Some uniqueness and growth theorems in the Cauchy problem forPu tt +Mu t +Nu=0 in Hilbert space

Mathematische Zeitschrift -     

LG/CY Correspondence Between tt* Geometries

The concept of $tt^∗$ geometric structure was introduced by physicists (see [4, 10] and references therein), and then studied firstly in mathematics by C. Hertling [28]. It is believed that the $tt^∗$ geometric structure contains the whole genus 0 information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^∗$ geometry and obtain the following result. Let $f ∈ \mathbb{C}[z_0,...,z_{n+1}]$ be a nondegenerate homogeneous polynomial of degree $nThe concept of $tt^∗$ geometric structure was introduced by physicists (see [4, 10] and references therein), and then studied firstly in mathematics by C. Hertling [28]. It is believed that the $tt^∗$ geometric structure contains the whole genus 0 information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^∗$ geometry and obtain the following result. Let $f ∈ \mathbb{C}[z_0,...,z_{n+1}]$ be a nondegenerate homogeneous polynomial of degree $nThe concept of tt* geometric structure was introduced by physicists(see[4,9]and references therein),and then studied firstly in mathematics by C.Hertling[26].It is believed that the tt* geometric structure contains the whole genus 0 information of a two dimensional topological field theory.In this paper,we propose the LG/CY correspondence conjecture for tt* geome-try and obtain the following result.Let f ∈?[z0,…,zn+1]be a nondegenerate homogeneous polynomial of degree n+2,then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface Xf in(CP)n+1 or a Landau-Ginzburg model represented by a hypersurface singularity( ?n+2,f),both can be written as a tt* structure.We proved that there exists a tt* substructure on Landau-Ginzburg side,which should correspond to the tt* structure from variation of Hodge structures in Calabi-Yau side.We build the isomorphism of almost all structures in tt* geometries between these two models except the isomorphism between real structures.     

Some constraints on Frobenius manifolds with a tt*-structure

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a K?hler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifold.     

On (q + t, t)-Arcs of Type (0, 2, t)

In this paper we construct an infinite series of (q + t, t)-arcs of type (0, 2, t). We show that this construction includes the Korchmáros-Mazzocca arcs, and we gain new infinite series of examples, too.     

Nonisomorphic solutions of (4t+3, 2t+1, t) designs

A technique has been developed to get a large number of nonisomorphic solutions of a (4t+3, 2t+1, t) design. Some results are proved on the conjecture for Hadamard matrices.     

Generalized hexagons of order (t,t)

A generalized hexagon of order (t,t) in which certain subsets are maximal may be characterized as the generalized hexagon associated with Dickson’s groupG ₂(t). From this geometric result, it follows that ifG is a group of automorphisms of a generalized hexagon of order (p,p) for a primep and ifG has rank 4 on points, thenG ⊵G ₂(p).     

On schematic orthogonal arraysOA(q t,k, q,t)

Am × k matrixA, with entries from a set ofq 2 elements, is called an orthogonal arrayOA(m, k, q, t) (t 2) if eachm × t submatrix ofA contains all possible 1 ×t row vectors with the same frequency(m = q ^t ). We call the array schematic if the set of rows ofA forms an association scheme with the relations determined by the Hamming distance. In this paper we determine the schematic orthogonal arraysOA(q ^t ,k, q, t) with2t – 1 > k.     

The directed packing numbersDD(t,v, v), t≧4

A directed packing is a maximal collection ofk-subsets, called blocks, of a set of cardinalityv having the property that no orderedt-subset occurs in more than one block. A block contains an orderedt-set if its symbols appear, left to right, in the block. The cardinality of such a maximal collection is denoted byDD(t, k, v). We consider the special case whenk=v and derive some results on the sizes of maximal collections.     

The Ramsey number R(3, t) has order of magnitude t2/log t

The Ramsey number R(s, t) for positive integers s and t is the minimum integer n for which every red-blue coloring of the edges of a complete n-vertex graph induces either a red complete graph of order s or a blue complete graph of order t. This paper proves that R(3, t) is bounded below by (1 – o(1))t/²/log t times a positive constant. Together with the known upper bound of (1 + o(1))t²/log t, it follows that R(3, t) has asymptotic order of magnitude t²/log t. © 1995 John Wiley & Sons, Inc.     

New star discrepancy bounds for (t,m,s)-nets and (t,s)-sequences

In this paper, we derive new general upper bounds on the star discrepancy of $(t,m,s)$ -nets and $(t,s)$ -sequences. These kinds of point sets are among the most widely used in quasi-Monte Carlo methods for numerical integration. By our new results, we improve on previous discrepancy bounds and prove a conjecture stated by the second author in an earlier paper.     

The dimension of the determinantal scheme vs (t,t,2) of the catalecticant matrix

Let T be a Gorenstein sequence of a graded artinian Gorenstein ring k[x ₀,x ₁,x ₂]/I We develop a dimension formula for PGor(T) in terms of the alignment character. Based on our formula, we find a very large component of V_s (t,t,2) when s=r_t-1+1 and t is large enough. This answers Diesel’s conjecture negatively. Further we show that V_s (t,t,2) is irreducible of dimension 3s-1 for s ≤t+1,t ≥ 2 and dim V_s (t,t,2)= 3s-1 for small s. Finally an algorithm to calculate dim V_s (t,t,2) is constructed, and we find the values of dim V_s (t,t,2) for t ≤ 16.     

$${\tt surveillance}$$: An R package for the monitoring of infectious diseases

Public health surveillance of emerging infectious diseases is an essential instrument in the attempt to control and prevent their spread. This paper presents the R package “surveillance”, which contains functionality to visualise routinely collected surveillance data and provides algorithms for the statistical detection of aberrations in such univariate or multivariate time series. For evaluation purposes, the package includes real-world example data and the possibility to generate surveillance data by simulation. To compare algorithms, benchmark numbers like sensitivity, specificity, and detection delay can be computed for a set of time series. Package motivation, use and potential are illustrated through a mixture of surveillance theory, case study and R code snippets.     

Asymptotic behavior of solutions to the damped quasilinear equation ∂2∂t2 u(x,t) + γ∂u∂t (x,t) − ∂∂xσ ∂u∂x (x,t)) = 0

Asymptotic lower bounds for the L² norms of solutions of initial-boundary value problems associated with the equation of the title are derived for a simple case in which the equation fails to exhibit strict hyperbolicity. It is shown that in such cases it can be expected that the norm of a solution will be bounded away from zero as t → +∞ even as the damping factor γ becomes infinitely large.