The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

Abstract: The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it correlates with a phase polynomial $\phi = e_\F(P)$ of degree at most $d-1$, thus $P: V \to \F$ is a polynomial of degree at most $d-1$. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case $\charac(F) \geq d$ from an ergodic theory counterpart, which was recently established by Bergelson and the authors. In low characteristic we obtain a partial result, in which the phase polynomial $\phi$ is allowed to be of some larger degree $C(d)$. The full inverse conjecture remains open in low characteristic; the counterexamples by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting can be avoided by a slight reformulation of the conjecture.


Characterization of Line Arrangement for which the Fundamental Group of the Complement is a Direct Product of Free Groups

Abstract: Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture


Uniform approximation on ideals of multilinear mappings

Abstract: For each ideal of multilinear mappings $\cal M$ we explicitly construct a corresponding ideal $^{a}{\cal M}$ such that multilinear forms in $^{a}{\cal M}$ are exactly those which can be approximated, in the uniform norm, by multilinear forms in ${\cal M}$. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence ${\cal M} \mapsto ^{a}{\cal M}$ is Aron-Berner stability preserving.


Exponential Squared Integrability for the Discrepancy Function in Two Dimensions

Abstract: Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On the other hand we use a digit scrambled version of the van der Corput set to show that this bound is tight in the case N=2^n, for some positive integer n. These results unify the corresponding classical results of Roth and Schmidt in a sharp fashion.


The Hall algebra of a spherical object

Abstract: We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object.


On the Stochastic Rank of Metric Functions

Abstract: For a class of integral operators with kernels metric functions on manifold we find some necessary and sufficient conditions to have finite rank. The problem we pose has a stochastic nature and boils down to the following alternative question. For a random sample of discrete points, what will be the probability the symmetric matrix of pairwise distances to have full rank? When the metric is an analytic function, the question finds full and satisfactory answer. As an important application, we consider a class of tensor systems of equations formulating the problem of recovering a manifold distribution from its covariance field and solve this problem for representing manifolds such as Euclidean space and unit sphere.


Topological regluing of rational functions

Abstract: Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichm\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop.


Well, Papa, can you multiply triplets?

Abstract: We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.


Limit Behaviour of Sequential Empirical Measure Processes

Abstract: In this paper, we obtain some uniform laws of large numbers and functional central limit theorems for sequential empirical measure processes indexed by classes of product functions satisfying appropriate Vapnik-Chervonenkis properties.


Homology of framed links embedded in thickened surfaces

Abstract: We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauffman bracket skein module of I-bundles over surfaces. This is accomplished by borrowing ideas from Bar-Natan's Khovanov homology theory for tangles and cobordisms and using embedded surfaces to generate the chain groups, instead of diagrams.


An example of Brunet-Derrida behavior for a branching-selection particle system on $\Z$

Abstract: We consider a branching-selection particle system on $\Z$ with $N \geq 1$ particles. During a branching step, each particle is replaced by two new particles, whose positions are shifted from that of the original particle by independently performing two random walk steps according to the distribution $p \delta_{1} + (1-p) \delta_{0}$, from the location of the original particle. During the selection step that follows, only the N rightmost particles are kept among the 2N particles obtained at the branching step, to form a new population of $N$ particles. After a large number of iterated branching-selection steps, the displacement of the whole population of $N$ particles is ballistic, with deterministic asymptotic speed $v_{N}(p)$. As $N$ goes to infinity, $v_{N}(p)$ converges to a finite limit $v_{\infty}(p)$. The main result is that, for every $0<p<1/2$, as $N$ goes to infinity, the order of magnitude of the difference $v_{\infty}(p)- v_{N}(p)$ is $\log(N)^{-2}$. This is called Brunet-Derrida behavior in reference to the 1997 paper by E. Brunet and B. Derrida "Shift in the velocity of a front due to a cutoff" (see the reference within the paper), where such a behavior is established for a similar branching-selection particle system, using both numerical simulations and heuristic arguments.


Stability of the vanishing of the $\overline\partial_b$-cohomology under small horizontal perturbations of the CR structure in compact abstract q-concave CR manifolds

Abstract: We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some $\overline\partial_b$-cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups.


A note on hyperbolic leaves and wild laminations of rational functions

Abstract: We study the affine orbifold laminations that were constructed by Lyubich and Minsky. An important question left open in their construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen is itself a question raised by Lyubich and Minsky.


Enriched spin curves on stable curves with two components

Abstract: L. Maino constructed a moduli space for enriched stable curves, by blowing-up the moduli space of Deligne-Mumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowing-up the moduli space of spin curves.


Touchscreen Voting Machines Cause Long Lines and Disenfranchise Voters

Abstract: Computerized touchscreen "Direct Recording Electronic" DRE voting systems have been used by over 1/3 of American voters in recent elections. In many places, insufficient DRE numbers in combination with lengthy ballots and high voter traffic have caused long lines and disenfranchised voters who left without voting. We have applied computer queuing simulation to the voting process and conclude that far more DREs, at great expense, would be needed to keep waiting times low. Alternatively, paper ballot-optical scan systems can be easily and economically scaled to prevent long lines and meet unexpected contingencies.


CM-stability of blow-ups and canonical metrics

Abstract: An asymptotic formula for the Tian-Paul CM-line of a flat family blown-up at a flat closed sub-scheme is given. As an application we prove that the blow-up of a polarized manifold along a (relatively) Chow-unstable submanifold admits no (extremal) constant scalar curvature Kahler metrics in classes making the exceptional divisors sufficiently small. Moreover a geometric characterization of relatively Chow-unstable configuration of points in the projective space is given. From this we get new examples of classes admitting no extremal Kahler metric also in the case of the projective plane blown-up at a finite set of points.


More Results on Regular Ultrafilters in ZFC

Abstract: We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.


Fields of Parametrization and Optimal Affine Reparametrization of Rational Curves

Abstract: In this paper we present three related results on the subject of fields of parametrization. Let C be a rational curve over a field of characteristic zero. Let K be a field finitely generated over Q, such that it is a field of definition of C but not a field of parametrization. It is known that there are quadratic extensions of K that parametrize C. First, we prove that there are infinitely many quadratic extensions of K that are fields of parametrization of C. As a consequence, we prove that the witness variety, that appear in the context of the parametric Weil's descente method, is always a special curve related to algebraic extensions, called hypercircle. It is possible that the witness variety is not a hypercircle for the given extension, but for an alternative one. We use these two facts to present an algorithm to solve the following optimal reparametrization problem. Given a birational parametrization f(t) of a curve C, compute the affine reparametrization at+b such f(at+b) has coefficients over a field as small as possible. The main advantage of this algorithm is that it does not need to compute any rational point on the curve.


