affine Kac-Moody Lie algebras at the critical level

Abstract: We present an explicit character formula for the irreducible highest weight representations of the non-twisted affine Kac-Moody Lie algebra at the critical level which are integrable over the corresponding finite-dimensional simple Lie algebra.


An elementary proof of global existence for nonlinear wave equations in an exterior domain

Abstract: The aim of this article is to present an elementary proof of a global existence result for nonlinear wave equations satifying the null condition in an exterior domain. The novelty of our proof is to avoid completely the scaling operator which would make the argument complicated in the mixed problem, by using a new weighted pointwise estimates of a tangential derivative to the light cone.


General concepts of graphs

Abstract: A little general abstract combinatorial nonsense delivered in this note is a presentation of some old and basic concepts, central to discrete mathematics, in terms of new words. The treatment is from a structural and systematic point of view. This note consists essentially of definitions and summaries.


Subgeometric ergodicity of Markov chains

Abstract: The goal of this paper is to give a short and self contained proof of general bounds for subgeometric rates of convergence, under practical conditions. The main result whose proof, based on coupling, provides an intuitive understanding of the results of Nummelin and Tuominen (1983) and Tuominen and Tweedie (1994). To obtain practical rates, a very general drift condition, recently introduced in Douc et al (2004) is used.


On the K\"ahler classes of constant scalar curvature metrics on blow ups

Abstract: In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and about the position of the blown up points are given.


Extreme-Value Analysis of Standardized Gaussian Increments

Abstract: Let $\{X_i,i=1,2,...\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately normalized, converges as $n\to\infty$ to the Gumbel distribution. In some sense, the the random variable $L_n$, being the maximum of $n(n+1)/2$ dependent standard gaussian variables, behaves like the maximum of $Hn \log n$ independent standard gaussian variables. Here, $H\in (0,\infty)$ is some constant. We also prove a version of the above result for the Brownian motion.


Optimal H2 order-one reduction by solving eigenproblems for polynomial equations

Abstract: A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included.


Contributions of Issai Schur to Analysis

Abstract: The name Schur is associated with many terms and concepts that are widely used in a number of diverse fields of mathematics and engineering. This survey article focuses on Schur's work in analysis. Here too, Schur's name is commonplace: The Schur test and Schur-Hadamard multipliers (in the study of estimates for Hermitian forms), Schur convexity, Schur complements, Schur's results in summation theory for sequences (in particular, the fundamental Kojima-Schur theorem), the Schur-Cohn test, the Schur algorithm, Schur parameters and the Schur interpolation problem for functions that are holomorphic and bounded by one in the unit disk. In this survey, we discuss all of the above mentioned topics and then some, as well as some of the generalizations that they inspired. There are nine sections of text, each of which is devoted to a separate theme based on Schur's work. Each of these sections has an independent bibliography. There is very little overlap. A tenth section presents a list of the papers of Schur that focus on topics that are commonly considered to be analysis. We begin with a review of Schur's less familiar papers on the theory of commuting differential operators.


Quelques math\'ematiques de la cryptographie \`a cl\'es publiques

Abstract: I present examples of mathematical objects that are of interest for public key cryptography. Text for the Journ\'ee Annuelle 2007 of the SMF.


A mathematical model for a copolymer in an emulsion

Abstract: In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with equal density. The emulsion consists of large blocks of oil and water, arranged in a percolation-type fashion. To make the model mathematically tractable, the copolymer is allowed to enter and exit a neighboring pair of blocks only at diagonally opposite corners. The energy of the copolymer in the emulsion is minus $\alpha$ times the number of hydrophobic monomers in oil minus $\beta$ times the number of hydrophilic monomers in water. Without loss of generality we may assume that the interaction parameters are restricted to the cone $\{(\alpha,\beta)\in \mathbb{R}^2\colon |\beta|\leq\alpha\}$. We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks percolate, there is a single critical curve in the cone separating a localized and a delocalized phase; (2) in the subcritical regime where the oil blocks do not percolate, there are three critical curves in the cone separating two localized phases and two delocalized phases, and meeting at two tricritical points. The different phases are characterized by different behavior of the copolymer inside the four neighboring pairs of blocks.


Homotopy classes of total foliations and bi-contact structures on three-manifolds

Abstract: On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures.


On Transformation of Potapov's Fundamental Matrix Inequality

Abstract: According to V.P.Potapov, a classical interpolation problem can be reformulated in terms of a so-called Fundamental Matrix Inequality (FMI). To show that every solution of the FMI satisfies the interpolation problem, we usualy have to transform the FMI in some special way. In this paper the number of of transformations of the FMI which come into play are motivated and demonstrated by simple, but typical examples.


An Abstract Interpolation Problem and the Extension Theory of Hermitian Operators

Abstract: The algebraic structure of V.P. Potapov's Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described using the Arov--Grossman formula. The extension theory of isometric operators is the proper language for treating interpolation problems of this type.


On the Theory of Matrix Valued Functions Belonging to the Smirnov Class

Abstract: A theory of matrix-valued functions from the matricial Smirnov class ${\goth N}_n^+({\Bbb D})$ is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class ${\goth N}_n^+({\Bbb D})$. We also consider a family $F_{\lambda} =F-\lambda I$ of functions belonging to the matricial Smirnov class which is indexed by a complex parameter $\lambda$. We show that with the exception of a ''very small'' set of such $\lambda$ the corresponding inner factor in the inner-outer factorization of the function $F_{\lambda}$ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class.


Survival Probabilities for N-ary Subtrees on a Galton-Watson Family Tree

Abstract: The family tree of a Galton-Watson branching process may contain N-ary subtrees, i.e. subtrees whose vertices have at least N>0 children. For family trees without infinite N-ary subtrees, we study how fast N-ary subtrees of height t disappear as t goes to infinity.


Nagata's embedding theorem

Abstract: In 1962-63, M. Nagata showed that an abstract variety could be embedded into a complete variety. Later, P. Deligne translated Nagata's proof into the language of schemes, but did not publish his notes. This paper, which is to appear as an appendix in a forthcoming book, gives an elaboration of Deligne's notes. It also contains some complementary results on extending divisors and vector sheaves to suitable completions.


How to disentangle two braided Hopf algebras

Abstract: We show how to define the tensor product of two braided Hopf algebras.


