Counting singular plane curves via Hilbert schemes

We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singularities for which our results suffice include the topological type with local equation x^a+y^b with ?a?3b. We work out the example of curves with the analytic type of singularity with local equation x²+yⁿ for 1<n<9.     

On the signless Laplacian index of cacti with a given number of pendant vertices

A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this article, we determine graphs with the largest signless Laplacian index among all the cacti with n vertices and k pendant vertices. As a consequence, we determine the graph with the largest signless Laplacian index among all the cacti with n vertices; we also characterize the n-vertex cacti with a perfect matching having the largest signless Laplacian index.     

Pointwise bornological spaces

With each metric space (X,d) we can associate a bornological space (X,Bd) where Bd is the set of all subsets of X with finite diameter. Equivalently, Bd is the set of all subsets of X that are contained in a ball with finite radius. If the metric d can attain the value infinite, then the set of all subsets with finite diameter is no longer a bornology. Moreover, if d is no longer symmetric, then the set of subsets with finite diameter does not coincide with the set of subsets that are contained in a ball with finite radius. In this text we will introduce two structures that capture the concept of boundedness in both symmetric and non-symmetric extended metric spaces.     

On the m-capacitated peripatetic salesman problem with capacity restrictions

We consider a special case of the problem of finding m Hamiltonian cycles with capacity restrictions on the number of usages of the edges (m-Capacitated Peripatetic Salesman Problem or m-CPSP): the minimization and maximization 2-CPSP with edge weights chosen from an integer segment {1, q} with the edges capacities given as independent identically distributed random variables equal to 2 with probability p and 1 with probability (1 ? p). Some polynomial algorithms are proposed for 2-CPSP_min and 2-CPSP_max with average performance guarantees. In particular, when the edge weights are equal to 1 and 2, the algorithms have approximation ratios (19 ? 5p)/12 and (25 + 7p)/36 for the minimization and the maximization problem correspondingly.     

Cohomology of Subregular Tilting Modules for Small Quantum Groups

Let U be a quantumgroup with divid d powers at root ofunity constructed froma rootsystem R .Let u U b th small quantumgroup.Th cohomologyof u with trivial coefficients was computed by Ginzburg and Kumar.It turns out to be isomorphic to the functions algebra of the nilpotent cone of a semisimpl algebraic group with root system R .In this not we calculate cohomology of u with coefficients in simplest reducible tilting modul with nontrivial cohomology.It appears to b isomorphic to th functions algebra of th closure of the subregular nilpotent orbit.     

2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

Average boundary conditions in Cauchy problems

We consider the general Cauchy problem with initial data in a Hilbert space and with a formal dissipative linear generator. A complete parametrization is known of the (abstract) boundary conditions which make this problem well set. We exhibit a distinguished subset $\text{B}$_E of the set $\text{B}$ of boundary conditions and demonstrate explicitly that the evolution associated with each B in $\text{B}$ can be represented as a (time independent) average over the evolutions associated with B′ in $\text{B}$_E. Applications are discussed to Schrödinger equations in bounded regions or with singular potentials.     

Two new sets of ellipse-related concyclic points

Two new circles (denoted by Γ _I and Γ _E ) are shown to be associated with any ellipse. Their analogies with two circles described by Barlotti are described. Two further new circles—denoted by Ω and Γ—are shown to be associated with any general point P of the ellipse. Tight relationships link the circles Ω and Γ with the circle K (previously introduced by the present author), as well as with Monge’s orthoptic circle, with Barlotti’s circles and with the circles Γ _I and Γ _E . In particular, the circle Ω is orthogonal to Monge’s circle. A new special point of the ellipse (the point T) is described. New properties of Fagnano’s point are described.     

Explicit Nordsieck methods with quadratic stability

We describe the construction of explicit Nordsieck methods with s stages of order p = s − 1 and stage order q = p with inherent quadratic stability and quadratic stability with large regions of absolute stability. Stability regions of these methods compare favorably with stability regions of corresponding general linear methods of the same order with inherent Runge–Kutta stability.     

Associative rings with large center

A ring R is called a ring with large center if any nonzero ideal of R has nonzero intersection with the center of R. We give some conditions for an ideal of a ring with large center to be itself a ring with large center, and also we provide an example of a ring with large center R and its ideal I ? R such that I is not a ring with large center.     

On the spectral radius of tricyclic graphs with a fixed diameter

A tricyclic graph is a connected graph with n vertices and n + 2 edges. In this article, we determine graphs with the largest spectral radius among the n-vertex tricyclic graphs with given diameter d.     

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally H?lder continuous with H?lder exponent b ? (0,1){\beta \in (0,1)} with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is H?lder continuous with exponent β with respect to the parabolic metric.     

Some Boolean Algebras with Finitely Many Distinguished Ideals I

We consider the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th_prin and Th_sa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg(T) be the Lindenbaum-Tarski algebra with respect to T, and we let intalg(α) be the interval algebra of α. Using rank diagrams, we show that Sentalg(Th_prin) ? intalg(ω⁴), Sentalg(Th_max) ? intalg(ω³) ? Sentalg(Th_ac), and Sentalg(Th_sa) ? intalg(ω² + ω²). For Th_max and Th_ac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic.     

Enumeration of rectangular arrays of zeros and ones

For r=1,2 the rectangular arrays of zeros and ones with r rows and n columns, with m_i zeros and r_i changes in the ith row, and with s_i vertical changes between the ith and (i+1)st rows, i=1,…,r-1, are enumerated. The number of arrays of zeros and ones with 3 rows and n columns, with r_i changes in the ith row, i=1,2,3, and with s_i vertical changes between the ith and (i+1)st rows, i=1,2, is also determined.     

Bi-orthogonality and zeros of transformed polynomials

Let D and E be two real intervals. We consider transformations that map polynomials with zeros in D into polynomials with zeros in E. A general technique for the derivation of such transformations is presented. It is based on identifying the transformation with a parametrised distribution φ (x, µ), x ∈ E, µ ∈ D, and forming the bi-orthogonal polynomial system with respect to φ. Several examples of such transformations are given.     

Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets

Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying p[`(p)]^T=1\mathbf {p}\overline{\mathbf{p}}^{T}=1 . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.     

Hardy type inequalities with exponential weights for a class of convolution operators

In this paper we consider operators of the form H=λ(-i∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in L ²(ℝ ⁿ ) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.