A mu-differentiable Lagrange multiplier rule

Abstract: We present some properties of the gradient of a mu-differentiable function. The Method of Lagrange Multipliers for mu-differentiable functions is then exemplified.


Normal numbers from Steinhaus viewpoint

Abstract: In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to Steinhaus, in 1923, is now used to introduce natural generalizations of the concept of normal numbers and explore their intrinsic probabilistic properties. We show that, in virtually all the cases, almost all real number in [0,1] is normal (with respect to this generalized concept). This procedure highlights some apparently hidden but interesting features of the Borel sigma-algebra and Lebesgue measure in [0,1].


The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

Abstract: We classify irreducible $\sigma$-twisted modules for the N=1 super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $\sigma$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $\sigma$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of generalized characters for $\mathcal{SW}(m)$. We are also able to relate irreducible $\mathcal{SW}(m)$ characters to characters for the triplet vertex algebra $\mathcal{W}(2m+1)$, studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857].


Stochastic relations of random variables and processes

Abstract: This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems.


Espace des modules de certains polyedres projectifs miroirs

Abstract: A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in $(0,\frac{\pi}{2}]$, and the union of $n$ copies of $\R^d$, when the polyhedron has the combinatorics of an \emph{ecimahedron}, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers $n$ and $d$ can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.


Secure two-dimensional tori are flat

Abstract: A riemannian manifold is secure if the geodesics between any pair of points in the manifold can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. The conjecture claims, in particular, that a riemannian torus of any dimension is secure if and only if it is flat. We prove this for two-dimensional tori.


Realisation of cycles by aspherical manifolds

Abstract: We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real symmetric tridiagonal (n=1)x(n+1) matrices. In particular, every homology class of every arcwise connected topological space can be realised by a continuous image of an aspherical manifold.


Carries, Shuffling and An Amazing Matrix

Abstract: The number of ``carries'' when $n$ random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of $n$ cards is repeatedly riffle shuffled. This gives new results for the statistics of carries and shuffling.


Improved Bounds on the Sizes of S.P Numbers

Abstract: A number which is S.P in base r is a positive integer which is equal to the sum of its base-r digits multiplied by the product of its base-r digits. These numbers have been studied extensively in The Mathematical Gazette. Recently, Shah Ali obtained the first effective bound on the sizes of S.P numbers. Modifying Shah Ali's method, we obtain an improved bound on the number of digits in a base-r S.P number. Our bound is the first sharp bound found for the case r=2.


The structural equations of Cartan and the wave solution Einstein's equation

Abstract: Work consists of introduction, two chapters, conclusion and four applications. In this work is examined the condition, with which the wave space metrics of Riemann- Cartan is the solution of Einstein equation in the void. Geometric structures were for this purpose studied on the differentiated variety: connectedness, curvature and twisting connectedness. For this was used this analytical apparatus of differential geometry as the calculation of exterior forms. In the first chapter the following concepts are examined: - variety; - vectors and 1- forms on the varieties; - metric tensor, connectedness and the covariant derivative; - form with the values in the vector spaces; - the structural equations of Cartan. The second chapter is dedicated to the presence of condition, during which wave certificate is the solution of Einstein equation in the void. Using the first and second structural equations of Cartan for the variety without twisting of that allotted by wave certificate, they were calculated: the 1- forms of connectedness and the coefficients of connectedness, and also Riemann tensor, Ricci tensor and scalar of curvature. In appendix 1 the derivation of formula for the coefficients of connected-ness in the space with the twisting is given. In appendix 2 the calculation of the components of the 1- forms of connectedness is carried out. In appendix 3 the calculations of the components of the 2- forms of curvature are presented. In appendix 4 the components of Ricci tensor are calculated.


Equivariant structure constants for ordinary and weighted projective space

Abstract: We compute the integral torus-equivariant cohomology ring for weighted projective space for two different torus actions by embedding the cohomology in a sum of polynomial rings $\oplus_{i=0}^n \Z[t_1, t_2,..., t_n]$. One torus action gives a result complementing that of Bahri, Franz, and Ray. For the other torus action, each basis class for weighted projective space is a multiple of the basis class for ordinary projective space; we identify each multiple explicitly. We also give a simple formula for the structure constants of the equivariant cohomology ring of ordinary projective space in terms of the basis of Schubert classes, as a sequence of divided difference operators applied to a specific polynomial.


Characteristic classes for Riemannian foliations

Abstract: The purpose of this paper is to both survey and offer some new results on the non-triviality of the characteristic classes of Riemannian foliations. We give examples where the primary Pontrjagin classes are all linearly independent. The independence of the secondary classes is also discussed, along with their total variation. Finally, we give a negative solution of a conjecture that the map of classifying spaces $\FRGq \to \FGq$ is trivial for codimension $q > 1$.


Hypergeometric formulas for lattice sums and Mahler measures

Abstract: We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and special values of $L$-series of elliptic curves.


Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class

Abstract: Using the harmonic map heat flow, we construct an energy class for wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that no stationary, travelling, self-similar, or degenerate wave maps exist in this energy class. These results form three of the five claims required in our earlier paper (arXiv:0805.4666) to prove global regularity for such wave maps. (The conditional claim of large data well-posedness is one of the remaining claims required in that paper.)


A Graph-Theoretic Approach to a Partial Order of Knots and Links

Abstract: We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knots and links with the crossing number less than or equal to six. The proofs are given by graph-theoretic methods.


Stabilization under shared communication with message losses and its limitations

Abstract: We consider a synthesis problem for a remotely controlled linear system where the communication is constrained because of the shared and unreliable nature of the channel. Modeling the constraints by a periodic transmission scheme and random message losses, we present an H-infinity design framework and study the limitations in the communication required for stabilization.


A Quiver Construction of Symmetric Crystals

Abstract: In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type $B$ are described by using the lower global basis of symmetric crystals of $V_\theta(\lambda)$. In this paper, we prove the existence of crystal bases and global bases of $V_\theta(0)$ for any symmetric quantized Kac-Moody algebra by using a geometry of quivers (with a Dynkin diagram involution). This is analogous to George Lusztig's geometric construction of $U_v^-$ and its lower global basis.


Chromogeometry

Abstract: Chromogeometry brings together Euclidean geometry (called blue) and two relativistic geometries (called red and green), in a surprising three-fold symmetry. We show how the red and green `Euler lines' and `nine-point circles' of a triangle interact with the usual blue ones, and how the three orthocenters form an associated triangle with interesting collinearities. This is developed in the framework of rational trigonometry using quadrance and spread instead of distance and angle. The former are more suitable for relativistic geometries.