Spectral Bounds for Dirac Operators on Open Manifolds

Abstract: We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.


Approximation by Lipschitz, analytic maps on certain Banach spaces

Abstract: We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets.


Evidence of Systematic Bias in 2008 Presidential Polling (preliminary report)

Abstract: Political polls achieve their results by sampling a small number of potential voters rather than the population as a whole. This leads to sampling error which most polling agencies dutifully report. But factors such as nonrepresentative samples, question wording and nonresponse can produce non-sampling errors. While pollsters are aware of such errors, they are difficult to quantify and seldom reported. When a polling agency, whether by intention or not, produces results with non-sampling errors that systematically favor one candidate over another, then that agency's poll is biased. We analyzed polling data for the (on-going) 2008 Presidential race, and though our methods do not allow us to identify which agencies' polls are biased, they do provide significant evidence that some agencies' polls are. We compared polls produced by major television networks with those produced by Gallup and Rasmussen. We found that, taken as a whole, polls produced by the networks were significantly to the left of those produced by Gallup and Rasmussen. We used the available data to provide a tentative ordering of the major television networks' polls from right to left. Our order was: FOX, CNN, NBC (which partners with the Wall Street Journal), ABC (which partners with the Washington Post), CBS (which partners with the New York Times). These results appear to comport well with the informal perceptions of the political leanings of these agencies. Our findings are preliminary, but they make a case for further research into the causes of and remedies for polling bias.


Q curvature prescription; forbidden functions and the GJMS null space

Abstract: On an even conformal manifold $(M,c)$, such that the critical GJMS operator has non-trivial kernel, we identify and discuss the role of a finite dimensional vector space $N(Q)$ of functions determined by the conformal structure. Using these we describe an infinite dimensional class of functions that cannot be the Q-curvature $Q^g$ for any $g$ in $c$. If certain functions arise in $N(Q)$ then $Q^g$ cannot be constant for any $g$ in $c$.


The Pentagram map: a discrete integrable system

Abstract: The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.


Conjugation-free geometric presentations of fundamental groups of arrangements

Abstract: We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union of disjoint cycles, has no line with more than two multiple points, and where the multiplicities of the multiple points are arbitrary. We also compute the exact group structure (by means of a semi-direct product of groups) of the arrangement of 6 lines whose graph consists of a cycle of length 3, and all the multiple points have multiplicity 3.


A rigidity theorem for quaternionic Kaehler structures

Abstract: We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space.


Jack deformations of Plancherel measures and traceless Gaussian random matrices

Abstract: We study random partitions $\lambda=(\lambda_1,\lambda_2,...,\lambda_d)$ of $n$ whose length is not bigger than a fixed number $d$. Suppose a random partition $\lambda$ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter $\alpha>0$. We prove that for all $\alpha>0$, in the limit as $n \to \infty$, the joint distribution of scaled $\lambda_1,..., \lambda_d$ converges to the joint distribution of some random variables from a traceless Gaussian $\beta$-ensemble with $\beta=2/\alpha$. We also give a short proof of Regev's asymptotic theorem for the sum of $\beta$-powers of $f^\lambda$, the number of standard tableaux of shape $\lambda$.


Conformally Osserman manifolds

Abstract: An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \ne 3, 4, 16$ is locally conformally equivalent either to a Euclidean space or to a rank-one symmetric space.


Simultaneous Binary Collisions for Collinear Four-Body Problem

Abstract: In this paper, we use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions for the collinear four- body problem in both the decoupled case and the coupled case. All the solutions are found and more importantly, we describe the relationship between the decoupled solutions and the coupled solutions.


A remark on the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities

Abstract: We consider the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. We deduce the uniqueness from the argument in the classical paper by Peletier and Serrin, thereby recovering a part of the uniqueness result of Ouyang and Shi.


Forcings constructed along morasses

Abstract: In a previous paper, we introduced a way of constructing a forcing along a simplified gap-1 morass such that the forcing satisfies a chain condition. Now, we generalize this to gap-2 morasses. As an application, we prove that GCH is consistent with the existence of a 0-dimensional Hausdorff topology on $\omega_3$ which has spread $\omega_1$.


On semilinear elliptic equations with global coupling

Abstract: We consider nonlinear elliptic equations which contains global coupling as a nonlinear term. We classify the existence of all possible positive solutions to this problem.


Relative Oscillation Theory for Jacobi Matrices

Abstract: We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by replacing nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices.


Demazure crystals of generalized Verma modules and a flagged RSK correspondence

Abstract: We prove that the Robinson-Schensted-Knuth correspondence is a $\gl_{\infty}$-crystal isomorphism between two realizations of the crystal graph of a generalized Verma module with respect to a maximal parabolic subalgebra of $\gl_{\infty}$. %This extends the previously known result that %the RSK correspondence is an isomorphism of bicrystals or double %crystals. A flagged version of the RSK correspondence is derived in a natural way by computing a Demazure crystal graph of a generalized Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition.


Lagrangian Floer theory on compact toric manifolds II : Bulk deformations

Abstract: This is a continuation of part I in the series of the papers on Lagrangian Floer theory on toric manifolds. Using the deformations of Floer cohomology by the ambient cycles, which we call bulk deformations, we find a continuum of non-displaceable Lagrangian fibers on some compact toric manifolds. We also provide a method of finding all fibers with non-vanishing Floer cohomology with bulk deformations in arbitrary compact toric manifolds, which we call bulk-balanced Lagrangian fibers.


Gibbs posterior for variable selection in high-dimensional classification and data mining

Abstract: In the popular approach of "Bayesian variable selection" (BVS), one uses prior and posterior distributions to select a subset of candidate variables to enter the model. A completely new direction will be considered here to study BVS with a Gibbs posterior originating in statistical mechanics. The Gibbs posterior is constructed from a risk function of practical interest (such as the classification error) and aims at minimizing a risk function without modeling the data probabilistically. This can improve the performance over the usual Bayesian approach, which depends on a probability model which may be misspecified. Conditions will be provided to achieve good risk performance, even in the presence of high dimensionality, when the number of candidate variables "$K$" can be much larger than the sample size "$n$." In addition, we develop a convenient Markov chain Monte Carlo algorithm to implement BVS with the Gibbs posterior.