Affine Buildings and Tropical Convexity

Abstract: The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SL$_d(K)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.


On a problem of Duke-Erdos-Rodl on cycle-connected subgraphs

Abstract: In this short note, we prove that for \beta < 1/5 every graph G with n vertices and n^{2-\beta} edges contains a subgraph G' with at least cn^{2-2\beta} edges such that every pair of edges in G' lie together on a cycle of length at most 8. Moreover edges in G' which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdos, and Rodl.


The Heegaard structure of Dehn filled manifolds

Abstract: We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and (4) two or more non-isotopic Heegaard surfaces become isotopic. We survey general results that give quite satisfactory restrictions to phenomena (1) and (2) and, in a parallel thread, give a complete classification of when all four phenomena occur when filling most torus knot exteriors. This latter thread yields sufficient (and perhaps necessary) conditions for the occurrence of phenomena (3) and (4).


Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics

Abstract: Functional limit theorem for continuous-time random walks (CTRW) are found in general case of dependent waiting times and jump sizes that are also position dependent. The limiting anomalous diffusion is described in terms of fractional dynamics. Probabilistic interpretation of generalized fractional evolution is given in terms of the random time change (subordination) by means of hitting times processes.


Proof of Riemann Hypothesis

Abstract: In this article, we will prove Riemann Hypothesis by using the mean value theorem of integrals. The function $ \xi(s) $ is introduced by Riemann, which zeros are identical equal to non-trivial zeros of zeta function.The function $ \xi(s) $ is an entire function, and its real part and imaginary part can be represented as infinite integral form. In the special condition, the mean value theorem of integrals is established for infinite integral. Using the mean value theorem of integrals and the isolation of zeros of analytic function, we determined that all zeros of the function $ \xi(s) $ have real part equal to$\frac{1}{2}$, namely, all non-trivial zeros of zeta function lies on the critical line. Riemann Hypothesis is true.


The characterization of the Carleson measures for analytic Besov spaces: a simple proof

Abstract: We give a simple proof of the characterization of the Carleson measures for the weighted analytic Besov spaces. Such characterization provides some information on the radial variation of an analytic Besov function.


On Jordan Derivations of Triangular Algebras

Abstract: In this short note we prove that every Jordan derivation of triangular algebras is a derivation.


The wreath product of Z with Z has Hilbert compression exponent 2/3

Abstract: Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all $\alpha\ge 0$ such that there exists a Lipschitz mapping $f:G\to L_2$ and a constant $c>0$ such that for all $x,y\in G$ we have $\|f(x)-f(y)\|_2\ge cd(x,y)^\alpha.$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\Z\bwr \Z $ is at most $\frac34$, and in \cite{NP07} was proved that this exponent is at least $\frac23$. Here we show that $\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.


Canonical representatives for residue classes of a polynomial ideal and orthogonality

Abstract: The aim of this paper is to unveil an unexpected relationship between the normal form of a polynomial with respect to a polynomial ideal and the more geometric concept of orthogonality. We present a new way to calculate the normal form of a polynomial with respect to a polynomial ideal I in the ring of multivariate polynomials over a field K, provided the field K is finite and the ideal I is a vanishing ideal. In order to use the concept of orthogonality, we introduce a symmetric bilinear form on a vector space over a finite field.


Generators of II_1 Factors

Abstract: In 2005, Shen introduced a new invariant, $\mathcal G(N)$, of a diffuse von Neumann algebra $N$ with a fixed faithful trace, and he used this invariant to give a unified approach to showing that large classes of ${\mathrm{II}}_1$ factors $M$ are singly generated. This paper focuses on properties of this invariant. We relate $\mathcal G(M)$ to the number of self-adjoint generators of a ${\mathrm{II}}_1$ factor $M$: if $\mathcal G(M)<n/2$, then $M$ is generated by $n+1$ self-adjoint operators, whereas if $M$ is generated by $n+1$ self-adjoint operators, then $\mathcal G(M)\leq n/2$. The invariant $\mathcal G(\cdot)$ is well-behaved under amplification, satisfying $\mathcal G(M_t)=t^{-2}\mathcal G(M)$ for all $t>0$. In particular, if $\mathcal G(\mathcal L\mathbb F_r)>0$ for any particular $r>1$, then the free group factors are pairwise non-isomorphic and are not singly generated for sufficiently large values of $r$. Estimates are given for forming free products and passing to finite index subfactors and the basic construction. We also examine a version of the invariant $\mathcal G_{\text{sa}}(M)$ defined only using self-adjoint operators; this is proved to satisfy $\mathcal G_{\text{sa}}(M)=2\mathcal G(M)$. Finally we give inequalities relating a quantity involved in the calculation of $\mathcal G(M)$ to the free-entropy dimension $\delta_0$ of a collection of generators for $M$.


SPM Bulletin 21

Abstract: Contents: 1. Editor's note; 2. Personal impressions from the SPM07 meeting; 3. Research announcements; 3.1. Coloring ordinals by reals; 3.2. Long Borel Hierarchies; 3.3. Rothberger's property in finite powers; 3.4. Special subsets of the reals and tree forcing notions; 3.5. All automorphisms of the Calkin algebra are inner; 3.6. Continuous selections and sigma-spaces; 3.7. On the closure of the diagonal of a T1-space; 3.8. Splitting families and Noetherian type; 3.9. Even more simple cardinal invariants; 3.10. A classification of CO spaces which are continuous images of compact ordered spaces; 4. Problem of the Issue


Spherical Continuous Wavelet Transforms arising from sections of the Lorentz group

Abstract: We consider the conformal group of the unit sphere $S^{n-1},$ the so-called proper Lorentz group Spin$^+(1,n),$ for the study of spherical continuous wavelet transforms (CWT). Our approach is based on the method for construction of general coherent states associated to square integrable group representations over homogeneous spaces. The underlying homogeneous space is an extension to the whole of the group Spin$^+(1,n)$ of the factorization of the gyrogroup of the unit ball by an appropriate gyro-subgroup. Sections on this homogeneous space are constituted by rotations of the subgroup Spin$(n)$ and M\"{o}bius transformations of the type $\phi_a(x)=(x-a)(1+ax)^{-1},$ where $a$ belongs to a given section on a homogeneous space of the unit ball. This extends in a natural way the work of Antoine and Vandergheynst to anisotropic conformal dilations.