Divisor and Totient Functions Estimates

Abstract: New unconditional estimates of the divisor and totient functions are contributed to the literature. These results are consistent with the Riemann hypothesis and seem to solve the Nicolas inequality for all sufficiently large integers.


A noncommutative extended de Finetti theorem

Abstract: The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and K\"ummerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.


On Lehner's `free' noncommutative analogue of De Finetti's theorem

Abstract: Inspired by Lehner's results on exchangeability systems we define `weak conditional freeness' and `conditional freeness' for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process.


A reduction method for noncommutative $L_p$-spaces and applications

Abstract: We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.


White Noise Calculus and Hamiltonian of a Quantum Stochastic Process

Abstract: A white noise quantum stochastic calculus is developped using classical measure theory as mathematical tool. Wick's and Ito's theorems have been established. The simplest quantum stochastic differential equation has been solved, unicity and the conditions for unitarity have been proven. The Hamiltonian of the associated one parameter strongly continuous group has been calculated explicitely.


Recurrence Relations for Strongly q-Log-Convex Polynomials

Abstract: We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.


Inverse zero-sum problems and algebraic invariants

Abstract: In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two.


The Slice Algorithm For Irreducible Decomposition of Monomial Ideals

Abstract: Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.


Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

Abstract: We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products $\Gamma\wr\SG_n$ and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon's multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.


On the total curvature of semialgebraic graphs

Abstract: We define the total curvature of a semialgebraic embedding of a graph in the 3-dimensional Euclidean space. We prove that it satisfies a Chern-Lashof type inequality and we describe when the equality holds. We also prove a generalization of a classical result of Fary and Milnor stating that certain graphs cannot be knotted if they are not too curved. The techniques employed are Morse theoretic.


Estimation of a diffusion model with trends taking in account the extremes. Application to temperature in France

Abstract: We built a model of the daily temperature based on a diffusion process and address to extreme values not taken into account in the literature on this kind of models. We first study, using non parametric tools, the trends on mean and variance. In a second step we estimate a stationary model first non parametrically and then using likelihood methods. Extreme values are taken into account in the estimation of model and to obtain a definitive estimation we use in a specific framework extreme theory for diffusions. A test of suitable model by simulation is done.


A matrix interpolation between classical and free max operations: I. The univariate case

Abstract: Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.


Quantum Group of Orientation preserving Riemannian Isometries

Abstract: We formulate a quantum group analogue of the group of orinetation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly $R$-twisted in the sense of a paper of one of the authors, and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any `good ' Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on $SU_\mu(2)$ and $S^2_{\mu 0}$ are dicussed.


Noncommutative independence from the braid group $B_\infty$

Abstract: We introduce `braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group $B_\infty$. It provides an extension of exchangeability which is tied to the symmetric group $S_\infty$. Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem (of C. K\"{o}stler). This endows the braid groups $B_n$ with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms (of R. Gohm) with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of $B_\infty$ and the irreducible subfactor with infinite Jones index in the non-hyperfinite $II_1$-factor $L(B_\infty)$ related to it. Our investigations reveal a new presentation of the braid group $B_\infty$, the `square root of free generator presentation' $F_\infty^{1/2}$. These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.


On the Briancon-Skoda theorem on a singular variety

Abstract: Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briancon-Skoda theorem for the local ring $\mathcal{O}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.


Optimal time and space regularity for solutions of degenerate differential equations

Abstract: We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.


Integral representation of renormalized self-intersection local times

Abstract: In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.


On arithmetic partitions of Z_n

Abstract: Generalizing a classical problem in enumerative combinatorics, Mansour and Sun counted the number of subsets of $\Z_n$ without certain separations. Chen, Wang, and Zhang then studied the problem of partitioning $\Z_n$ into arithmetical progressions of a given type under some technical conditions. In this paper, we improve on their main theorems by applying a convolution formula for cyclic multinomial coefficients due to Raney-Mohanty.


Principal bundles, quasi-abelian varieties and structure of algebraic groups

Abstract: We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global functions). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes.


Distinguished Orbits of Reductive Groups

Abstract: We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\mathbb C$ which acts on $V^\mathbb C := V \otimes \mathbb C$. Borel-Harish-Chandra show that $G^\mathbb C \cdot v \cap V$ is a finite union of $G$-orbits; moreover, $G^\mathbb C \cdot v$ is closed if and only if $G\cdot v$ is closed. We show that the same result holds not just for closed orbits but for the so-called distinguished orbits. An orbit is called distinguished if it contains a critical point of the norm squared of the moment map on projective space. Our main result compares the complex and real settings to show $G\cdot v$ is distinguished if and only if $G^\mathbb C \cdot v$ is distinguished. In addition, we show that if an orbit is distinguished, then under the negative gradient flow of the norm squared of the moment map the entire $G$-orbit collapses to a single $K$-orbit. This result holds in both the complex and real settings. We finish with some applications to the study of the left-invariant geometry of Lie groups.


On the difference between solutions of discrete tomography problems

Abstract: We consider the problem of reconstructing binary images from their horizontal and vertical projections. We present a condition that the projections must necessarily satisfy when there exist two disjoint reconstructions from those projections. More generally, we derive an upper bound on the symmetric difference of two reconstructions from the same projections. We also consider two reconstructions from two different sets of projections and prove an upper bound on the symmetric difference in this case.


Ricci-flat K\"ahler metrics on crepant resolutions of K\"ahler cones

Abstract: We prove that a crepant resolution of a Ricci-flat K\"ahler cone X admits a complete Ricci-flat K\"ahler metric asymptotic to the cone metric in every K\"ahler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat K\"ahler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat K\"ahler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.


An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number

Abstract: We give a simple axiomatic definition of a rational-valued invariant s(W,V,e) of triples (W,V,e), where W is a (smooth, oriented, closed) 6-manifold and V is a 3-submanifold of W, and where e is a second rational cohomology class of the complement of V satisfying a certain condition. The definition is stated in terms of cobordisms of such triples and the signature of 4-manifolds. When W = S^6 and V is a smoothly embedded 3-sphere, and when e/2 is the Poincare dual of a Seifert surface of V, the invariant coincides with -8 times Haefliger's embedding invariant of (S^6,V). Our definition recovers a more general invariant due to Takase, and contains a new definition for Milnor's triple linking number of algebraically split 3-component links in R^3 that is close to the one given by the perturbative series expansion of the Chern-Simons theory of links in R^3.


A sharp bound for the reconstruction of partitions

Abstract: Answering a question of Cameron, Pretzel and Siemons proved that every integer partition of $n\ge 2(k+3)(k+1)$ can be reconstructed from its set of $k$-deletions. We describe a new reconstruction algorithm that lowers this bound to $n\ge k^2+2k$ and present examples showing that this bound is best possible.