The Packing Measure of the Range of Super-Brownian Motion

Abstract: We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d\geq 5$. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the $g$-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension $d$.


G-Structures defined on pseudo-Riemannian manifolds

Abstract: Concepts and techniques from the theory of G-structures of higher order are applied to the study of certain structures (volume forms, conformal structures, linear connections and projective structures) defined on a pseudo-Riemanniann manifold. Several relationships between the structures involved have been investigated. The operations allowed on G-structures, such as intersection, inclusion, reduction, extension and prolongation, were used for it.


Locally well generated homotopy categories of complexes

Abstract: We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring R for B = Mod-R.


Liaison invariants and the Hilbert scheme of codimension 2 subschemes in P^{n+2}

Abstract: In this paper we study the Hilbert scheme, Hilb(P), of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in P^4 and 3-folds in P^5, and the Hilbert scheme stratification H_{c} of constant cohomology. For every (X) in Hilb(P) we define a number \delta_X in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + \delta_X - \dim_{(X)} H_{c} and 1 + \delta_X - \dim T_{c} are CI-biliaison invariants where T_{c} is the tangent space of H_{c} at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilb(P) in terms of \delta_X and the CI-biliaison invariant. Both invariants are equal in this case. Recall that, for space curves C, Martin-Deschamps and Perrin have proved the smoothness of the ``morphism'', H_{c} -> E = isomorphism classes of graded artinian modules, given by sending C onto its Rao-module. For surfaces X in P^4 we have two Rao-modules M_i and an induced extension b in Ext^2(M_2,M_1) and a result of Horrocks and Rao saying that a triple D := (M_1,M_2,b) of modules M_i of finite length and an extension b as above determine a surface X up to biliaison. We prove that the corresponding ``morphism'', H_{c} -> V = isomorphism classes of graded artinian modules M_i commuting with b, is smooth, and we get a smoothness criterion for H_{c}. Moreover we get some smoothness results for Hilb(P), valid also for 3-folds, and we give examples of obstructed surfaces and 3-folds. The linkage result we prove in this paper turns out to be useful in determining the structure and dimension of H_{c}, and for proving the main biliaison theorem above.


Niceness theorems

Abstract: Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, ... . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.


Detecting orbits along subvarieties via the moment map

Abstract: Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the criterion are easily verified in practice and are used to construct continuous families of (non-isomorphic) nilpotent Lie groups which do not admit left-invariant Ricci soliton metrics. Other applications to the left-invariant geometry of Lie groups are also given. The note finishes by applying our techniques to the adjoint representation. The classical result of finiteness of nilpotent orbits is reproven and it is shown that each of these orbits contains a critical point of the norm squared of the moment map.


Remarks on the non-commutative Khintchine inequalities for $0<p<2$

Abstract: We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to $R_k+C_k$ with $k\sim n^{\frac12}$. This naturally raises several interesting questions concerning the best possible such $k$. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for $0<p<1$ but we can prove several would be corollaries. For instance, given an infinite scalar matrix $[x_{ij}]$, we give a necessary and sufficient condition for $[\pm x_{ij}]$ to be in the Schatten class $S_p$ for almost all (independent) choices of signs $\pm 1$. We also characterize the bounded Schur multipliers from $S_2$ to $S_p$. The latter two characterizations extend to $0<p<1$ results already known for $1\le p\le2$. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting.


Conditional Limits of W_p scale Mixture Distributions

Abstract: In this paper we introduce the class of W_p scale mixture random vectors with a particular radial decomposition and a independent splitting property specified by some random variable W_p, and a positive constant p. We derive several conditional limit results assuming that the distribution of the random radius is in the max-domain of attraction of a univariate extreme value distribution and W_p has a certain tail asymptotic behaviour. As an application we obtain the joint asymptotic distribution of concomitants of order statics considering certain bivariate W_p scale miture samples.


A few things I learnt from Jurgen Moser

Abstract: A few remarks on integrable dynamical systems inspired by discussions with Jurgen Moser and by his work.


Simple Compact Quantum Groups I

Abstract: The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for $Q \in GL(n, {\mathbb C})$ satisfying $Q \bar{Q} = \pm I_n$, $n \geq 2$; (b) The quantum automorphism groups $A_{aut}(B, \tau)$ of finite dimensional $C^*$-algebras $B$ endowed with the canonical trace $\tau$ %endowed with a tracial functional $tr$ when $\dim(B) \geq 4$, including the quantum permutation groups $A_{aut}(X_n)$ on $n$ points ($n \geq 4$); (c) The standard deformations $K_q$ of simple compact Lie groups $K$ and their twists $K_q^u$, as well as Rieffel's deformation $K_J$.


A Rigidity Condition for Holomorphic Generator in Strongly Convex Domains

Abstract: In this paper I give a condition, in terms of boundary behaviour, that forces the infinitesimal generator of a one-parameter semigroup of holomorphic maps to vanish.


Period and index of genus one curves over number fields

Abstract: The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.


Local energy estimate on Kerr black hole backgrounds

Abstract: We study dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of thispaper is to establish uniform energy bounds and local energy decay for such backgrounds.


Large gaps between random eigenvalues

Abstract: We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\ kappa_{\beta}+o(1)\bigr)\lambda^{\gamma_{\beta}}\exp\biggl(-{\bet a}{64}\lambda^2+\biggl({\beta}{8}-{1}{4}\biggr)\lambda\biggr)\] as $\lambda\to\infty$, where \[\gamma_{\beta}={1}{4}\biggl({\beta}{2}+{2}{\beta}-3\biggr)\] and $\kappa_{\beta}$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus.


A note on the eigenvalues of double band matrices

Abstract: We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form $r \zeta$, where $r$ is a nonnegative real number and $\zeta$ is a $p$th root of unity, where $p$ is the period of the matrix, which is computed from the distance between the bands.


Cartan--Helgason theorem, Poisson transform, and Furstenberg--Satake compactifications

Abstract: The connections between the objects mentioned in the title are used to give a short proof of the Cartan--Helgason theorem and a natural construction of the compactifications.


Riemannian metrics having common geodesics with Berwald metrics

Abstract: In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; this statement is a "Berwald" version of Hilbert's 4th problem. Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwald metric admits a Riemannian metric that is (nontrivially) geodesically equivalent to it. The system of PDE is linear and of Cauchy-Frobenius type, i.e., the derivatives of unknown functions are explicit expressions of the unknown functions. As an application (Corollary 2), we obtain that geodesic equivalence of an essentially Berwald metric and a Riemannian metric is always affine equivalence provided both metrics are complete.