Abstract: In this note we study Fano threefolds with noncyclic torsion in the divisor class group. Since they can all be obtained as quotients of Fano threefolds, we get also all examples that can be obtained as quotients of low codimension Fanos in the weighted projective space.


Kleinian groups with ubiquitous surface subgroups

Abstract: We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.


The Riemann hypothesis - an elementary analytic approach based on complex Laplace transform

Abstract: An elementary analytic proof of the famous Riemann hypothesis is given. The main "accent" of the proof is a both using of the 2-dimensional double real and complex Laplace integral representations of the Green function $\mid z \mid^{-2}$.


Formation of singularities for a transport equation with nonlocal velocity

Abstract: We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.


Skew-Hadamard matrices of orders 436, 580 and 988 exist

Abstract: We construct two difference families on each of the cyclic groups of order 109, 145 and 247, and use them to construct skew-Hadamard matrices of orders 436, 580 and 988. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array.


On pathwise uniqueness for reflecting Brownian motion in $C^{1+\gamma}$ domains

Abstract: Pathwise uniqueness holds for the Skorokhod stochastic differential equation in $C^{1+\gamma}$ domains in $\mathbb{R}^d$ for $\gamma >1/2$ and $d\geq3$.


A transient Markov chain with finitely many cutpoints

Abstract: We give an example of a transient reversible Markov chain that almost surely has only a finite number of cutpoints. We explain how this is relevant to a conjecture of Diaconis and Freedman and a question of Kaimanovich. We also answer Kaimanovich's question when the Markov chain is a nearest-neighbor random walk on a tree.


Twisted Alexander polynomial of links in the projective space

Abstract: We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.


Quantum Teichm\"uller spaces and Kashaev's 6j-symbols

Abstract: The Kashaev invariants of 3-manifolds are based on $6j$-symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of $U_q(sl(2,\C))$. In this paper, we show that Kashaev's $6j$-symbols are intertwining operators of local representations of quantum Teichm\"uller spaces. This relates Kashaev's work with the theory of quantum Teichm\"uller space, which was developed by Chekhov-Fock, Kashaev and continued by Bonahon-Liu.


Eigenvalues and lambda constants on Riemannian submersions

Abstract: Given a Riemannian submersion, we study the relation between lambda constants introduced by G.Perelman on the base manifold and the total space of a Riemannian submersion. We also discuss the relationship between the first eigenvalues of Laplacians on the base manifold and that of the total space. The quantities on warped products are discussed in detail.


Heegaard Floer homology and fibred 3--manifolds

Abstract: Given a closed 3--manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author.


Properties of positive solutions of an Elliptic Equation with negative exponents

Abstract: In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local gradient estimate and L1 lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.


Lines of minima are uniformly quasi-geodesic

Abstract: We continue the comparison between lines of minima and Teichmueller geodesics begun in [CRS1]. We show that in the Teichmueller space of a surface S, lines of minima are quasi-geodesic with respect to the Teichmueller metric. The quasi-geodesic constants depend only on the topological type of S.


Geometry of curves with exceptional secant planes: linear series along the general curve

Abstract: We study linear series on a general curve of genus $g$, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of $d$-secant $(d-2)$-planes to $(2d-1)$-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in $d$.


Collapse of unit horizontal bundles equipped with a metric of Cheeger-Gromoll type

Abstract: We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse theorem for unit horizontal bundles associated with a sequence of warped Riemannian submersions.


Marcus-Lushnikov processes, Smoluchowski's and Flory's models

Abstract: The Marcus-Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses $x$ and $y$ merges into a single particle at a given rate $K(x,y)$. We consider a {\it strongly gelling} kernel behaving as $K(x,y)=x^\alpha y + x y^\alpha$ for some $\alpha\in (0,1]$. In such a case, it is well-known that {\it gelation} occurs, that is, giant particles emerge. Then two possible models for hydrodynamic limits of the Marcus-Lushnikov process arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with finite ones. We show that, when using a suitable cut-off coagulation kernel in the Marcus-Lushnikov process and letting the number of particles increase to infinity, the possible limits solve either the Smoluchowski equation or the Flory equation. We also study the asymptotic behaviour of the largest particle in the Marcus-Lushnikov process without cut-off and show that there is only one giant particle. This single giant particle represents, asymptotically, the lost mass of the solution to the Flory equation.


Weak collapsing and geometrisation of aspherical 3-manifolds

Abstract: Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman's proof of the Geometrisation Conjecture for aspherical 3-manifolds.


Preserving zeros of a polynomial

Abstract: We study non-linear surjective mappings on subsets of ${\cal M}_n(F)$, which preserve the zeros of some fixed polynomials in noncommuting variables. Keywords: Matrix algebra, Multilinear polynomials, Preservers.


Homogenization of nonlinear scalar conservation laws

Abstract: We study the limit as $\e\to 0$ of the entropy solutions of the equation $\p_t \ue + \dv_x[A(\frac{x}{\e},\ue)] =0$. We prove that the sequence $\ue$ two-scale converges towards a function $u(t,x,y)$, and $u$ is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in $L^1_{\text{loc}}$.


The size of the largest component below phase transition in inhomogeneous random graphs

Abstract: We study the "rank 1 case" of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result is new, it completes the corresponding known result in the supercritical case. We provide some examples of application of a new formula.


Constrained Ramsey Numbers

Abstract: For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.


On the strong chromatic number of random graphs

Abstract: Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior of this parameter for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove that the strong chromatic number is a.s. concentrated on one value \Delta+1, where \Delta is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.


Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

Abstract: Let $\F_q$ ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in $\F_q[x]$ with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in $\F_{q^3}$ with prescribed trace and norm over $\F_q$ improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over $\F_{2^r}$ divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case $r$ odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field $\F_{3^r}$, first proved by Katz and Livne, is given.


A family of acyclic functors

Abstract: We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset is used. Also the dual version for injective functors and higher inverse limits is included. Graded posets include simplicial complexes, subdivision categories and simplex-like posets.


Linear precision for parametric patches

Abstract: We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas' toric patches, which we show is equivalent to a certain rational map on CP^d being a birational isomorphism. Lastly, we establish the connection between linear presision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches.


A method for integral cohomology of posets

Abstract: We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb's conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups.