Cohomology and Support Varieties for Lie Superalgebras of Type W(n)

Abstract: Boe, Kujawa and Nakano recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper we calculate the cohomology ring of the Cartan type Lie superalgebra W(n) relative to the degree zero component W(n)_{0} and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite dimensional W(n)-supermodules which are completely reducible over W(n)_{0}. We calculate the support varieties of all simple supermodules in this category. Remarkably our computations coincide with the prior notion of atypicality for Cartan type superalgebras due to Serganova. We also present new results on the realizability of support varieties which hold for both classical and Cartan type superalgebras.


Regularity conditions for arbitrary Leavitt path algebras

Abstract: We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field $K$. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph $E$: (1) $L_K(E)$ is von Neumann regular. (2) $L_K(E)$ is $\pi$-regular. (3) $E$ is acyclic. (4) $L_K(E)$ is locally $K$-matricial. (5) $L_K(E)$ is strongly $\pi$-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.


Improved testing inference in mixed linear models

Abstract: Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Oftentimes, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test and also to a test obtained from a modified profile likelihood function. Our results generalize those in Zucker et al. (Journal of the Royal Statistical Society B, 2000, 62, 827-838) by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report numerical evidence which shows that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed.


Survey on geometric group theory

Abstract: This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for proper actions, measurable group theory, some open problems.


Observations regarding compactness in the $\overline{\partial}$-Neumann problem

Abstract: We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\mathbb{C}^{n}$ and for complete Hartogs domains in $\mathbb{C}^{2}$.


Convexity in semi-algebraic geometry and polynomial optimization

Abstract: We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.


Prescription of Q-curvature on closed Riemannian manifolds

Abstract: In this paper it is hown that given any smooth, positive function f on a closed, smooth manifold of dimension greater than four and with positive Paneitz invariant, there exists a metric on M such that $Q_g$ = f.


Connected sums of closed Riemannian manifolds and fourth order conformal invariants

Abstract: In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities change, analogous to how the Yamabe constants and the Yamabe invariants do, under the connected sum operations.


A structure theorem of Dirac-harmonic maps between spheres

Abstract: For an arbitrary Dirac-harmonic map $(\phi,\psi)$ between compact oriented Riemannian surfaces, we shall study the zeros of $|\psi|$. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of $|\psi|$ and the genus of $M$ and $N$. On the basis, we could clarify all of nontrivial Dirac-harmonic maps from $S^2$ to $S^2$.


Multiple fibers of del Pezzo fibrations

Abstract: We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity $\ge 6$. We also obtain a rough classification possible configurations of singular points on multiple fibers and give some examples.


Gap conjecture for 3-dimensional canonical thresholds

Abstract: We prove that the interval $(5/6, 1)$ contains no 3-dimensional canonical thresholds.


A Generalization of Fulton-MacPherson Configuration Spaces

Abstract: Presented is a wonderful compactification of n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it is the Fulton-MacPherson configuration space.


Cohomogeneity one disk bundles with normal homogeneous collars

Abstract: We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in $\{2,3,4,6,8\}$. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.


A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

Abstract: Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection $D$ is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D is D-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.


Approximation of conformal mappings by circle patterns

Abstract: A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in $(0,\pi)$. Two sequences of circle patterns are employed to approximate a given conformal map $g$ and its first derivative. For the domain of $g$ we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of $g'$ ($|g'|$ or $\arg g'$). For quasicrystallic circle patterns the convergence result is strengthened to $C^\infty$-convergence on compact subsets.


On Pseudo Algebraically Closed Extensions of Fields

Abstract: The notion of `Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had new applications. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. This vastly generalizes earlier works of Jarden-Razon, Jarden, and Jarden and the author. This also leads to a classification of all finite PAC extensions which in turn proves the `bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g.\ transitiveness of PAC extensions.


On reduction integer programs to knapsack problem

Abstract: Let $A$ be an integral nonnegative $m\times n$ matrix, $b$ be an integral nonnegative vector. It is suggested new method for reduction of integer program $\max \{cx| Ax=b, x\ge 0, x\in\mathbf{Z}^n\}$ to knapsack problem $\max \{c'x| fAx=fb, x\ge 0, x\in\mathbf{Z}^n\}$.


Almost Paracontact Manifolds

Abstract: In this paper eleven basic classes of almost paracontact manifolds are introduced and some examples are constructed.


G\'eom\'etrie et cognition; l'exemple du continu

Abstract: In this paper I propose the idea to establish a clear distinction between the foundations of truth and the foundations of meaning in Mathematics. I explore on the most basic example, the mathematical line, the possibility that the foundations of its meaning are provided by a protomathematical object resulting from the identification by our perceptual system of the visual line and the vestibular line, a point of view suggested by recent results of neurophysiology.


Special Examples of Diffusions in Random Environment

Abstract: In this note we present some examples of diffusions in random environment whose asymptotic behavior is rather surprising. We construct a family of diffusions that are small perturbations of Brownian motion with non-vanishing expected local drift under the static measure of the environment but where the ballistic behavior is lost. As slight modifications of this collection of diffusions we also provide examples with ballistic behavior where the non-vanishing limiting velocity points to a direction opposite to the expected local drift under the static measure.


A prolongation-projection algorithm for computing the finite real variety of an ideal

Abstract: We provide a real algebraic symbolic-numeric algorithm for computing the real variety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may not be. Our approach uses sets of linear functionals on $R[X]$, vanishing on a given set of polynomials generating $I$ and their prolongations up to a given degree, as well as on polynomials of the real radical ideal of $I$, obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm. This algorithm, based on standard numerical linear algebra routines and semidefinite optimization, combines techniques from previous work of the authors together with an existing algorithm for the complex variety. This results in a unified methodology for the real and complex cases.


Small overlap monoids II: automatic structures and normal forms

Abstract: We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene's theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest.


Combinatorics of line arrangements and characteristic varieties

Abstract: This paper has been withdrawn by the author due to a sheaf-theoretic error, in the end of the proof of the main theorem.


Crossed products by twisted partial actions and graded algebras

Abstract: For a twisted partial action \Theta of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_\Theta G is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_\Theta G for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.


On the mean square of the error term for the two-dimensional divisor problem (I)

Abstract: Suppose $a$ and $b$ are two fixed positive integers such that $(a,b)=1.$ In this paper we shall establish an asymptotic formula for the mean square of the error term $\Delta_{a,b}(x)$ of the general two-dimensional divisor problem.


The topological cyclic Deligne conjecture

Abstract: Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2-discs operad.


What does a random contingency table look like?