Very Twisted Stable Maps

Abstract: Let X be a smooth projective Deligne-Mumford stack over an algebraically closed field k of characteristic 0. In this paper we construct the moduli stack of very twisted stable maps, extending the notion of twisted stable maps by Abramovich and Vistoli to allow for generic stabilizers on the source curves. We also consider the Gromov-Witten theory given by this construction.


A sufficient condition for intrinsic knotting of bipartite graphs

Abstract: We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant C_n such that a bipartite graph with exactly n \geq 5 vertices in one part and |E(G)| \geq 4 |V(G)| + C_n is intrinsically knotted. Finally, we classify bipartite graphs with ten or fewer vertices with respect to intrinsic knotting.


Summation of Hyperharmonic Series

Abstract: We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series.


About the Non-Integer Property of Hyperharmonic Numbers

Abstract: It was proven in 1915 by Leopold Theisinger that the $H_n$ harmonic numbers are never integers. In 1996 Conway and Guy have defined the concept of hyperharmonic numbers. The question naturally arises: are there any integer hyperharmonic numbers? The author gives a partial answer to this question and conjectures that the answer is "no".


On the Harary-Kauffman Conjecture and Turk's Head Knots

Abstract: The m,n Turk's Head Knot, THK(m,n), is an "alternating (m,n) torus knot." We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m \geq 5 is odd and n \geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m,2) when P_m is a Pell prime.


Can lattices in SL(n,R) act on the circle?

Abstract: This expository paper describes the various methods that have yielded partial results on the conjecture that if n > 2, then no lattice in SL(n,R) has a faithful action on the circle (by homeomorphisms). Topics include amenability, Kazhdan's property (T), bounded cohomology, bounded generation, and the Reeb-Thurston Stability Theorem.


On the Exel crossed product of topological covering maps

Abstract: For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^*$-algebras $\cros$ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of $C(X)$ into $\cros$ is a maximal abelian $C^*$-subalgebra of $\cros$; any nontrivial two sided ideal of $\cros$ has non-zero intersection with the imbedded copy of $C(X)$; a certain natural representation of $\cros$ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product $C^*$-algebras of homeomorphism dynamical systems.


Product-type non-commutative polynomial states

Abstract: In math/0702157, arXiv:0712.4185, we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on non-commutative polynomials, namely for various product states. In particular, we introduce a notion of a product-type state on polynomials, which covers all the non-commutative universal products and excludes some other familiar non-commutative products, and which guarantees a number of nice properties for the corresponding polynomials.


Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations

Abstract: In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations, by means of tools from nonlinear control theory. Lyapunov-based and Small-Gain feedback stabilization methods are exploited and numerous illustrating applications are presented for systems with a globally asymptotically stable equilibrium point. The obtained results can be used for the control of the global discretization error as well.


The complex Lorentzian Leech lattice and the bimonster (II)

Abstract: Let $D$ be the incidence graph of the projective plane over $\FF_3$. The Artin group of the graph $D$ maps onto the bimonster and a complex hyperbolic reflection group $\Gamma$ acting on 13 dimensional complex hyperbolic space $Y$. The generators of the Artin group are mapped to elements of order 2 (resp. 3) in the bimonster (resp. $\Gamma$). Let $Y^{\circ} \subseteq Y$ be the complement of the union of the mirrors of $\Gamma$. Daniel Allcock has conjectured that the orbifold fundamental group of $Y^{\circ}/\Gamma$ surjects onto bimonster. In this article we study the reflection group $\Gamma$. Our main result shows that there is homomorphism from the Artin group of $D$ to the orbifold fundamental group of $Y^{\circ}/\Gamma$, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in $\Gamma$. This answers a question in Allcock's article "A monstrous proposal" and takes a step towards the proof of Allcock's conjecture. The finite group $\op{PGL}(3, \FF_3) \subseteq \Aut(D)$ acts on $Y$ and fixes a complex hyperbolic line pointwise. We show that the restriction of $\Gamma$-invariant meromorphic automorphic forms on $Y$ to the complex hyperbolic line fixed by $\op{PGL}(3, \FF_3)$ gives meromorphic modular forms of level 13.


Calculation of the Norm of the Error Functional of Optimal Quadrature Formulas in the Space $W_2^{(2,1)}(0,1)$

Abstract: In this paper in the space $W_2^{(2,1)}(0,1)$ square of the norm of the error functional of a optimal quadrature formula is calculated.


Einstein metrics and GIT stability

Abstract: In this expository article we review the problem of finding Einstein metrics on compact K\"ahler manifolds and Sasaki manifolds. In the former half of this article we see that, in the K\"ahler case, the problem fits better with the notion of stability in Geometric Invariant Theory if we extend the problem to that of finding extremal K\"ahler metrics or constant scalar curvature K\"ahler (cscK) metrics. In the latter half of this paper we see that most of ideas in K\"ahler geometry extend to Sasaki geometry as transverse K\"ahler geometry. We also summarize recent results about the existence of toric Sasaki-Einstein metrics.


Aleph-zero-categorical groups and their completions

Abstract: We embed a countably categorical group G into a locally compact group c(G) with a non-trivial topology and study how topological properties of c(G) are connected with the structure of definable subgroups of G.


Generalization of l1 constraints for high dimensional regression problems

Abstract: We focus on the high dimensional linear regression $Y\sim\mathcal{N}(X\beta^{*},\sigma^{2}I_{n})$, where $\beta^{*}\in\mathds{R}^{p}$ is the parameter of interest. In this setting, several estimators such as the LASSO and the Dantzig Selector are known to satisfy interesting properties whenever the vector $\beta^{*}$ is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into $\mathcal{DC}(s)=\{\beta\in\mathds{R}^{p},\|X'(Y-X\beta)\|_{\infty}\leq s\}$ - using an $\ell_{1}$ distance for the Dantzig Selector and $\ell_{2}$ for the LASSO. For a well chosen $s>0$, this set is actually a confidence region for $\beta^{*}$. In this paper, we investigate the properties of estimators defined as projections on $\mathcal{DC}(s)$ using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives.


Logical tools for handling change in agent-based systems

Abstract: We give a unified approach to various results and problems of nonclassical logics


Defeasible inheritance systems and reactive diagrams

Abstract: We give an analysis of defeasible inheritance diagrams, also from the perspective of reactive diagrams.


