Minimal sections of conic bundles

Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X). The general curve C of any of these two families is a section of the conic bundle projection, and our approach relates such C to a maximal subbundle of a rank 2 vector bundle E(C) on C, or -- to a minimal section of the ruled surface P(E(C)). The families C(+) and C(-) correspond to the two possible types of versal deformations of ruled surfaces over curves of fixed genus g(C). As an application, we find parameterizations of J(X) and (Theta) for certain classes of Fano threefolds, and study the sets Sing(Theta) of the singularities of (Theta).Comment: Duke preprint, 29 pages. LaTex 2.0

Toward Clemens' Conjecture in degrees between 10 and 24

We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex \IP^4, the Hilbert scheme of rational, smooth and irreducible curves $C$ of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each $C$ is embedded in $F$ with balanced normal sheaf \O(-1)\oplus\O(-1), and in \IP^4 with maximal rank.Comment: Plain Tex, This eleven page paper is a joint manuscript, produced in connection with the first author's participation in the conference "Geometry and Physics", Zlatograd, Bulgaria, August 28 - Sept.2, 1995." This version contains a small change in Remark (3.3); the hope expressed there has been refine

Finite symmetric functions with non-trivial arity gap

Given an $n$-ary $k-$valued function $f$, $gap(f)$ denotes the essential arity gap of $f$ which is the minimal number of essential variables in $f$ which become fictive when identifying any two distinct essential variables in $f$. In the present paper we study the properties of the symmetric function with non-trivial arity gap ($2\leq gap(f)$). We prove several results concerning decomposition of the symmetric functions with non-trivial arity gap with its minors or subfunctions. We show that all non-empty sets of essential variables in symmetric functions with non-trivial arity gap are separable.Comment: 12 page

Normal and normally outer automorphisms of free metabelian nilpotent Lie algebras

Let L be the free m-generated metabelian nilpotent of class c Lie algebra over a field of characteristic 0. An automorphism f of L is called normal if f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal subgroup N(L) of Aut(L) containing the group of inner automorphisms. We describe the group of normal automorphisms of L and the quotient group of Aut(L) modulo N(L).Comment: to appear in Serdica Mathematical Journa

Algebras, dialgebras, and polynomial identities

This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.Comment: 32 page

Constructing 77-clusters

A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent $7$-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 different $7$-clusters, some of them having huge integer edge lengths. On the way, we exhaustively determined all Heronian triangles with largest edge length up to $6\cdot 10^6$.Comment: 18 pages, 2 figures, 2 table

Local energy decay in even dimensions for the wave equation with a time-periodic non-trapping metric and applications to Strichartz estimates

We obtain local energy decay as well as global Strichartz estimates for the solutions $u$ of the wave equation $\partial_t^2 u-div_x(a(t,x)\nabla_xu)=0,\ t\in{\R},\ x\in{\R}^n,$with time-periodic non-trapping metric$a(t,x)$equal to$1$outside a compact set with respect to$x$. We suppose that the cut-off resolvent$R_\chi(\theta)=\chi(\mathcal U(T, 0)-e^{-i\theta})^{-1}\chi$, where$\mathcal U(T, 0)$is the monodromy operator and$T$the period of$a(t,x)$, admits an holomorphic continuation to$\{\theta\in\mathbb{C}\ :\ \textrm{Im}(\theta) \geq 0\}$, for$n \geq 3$, odd, and to$\{ \theta\in\mathbb C\ :\ \textrm{Im}(\theta)\geq0,\ \theta\neq 2k\pi-i\mu,\ k\in\mathbb{Z},\ \mu\geq0\}$for$n \geq4$, even, and for$n \geq4$even$R_\chi(\theta)$is bounded in a neighborhood of$\theta=0$

Generalized backscattering and the Lax-Phillips transform

Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle $\omega =S\theta$ in terms of the incoming angle with $S$ orthogonal and \Id-S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials.Comment: Minor changes, typos corrected, references adde

Some examples of rigid representations

Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ (resp. $c_j\subset gl(n,{\bf C})$) so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ (resp. $A_j\in c_j$) satisfying the equality $M_1... M_{p+1}=I$ (resp. $A_1+... +A_{p+1}=0$)}. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere. We give new examples of existence of such $(p+1)$-tuples of matrices $M_j$ (resp. $A_j$) which are {\em rigid}, i.e. unique up to conjugacy once the classes $C_j$ (resp. $c_j$) are fixed. For rigid representations the sum of the dimensions of the classes $C_j$ (resp. $c_j$) equals $2n^2-2$

Links and Hubs of Scientific Information - the Case of the MTMT

We present the Hungarian National Scientific Bibliography project: the MTMT. We argue that presently available commercial systems cannot be used as a comprehensive national bibliometric tool. The new database was created from existing databases of the Hungarian Academy of Sciences, but expected to be re-engineered in the future. The data curation model includes harvesting, the work of expert bibliographers and author feedback. MTMT will work together with the other services in the web of scientific information, using standard protocols and formats, and act as a hub. It will present the scientific output of Hungary together with the repositories containing the full text, wherever available. The database will be open, but not freely harvestable, and only for non-commercial use