Abstract: Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer vectors such that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of non-negative mxn integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D in Sigma(R,C) is close with high probability to a particular matrix ("typical table'') Z defined as follows. We let g(x)=(x+1) ln(x+1)-x ln x for non-negative x and let g(X)=sum_ij g(x_ij) for a non-negative matrix X=(x_ij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative mxn matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.


Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials

Abstract: Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $\deg_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \geq 0, P \leq 0, P \in {\mathcal P} \cup {\mathcal Q}$. We describe an algorithm for computing the the Betti numbers of $S$. The complexity of the algorithm is bounded by $(\ell s m d)^{2^{O(m+k)}}$. The complexity of the algorithm interpolates between the doubly exponential time bounds for the known algorithms in the general case, and the polynomial complexity in case of semi-algebraic sets defined by few quadratic inequalities known previously. Moreover, for fixed $m$ and $k$ this algorithm has polynomial time complexity in the remaining parameters.


Harmonic measures versus quasiconformal measures for hyperbolic groups

Abstract: We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \qc measures on the boundary of the group.


Nested iterative algorithms for convex constrained image recovery problems

Abstract: The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be non-smooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.


On elliptic modular foliations

Abstract: In this article we consider the three parameter family of elliptic curves $E_t: y^2-4(x-t_1)^3+t_2(x-t_1)+t_3=0, t\in\C^3$ and study the modular holomorphic foliation $\F_{\omega}$ in $\C^3$ whose leaves are constant locus of the integration of a 1-form $\omega$ over topological cycles of $E_t$. Using the Gauss-Manin connection of the family $E_t$, we show that $\F_{\omega}$ is an algebraic foliation. In the case $\omega=\frac{xdx}{y}$, we prove that a transcendent leaf of $\F_{\omega}$ contains at most one point with algebraic coordinates and the leaves of $\F_{\omega}$ corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family $E_t$, we find a uniformization of $\F_{\omega}$ in $T$, where $T\subset \C^3$ is the locus of parameters $t$ for which $E_t$ is smooth. We find also a real first integral of $\F_\omega$ restricted to $T$ and show that $\F_{\omega}$ is given by the Ramanujan relations between the Eisenstein series.


$q$-Partition Algebra Combinatorics

Abstract: We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a polynomial in $q$ that specializes as $d_{n,r}(1) = n^r$ and $d_{n,r}(0) = B(r)$, the $r$th Bell number. We compute $d_{n,r}(q)$ in two ways. The first is purely combinatorial. We show that $d_{n,r}(q) = \sum_\lambda f^\lambda(q) m_r^\lambda$, where $f^\lambda(q)$ is the $q$-hook number and $m_r^\lambda$ is the number of $r$-vacillating tableaux. Using a Schensted bijection, we write this as a sum over integer sequences which, when $q$-counted by inverse major index, gives $d_{n,r}(q)$. The second way is algebraic. We find a basis of $\IR_q^r$ that is indexed by $n$-restricted $q$-set partitions of $\{1,..., r\}$, and we show that there are $d_{n,r}(q)$ of these.


A note on palindromic $\delta$-vectors for certain rational polytopes

Abstract: Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart $\delta$-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.


Sums of squares and orthogonal integral vectors

Abstract: Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible.


On primitive Dirichlet characters and the Riemann hypothesis

Abstract: For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-\gamma} \phi(2n_k)/\log\log(2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $\gamma$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $\phi(n)$ is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many $k$ for which the same inequality holds and infinitely many $k$ for which it fails to hold.


Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Abstract: We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection $q: A\to A$ whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that $q$ is not a local homeomorphism precisely at $\Ok$. We suppose also that there is a rational function $p$ with the Julia set $J$ such that $(A,q)$ and $(J,p)$ are conjugate. We prove that if $A$ has bounded turning and $p$ has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS $\{\lambda z,\lambda z+1\}$ where $\lambda$ is a complex parameter in the unit disk, such that its attractor $A_\lam$ is a dendrite, which happens whenever $\Ok$ is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map $q_\lam$ on $A_\lam$. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map $p_c(z)=z^2+c$, with the Julia set $J_c$ such that $(A_\lam,q_\lam)$ and $(J_c,p_c)$ are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.


The Planar Rook Algebra and Pascal's Triangle

Abstract: We study the combinatorial representation theory of the ``planar rook algebra" $P_n$. This algebra has a basis consisting of planar rook diagrams and multiplication given by diagram concatenation. For each integer $0 \le k \le n$, we construct natural representations $V^n_k$ which form a complete set of non-isomorphic, irreducible $P_n$-representations. We explicitly decompose the regular representation of $P_n$ into a direct sum of irreducible modules. We compute the Bratteli diagram for the tower of algebras $P_0 \subseteq P_1 \subseteq P_2 \subseteq ...$ and show that this Bratteli diagram is Pascal's triangle. In fact, we show that many of the binomial identities, both additive and multiplicative, have interpretations in terms of the representation theory of the planar rook algebra.


Short note on additive sequences and on recursive processes

Abstract: Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver ratio, but a dense set of ratio limits that corresponds to an infinity of conceivable recursive additive rules. We shall show that some of these limits have nice properties. Identities involving Fibonacci and Lucas sequences will be viewed as special cases of more general identities. We shall show that some properties of the Pascal Triangle belong also to other similar objects. In Dynamical Systems and Chaos Theory we shall encounter weird orbits, whose order is higher than the number of its distinct elements and, beyond the chaos point, a rather unexpected belated convergence to 0, after a pseudo chaotic behaviour during as many terms as one may wish. Time and again, we shall find here the Feigenbaum constant. In Formal Grammars we shall see that recursive rules applied to concatenation are sometimes equivalent to formal grammars although generally more restrictive.


Localization of algebras over coloured operads

Abstract: We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.


On localization in Kronecker's diophantine theorem

Abstract: We study the localization problem appearing in Kronecker's diophantine theorem. We introduce a probabilistic approach allowing to extend for general $\Q$-linearly independent sequences a result of T\'uran concerning the sequence $ (\log p_\ell)$, $p_\ell$ being the $\ell$-th prime.


Metric connections in projective differential geometry

Abstract: We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.


Gradient estimates for a nonlinear parabolic equation under Ricci flow

Abstract: Let $(M,g(t))$, $0\le t\le T$, be a n-dimensional complete noncompact manifold, $n\ge 2$, with bounded curvatures and metric $g(t)$ evolving by the Ricci flow $\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$. We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation $\frac{\1 u}{\1 t}=\Delta u-au\log u-qu$ where $a\in\R$ is a constant and $q$ is a smooth function on $M\times [0,T]$.


