The k - Very Ampleness and k - Spannedness on Polarized Abelian Surfaces

We give a numerical criterion for an ample line bundle on an abelian surface to be it-very ample for a nonnegative integer k. This result implies the equivalence of k-very ampleness and k-spannedness. We also give a complete classification of polarized abelian surfaces with k - very ample polarizations. Our results extend those of Bauer and Szemberg [BaSzl] and Ramanan [Ra].     

Generalized Adjunction of k‐Very Ample Vector Bundles on Algebraic Surfaces II

We study the k‐very ampleness of the adjoint bundle K_S + det E associated to a (k — 1)‐very ample vector bundle E with degree greater than or equal to 4k + 5 on an algebraic surface S. We classify polarized surfaces (S, E) which the k‐very ampleness of K_S + det E fails.     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

List Colorings of K5‐Minor‐Free Graphs With Special List Assignments

The following question was raised by Bruce Richter. Let G be a planar, 3‐connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K₅‐minor‐free r‐connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[S_k], where S_k: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(S_k) in G between the components of G[S_k]. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:18–30, 2012     

Improvement of one inequality for algebraic polynomials

We prove that the inequality ||g (·/ n ) ||_L1[-1,1] ||P_n+k||_L1[-1,1] ￡ 2 ||gP_n+k||_L1[-1,1]\vert\vert g (\cdot / n ) \vert\vert_{L_{1}[-1,1]} \vert\vert P_{n+k}\vert\vert_{L_{1}[-1,1]} \leq 2 \vert\vert gP_{n+k}\vert\vert_{L_{1}[-1,1]}, where g : [-1, 1]→ℝ is a monotone odd function and P _n+k is an algebraic polynomial of degree not higher than n + k, is true for all natural n for k = 0 and all natural n ≥ 2 for k = 1. We also propose some other new pairs (n, k) for which this inequality holds. Some conditions on the polynomial P _n+k under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.     

On isomorphic factorizations of circulant graphs

We investigate the conjecture that every circulant graph X admits a k‐isofactorization for every k dividing |E(X)|. We obtain partial results with an emphasis on small values of k. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 406–414, 2006     

The Binomial k‐Clique

A finite collection C of k‐sets, where is called a k‐clique if every two k‐sets (called lines) in C have a nonempty intersection and a k‐clique is a called a maximal k‐clique if and C is maximal with respect to this property. That is, every two lines in C have a nonempty intersection and there does not exist A such that , and for all . An elementary example of a maximal k‐clique is furnished by the family of all the k‐subsets of a ‐set. This k‐clique will be called the binomial k‐clique. This paper is intended to give some combinatorial characterizations of the binomial k‐clique as a maximal k‐clique. The techniques developed are then used to provide a large number of examples of mutually nonisomorphic maximal k‐cliques for a fixed value of k.     

Coloring graphs from lists with bounded size of their union

A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 2^4k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Toughness and the existence of k-factors

A connected graph G is called t-tough if t · w(G - S) ? |S| for any subset S of V(G) with w(G - S) > 1, where w(G - S) is the number of connected components of G - S. We prove that every k-tough graph has a k-factor if k|G| is even and |G| ? k + 1. This result, first conjectured by Chvátal, is sharp in the following sense: For any positive integer k and for any positive real number ε, there exists a (k - ε)-tough graph G with k|G| even and |G| ? k + 1 which has no k-factor.     

Real line bundles on k-gonal real curves

For a generalk-gonal complex curve of genusg its variety of special line bundlesL with deg(L) =d andh ⁰(L) >r is known to contain an irreducible component of the expected dimension ρ_g (d, r) provided that the Brill-Noether number ρ_g (d, r) is non-negative andr ≤ k - 2. It is the purpose of this note to transfer this result of Brill-Noether type to the case ofk-gonal real curves, for real line bundles.     

Representation of the space of polyanalytic functions as a direct sum of orthogonal subspaces. Application to rational approximations

Suppose thatD={z:|z|<1}, L ₂ (D) is the space of functions square-integrable over area inD,A _k (D) is the set of allk-analytic functions inD, (A ₁ (D)=A(D) is the set of all analytic functions inD),A _k L ₂ (D)=L ₂ (D)∩A _k (D),A ₁ L ₂ (D)=AL ₂ (D),

On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension

Let X be a smooth projective variety over ? and L be a nef-big divisor on X. Then (X, L) is called a quasi - polarized manifold. Then we conjecture that g(L) ≥ q(X), where g(L) is the sectional genus of L and q(X) = dim H¹(O_x) is the irregularity of X. In general it is unknown that this conjecture is true or not even in the case of dim X = 2. For example, this conjecture is true if dim X = 2 and dim H≥(L) > 0. But it is unknown if dim X ≥ 3 and dim H⁰(L) > 0. In this paper, we consider a lower bound for g(L) if dim X = 2, dim H⁰(L) ≥ 2, and k(X) ≥ 0. We obtain a stronger result than the above conjecture if dim Bs|L| ≤ 0 by a new method which can be applied to higher dimensional cases. Next we apply this method to the case in which dim X = n ≥ 3 and we obtain a lower bound for g(L) if dim X = 3, dim H⁰(L) ≥ 2, and k(X) ≥ 0.     

Latin squares and superqueens

Let L be a Latin square of order n with entries from {0, 1,…, n ? 1}. In addition, L is said to have the (n, k) property if, in each right or left wrap around diagonal, the number of cells with entries smaller than k is exactly k. It is established that a necessary and sufficient condition for the existence of Latin squares having the (n, k) property is that of (2|n ? 2| k) and (3|n ? 3| k). Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.     

On the Homological Dimension of Lattices

Let L be a finite lattice and let . It is shown that if the order complex satisfies then |L| ≥ 2 ^k . Equality |L| = 2 ^k holds iff L is isomorphic to the Boolean lattice {0,1} ^k . Research supported by the Israel Science Foundation.     

Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

When linear and weak discrepancy are equal

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable, then |h_L(x)−h_L(y)|≤k, whereas the weak discrepancy is the least k such that there is a weak extension W of P such that if x and y are incomparable, then |h_W(x)−h_W(y)|≤k. This paper resolves a question of Tanenbaum, Trenk, and Fishburn on characterizing when the weak and linear discrepancy of a poset are equal. Although it is shown that determining whether a poset has equal weak and linear discrepancy is -complete, this paper provides a complete characterization of the minimal posets with equal weak and linear discrepancy. Further, these minimal posets can be completely described as a family of interval orders.     

Ample subvarieties and q-ample divisors

We introduce a notion of ampleness for subschemes of any codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti–Grauert vanishing theorem.