On complex n-folds polarized by an ample line bundle L with .

Let (X L) be a polarized manifold with dim X = n≥3 and dim Bs |L|≤0. In this paper, we classify (X,L) with _g(L) = q(X) +m and h^o(L) ≥ n + m.     

On Polarized 3-Folds (X,L) Such That <lowercase > h < /lowercase >0(L) = 2 and the Sectional Genus of (X,L) is Equal to the Irregularity of X

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this article, we give a characterization of (X, L) with g(X, L) = q(X) and h⁰(L) = 2, where g(X, L) (resp. q(X)) denotes the sectional genus of (X, L) (resp. the irregularity of X).     

A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and . In this paper, we treat the case . First we prove that this conjecture is true for , and we classify (X,L) withg(L)=q(X), where is the Kodaira dimension of X. Next we study some special cases of .     

On polarized 3-folds (X, L) withg(L)=q(X)+1 andh 0(L)≥4

Let (X, L) be a polarized 3-fold over the complex number field. In [Fk3], we proved thatg(L)≥q(X) ifh ⁰(L)≥2 and moreover we classified (X, L) withh ⁰(L)≥3 andg(L)=q(X), whereg(L) is the sectional genus of (X, L) andq(X)=dimH ¹(O _X ) the irregularity ofX. In this paper we will classify polarized 3-folds (X, L) withh ⁰(L)≥4 andg(L)=q(X)+1 by the method of [Fk3].     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(K _X L))/2=3 are classified by means of the main techniques of adjunction theory.     

Addendum to “On the Sectional Geometric Genus of Quasi-Polarized Varieties,I” by Yoshiaki Fukuma

Let (X, L) be a polarized manifold defined over the complex number field with dim X = n such that L is very ample. In this article, we will improve the classification of (X, L) with g ₂(X, L) = h ²(𝒪 _X ) + 1 which was obtained in Fukuma (Fukuma 2004 Fukuma , Y. ( 2004 ). On the sectional geometric genus of quasi-polarized varieties, I . Comm. Alg. 32 : 1069 – 1100 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 3.6), where g ₂(X, L) denotes the second sectional geometric genus of (X, L).     

The (Lp, Lq) bilinear Hardy-Littlewood function for the tail

Let (X, B, μ, T) be a measure preserving dynamical system on a finite measure space. Consider the maximal function

Cyclotomic numbers and primitive idempotents in the ring GF(q)[x]/(x−1)

Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f, and q^f=1+pλ. Here expressions for all the primitive idempotents in the ring R_pⁿ=GF(q)[x]/(x^pn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pⁿ over GF(q) are also discussed.     

Relative ( pn , p , pn , n )-difference sets with GCD ( p , n )=1

Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)=1 through a group ring approach. We then give a construction of (p(p+1),p,p(p+1),p+1) relative difference sets with p a Mersenne prime. Supported by National Natural Science Foundation of China (10331030).     

Growth of transcendental entire functions on algebraic varieties

LetX be a complete intersection algebraic variety of codimensionm>1 in ℂ ^m+n . We define the notion of (p,q)-order and (p,q)-K-type for transcendental entire functionsfεO(ℂ ^m+n ) whereK is a non-pluripolar compact subset of ℂ ^m+n . Further, we consider the analogues of (p,q)-order and (p,q)-K-type inO(X). We discuss the series expansions of the functions inO(X) in terms of an orthogonal basis in a Hilbert spaceL ²(X, μ), where μ is a capacitary extremal measure onK. Author is grateful to the NSA for partial support during the period of this research.     

Asymptotic Limits for a Two-Dimensional Recursion

In this paper we consider the general two-dimensional recursion g(n + 1, r + 1) = g(n + 1, r) + bg(n, r) + g(n, r + 1) with boundary conditions g(n, 0) = g(0, r) = 1 We develop various exact and asymptotic formulas for the g(n, r).     