Recognizing indecomposable subcontinua of surfaces from their complements

Abstract: We prove two theorems which allow one to recognize indecomposable subcontinua of closed surfaces without boundary. If $X$ is a subcontinuum of a closed surface $S$, we call the components of $S \setminus X$ the complementary domains of $X$. We prove that a continuum is either indecomposable or the union of two indecomposable continua whenever it has a sequence of distinct complementary domains whose boundaries limit to the continuum in the Hausdorff metric. We define a slightly stronger condition on the complementary domains of a continuum, called the double-pass condition, which we conjecture is equivalent to indecomposability of the continuum. We prove that this is so for continua which are not the boundary of one of their complementary domains.


Relations on Generalized Degree Sequences

Abstract: We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley.


Isometry types of profinite groups

Abstract: Let T be a rooted tree and Iso(T) be the group of isometries of T. Using model-theoretic tools we study closed subgroups G of Iso(T) with respect to the number of conjugacy classes of Iso(T) having representatives in G.


Infinite sequence of fixed point free pseudo-Anosov homeomorphisms

Abstract: We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy-Veech induction strategy.


On Unitary Representations of GL2n Distinguished by the Symplectic Group

Abstract: We provide a family of representations of GL(2n) over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp(2n)- distinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, re- lies on the theory of automorphic forms. The results of this paper together with later works by the authors imply that the family of representations studied in this paper contains all irreducible, unitary representations of the general linear group that are distin- guished by the symplectic group.


Index Theory for Coverings

Abstract: We establish the basics of the analysis of operators on coverings of manifolds with cylindrical ends with a group of deck transformations $\Gamma$. We prove the $\Gamma$-analogue of the Atiyah-Patodi-Singer formula for Dirac operators on such coverings.


Invariant tensors and cellular categories

Abstract: Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U. Then the centraliser algebra of W has a basis called the dual canonical basis which gives an integral form. We show that this integral form is cellular by using results due to Lusztig.


About the maximal rank of 3-tensors over the real and the complex number field

Abstract: High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results.


Herleitung von Skalarprodukten aus Symmetrieprinzipien

Abstract: This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with a half-line on its boundary. From there the existence of an invariant scalar product is deduced, which then also implies Pythagoras theorem in a quite precise form. This is in contrast to the usual procedure to model ambient space by asking for a distinguished scalar product and using Pythagoras theorem as known from high school to connect with reality.


Mackey functors and bisets

Abstract: For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.


Lower bound for the remainder in the prime-pair conjecture

Abstract: For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an explicit positive constant C_{2r}. A heuristic argument indicates that the remainder e_{2r}(x) in this approximation cannot be of lower order than x^beta, where beta is the supremum of the real parts of zeta's zeros. The argument also suggests an approximation for pi_{2r}(x) similar to one of Riemann for pi(x).


On the number of components of a complete intersection of real quadrics

Abstract: Our main results concern complete intersections of three real quadrics. We prove that the maximal number $B^0_2(N)$ of connected components that a regular complete intersection of three real quadrics in $\Bbb{P}^N$ can have differs at most by one from the maximal number of ovals of the submaximal depth $[(N-1)/2]$ of a real plane projective curve of degree $d=N+1$. As a consequence, we obtain a lower bound \smash{$\frac14 N^2+O(N)$} and an upper bound \smash{$\frac38 N^2+O(N)$} for $B^0_2(N)$.


Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data

Abstract: The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the B\'enard system are also discussed.


The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity

Abstract: We are concerned with the so-called Boussinesq equations with partial viscosity. These equations consist of the ordinary incompressible Navier-Stokes equations with a forcing term which is transported {\it with no dissipation} by the velocity field. Such equations are simplified models for geophysics (in which case the forcing term is proportional either to the temperature, or to the salinity or to the density). In the present paper, we show that the standard theorems for incompressible Navier-Stokes equations may be extended to Boussinesq system despite the fact that there is no dissipation or decay at large time for the forcing term. More precisely, we state the global existence of finite energy weak solutions in any dimension, and global well-posedness in dimension $N\geq3$ for small data. In the two-dimensional case, the finite energy global solutions are shown to be unique for any data in $L^2(\R^2).$


Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces

Abstract: This paper is devoted to the study of the Cauchy problem for the Boussinesq system with partial viscosity in dimension $N\geq3.$ First we prove a global existence result for data in Lorentz spaces satisfying a smallness condition which is at the scaling of the equations. Second, we get a uniqueness result in Besov spaces with {\it negative} indices of regularity (despite the fact that there is no smoothing effect on the temperature). The proof relies on a priori estimates with loss of regularity for the nonstationary Stokes system with convection. As a corollary, we obtain a global existence and uniqueness result for small data in Lorentz spaces.


Hochschild two-cocycles and the good triple $(As,Hoch,Mag^\infty)$

Abstract: Hochschild two-cocycles play an important role in the deformation \`a la Gerstenhaber of associative algebras. The aim of this paper is to introduce the category of Hoch-algebras whose objects are associative algebras equipped with an extra magmatic operation \succ verifying the Hochschild two-cocycle relation: (x \succ y)*z+ (x*y)\succ z= x\succ (y*z)+ x*(y\succ z). The free Hoch-algebra over a K-vector space is given in terms of planar rooted trees and the triples of operads (As,Hoch, Mag^\infty) endowed with the infinitesimal relations are shown to be good. We then obtain an equivalence of categories between connected infinitesimal Hoch-bialgebras and Mag^\infty-algebras.


On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces

Abstract: We give a definition of the spectral flow for paths of bounded essentially hyperbolic operators on a Banach space. The spectral flow induces a group homomorphism on the fundamental group of every connected component of the space of essentially hyperbolic operators. We prove that this homomorphism completes the exact homotopy sequence of a Serre fibration. This allows us to characterise its kernel and image and to produce examples of spaces where it is not injective or not surjective, unlike what happens for Hilbert spaces. For a large class of paths, namely the essentially splitting, the spectral flow of $ A $ coincides with $ -\ind(F_A) $, the Fredholm index of the differential operator $ F_A (u) = u' - A u $.


Formality of cyclic cochains

Abstract: We prove Kontsevich's cyclic formality conjecture.


Finding the sum of any series from a given general term

Abstract: Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor expansion of y about x. In sections 21 to 23 Euler uses the formula to find expressions for the sums of the nth powers of the first x integers. He gives the general formula for this, and works it out explicitly up to n=16. In sections 25 to 28 he applies the summation formula to getting approximations to partial sums of the harmonic series, and in sections 29 to 30 to partial sums of the reciprocals of the odd positive integers. In sections 31 to 32, Euler gets an approximation to zeta(2); in section 33, approximations for zeta(3) and zeta(4). I found David Pengelley's paper "Dances between continuous and discrete: Euler's summation formula", in the MAA's "Euler at 300: An Appreciation", edited by Robert E. Bradley, Lawrence A. D'Antonio, and C. Edward Sandifer, very helpful and I recommend it if you want to understand the summation formula better.