Diagonalization and representation results for nonpositive sesquilinear form measures

Abstract: We study decompositions of operator measures and more general sesquilinear form measures $E$ into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent $E$ as a trace class valued measure of bounded variation on a new Hilbert space related to $E$. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain properties, but this choice itself is not canonical. We present relations to Naimark type dilations and direct integrals.


Independent transversals in locally sparse graphs

Abstract: Let G be a graph with maximum degree \Delta whose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta), then G has an independent transversal for sufficiently large \Delta. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.


The role of the cotangent bundle in resolving ideals of fat points in the plane

Abstract: We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them.


On elliptic Dunkl operators

Abstract: We attach elliptic Dunkl operators to an abelian variety with a finite group action. This generalizes elliptic Dunkl operators for Weyl groups, defined by Buchstaber, Felder, and Veselov in 1994. We show that these operators commute, and use them to define representations from category O of elliptic Cherednik algebras. We also consider the monodromy representations of differential equations defined by elliptic Dunkl operators, and show that they yield finite dimensional rrepresentations of generalized double affine Hecke algebras.


Vector invariants of a class of pseudo-reflection groups and multisymmetric syzygies

Abstract: First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field. Special case of the result is a finite presentation of the algebra of multisymmetric polynomials. Reducedness of the invariant commuting scheme is proved as a by-product. The algebra of multisymmetric polynomials over an arbitrary base ring is revisited.


Minimality in topological groups and Heisenberg type groups

Abstract: We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over "good" rings. By "good" rings we mean archimedean absolute valued (not necessarily associative) division rings. Some of the classical rings which we consider besides the field of reals are the ring of quaternions and the ring of octonions. This way we generalize in part a previous result which was obtained by Dikranjan and Megrelishvili and involved the Heisenberg group.


Canonical heights and the arithmetic complexity of morphisms on projective space

Abstract: Let F and G be morphisms of degree at least 2 from P^N to P^N that are defined over the algebraic closure of Q. We define the arithmetic distance d(F,G) between F and G to be the supremum over all algebraic points P of |h_F(P)-h_G(P)|, where h_F and h_G are the canonical heights associated to the morphisms F and G, respectively. We prove comparison theorems relating d(F,G) to more naive height functions and show that for a fixed G, the set of F satisfying d(F,G) < B is a set of bounded height. In particular, there are only finitely many such F defined over any given number field.


Nonarchimedean Green functions and dynamics on projective space

Abstract: Let F: P^N_K --> P^N_K be a morphism of degree d > 1 defined over a field K that is algebraically closed and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function G_F associated to F is Holder continuous on P^N(K) and that the Fatou set F is equal to the set of points at which G_F is locally constant. Further, G_F vanishes precisely on the set of points P such that F has good reduction at every point in the forward orbit of P. We also prove that the iterates of F are locally uniformly Lipschitz on the Fatou set of F.


A resolution of the K(2)-local sphere at the prime 3

Abstract: We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^hF where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic.


Mapping Class Groups do not have Kazhdan's Property (T)

Abstract: We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan's property (T).


Operads of compatible structures and weighted partitions

Abstract: In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and pre-Lie algebras.


Gorenstein Multiple Structures on Smooth Algebraic Varieties

Abstract: We characterize the Gorenstein nilpotent scheme structures on a smooth algebraic variety as support, in terms of a duality property of the graded objects associated to two canonical filtrations.


Non-abelian Reidemeister torsion for twist knots

Abstract: This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.


Abstract: The analog of the Schauder inequality for closed surfaces in Euclidean spaces is obtained in this article.


On the derivative of the Minkowski question mark function $?(x)$

Abstract: Let $ x = [0;a_1,a_2,...]$ be the decomposition of the irrational number $x \in [0,1]$ into regular continued fraction. Then for the derivative of the Minkowski function $?(x)$ we prove that $?'(x) = +\infty$ provided $ \limsup_{t\to \infty}\frac{a_1+...+a_t}{t} <\kappa_1 =\frac{2\log \lambda_1}{\log 2} = 1.388^+$, and $?'(x) = 0$ provided $ \liminf_{t\to \infty}\frac{a_1+...+a_t}{t} >\kappa_2 = \frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$ (here $ L_j = \log (\frac{j+\sqrt{j^2+4}}{2}) - j\cdot\frac{\log 2}{2}$). Constants $\kappa_1,\kappa_2$ are the best possible. Also we prove that $?'(x) = +\infty$ holds for all $x$ with partial quotients bounded by 4.


Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs

Abstract: We address the following question: When a randomly chosen regular bipartite multi--graph is drawn in the plane in the ``standard way'', what is the distribution of its maximum size planar matching (set of non--crossing disjoint edges) and maximum size planar subgraph (set of non--crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of r-regular bipartite multi--graphs with maximum planar matching (maximum planar subgraph) of at most d edges to a signed sum of restricted lattice walks in Z^d, and to the number of pairs of standard Young tableaux of the same shape and with a ``descend--type'' property. Our results are obtained via generalizations of two combinatorial proofs through which Gessel's identity can be obtained (an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of LISs, and key to the analytic attack on Ulam's problem). We also initiate the study of pattern avoidance in bipartite multigraphs and derive a generalized Gessel identity for the number of bipartite 2-regular multigraphs avoiding a specific (monotone) pattern.


Existence of Kirillov-Reshetikhin crystals for nonexceptional types

Abstract: Using the methods of Kang et al. and recent results on the characters of Kirillov-Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov-Reshetikhin crystals B^{r,s} is established for all nonexceptional affine types. We also prove that the crystals B^{r,s} of type B_n^{(1)}, D_n^{(1)}, and A_{2n-1}^{(2)} are isomorphic to recently constructed combinatorial crystals for r not a spin node.


Periodic harmonic functions on lattices and points count in positive characteristic

Abstract: This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety.


AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups

Abstract: Let $X$ be a compact metric space and let $\Lambda$ be a $\Z^k$ ($k\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\rtimes_{\Lambda}\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $\Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $\Lambda: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_{\Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $\Lambda$-invariant tracial state.


Characterization of intrinsically harmonic forms

Abstract: Let $M$ be a closed oriented manifold of dimension $n$ and $\omega$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $\omega$ is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answer in the general case, i.e. not making any assumptions on the zero set of $\omega$.