Galois groups over function fields of positive characteristic

Abstract: We describe examples motivated by the work of Serre and Abhyankar.


Adequacy of Link Families

Abstract: Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links whose generating link has at most $n=12$ crossings.


On central extensions and definably compact groups in o-minimal structures

Abstract: We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).


Product formula for Atiyah-Patodi-Singer index classes and higher signatures

Abstract: We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.


Lineability of the set of bounded linear non-absolutely summing operators

Abstract: In this note we solve, except for extremely pathological cases, a question posed by Puglisi and Seoane-Sepulveda on the lineability of the set of bounded non-absolutely summing linear operators. We also show how the idea of the proof can be adapted to several related situations.


Universality of Crystallographic Pinning

Abstract: We study traveling waves for reaction diffusion equations on the spatially discrete domain $\Z^2$. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. Mallet-Paret has shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.


Quantum symmetries and quantum isometries of compact metric spaces

Abstract: We prove that a compact quantum group with faithful Haar state which has a faithful action on a compact space must be a Kac algebra, with bounded antipode and the square of the antipode being identity. The main tool in proving this is the theory of ergodic quantum group action on $C^*$ algebras. Using the above fact, we also formulate a definition of isometric action of a compact quantum group on a compact metric space, generalizing the definition given by Banica for finite metric spaces, and prove for certain special class of metric spaces the existence of the universal object in the category of those compact quantum groups which act isometrically and are `bigger' than the classical isometry group.


A Criterion for the Viability of Stochastic Semilinear Control Systems via the Quasi-Tangency Condition

Abstract: In this paper we study a criterion for the viability of stochastic semilinear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control systems. We obtain Nagumo's stochastic theorem and we present a method allowing to provide explicit criteria for the viability of smooth sets. We analyze the conditions characterizing the viability of the unit ball. The paper generalizes recent results from the deterministic framework.


H^1 and BMO for certain locally doubling metric measure spaces of finite measure

Abstract: In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,\rho,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and "isoperimetric" property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R^n,\rho_\phi, m_\phi), where dm_\phi=e^{-\phi} dx and \rho_\phi$ is the Riemannian metric corresponding to the length element ds^2=(1+ |grad\phi|)^2(dx_1^2+...+dx_n^2)$. This generalizes previous work of the last two authors for the Gauss space.


Computing the Newton polygon of the implicit equation

Abstract: We consider polynomially and rationally parameterized curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of its implicit polygon, i.e. of the implicit equation's Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations of point sets defined by Cayley's trick. We distinguish polynomial and rational parameterizations, where the latter may have the same or different denominators; the implicit polygon is shown to have, respectively, up to 4, 5, or 6 vertices.


Estimates for functions of the Laplacian on manifolds with bounded geometry

Abstract: In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel.


Entire solutions to equivariant elliptic systems with variational structure

Abstract: In the present paper we consider the system {\Delta}u - W_u (u) = 0, where u: R^n to R^n, for a class of potentials W: R^n to R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity.


Stability of invariant measures

Abstract: We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and it also differs from topologies induced by the Riesz Representation Theorem. It turns out that the constructed topology is a solution of a limit case of a $p$-optimal transport problem, for $p=\infty$.


Convergence of a Family of Series

Abstract: In this article we construct a family of expressions $\varepsilon(n)$. For each element E(n) from $\varepsilon(n)$, the convergence of the series $\sum_{n \ge n_E}{E(n)}$ can be determined in accordance to the theorems of this article. Some applications are also presented.


Spectral Connectivity Analysis

Abstract: Spectral kernel methods are techniques for transforming data into a coordinate system that efficiently reveals the geometric structure - in particular, the "connectivity" - of the data. These methods depend on certain tuning parameters. We analyze the dependence of the method on these tuning parameters. We focus on one particular technique - diffusion maps - but our analysis can be used for other methods as well. We identify the population quantities implicitly being estimated, we explain how these methods relate to classical kernel smoothing and we define an appropriate risk function for analyzing the estimators. We also show that, in some cases, fast rates of convergence are possible even in high dimensions.


Doubling property for biLipschitz homogeneous geodesic surfaces

Abstract: In this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups.


Multi-parameter Quantum Groups and Quantum Shuffles, (I)

Abstract: In this article, we study the multi-parameter quantum groups defined by generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category $\mathcal O$. This presentation will be convenient for our later discussions. We present two explicit descriptions here: as a Hopf 2-cocycle deformation, and as the multi-parameter quantum shuffle realization of the positive part.


Envelopes of holomorphy and holomorphic discs

Abstract: The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the continuity principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.


Formulae of Partial Reduction for Linear Systems of First Order Operator Equations

Abstract: This paper deals with reduction of non-homogeneous linear systems of first order operator equations with constant coefficients. An equivalent reduced system, consisting of higher order linear operator equations having only one variable and first order linear operator equations in two variables, is obtained by using the rational canonical form.


Differential transcendency in the theory of linear differential systems with constant coefficients

Abstract: In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant coefficients and to the question of differential transcendency.


Plans D'Experiences D'Information De Kullback-Leibler Minimale

Abstract: Experimental designs are tools which can dramatically reduce the number of simulations required by time-consuming computer codes. Because we don't know the true relation between the response and inputs, designs should allow one to fit a variety of models and should provide information about all portions of the experimental region. One strategy for selecting the values of the inputs at which to observe the response is to choose these values so they are spread evenly throughout the experimental region, according to "space-filling designs". In this article, we suggest a new method based on comparing the empirical distribution of the points in a design to the uniform distribution with the Kullback-Leibler information. The considered approach consists in estimating this difference or, reciprocally, the Shannon entropy. The entropy is estimated by a Monte Carlo method where the density function is replaced by its kernel density estimator or by using the nearest neighbor distances


Recovery of Missing Samples in Oversampling Formulas for Band Limited Functions

Abstract: In a previous paper, the author constructed frames and oversampling formulas for band-limited functions, in the framework of the theory of shift-invariant spaces. In this article we study the problem of recovering missing samples. We find a sufficient condition for the recovery of a finite set of missing samples. The condition is expressed as a linear independence of the components of a vector W over the space of trigonometric polynomials determined by the frequencies of the missing samples. We apply the theory to the derivative sampling of any order and we illustrate our results with a numerical experiment.