On the fundamental solution of an elliptic equation in nondivergence form

Abstract: We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, ${\mathcal L}(x,\del_x)=a_{ij}(x)\del_i\del_i$, for $n\geq 3$. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed $y$, we construct a solution of ${\mathcal L}Z_y(x)=0$ for $0<|x-y|<\e$ with explicit leading order term which is $O(|x-y|^{2-n}e^{I(x,y)})$ as $x\to y$, where $I(x,y)$ is given by an integral and plays an important role for the fundamental solution: if $I(x,y)$ approaches a finite limit as $x\to y$, then we can solve ${\mathcal L}(x,\del_x)F(x,y)=\de(x-y)$, and $F(x,y)$ is asymptotic as $x\to y$ to the fundamental solution for the constant coefficient operator ${\mathcal L}(y,\del_x)$. On the other hand, if $I(x,y)\to -\infty$ as $x\to y$ then the solution $Z_y(x)$ violates the "extended maximum principle" of Gilbarg & Serrin \cite{GS} and is a distributional solution of ${\mathcal L}(x,\del_x)Z_y(x)=0$ for $|x-y|<\e$ although $Z_y$ is not even bounded as $x\to y$.


Topological complexity of configuration spaces

Abstract: The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m>1$ and n>1; the answer was previously known in the cases m=2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.


Motion planning in spaces with small fundamental group

Abstract: We establish sharp upper bounds for the topological complexity of motion planning problem in spaces with small fundamental group, i.e. when it is finite of small order or has small cohomological dimension.


High-dimensional additive modeling

Abstract: We propose a new sparsity-smoothness penalty for high-dimensional generalized additive models. The combination of sparsity and smoothness is crucial for mathematical theory as well as performance for finite-sample data. We present a computationally efficient algorithm, with provable numerical convergence properties, for optimizing the penalized likelihood. Furthermore, we provide oracle results which yield asymptotic optimality of our estimator for high dimensional but sparse additive models. Finally, an adaptive version of our sparsity-smoothness penalized approach yields large additional performance gains.


Theoretical properties of Cook's PFC dimension reduction algorithm for linear regression

Abstract: We analyse the properties of the Principal Fitted Components (PFC) algorithm proposed by Cook. We derive theoretical properties of the resulting estimators, including sufficient conditions under which they are $\sqrt{n}$-consistent, and explain some of the simulation results given in Cook's paper. We use techniques from random matrix theory and perturbation theory. We argue that, under Cook's model at least, the PFC algorithm should outperform the Principal Components algorithm.


Irrevocable Multi-Armed Bandit Policies

Abstract: This paper considers the multi-armed bandit problem with multiple simultaneous arm pulls. We develop a new `irrevocable' heuristic for this problem. In particular, we do not allow recourse to arms that were pulled at some point in the past but then discarded. This irrevocable property is highly desirable from a practical perspective. As a consequence of this property, our heuristic entails a minimum amount of `exploration'. At the same time, we find that the price of irrevocability is limited for a broad useful class of bandits we characterize precisely. This class includes one of the most common applications of the bandit model, namely, bandits whose arms are `coins' of unknown biases. Computational experiments with a generative family of large scale problems within this class indicate losses of up to 5 to 10% relative to an upper bound on the performance of an optimal policy with no restrictions on exploration. We also provide a worst-case theoretical analysis that shows that for this class of bandit problems, the price of irrevocability is uniformly bounded: our heuristic earns expected rewards that are always within a factor of 1/8 of an optimal policy with no restrictions on exploration. In addition to being an indicator of robustness across all parameter regimes, this analysis sheds light on the structural properties that afford a low price of irrevocability.


On some block ciphers and imprimitive groups

Abstract: The group generated by the round functions of a block ciphers is a widely investigated problem. We identify a large class of block ciphers for which such group is easily guaranteed to be primitive. Our class includes the AES and the SERPENT.


On the Allen-Cahn equation in the Grushin plane: a monotone entire solution that is not one-dimensional

Abstract: We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate. However, they are not necessarily one-dimensional, as a counter-example shows.


Optimal oracle inequalities for model selection

Abstract: Model selection is often performed by empirical risk minimization. The quality of selection in a given situation can be assessed by risk bounds, which require assumptions both on the margin and the tails of the losses used. Starting with examples from the 3 basic estimation problems, regression, classification and density estimation, we formulate risk bounds for empirical risk minimization under successively weakening conditions and prove them at a very general level, for general margin and power tail behavior of the excess losses.


Noncommutative Cartan sub-algebras of C*-algebras

Abstract: J. Renault has recently found a generalization of the caracterization of C*-diagonals obtained by A. Kumjian in the eighties, which in turn is a C*-algebraic version of J. Feldman and C. Moore's well known Theorem on Cartan subalgebras of von Neumann algebras. Here we propose to give a version of Renault's result in which the Cartan subalgebra is not necessarily commutative [sic]. Instead of describing a Cartan pair as a twisted groupoid C*-algebra we use N. Sieben's notion of Fell bundles over inverse semigroups which we believe should be thought of as "twisted etale groupoids with noncommutative unit space". En passant we prove a theorem on uniqueness of conditional expectations.


Smooth self-similar blow-up profiles for the wave map equation

Abstract: Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold which has an equator (example: the sphere). In dimension 3, this article gives a necessary and sufficient condition for the existence of a smooth self-similar blow up profile. More generally, we study the relation between 1. the minimizing properties of the equator map for the (elliptic) Dirichlet energy and 2. the existence of a smooth blow-up profile for the (hyperbolic) wave map problem. Several applications of this approach are described.


Geometrically Constructed Bases for Homology of Non-Crossing Partition Lattices

Abstract: For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Bj\"{o}rner and Wachs in \cite{BW} using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.


Dimension of automorphisms with fixed degree for polynomial algebras

Abstract: Let $K[x,y]$ be the polynomial algebra in two variables over an algebraically closed field $K$. We generalize to the case of any characteristic the result of Furter that over a field of characteristic zero the set of automorphisms $(f,g)$ of $K[x,y]$ such that $\max\{\text{deg}(f),\text{deg}(g)\}=n\geq 2$ is constructible with dimension $n+6$. The same result holds for the automorphisms of the free associative algebra $K< x,y>$. We have also obtained analogues for free algebras with two generators in Nielsen -- Schreier varieties of algebras.