Pointwise Best Approximation in the Space of Strongly Measurable Functions with Applications to Best Approximation in L(μ,X)

The object of this paper is to prove the following theorem: Let Y be a closed subspace of the Banach space X, (S,Σ,μ) a σ-finite measure space, L(S,Y) (respectively, L(S, X)) the space of all strongly measurable functions from S to Y (respectively, X), and p a positive number. Then L(S,Y) is pointwise proximinal in L(S,X) if and only if L^p(μ,Y) is proximinal in L^p(μ,X). As an application of the theorem stated above, we prove that if Y is a separable closed subspace of the Banach space X, p is a positive number, then L^p(μ,Y) is proximinal in L^p(μ,X) if and only if Y is proximinal in X. Finally, several other interesting results on pointwise best approximation are also obtained.     

The Cardinality of Sets of k -Independent Vectors over Finite Fields

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind _q (n, k) of a k-independent set of vectors in the n-dimensional vector space F _q ⁿ over the finite field F _q of order q. Namely, we give a necessary and sufficient condition for Ind _q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.     

Best approximation in L(μ, X), II

The object of this paper is to prove the following theorem: If Y is a closed subspace of the Banach space X, then L¹(μ, Y) is proximinal in L¹(μ, X) if and only if L^p(μ, Y) is proximinal in L^p(μ, Y) for every p, 1 < p < ∞. As an application of this result we prove that if Y is either reflexive or Y is a separable proximinal dual space, then L¹(μ, Y) is proximinal in L¹(μ, X).     

On Dirichlet Series with Periodic Coefficients

We investigate Dirichlet series L(s, f) = _n=1 with q-periodic coefficients f(n), i.e. f(n+q) = f(n) for all integers n and some fixed integer q, and we prove an asymptotic formula for the number of nontrivial zeros of L(s, f). Further, we give a necessary condition for L(s, f) to have a distribution of the nontrivial zeros symmetrical with respect to the critical line.     

Families of finite sets with three intersections

LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n ²), and show that in some casesm(n, k, L)=O(n ^3/2), which is quite surprising.     

Schur functions and the invariant polynomials characterizing U(n) tensor operators

We give a direct formulation of the invariant polynomials _μG_q⁽ⁿ⁾(, Δ_i,;, x_{i,i + 1},) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions S_λ known as Schur functions. To this end, we show after the change of variables Δ_i = γ_i − δ_i and x_{i, i + 1} = δ_i − δ_{i + 1} that_μG_q⁽ⁿ⁾(,Δ_i;, x_{i, i + 1},) becomes an integral linear combination of products of Schur functions S_α(, γ_i,) · S_β(, δ_i,) in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}, respectively. That is, we give a direct proof that _μG_q⁽ⁿ⁾(,Δ_i,;, x_{i, i + 1},) is a bisymmetric polynomial with integer coefficients in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials _μ^mG_q⁽ⁿ⁾(γ₁,…, γ_n; δ₁,…, δ_m). These new symmetries enable us to give an explicit formula for both _μ^mG₁⁽ⁿ⁾(γ; δ) and ₁G₂⁽ⁿ⁾(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for _μ^mG_q⁽ⁿ⁾(γ; δ).     

Characterization of abelian varieties

We prove that any smooth complex projective variety X with plurigenera P ₁(X)=P ₂(X)=1 and irregularity q(X)=dim(X) is birational to an abelian variety. Oblatum 26-V-1999 & 13-VI-2000?Published online: 11 October 2000     

Rational functions with partial quotients of small degree in their continued fraction expansion

For a rational functionf/g=f(x)/g(x) over a fieldF with ged (f,g)=1 and deg (g)1 letK(f/g) be the maximum degree of the partial quotients in the continued fraction expansion off/. ForfF[x] with deg (f)=k1 andf(O)O putL(f)=K(f(x)/x ^k ). It is shown by an explicit construction that for every integerb with 1bk there exists anf withL(f)=b. IfF=F ₂, the binary field, then for everyk there is exactly onefF ₂[x] with deg (f)=k,f(O)O, andL(f)=1. IfF _q is the finite field withq elements andgF _q [x] is monic of degreek1, then there exists a monic irreduciblefF _q [x] with deg (f)=k, gcd (f,g)=1, andK(f/g)<2+2 (logk)/logq, where the caseq=k=2 andg(x)=x ²+x+1 is excluded. An analogous existence theorem is also shown for primitive polynomials over finite fields. These results have applications to pseudorandom number generation.