Calabi flow and projective embeddings

Abstract: Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X, which attempts to deform the given embedding into a balanced one. If L->X is an ample line bundle, considering embeddings via H^0(L^k) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldson's theorem relating balanced embeddings to metrics with constant scalar curvature [JDG 59(3):479-522, 2001]. In our proof we combine Donaldson's techniques with an asymptotic result of Liu-Ma [arXiv:math/0601260v2] which, as we explain, describes the asymptotic behaviour of the derivative of the map FS\circ Hilb whose fixed points are balanced metrics.


A strict non-standard inequality .999... < 1

Abstract: Is .999... equal to 1? Lightstone's decimal expansions yield an infinity of numbers in [0,1] whose expansion starts with an unbounded number of digits "9". We present some non-standard thoughts on the ambiguity of an ellipsis, modeling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value. In our model, the student resistance to the unital evaluation of .999... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students' hunch that .999... can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.


Transference principles and locally symmetric spaces

Abstract: We explain how the Transference Principles from Diophantine approximation can be interpreted in terms of geometry of the locally symmetric spaces $T_n=SO(n) \backslash SL(n,R) /SL(n,Z)$ with $n>1$, and how, via this dictionary, they become transparent geometric remarks and can be easily proved. Indeed, a finite family of linear forms is naturally identified to a locally geodesic ray in a space $T_n$ and the way this family is approximated is reflected by the heights at which the ray rises in the cuspidal end. The only difference between the two types of approximation appearing in a Transference Theorem is that the height is measured with respect to different rays in $W$, a Weyl chamber in $T_n$. Thus the Transference Theorem is equivalent to a relation between the Busemann functions of two rays in $W$. This relation is easy to establish on $W$, because restricted to it the two Busemann functions become two linear forms. Since $T_n$ is at finite Hausdorff distance from $W$, the same relation is satisfied up to a bounded perturbation on the whole of $T_n$.


New asymptotic estimates for spherical designs

Abstract: Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4, a_4 <= 7, a_5 <= 9, a_6 <= 11, a_7 <= 12, a_8 <= 16, a_9 <= 19, a_10 <= 22, and a_n < n/2*log_2(2n), n > 10.


On sutured Floer homology and the equivalence of Seifert surfaces

Abstract: We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopic minimal genus Seifert surfaces for the knot 8_3. A key ingredient for this technique is finding appropriate Heegaard diagrams for the sutured manifold associated to the complement of a Seifert surface.


On the $8n^2$-inequality

Abstract: We give a complete proof of the so called $8n^2$-inequality, a local inequality for the self-intersection of a movable linear system at an isolated centre of a non canonical singularity. The inequality was suggested and several times published by I.Cheltsov but some of his arguments are faulty. We explain the mistake and replace the faulty piece by a correct argument.


Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions

Abstract: This paper considers Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-\gamma} < C_\gamma \int_{R^d} V(x)_+^{\gamma + d/4} dx for gamma \geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_\gamma > 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained.


Good formal structures on flat meromorphic connections, I: Surfaces

Abstract: We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators, and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface, in order to verify the numerical criterion.


Formality of the constructible derived category for spheres: A combinatorial and a geometric approach

Abstract: We describe the constructible derived category of sheaves on the $n$-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods: As a combinatorial approach, we reformulate the problem in terms of representations of quivers and prove formality for the 2-sphere, for coefficients in a principal ideal domain. We give a suitable generalization of this formality result for the 2-sphere stratified in several points and their complement. As a geometric approach, we give a description of the underlying dg algebra in terms of differential forms, which allows us to prove formality for $n$-spheres, for real or complex coefficients.


Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function

Abstract: We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup.


Linear systems over P1xP1 with base points of multiplicity bounded by three

Abstract: We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general positions. As an application we obtain a classification of special linear systems on P1xP1 for which the multiplicities do not exceed 3.


Tori Embedded in R3 with Dense Principal Lines

Abstract: In this paper are given examples of tori T2 embedded in R3 with all their principal lines dense. These examples are obtained by stereographic projection of deformations of the Clifford torus in S3.


Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

Abstract: We prove that if U\subset\R^n is an open domain whose closure \overline{U} is compact in the path metric, and F is a Lipschitz function on \partial{U}, then for each \beta\in\R there exists a unique viscosity solution to the \beta-biased infinity Laplacian equation \beta |\nabla u| + \Delta_\infty u=0 on U that extends F, where \Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the \beta-biased \eps-game as follows. The starting position is x_0 \in U. At the k^\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \exp(\beta\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in \partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to 0 as \eps \to 0, where u is the unique extension of F to \overline{U} that satisfies comparison with \beta-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with \beta-exponential cones if and only if it is a viscosity solution to the \beta-biased infinity Laplacian equation.


Convex PBW-type Lyndon Basis and Restricted Two-parameter Quantum Group of Type G_2

Abstract: We construct finite-dimensional pointed Hopf algebras \mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \cite{HS}, which turn out to be of Drinfel'd doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G_2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right integrals. Besides these, we determine all of the Hopf algebra isomorphisms of u_{r,s}(G_2) in terms of the description of the sets of its left (right) skew-primitive elements.


Scale-invariant groups

Abstract: Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G_n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F\wr\Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS(1,m); the affine groups A\ltimes\Z^d, for any A\leq GL(\Z,d). However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.


Cube diagrams and 3-dimensional Reidemeister-like moves for knots

Abstract: In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.


On Volumes of Arithmetic Line Bundles

Abstract: We show an arithmetic generalization of the recent work of Lazarsfeld-Mustata which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.


Periodic Solutions with Singularities in Two Dimensions in the $n$-body Problem

Abstract: Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily extends to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated.


The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited

Abstract: In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF.


On the mod - p cohomology of Out(F_{2(p-1)}

Abstract: We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n).


Competitive or Weak Cooperative Stochastic Lotka-Volterra Systems Conditioned to Non-Extinction

Abstract: We are interested in the long time behavior of a two-type density-dependent biological population conditioned to non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka-Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned to non extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a $d$-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.


A note on the Poincar'e series of the invariants of ternary forms

Abstract: Analogue of Springer's formula for the Poincar\'e series of the algebra invariants of ternary form is found.


Stability of mean convex cones under mean curvature flow

Abstract: We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers.


Asymptotics for the survival probability in a killed branching random walk

Abstract: Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\gamma-\epsilon$, where $\gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $\epsilon\to 0$, the probability in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}$, where $\beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.


Quasi-alternating Montesinos links

Abstract: The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as provided bounds on unknotting number and slice genus, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.


Dispersion of volume under the action of isotropic Brownian flows

Abstract: We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive.