Infinitesimal Castelnuovo Theory in Abelian Varieties

Abstract: The purpose of this article is to show that the Castelnuovo theory for abelian varieties, developed by G. Pareschi and M. Popa, can be infinitesimalized. More precisely, we prove that an irreducible principally polarized abelian variety has a finite scheme in extremal position, in the sense of Castelnuovo theory for abelian varieties, if, and only if, it is a Jacobian and the scheme is contained in a unique Abel-Jacobi curve.


A refinement of Khovanov-Rozansky link homology

Abstract: This paper has been withdrawn by the author due to an error in the proof of Theorem 1.


Sign lemma for dimension shifting

Abstract: There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the lemma by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.


Algebraic K-theory and cubical descent

Abstract: In this note we apply Guillen-Navarro descent theorem, \cite{GN02}, to define a descent variant of the algebraic $K$-theory of varieties over a field of characteristic zero, $\mathcal{KD}(X)$, which coincides with $\mathcal{K}(X)$ for smooth varieties. After a result of Haesemeyer, this new theory is equivalent to the homotopy algebraic $K$-theory introduced by Weibel. We also prove that there is a natural weight filtration on the groups $KH_\ast(X)$.


Uniqueness of $\bf C^*$- and $\bf C_+$-actions on Gizatullin surfaces

Abstract: A Gizatullin surface is a normal affine surface $V$ over $\bf C$, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $\bf C^*$-actions and $\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\bf C_+$-actions on $V$ considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one $\bf A^1$-fibration $V\to S$ up to an isomorphism of the base $S$. Moreover an effective $\bf C^{*}$-action on them, if it does exist, is unique up to conjugation and inversion $t\mapsto t^{-1}$ of $\bf C^*$. Obviously uniqueness of $\bf C^*$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of $\bf A^1$-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $\bf C^*$-actions and $\bf A^1$-fibrations. In the present paper we obtain a criterion as to when $\bf A^1$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of $V$ and the base $S$. We exhibit as well a large subclasses of Gizatullin $\bf C^{*}$-surfaces for which a $\bf C^*$-action is essentially unique and for which there are at most two conjugacy classes of $\bf A^1$-fibrations over $\bf A^1$.


Expanders and the Affine Building of ${\rm Sp}_n$

Abstract: For $n \geq 2$ and a local field $K$, let $\Delta_n$ denote the affine building naturally associated to the symplectic group ${\rm Sp}_n(K)$. We compute the spectral radius of the subgraph $Y_n$ of $\Delta_n$ induced by the special vertices in $\Delta_n$, from which it follows that $Y_n$ is an analogue of a family of expanders and is non-amenable.


Partition Identities and the Coin Exchange Problem

Abstract: The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.


Existence theorem and blow-up criterion of the strong solutions to the Magneto-micropolar fluid equations

Abstract: In this paper we study the magneto-micropolar fluid equations in $\R^3$, prove the existence of the strong solution with initial data in $H^s(\R^3)$ for $s> {3/2}$, and set up its blow-up criterion. The tool we mainly use is Littlewood-Paley decomposition, by which we obtain a Beale-Kato-Majda type blow-up criterion for smooth solution $(u,\omega,b)$ which relies on the vorticity of velocity $\nabla\times u$ only.


The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem

Abstract: Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovsky's theorem, for every positive integer n with m \prec n in the Sharkovsky's ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question.


The periodic table of $n$-categories for low dimensions II: degenerate tricategories

Abstract: We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between \bfseries{BrMonCat} and \bfseries{Tricat}. We show how to alter the natural structure of \bfseries{Tricat} in two different ways to provide a comparison, but show that only the more brutal alteration yields an equivalence. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure \bfseries{MonBicat}.


Cohomology of diffeological spaces and foliations

Abstract: Let (M,F) be a foliated manifold. We study the relationship between the basic cohomology Hb(M,F) of the foliation and the De Rham cohomology H(DF) of the space of leaves M/F as a quotient diffeological space. We prove that for an arbitrary foliation there is a morphism from H(DF) to Hb(M,F). It is an isomorphism when F is a Q-foliation.


Hypersurfaces in H^{n+1} and conformally invariant equations: the generalized Christoffel and Nirenberg problems

Abstract: Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on $\S^n$ for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result into a more general framework. Specifically, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in $H^{n+1}$ is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on $\S^n$ and the theory of hypersurfaces with prescribed hyperbolic Gauss map in $\H^{n+1}$. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals $W(\k_1,..., \k_n)$, the only compact immersed hypersurfaces in $H^{n+1}$ on which $W$ is constant are round spheres.


Stable Border Bases for Ideals of Points

Abstract: Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $\widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $\widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(\widetilde X)$.


On large automorphism groups of algebraic curves in positive characteristic

Abstract: In his investigation on large $K$-automorphism groups of an algebraic curve, Stichtenoth obtained an upper bound on the order of the first ramification group of an algebraic curve $\cX$ defined over an algebraically closed field of characteristic $p$. Stichtenoth's bound has raised the problem of classifying all $\K$-automorphism groups $G$ of $\cX$ with the following property: There is a point $P\in \cX$ for which \begin{equation} |G_P^{(1)}|>\frac{p}{p-1}g. \end{equation} Such a classification is obtained here by proving Theorem 1.3


Lower bounds for moments of zeta prime rho

Abstract: Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan that provides analogous bounds for moments of $L$-functions at the central point, averaged over families.


Special isogenies and tensor product multiplicities

Abstract: We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann's Path Model, and geometrically, using isogenies between algebaric groups defined over an algebraically closed field of positive characteristic.


Hodge-Stickelberger polygons for L-functions of exponential sums of P(x^s)

Abstract: Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger theory). We prove an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. We prove this Hodge-Stickelberger polygon is the limit of generic Newton polygons of P(x^s) in a sense that was made explicit in the paper.


Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms

Abstract: Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a unique formal "correction" of the nonlinear part so that the "corrected" system is formally linearizable. Normal forms of these systems are found, providing also their classification.


On the cyclicity of weight-homogeneous centers

Abstract: Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.


Analytic linearization of nonlinear perturbations of Fuchsian systems

Abstract: Nonlinear perturbation of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable.