Finite energy scattering for the Lorentz-Maxwell equation

Abstract: In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz-Maxwell equation for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation.


First integrals of linear differential systems

Abstract: We investigate the problem of the existence of first integrals for multidimensional and ordinary linear differential systems with constant coefficients. The spectral method of the first integrals basis construction for these systems of linear differential equations is developed.


Non-simple purely infinite rings

Abstract: In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-)unital $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many nonsimple Leavitt path algebras is purely infinite.


Orbifolds as stacks?

Abstract: The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" definition of orbifolds as proper etale Lie groupoids up to Morita equivalence. The second goal is to describe two complementary ways of thinking of orbifolds as a 2-category: 1. the weak 2-category of foliation Lie groupoids, bibundles and equivariant maps between bibundles and 2. the strict 2-category of Deligne-Mumford stacks over the category of smooth manifolds.


Asymptotique des nombres de Betti des vari\'et\'es arithm\'etiques

Abstract: We study the question of the growth of Betti numbers of certain arithmetic varieties in tower of congruence coverings. In fact, our results are about Siegel varieties and varieties associated to orthogonal groups. We explain how a theorem of Waldspurger can be used to obtain lower and upper bound. Our results are in the direction of conjectures made by Sarnak and Xue.


Computing invariants and semi-invariants by means of Frobenius Lie algebras

Abstract: Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its semicenter. A sufficient condition is given in order for Sz(U(L)) to be a polynomial algebra over k. Surprisingly, this condition holds for many Lie algebras, especially among those for which the radical is nilpotent, in which case Sz(U(L))=Z(U(L)). In particular, it allows the explicit description of Z(U(L)) for more than half of all complex, indecomposable nilpotent Lie algebras of dimension at most 7.


Solving period problems for minimal surfaces with the support function

Abstract: In this paper we show how to bypass the usual difficulties in the analysis of elliptic integrals that arise when solving period problems for minimal surfaces. The method consists of replacing period problems with ordinary Sturm-Liouville problems involving the support function. We give a practical application by proving existence of the sheared Scherk-Karcher family of surfaces numerically described by Wei. Moreover, we show that this family is continuous, and both of its limit-members are the singly periodic genus-one helicoid.


Jet and prolongation spaces

Abstract: The notion of prolongation of an algebraic variety is developed in an abstract setting that generalises the difference and (Hasse) differential contexts. An interpolating map that compares the prolongation spaces with algebraic jet spaces is introduced and studied.


Some examples of compact composition operators on $H^2$

Abstract: To appear in J. Functional Analysis


A criterion of weak compactness for operators on subspaces of Orlicz spaces

Abstract: To appear in J. Funct. Spaces and Appl.


Compact composition operators on $H^2$ and Hardy-Orlicz spaces

Abstract: We compare the compactness of composition operators on $H^2$ and on Orlicz-Hardy spaces $H^\Psi$. We show in particular that exists an Orlicz function $\Psi$ such that $H^{3+\eps} \subseteq H^\Psi \subseteq H^3$ for every $\eps >0$, and a composition operator $C_\phi$ which is compact on $H^3$ and on $H^{3+\eps}$, but not compact on $H^\Psi$.


More Constructions for Tur\'an's (3, 4)-Conjecture

Abstract: For Tur\'an's (3, 4)-conjecture, in the case of n = 3k+1 vertices, (.5)6^{k-1} non-isomorphic complexes are constructed that attain the conjecture. In the case of n = 3k+2 vertices, 6^{k-1} non-isomorphic complexes are constructed that attain the conjecture.


The Charney-Davis conjecture for certain subdivisions of spheres

Abstract: Notions of sesquiconstructible complexes and odd iterated stellar subdivisions are introduced, and some of their basic properties are verified. The Charney-Davis conjecture is then proven for odd iterated stellar subdivisions of sesquiconstructible balls and spheres.


A strong antidiamond principle compatible with CH

Abstract: A strong antidiamond principle (*c) is shown to be consistent with CH. This principle can be stated as a "P-ideal dichotomy": every P-ideal on omega-1 (i.e. an ideal that is sigma-directed under inclusion modulo finite) either has a closed unbounded subset of omega-1 locally inside of it, or else has a stationary subset of omega-1 orthogonal to it. We rely on Shelah's theory of parameterized properness for NNR iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application of the NNR iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic.


Online data processing: comparison of Bayesian regularized particle filters

Abstract: The aim of this paper is to compare three regularized particle filters in an online data processing context. We carry out the comparison in terms of hidden states filtering and parameters estimation, considering a Bayesian paradigm and a univariate stochastic volatility model. We discuss the use of an improper prior distribution in the initialization of the filtering procedure and show that the regularized Auxiliary Particle Filter (APF) outperforms the regularized Sequential Importance Sampling (SIS) and the regularized Sampling Importance Resampling (SIR).


Convex functions on Grassmannian manifolds and Lawson-Osserman problem

Abstract: We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.


Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III

Abstract: We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. Our main technical tools are the structural analysis of type III factors and the theory of canonical extension of endomorphisms introduced by Izumi.


The notion of $\psi$-weak dependence and its applications to bootstrapping time series

Abstract: We give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.


On the vanishing of Selmer groups for elliptic curves over ring class fields

Abstract: Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character \chi of G. Our main result is that if the special value of the \chi-twisted L-function of E/K is non-zero then the tensor product (with respect to \chi) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite extension of Z_p containing the values of \chi. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a \chi-twisted version of the Birch and Swinnerton-Dyer conjecture for E over H (Bertolini and Darmon) and of the vanishing of the p-Selmer group of E/K for almost all p (Kolyvagin) in the case of analytic rank zero.


Complex Spaces and Nonstandard Schemes

Abstract: We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of such schemes, sitting in between usual algebraic schemes (over the complex numbers) and that of complex spaces. We clarify the structure of prime ideals in a Stein algebra, coming from nonstandard points and show in particular that ANY maximal and minimal prime ideal in a Stein algebra is the vanishing ideal of a nonstandard point. Other applications of our point of view are given for differential forms (a la Leibniz), generic points (as nonstandard ones), meromorphic functions, hyperbolicity. The essential tools taken from nonstandard mathematics and adapted for our purposes, are summarized in the appendix.


Dynamic programming for infinite horizon boundary control problems of PDE's with age structure

Abstract: We develop the dynamic programming approach for a family of infinite horizon boundary control problems with linear state equation and convex cost. We prove that the value function of the problem is the unique regular solution of the associated stationary Hamilton--Jacobi--Bellman equation and use this to prove existence and uniqueness of feedback controls. The idea of studying this kind of problem comes from economic applications, in particular from models of optimal investment with vintage capital. Such family of problems has already been studied in the finite horizon case by Faggian. The infinite horizon case is more difficult to treat and it is more interesting from the point of view of economic applications, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is here performed through a nontrivial limiting procedure from the corresponding finite horizon problem.