Biorthonormal Systems in Freud-type Weighted Spaces with Infinitely Many Zeros - An Interpolation Problem

Abstract: In a Freud-type weighted ($w$) space, introducing another weight ($v$) with infinitely many roots, we give a complete and minimal system with respect to $vw$, by deleting infinitely many elements from the original orthonormal system with respect to $w$. The construction of the conjugate system implies an interpolation problem at infinitely many nodes. Besides the existence, we give some convergence properties of the solution.


On Unique Additive Representations of Positive Integers and Some Close Problems

Abstract: Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion relation for Moser's numbers. We also study interesting properties of the sequence (rm_r(n-1)+1),n>=1, and its connection with some important problems. In particular, in the case of r=2 this sequence is surprisingly connected with the numbers solving the combinatorial Josephus-Groer problem. We pose also some open questions.


Constructing elliptic curves over finite fields with prescribed torsion

Abstract: We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N).


On The Negative K-Theory of Schemes in Finite Characteristic

Abstract: We study the negative $K$-theory of singular varieties over a field of positive characteristic and in particular, prove the vanishing of $K_i(X)$ for $i < -d-2$ for a $k$-variety of dimension $d$.


Greedy Polyominoes and first-passage times on random Voronoi tilings

Abstract: Let N be distributed as a Poisson random set on R^d with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v\in N}, where C_v is composed by points x in R^d that are closer to v than to any other v' in N. A polyomino P of size n is a connected union (in the usual R^d topological sense) of n tiles, and we denote by Pi_n the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pi_n for some functionals F, mainly when F(C_v) is the number of faces of C_v. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance.


Representations and characterizations of polynomial functions on chains

Abstract: We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [arXiv 0901.4888, arXiv 0808.2619] and by considering further conditions, namely comonotonic minitivity and maxitivity.


Categories. Beginning Course (in Russian)

Abstract: This is a short textbook on Category Theory for Russian speaking students. It consists of three chapters: Categories and Functors, Representable Functors (including Adjoint Functors and (Co)limits) and Tensor Categories.


A local optimal diastolic inequality on the two-sphere

Abstract: Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over the square of the length of a shortest closed geodesic. Our diastolic inequality asserts that this conjecture is to some extent locally true.


A Pentagonal Crystal, the Golden Section, alcove packing and aperiodic tilings

Abstract: A Lie theoretic interpretation is given to a pattern with five-fold symmetry occurring in aperiodic Penrose tiling based on isosceles triangles with length ratios equal to the Golden Section. Specifically a $B(\infty)$ crystal based on that of Kashiwara is constructed exhibiting this five-fold symmetry. It is shown that it can be represented as a Kashiwara $B(\infty)$ crystal in type $A_4$. Similar crystals with $(2n+1)$-fold symmetry are represented as Kashiwara crystals in type $A_{2n}$. The weight diagrams of the latter inspire higher aperiodic tiling. In another approach alcove packing is seen to give aperiodic tiling in type $A_4$. Finally $2m$-fold symmetry is related to type $B_m$.


Lower bounds for the principal genus of definite binary quadratic forms

Abstract: We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.


Generalization of the Logarithm Function and of the Exponential Function with Arbitrary Base

Abstract: The logarithm function and the exponential function are, by nature, base dependent. Thus, in this paper I introduces an arbitrary base in the logarithm and exponential functions, both dependent on $q$, in order to have $\log_a(x;q)$ and $a_q^x$. Some of the properties of these functions had been analyzed. The logarithm function was applied to the entropy which resulted in the $S_q = k[1 - \sum_{i = 1}^Wp_i^{q}]/[1 - e^{1 - q}]$ expression.


Extending Isotopies of Planar Continua

Abstract: In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$ is an isotopy starting at the identity. Can $h$ be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum $Z$ and use it to show that an accessible point is preserved during the isotopy. We show next that the isotopy can be extended over hyperbolic crosscuts. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an isotopy.


Remarks on Springer's representations

Abstract: We give an apriori description of a set of irreducible representations of a Weyl group which parametrize the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre concerning the conjugacy class of a power of a unipotent element in a connected reductive group.


Sharp Transitions in Making Squares

Abstract: In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to rapidly determine a subsequence whose product is a square (which we call a square product). In his lecture at the 1994 International Congress of Mathematicians, Pomerance observed that the following problem encapsulates all of the key issues: Select integers a_1, a_2, >... at random from the interval [1,x], until some (non-empty) subsequence has product equal to a square. Find good estimate for the expected stopping time of this process. A good solution to this problem should help one to determine the optimal choice of parameters for one's factoring algorithm, and therefore this is a central question. Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability 1-o(1) the first subsequence whose product equals a square occurs after at least J_0^{1-o(1)} integers have been selected, but no more than J_0, for an appropriate (explicitly determined) J_0=J_0(x). Herein we determine this expected stopping time up to a constant factor, tightening Pomerance's interval to $$[ (\pi/4)(e^{-\gamma} - o(1))J_0, (e^{-\gamma} + o(1)) J_0],$$ where $\gamma = 0.577...$ is the Euler-Mascheroni constant. We will also confirm the well established belief that, typically, none of the integers in the square product have large prime factors. We believe the upper of the two bounds to be asymptotically sharp.


On surfaces with p_g=2q-3

Abstract: We study minimal complex surfaces S of general type with q(S)=q and p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a fibration onto a curve of genus >=2. For these surfaces K^2=8\chi. In general we prove that K^2>=7\chi-1 and that the stronger inequality K^2\ge 8\chi holds under extra assumptions (e.g., if the canonical system has no fixed part or the canonical map has even degree). We also describe the Albanese map of S.


Hom-Algebras and Hom-Coalgebras

Abstract: The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.


Cohomology of graph hypersurfaces associated to certain Feynman graphs

Abstract: To any Feynman graph (with 2n edges) we can associate a hypersurface X\subset\PP^{2n-1}. We study the middle cohomology H^{2n-2}(X) of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer (Commun. Math. Phys. 267, 2006) have computed this cohomology for the first series of examples, the wheel with spokes graphs WS_n, n\geq 3. Using the same technique, we introduce the generalized zigzag graphs and prove that W_5(H^{2n-2}(X))=\QQ(-2) for all of them (with W_{*} the weight filtration). Next, we study primitively log divergent graphs with small number of edges and the behavior of graph hypersurfaces under the gluing of graphs.


On perfectly generating projective classes in triangulated categories

Abstract: We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman's Freyd--style representability theorem we give a new proof of Brown Representability Theorem.