Kummer subfields of tame division algebras over Henselian valued fields

Abstract: By generalizing the method used by Tignol and Amitsur in [TA85], we determine necessary and sufficient conditions for an arbitrary tame central division algebra D over a Henselian valued field E to have Kummer subfields [Corollary 2.11 and Corollary 2.12]. We prove also that if D is a tame semiramified division algebra of prime power degree p^n over E such that p\neq char(\bar E) and rk(\Gamma_D/\Gamma_E)\geq 3 [resp., such that p\neq char(\bar E) and p^3 divides exp(\Gamma_D/\Gamma_E)], then D is non-cyclic [Proposition 3.1] [resp., D is not an elementary abelian crossed product [Proposition 3.2]].


FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

Abstract: We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.


A Note on Carleman's Inequality

Abstract: We study a weighted version of Carleman's inequality via Carleman's original approach. As an application of our result, we prove a conjecture of Bennett.


Koszul duality in deformation quantization, I

Abstract: Let $\alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $\mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $f\star g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $\alpha$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_i\otimes x_j-x_j\otimes x_i=R_{ij}(\hbar)$ where $R_{ij}(\hbar)\in T(V^*)\otimes\hbar \mathbb{C}[[\hbar]]$, $R_{ij}=\hbar \Sym(\alpha_{ij})+\mathcal{O}(\hbar^2)$, with one relation for each pair of $i,j=1...\dim V$. We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincar\'{e}-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum $R$-matrix on $V$. At the same time we get a free resolution of the deformed algebra (for an arbitrary $\alpha$). The construction of this PBW algebra is rather simple, as well as the proof of the PBW property. The major efforts should be undertaken to prove the conjecture that in this way we get an algebra isomorphic to the Kontsevich star-algebra.


Galois theory of iterated endomorphisms

Abstract: Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of $A$ we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes $\p$ in the ring of integers of $F$ such that the order of $(\alpha \bmod{\p})$ is prime to $\ell$. We compute this density in the general case for several classes of $A$, including elliptic curves and one-dimensional tori. For example, if $F$ is a number field, $A/F$ is an elliptic curve with surjective 2-adic representation and $\alpha \in A(F)$ with $\alpha \not\in 2A(F(A[4]))$, then the density of $\mathfrak{p}$ with ($\alpha \bmod{\p}$) having odd order is 11/21.


Stochastic Parabolic Equations of Full Second Order

Abstract: A procedure is described for defining a generalized solution for stochastic differential equations using the Cameron-Martin version of the Wiener Chaos expansion. Existence and uniqueness of this Wiener Chaos solution is established for parabolic stochastic PDEs such that both the drift and the diffusion operators are of the second order.


From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration

Abstract: The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two such extensions, corresponding to the \Ito-Skorokhod and the Stratononovich integrals, and provides an efficient tool to study these integrals, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields.


Inversely Unstable Solutions of Two-Dimensional Systems on Genus-p Surfaces and the Topology of Knotted Attractors

Abstract: In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.


Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions

Abstract: Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities.


Commuting linear operators and algebraic decompositions

Abstract: For commuting linear operators $P_0,P_1,..., P_\ell$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P=P_0P_1... P_\ell$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition $P$ may be derived from generalised symmmetries of the component operators $P_i$ in the system.


The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers

Abstract: The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that.


Higher K-theory via universal invariants

Abstract: Using the formalism of Grothendieck's derivators, we construct `the universal localizing invariant of dg categories'. By this, we mean a morphism U_l from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator M^loc such that U_l commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle and is universal for these properties. Similary, we construct the `the universal additive invariant of dg categories', i.e. the universal morphism of derivators U_a to a strong triangulated derivator M^add which satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen K-theory appears as a mapping space in the target of the universal additive invariant. This is the first conceptual characterization of Quillen-Waldhausen's K-theory since its definition in the early 70's. As an application we obtain for free the higher Chern characters from K-theory to cyclic homology.


A Simple Method Which Generates Infinitely Many Congruence Identities

Abstract: A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the number of periodic points of minimal period m must be divisible by m, we obtain infinitely many congruence identities.


On the Cordial Deficiency of Complete Multipartite Graphs

Abstract: We calculate the cordial edge deficiencies of the complete multipartite graphs and find an upper bound for their cordial vertex deficiencies. We also give conditions under which the tensor product of two cordial graphs is cordial.


A system for constructing relatively small polyhedra from Sonob\'e modules

Abstract: We develop a quite elementary graph theoretic system for designing small-size augmented origami polyhedra out of Sonob\'e modules beginning with a (convex or not) deltahedron.


Analysis of the expected number of bit comparisons required by Quickselect

Abstract: When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank.


Adelic amoebas disjoint from open halfspaces

Abstract: We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropicalization, and a theorem of Zhang on torsion points in subvarieties of tori.


Minimal intersection of curves on surfaces

Abstract: In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words the bracket with a simple element has no cancellation and determines minimal intersection numbers. We show that analogous results hold for the Lie bracket (also discovered by Goldman) of unoriented curves. We give three applications: a factorization of Thurston's map defining the boundary of Teichmuller space, various decompositions of the underlying vector space of conjugacy classes into ad invariant subspaces and a connection between bijections of the set of conjugacy classes of curves on a surface preserving the Goldman bracket and the mapping class group.


Reducible Families of Curves with Ordinary Multiple Points on Surfaces in Projective Three-Space

Abstract: In math.AG/0108089, math.AG/0212090 and math.AG/0308247 we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results by Greuel, Lossen and Shustin and an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves on surfaces in projective three-space with only ordinary multiple points which are reducible and where at least one component does not have the expected dimension. The examples show that for families of curves with ordinary multiple points the conditions for T-smoothness in math.AG/0308247 have the right asymptotics.


Some obstructed equisingular families of curves on surfaces in projective three-space

Abstract: Very few examples of obstructed equsingular families of curves on surfaces other than the projective plane are known. Combining results from Westenberger and Hirano with an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves with simple singularities on surfaces in projective three-space which are not T--smooth, i.e. do not have the expected dimension, and we compare this with conditions (showing the same asymptotics) which ensure the existence of a T--smooth component.


On the topological stable rank of non-selfadjoint operator algebras

Abstract: We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.


Transitive spaces of operators

Abstract: We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson's result on the weak-operator density of transitive spaces that are masa bimodules.