Entropy of meromorphic maps and dynamics of birational maps

Abstract: We study the dynamics of meromorphic maps for a compact Kaehler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of P^k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic.


Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen

Abstract: Pranab K. Sen has contributed extensively to many areas of Statistics including order statistics, nonparametrics, robust inference, sequential methods, asymptotics, biostatistics, clinical trials, bioenvironmental studies and bioinformatics. His long list of over 600 publications and 22 books and volumes along with numerous citations during the past 5 decades bear testimony to his work. All three of us have had the good fortune of being associated with him in different capacities. He has given professional and personal advice on many occasions to all of us, and we feel that our lives have certainly been enriched by our association with him. He has been over the years a friend, philosopher and a guide to us, and still continues to be one! While parametric statistical inference remains ever so popular, semi-parametric, Bayesian and nonparametric inferential methods have attracted great attention from numerous applied scientists because of their weaker assumptions, which make them naturally robust and so more appropriate in real-life applications. This clearly signals for ``beyond parametrics'' approaches which include nonparametrics, semi-parametrics, Bayes methods and many others. Motivated by this feature, and his drive in the ``beyond parametrics'' area, we thought that it will be only appropriate for a volume in honor of Pranab Kumar Sen to focus on this aspect of statistical inference and its applications. With this in mind, we have put together this volume in order to (i) review some of the recent developments in this direction, (ii) focus on some new methodologies and highlight their applications, and (iii) suggest some interesting open problems and possible new directions for further research.


Proof of Ira Gessel's Lattice Path Conjecture

Abstract: We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .


Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras

Abstract: If a vertex operator algebra $V=\oplus_{n=0}^{\infty}V_n$ satisfies $\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge $c$ and its Griess algebra is isomorphic to $Sym_d(\C)$ for any complex number $c$.


The doubly periodic Scherk-Costa surfaces

Abstract: We present a new family of embedded doubly periodic minimal surfaces, of which the symmetry group does not coincide with any other example known before.


The Quasi-Holonomic Ansatz and Restricted Lattice Walks

Abstract: The great enumerator Germain Kreweras empirically discovered this intriguing fact, and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators, for example, Heinrich Niederhausen[N], Ira Gessel[G1], and Mireille Bousquet-M\'elou[B], found other ingenious, ``simpler'' proofs. Yet none of them is as simple as ours! Our proof (with the generous help of our faithful computers) is ``ugly'' in the traditional sense, since it would be painful for a lowly human to follow all the steps. But according to our humble aesthetic taste, this proof is much more elegant, since it is (conceptually) one-line. So what if that line is rather long (a huge partial-recurrence equation satisfied by the general counting function), it occupies less storage than a very low-resolution photograph.


Heat-flow monotonicity related to the Hausdorff--Young inequality

Abstract: It is known that if $q$ is an even integer then the $L^q(\mathbb{R}^d)$ norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres "simultaneously slide" to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if $q > 2$ is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff--Young inequality.


Sums and Differences of Three k-th Powers

Abstract: Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take \theta strictly less than 1. We also prove the estimate O(B^{10/k}), (subject to N << B) which is better for large k. The results extend to representations by an arbitrary fixed nonsingular ternary from. However ``non-trivial'' must then be suitably defined. Consideration of the singular form x^{k-1}y-z^k allows us to establish an asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes, answering a problem raised by Hooley.


The Ratio Monotonicity of the Boros-Moll Polynomials

Abstract: In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the $\infty$-log-concavity.


On cotilting cotorsion and torsion pairs

Abstract: In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the $\Sigma$-pure injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a $\Sigma$-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a $\Sigma$-pure injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by $\Sigma$-pure injective cotilting modules.


Invariants associated with linear ordinary differential equations

Abstract: We apply a novel method for the equivalence group and its infinitesimal generators to the investigation of invariants of linear ordinary differential equations. First, a comparative study of this method is illustrated by an example. Next, the method is used to obtain the invariants of low order linear ordinary differential equations, and the structure invariance group for an arbitrary order of these equations. Other properties of these equations are also discussed, including the exact number of their invariants.


On submanifolds with recurrent second fundamental form in spaces of constant curvature

Abstract: The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.


The triangle-free process

Abstract: Consider the following stochastic graph process. We begin with the empty graph on n vertices and add edges one at a time, where each edge is chosen uniformly at random from the collection of potential edges that do not form triangles when added to the graph. The process terminates at a maximal traingle-free graph. Here we analyze the triangle-free process, determining the likely order of magnitude of the number of edges in the final graph. As a corollary we show that the triangle-free process is very likely to produce a Ramsey R(3,t) graph; that is, our analysis of the triangle-free process gives a new proof of the lower bound on R(3,t) previously established by Jeong Han Kim. The techniques introduced extend to the K_4-free process thereby establishing a small improvement in the best known lower bound on the Ramsey number R(4,t).


Nice inducing schemes and the thermodynamics of rational maps

Abstract: We consider the thermodynamic formalism of a complex rational map $f$ of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter $t$ we study the (non-)existence of equilibrium states of $f$ for the potential $-t \ln |f'|$, and the analytic dependence on $t$ of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism of a rational map that is "expanding away from critical points" and that has arbitrarily small "nice sets" with some additional properties. Our results apply in particular to non-renormalizable polynomials without indifferent periodic points, infinitely renormalizable quadratic polynomials with a priori bounds, real quadratic polynomials, topological Collet-Eckmann rational maps, and to backward contracting rational maps. As an application, for these maps we describe the dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results.


On logarithmic extension of overconvergent isocrystals

Abstract: In this paper, we establish a criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor. This is a generalization of a result of Kedlaya, who treated the case of unipotent monodromy. Our result is regarded as a $p$-adic analogue of the theory of canonical extension of regular singular integrable connections on smooth varieties of characteristic 0.


Parabolic, hyperbolic and elliptic Poincar\'e series

Abstract: Following Petersson, we study the parabolic, hyperbolic and elliptic expansions of holomorphic cusp forms and the associated Poincar\'e series. We show how these ideas extend to the space of second-order cusp forms.


The Semigroup of Betti Diagrams

Abstract: The recent proof of the Boij-Soederberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup of Betti diagrams. We prove that this semigroup is finitely generated, and we answer several other fundamental questions about this semigroup.


Normal Families of Bicomplex Holomorphic Functions

Abstract: In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice $\bold{j}=0$.