Discrete multivariate distributions

Abstract: This article brings in two new discrete distributions: multidimensional Binomial distribution and multidimensional Poisson distribution. Those distributions were created in eventology as more correct generalizations of Binomial and Poisson distributions. Accordingly to eventology new laws take into account full distribution of events. Also, in article its characteristics and properties are described


Puiseux power series solutions for systems of equations

Abstract: We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.


Eventological theory of decision making for stock markets

Abstract: The eventological theory of decision-making, the theory of eventfull decision-making is a theory of decision-making based on eventological principles and using results of mathematical eventology; a theoretical basis of the practical eventology. The beginnings of this theory which have arisen from eventfull representation of the reasonable subject and his decisions in the form of eventological distributions (E-distributions) of sets of events and which are based on the eventological H-theorem are offered. The illustrative example of the eventological decision-making by the reasonable subject on his own eventfull behaviour in the financial or share market is considered.


On the size of minimal unsatisfiable formulas

Abstract: An unsatisfiable formula is called minimal if it becomes satisfiable whenever any of its clauses are removed. We construct minimal unsatisfiable $k$-SAT formulas with $\Omega(n^k)$ clauses for $k \geq 3$, thereby negatively answering a question of Rosenfeld. This should be compared to the result of Lov\'asz which asserts that a critically 3-chromatic $k$-uniform hypergraph can have at most $\binom{n}{k-1}$ edges.


Large deviations for random walks in random environment on a Galton-Watson tree

Abstract: Consider a random walk in random environment on a supercritical Galton--Watson tree, and let $\tau_n$ be the hitting time of generation $n$. The paper presents a large deviation principle for $\tau_n/n$, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time.


More results on greedy defining sets

Abstract: The greedy defining sets of graphs were appeared first time in [M. Zaker, Greedy defining sets of graphs, Australas. J. Combin, 2001]. We show that to determine the greedy defining number of bipartite graphs is an NP-complete problem. This result answers affirmatively the problem mentioned in the previous paper. It is also shown that this number for forests can be determined in polynomial time. Then we present a method for obtaining greedy defining sets in Latin squares and using this method, show that any $n\times n$ Latin square has a GDS of size at most $n^2-(n\log n)/4$. Finally we present an application of greedy defining sets in designing practical secret sharing schemes.


Spurious caustics of Dispersion Relation Preserving schemes

Abstract: A linear dispersive mechanism leading to a burst in the $L_\infty$ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated to interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.


On the branch curve of a general projection of a surface to a plane

Abstract: In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps.


On the moments and distribution of discrete Choquet integrals from continuous distributions

Abstract: We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [1] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions.


Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations

Abstract: We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the non variational version of the approach, consisting of simply solving the first order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research.


The geometric correspondence in some special cases

Abstract: The geometric Langlands correspondence for function fields over finite fields has been proved by Frenkel, Gaitsgory, Vilonen. The aim of this article is to write translation for curves over the complex field and prove the correspondence in some special cases and some new cases, outside the frame of the usual geometric Langlands correspondence.


Gaussian Correlation Conjecture for Symmetric Convex Sets

Abstract: Gaussian correlation conjecture states that the Gaussian measure of the intersection of two symmetric convex sets is greater or equal to the product of the measures.


Trimming and likelihood: Robust location and dispersion estimation in the elliptical model

Abstract: Robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated Mahalanobis distance. Here we analyze some one (or $k$)-step Maximum Likelihood Estimators computed on a subsample obtained with such a procedure. We introduce different models which arise naturally from the ways in which the discarded data can be treated, leading to truncated or censored likelihoods, as well as to a likelihood based on an only outliers gross errors model. Results on existence, uniqueness, robustness and asymptotic properties of the proposed estimators are included. A remarkable fact is that the proposed estimators generally keep the breakdown point of the initial (robust) estimators, but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate $n^{1/2}$, independently of the rate of convergence of the initial estimator.


First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes

Abstract: We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the $W$-invariant Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible root systems of types $A$, $B$, $D$. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms.


Sharp ill-posedness result for the periodic Benjamin-Ono equation

Abstract: We prove the discontinuity for the weak $ L^2(\T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(\T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author.


Generalized Bessel function of Type D

Abstract: We write down the generalized Bessel function associated with the root system of type $D$ by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type $D$.


Torus invariant divisors

Abstract: Using the language of polyhedral divisors and divisorial fans we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture X is given by a divisorial fan on a smooth projective curve Y. Cartier divisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one dimensional faces of it. Furthermore we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(D_h) will be determined by a subset of a weight polytope associated to h, and global sections of specific line bundles on the underlying curve Y.


Twisted convolution, pseudo-differential operators and Fourier modulation spaces

Abstract: We discuss continuity of the twisted convolution on (weighted) Fourier modulation spaces. We use these results to establish continuity results for the twisted convolution on Lebesgue spaces. For example we prove that if $\omega$ is an appropriate weight and $1\le p\le 2$, then $L^p_{(\omega)}$ is an algebra under the twisted convolution.


The curve of lines on a prime Fano threefold of genus 8

Abstract: We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(\Gamma,L)$, where $\Gamma$ is its Fano curve of lines and $L=O_{\Gamma}(1)$ is the theta-characteristic which gives a natural embedding $\Gamma \subset \matbb{P}^5$.


Rook placements in Young diagrams and permutation enumeration

Abstract: Given two operators $\hat D$ and $\hat E$ subject to the relation $\hat D\hat E -q \hat E \hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where $p=(1-q)/q^2$. This case naturally arises in the context of the PASEP, a random process whose partition function and stationary distribution are expressed using two operators $D$ and $E$ subject to the relation $DE-qED=D+E$ (matrix Ansatz). Using the link obtained by Corteel and Williams between the PASEP, permutation tableaux and permutations, we prove a conjecture of Corteel and Rubey about permutation enumeration. This result gives the generating function for permutations of given size with respect to the number of ascents and occurrences of the pattern 13-2, this is also the moments of the $q$-Laguerre orthogonal polynomials.


Differences of random Cantor sets and lower spectral radii

Abstract: We investigate the question under which conditions the algebraic difference between two independent random Cantor sets $C_1$ and $C_2$ almost surely contains an interval, and when not. The natural condition is whether the sum $d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that \emph{generically} it should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities $(p_0,p_1)$ the interior of the region where the Palis conjecture does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1>\sqrt{2}$ and $p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of $2\times 2$ matrices.