Transitive decompositions of graphs and their links with geometry and origami

Abstract: A transitive decomposition of a graph is a partition of the edge or arc set giving a set of subgraphs which are preserved and permuted transitively by a group of automorphisms of the graph. In this paper we give some background to the study of transitive decompositions and highlight a connection with partial linear spaces. We then describe a simple method for constructing transitive decompositions using graph quotients, and we show how this may be used in an application to modular origami.


Les classes d'Eisenstein des varietes de Hilbert-Blumenthal

Abstract: This article deals with the Eisenstein classes of Hilbert-Blumenthal families of abelian varieties. We first give a coordinate expression of these one at the topological level, using currents defined by Levin. Then we study the degeneration of these Eisenstein classes at a cusp of the Baily-Borel compactification of the Hilbert-Blumenthal variety. We show, using the explicit description of the Eisenstein classes obtained previously, that these classes degenerate in special values of an $L$-function associated to the underlying totally real number field. We deduce then both a geometric proof the Klingen-Siegel Theorem and a non vanishing result for some of these Eisenstein classes .


An extension of a Bourgain--Lindenstrauss--Milman inequality

Abstract: Let || . || be a norm on R^n. Averaging || (\eps_1 x_1, ..., \eps_n x_n) || over all the 2^n choices of \eps = (\eps_1, ..., \eps_n) in {-1, +1}^n, we obtain an expression ||| . ||| which is an unconditional norm on R^n. Bourgain, Lindenstrauss and Milman showed that, for a certain (large) constant \eta > 1, one may average over (\eta n) (random) choices of \eps and obtain a norm that is isomorphic to ||| . |||. We show that this is the case for any \eta > 1.


Congruence Identities Arising From Dynamical Systems

Abstract: By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory.


The Intrinsic Fundamental Group of a Linear Category

Abstract: We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.


Robot motion planning, weights of cohomology classes, and cohomology operations

Abstract: The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik - Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result.


Alexander polynomials: Essential variables and multiplicities

Abstract: We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give sharp upper bounds for the twisted Betti ranks of a group, in terms of multiplicities constructed from the Alexander polynomial. For Seifert links in homology 3-spheres, these bounds become equalities, and our formula shows explicitly how the Alexander polynomial determines all the characteristic varieties.


Matching polytopes, toric geometry, and the non-negative part of the Grassmannian

Abstract: In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\geq 0}. This is a cell complex whose cells Delta_G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Delta_G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then use the data of P(G) to define an associated toric variety X_G. We use our technology to prove that the cell decomposition of (Gr_{kn})_{\geq 0} is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr_{kn})_{\geq 0} is 1.


Varieties with very little transcendental cohomology

Abstract: Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most of this paper is concerned with giving estimates on this number, along with examples where it is small. As an application, we check or recheck the Hodge conjectue in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes of a small number points on surfaces with p_g=0, and generic hypersurfaces.


Logarithmic comparison theorem versus Gauss-Manin system for isolated singularities

Abstract: For quasihomogeneous isolated hypersurface singularities, the logarithmic comparison theorem has been characterized explicitly by Holland and Mond. In the non quasihomogeneous case, we give a necessary condition for the logarithmic comparison theorem in terms of the Gauss-Manin system of the singularity. It shows in particular that the logarithmic comparison theorem can hold for a non quasihomogeneous singularity only if 1 is an eigenvalue of the monodromy.


The Harmonic Series and the nth Term Test for Divergence

Abstract: The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and sum over k, 1<=k<=n-1.


Big-Pieces-of-Lipschitz-Images Implies a Sufficient Carleson Estimate in a Metric Space

Abstract: This note is intended to be a supplement to the bi-Lipschitz decomposition of Lipschitz maps shown in [Sch]. We show that in the case of 1-Ahlfors-regular sets, the condition of having `Big Pieces of bi-Lipschitz Images' (BPBI) is equivalent to a Carleson condition.


Newton-Hodge Filtration for Self-Dual F-Crystals

Abstract: In this paper we study F-crystals with self-dual structure over base schemes of characteristic p. We generalize Katz's Newton-Hodge Filtration Theorem to F-crystals with self-dual structure.


The noncommutative residue and canonical trace in the light of Stokes' and continuity properties

Abstract: We show that the noncommutative residue density, resp. the cut-off regularised integral are the only closed linear, resp. continuous closed linear forms on certain classes of symbols. This leads to alternative proofs of the uniqueness of the noncommutative residue, resp. the canonical trace as linear, resp. continuous linear forms on certain classes of classical pseudodifferential operators which vanish on brackets. The uniqueness of the canonical trace actually holds on classes of classical pseudodifferential with vanishing residue density which include non integer order operators in all dimensions and odd-class (resp. even-class) operators in odd (resp. even) dimensions. The description of the canonical trace for non integer order operators as an integrated global density on the manifold is extended to odd-class (resp. even-class) operators in odd (resp. even) dimensions on the grounds of defect formulae for regularised traces of classical pseudodifferential operators.


Chern-Weil calculus extended to a class of infinite dimensional manifolds

Abstract: We discuss possible extensions of the classical Chern-Weil formalism to an infinite dimensional setup. This is based on joint work with Steven Rosenberg, joint work with Simon Scott and joint work with Jouko Mickelsson.


Gray Curvature Identities for Almost Contact Metric Manifolds

Abstract: The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold.


A Groshev Theorem for Small Linear Forms

Abstract: In this paper the absolute value or distance from the origin analogue of the classical Khintchine-Groshev theorem is established for a single linear form with a `slowly decreasing' error function.


Prescribing the behaviour of geodesics in negative curvature

Abstract: Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.


Planar maps whose second iterate has a unique fixed point

Abstract: Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of either (c1) the union of the number 0 with the intervals ]-\infty, -1] and [1,\infty[, or (c2) the interval [-1-a, 1+a]. Conditions under which Fix(F^n), with n>1, is at most unitary are considered.


Betti numbers for fat point ideals in the plane: a geometric approach

Abstract: We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of the plane. We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.


The Kashiwara-Vergne conjecture

Abstract: This is a Bourbaki's seminar text. We introduce the combinatorial Kashiwara-Vergne conjecture on the Baker-Campbell-Hausdorff serie. After recalling previous results and consequences, we explain the Alekseev-Meinrenken's proof [math.QA/0506499]


On the genealogy on conditioned stable L\'evy forest

Abstract: We give a realization of the stable L\'evy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering $k$ independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to $n$. We prove that when $n$ and $k$ tend towards $+\infty$, under suitable rescaling, the associated coding random walk, the contour and height processes converge in law on the Skorokhod space respectively towards the "first passage bridge" of a stable L\'evy process with no negative jumps and its height process.


