On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

A bounded compactness theorem forL 1-embeddability of metric spaces in the plane

LetM=(W, d) be a metric space. LetL ¹ denote theL ¹ metric. AnL ¹-embedding ofM into Cartesiank-space ℝ ^k is a distance-preserving map from (W, d) into (ℝ ^k ,L ¹). Letc(k) be the smallest integer such that for every metric spaceM, M isL ¹-embeddable inR ^k iff everyc(k)-sized subspace ofM isL ¹-embeddable inR ^k. A special case of a theorem of Menger (see p. 94 of [5]) says thatc(1) exists and equals 4. We show thatc(2) exists and satisfies 6≦c(2)≦11. Whether or notc(k) exists for anyk≧3 is an open question. The research of S. M. Malitz was partially supported by NSF Grant CCR-8909953.     

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.     

Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.     

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].     

Uniqueness of Solutions of Some Nonlocal Boundary-Value Problems for Operator-Differential Equations on a Finite Segment

For the equation L ₀ x(t) + L ₁ x ⁽¹⁾(t) + ... + L _n x ⁽ⁿ⁾(t) = 0, where L _k, k = 0, 1, ... , n, are operators acting in a Banach space, we formulate conditions under which a solution x(t) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.     

Continuity andkth order differentiability in Orlicz-Sobolev spaces:W kLA

The paper gives a necessary and sufficient condition for the embedding of the Orlicz-Sobolev spaceW ^kL_A (Ω) inC(Ω). The same condition is also found to be necessary and sufficient so that a continuous function inW ^kL_A (Ω) be differentiable of orderk almost everywhere in Ω.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

The Weights of Irreducible SL 3(q)-Modules in the Defining Characteristic

This is a final step in solving the problem of recognition of the simple groups L ₃(p ^k) by element orders. It is proven that when L ₃(p ^k) acts on an elementary abelian p-group, there always appears an element of new order. A model is proposed for constructing the absolutely irreducible p-modular representations of L ₃(p ^k) in polynomial spaces.     

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L _p,k, and the finite field F(p ^k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ₁ of the variety V(L _p,k) generated by L _p,k into the variety V(F(p ^k)) generated by F(p ^k) and an interpretation Φ₂ of V(F(p ^k)) into V(L _p,k) such that Φ₂Φ₁(B) = B for every B ε V(L _p,k) and Φ₁Φ₂(R) = R for every R ε V(F(p ^k)).     

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.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

On the spaces C k (ℝ) ∩ L p (ℝ)

We show that the Fréchet-Sobolev spaces C(ℝ) ∩ L ^p (ℝ) and C ^k (ℝ) ∩ L ^p (ℝ) are not isomorphic for p ≠ 2 and k ∈ ℕ. Research supported by the Italian MURST.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

Random k-dimensional orders: Width and number of linear extensions

We consider the width W _k (n) and number L _k (n) of linear extensions of a random k-dimensional order P _k (n). We show that, for each fixed k, almost surely W _k (n) lies between (k/2–C)n _1–1/k and 4kn ^1-1/k , for some constant C, and L _k (n) lies between (e ^-2 n ^1-1/k ) ⁿ and (2kn ^1-1/k ) ⁿ . The bounds given also apply to the expectations of the corresponding random variables. We also show that W _k (n) and log L _k (n) are sharply concentrated about their means.     

On multivalued topologies on <Emphasis Type="Italic">L</Emphasis>-powersets of multivalued sets

Given an M-valued equality E: X×X → M on a set X, we extend it to the M-valued equality ε: L ^X × L ^X → M on the L-powerset L ^X of X, where L is a complete sublattice of a GL-monoid M. As a result, we come to a category SET(M,L) whose objects are quadruples (X,E,L ^X , ε). This category serves as a ground category for the category L-TOP(M) of (L,M)-valued topological spaces and some of its subcategories, which are the main subject of this paper. In particular, as special cases, we obtain here Chang-Goguen, Lowen, Kubiak-Šostak, and some other known categories related to fuzzy topology. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 237–247, 2005.     

Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

Weighted estimates for the <Emphasis Type="Italic">k</Emphasis>-plane transform of radial functions on euclidean spaces

We discuss the L ^p − L ^q mapping property of k-plane transforms acting on radial functions in certain weighted L ^p spaces with power weight. We show that for all admissible power weights it is not always possible to get strong (p, q) boundedness of the k-plane transform. However, we prove the best possible estimates with respect to the Lorentz norms.     

On the number of list-colorings

Given a list of boxes L for a graph G (each vertex is assigned a finite set of colors that we call a box), we denote by f(G, L) the number of L-colorings of G (each vertex must be colored wiht a color of its box). In the case where all the boxes are identical and of size k, f(G, L) = p(G, k), where P=G, k) is the chromatic polynominal of G. We denote by F(G, k) the minimum of f(G, L) over all the lists of boxes such that each box has size at least k. It is clear that F(G, k) ≤ P(G, k) for all G, k, and we will see in the introduction some examples of graphs such that F(G, k) < P(G, k) for some k. However, we will show, in answer to a problem proposed by A. Kostochka and A. Sidorenko (Fourth Czechoslovak Symposium on Combinatorics, Prachatice, Jin, 1990), that for all G, F(G, k) = P(G, k) for all k sufficiently large. It will follow in particular that F(G, k) is not given by a polynominal in k for all G. The proof is based on the analysis of an algorithm for computing f(G, L) analogous to the classical one for computing P(G, k).     

The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).