A refinement of multi-dimensional persistence

Abstract: We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes.


Minimal data rate stabilization of nonlinear systems over networks with large delays

Abstract: Control systems over networks with a finite data rate can be conveniently modeled as hybrid (impulsive) systems. For the class of nonlinear systems in feedfoward form, we design a hybrid controller which guarantees stability, in spite of the measurement noise due to the quantization, and of an arbitrarily large delay which affects the communication channel. The rate at which feedback packets are transmitted from the sensors to the actuators is shown to be arbitrarily close to the infimal one.


A Dynamical Systems Approach to the Kadison-Singer Problem

Abstract: In these notes we develop a link between the Kadison-Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the state. We prove that if any state corresponding to a minimal idempotent point extends uniquely to the von Neumann algebra of the group, then every state extends uniquely to the von Neumann algebra of the group. We prove that if any state arising in the Kadsion-Singer problem has a unique extension, then the injective envelope of a C*-crossed product algebra associated with the state necessarily contains the full von Neumann algebra of the group. We prove that this latter property holds for states arising from rare ultrafilters and $\delta$-stable ultrafilters, independent, of the group action and also for states corresponding to non-recurrent points in the corona of the group.


Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

Abstract: We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by any approximation method using an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is $n^{-H-1/2},$ where $n$ denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.


Some applications of the Mellin transform to branching processes

Abstract: We introduce a Mellin transform of functions which live on all of $\bR$ and discuss its applications to the limiting theory of Bellman-Harris processes, and specifically Luria-Delbr\"uck processes. More precisely, we calculate the life-time distribution of particles in a Bellman-Harris process from their first-generation offspring and limiting distributions, and prove a formula for the Laplace transform of the distribution of types in a Luria-Delbr\"uck process in the Mittag-Leffler limit. As a by-product, we show how to easily derive the (classical) Mellin transforms of certain stable probability distributions from their Fourier transform.


The twisted Mellin transform

Abstract: The "twisted Mellin transform" is a slightly modified version of the usual classical Mellin transform on $L^2(\mathbb R)$. In this short note we investigate some of its basic properties. From the point of views of combinatorics one of its most important interesting properties is that it intertwines the differential operator, $df/dx$, with its finite difference analogue, $\nabla f= f(x)-f(x-1)$. From the point of view of analysis one of its most important properties is that it describes the asymptotics of one dimensional quantum states in Bargmann quantization.


Some new equivalences of Anderson's paving conjectures

Abstract: Anderson's paving conjectures are known to be equivalent to the Kadison-Singer problem. We prove some new equivalences of Anderson's conjectures that require the paving of smaller sets of matrices. We prove that if the strictly upper triangular operatorss are pavable, then every 0 diagonal operator is pavable. This result follows from a new paving condition for positive operators. In addition, we prove that if the upper triangular Toeplitz operators are paveable, then all Toeplitz operators are paveable.


Compactness of the Complex Green Operator

Abstract: Let $\Omega\subset\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $b\Omega$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $\Omega$. We prove that if $1 \leq q \leq n-2$ and $b\Omega$ satisfies $(P_q)$ and $(P_{n-q-1})$, then $G_{q}$ is a compact operator (and so is $G_{n-1-q}$). Our method relies on a jump type formula to represent forms on the boundary, and we prove an auxiliary compactness result for an `annulus' between two pseudoconvex domains. Our results, combined with the known characterization of compactness in the $\bar{\partial}$-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.


A quantitative version of the Besicovitch projection theorem via multiscale analysis

Abstract: By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but weak) upper bound on the average projection of the $n^{th}$ generation of a product Cantor set.


A note for Gromov's distance functions on the space of mm-spaces

Abstract: This is just a note for \cite[Chapter$3{1/2}_+$]{gromov}. Maybe this note is obvious for a reader who knows metric geometry. I wish that someone study further in this direction.


Harder-Narasimhan categories

Abstract: We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by $\mathbb R$), which can not be stated in the classical setting of Harder and Narasimhan's formalism.


Convergence of Harder-Narasimhan polygons

Abstract: We establish in this article convergence results of normalized Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by introducing the Harder-Narasimhan filtration indexed by $\mathbb R$ and the associated Borel probability measure.


On the connected component of compact composition operators on the Hardy space

Abstract: We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space $H^2$ on the unit disc. This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures.


On singular cubic surfaces

Abstract: We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.


Le th\'eor\`eme de Riemann-Roch et ses applications

Abstract: The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this article is to present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves.


Bounds on the concentration function in terms of Diophantine approximation

Abstract: We demonstrate a simple analytic argument that may be used to bound the Levy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its multidimensional generalisation.


Singular Loci of Hibi toric varieties

Abstract: We first construct explicit bases for the cotangent spaces at singular points on Hibi toric varieties, i.e., toric varieties associated to distributive lattices. We then determine the singular loci of these toric varieties.


Universal flattening of Frobenius

Abstract: For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme.


Holonomic rank of A-hypergeometric differential-difference equations

Abstract: We introduce A-hypergeometric differential-difference equation and prove that its holonomic rank is equal to the normalized volume of A with giving a set of convergent series solutions.


The p-modular Descent Algebra of the Symmetric Group

Abstract: The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.


On the Descent Algebra of Type $D$

Abstract: Here we give a combinatorial interpretation of Solomon's rule for multiplication in the descent algebra of Weyl groups of type $D$, $\Sigma D_n$. From here we show that $\Sigma D_n$ is a homomorphic image of the descent algebra of the hyperoctahedral group, $\Sigma B_{n-2}$.


Contraction groups in complete Kac-Moody groups

Abstract: Let $G$ be an abstract Kac-Moody group over a finite field and $\bar{G}$ be the closure of the image of $G$ in the automorphism group of its positive building. We show that if the Dynkin diagram associated to $G$ is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in $\bar{G}$ which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)


A Proof of Solomon's Rule

Abstract: We put forward a proof of Solomon's rule, in terms of matrices, for multiplication in the descent algebra of the symmetric group. Our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions, $\{(i,i+1)\}_{i=1}^{n-1}$.