Special Linear Systems Research Articles

Very special divisors on real algebraic curves

We study special linear systems called ‘very special’ whose dimension does not satisfy a Clifford-type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of very special linear systems that are simple are also given.

Special Effect Varieties and $(-1)$-Curves

Here we introduce the concept of special effect curve which permits to study, from a different point of view, special linear systems in P^2, i.e. linear system with general multiple base points whose effective dimension is strictly greater than the expected one. In particular we study two different kinds of special effect: the \alpha-special effect is defined by requiring some numerical conditions, while the definition of h^1-special effect concerns cohomology groups. We state two new conjectures for the characterization of special linear systems and we prove they are equivalent to the Segre and the Harbourne-Hirschowitz ones.

Iterative Solution of Piecewise Linear Systems and Applications to Flows in Porous Media

The correct numerical modeling of free-surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In doing so, one may fulfill relevant physical constraints. The existence, the uniqueness, and two constructive iterative methods to solve a piecewise linear system of the form $\max[\boldsymbol{l},\min(\boldsymbol{u},\mathbf{x})]+T\mathbf{x}=\mathbf{b}$ are analyzed. The methods are shown to have a finite termination property; i.e., they converge to an exact solution in a finite number of steps and, actually, they converge very quickly, as confirmed by a few numerical tests, which are derived from the mathematical modeling of flows in porous media.

Special linear systems and syzygies

Let X be the base locus of a linear system L of hypersurfaces in P^r(C). In this paper it is showed that the existence of linear syzygies for the ideal of X has strong consequences on the fibres of the rational map associated to L. The case of hyperquadrics is particularly addressed. The results are applied to the study of rational maps and to the Perazzo's map for cubic hypersurfaces.

Curves Having Schemes of Special Linear Systems with Multiple Components

In this article, we show that it is of frequent occurrence that smooth curves do have multiple components for some of their schemes of special divisors. For a smooth space curve C of degree d, we give sufficient conditions implying that has a multiple component, and we prove the existence of many space curves satisfying those conditions. As an example of a more general result for curves in ℙ r , we prove that general complete intersection curves of degree d in ℙ4 do have multiple components for the schemes and .

Iterative Solution of Piecewise Linear Systems

The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.

On a class of special linear systems of $\mathbb{P}^3$

In this paper we deal with linear systems of P 3 through fat points. We consider the behavior of these systems under a cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones.

On double coverings of hyperelliptic curves

We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety W d r for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h ≥ 3 are also presented.

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.

A solution to certain polynomial equations with applications to nonlinear fitting

We present a combinatorial method for solving a certain system of polynomial equations of Vandermonde type in 2N variables by reducing it to the problem of solving two special linear systems of size N and rooting a single univariate polynomial of degree N. Over C, all solutions can be found with fixed precision using, up to polylogarithmic factors, O(N 2 ) bitwise operations in the worst case. Furthermore, if the data is well conditioned, then this can be reduced to O(N) bit operations, up to polylogarithmic factors. As an application, we show how this can be used to fit data to a complex exponential sum with N terms in the same, nearly optimal, time.

A counterexample to a conjecture on linear systems on ℙ3

In his paper (1) Ciliberto proposes a conjecture in order to characterize special linear systems of P n through multiple base points. In this note we give a counterexample to this con- jecture by showing that there is a substantial dierence between the speciality of linear systems on P 2 and those of P 3 . Let us take the projective space P n and let us consider the linear system of hyper- surfaces of degree d having some points of fixed multiplicity. The virtual dimension of such systems is the dimension of the space of degree d polynomials minus the con- ditions imposed by the multiple points and the expected dimension is the maximum between the virtual one and � 1. The systems whose dimension is bigger than the ex- pected one are called special systems. There exists a conjecture due to Hirschowitz (see (5)), characterizing special linear systems on P 2 , which has been proved in some special cases (2), (3), (7), (6). Concerning linear systems on P n , in (1) Ciliberto gives a conjecture based on the classification of special linear systems through double points. In this note we describe a linear system on P 3 that we found in a list of special systems generated with the help of Singular (4) and which turns out to be a counterexample to that conjecture. The paper is organized as follows: in Section 1 we fix some notation and state Ciliberto's conjecture, while Section 2 is devoted to the counterexample. In Section 3 we try to explain speciality of some systems by the Riemann-Roch formula, and we conclude the note with an appendix containing some computations.

On the variety Wdr( C) whose dimension is at least d−3 r−2

Let C be a smooth projective algebraic curve which is not a curve of even gonality admitting an automorphism of order 2. Under this rather mild restriction on C, we establish a stronger dimension theoretic statement concerning the variety of special linear systems; we prove that if dim W d r(C)⩾d−3r−2⩾0 for some d, r>0 with d⩽ g−5 and g⩾21, then C is a k-gonal curve with k⩽6, a 4-sheeted covering of an elliptic curve or a plane curve of degree 8.

POSITIVE SEMIDEFINITENESS OF DISCRETE QUADRATIC FUNCTIONALS

AbstractWe consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

Some algorithms for solving special tridiagonal block Toeplitz linear systems

This paper is focused on different methods and algorithms for solving tridiagonal block Toeplitz systems of linear equations. We consider the El-Sayed method (Ph.D. Thesis, 1996) for such systems and propose several modifications that lead to different algorithms, which we discuss in detail. Our algorithms are then compared with some classical techniques as far as implementation time is concerned, number of operations and storage. Comments and conclusions for computing efficiency of the proposed new algorithms are given. Numerical experiments corroborating the theoretical results are also presented.

MARTENS' DIMENSION THEOREM FOR CURVES OF EVEN GONALITY

For a smooth projective irreducible algebraic curve C of odd gonality, the maximal possible dimension of the variety of special linear systems (C) is d－3r by a result of M. Coppens et at. [4]. This bound also holds if C does not admit an involution. Furthermore it is known that if dim d-3r-1 for a curve C of odd gonality, then C is of very special type of curves by a recent progress made by G. Martens [11] and Kato-Keem [9]. The purpose of this paper is to pursue similar results for curves of even gonality which does not admit an involution.

ON THE VARIETY OF SPECIAL LINEAR SYSTEMS OF DEGREE g-1 ON SMOOTH ALGEBRAIC CURVES

We classify smooth projective algebraic curves C of genus g such that the variety of special linear systems [Formula: see text] has dimension g- 7. We first prove that if [Formula: see text] has dimension g-7≥0 then C is either trigonal, tetragonal, a double covering of a curve of genus 2 or a smooth plane sextic. This result establishes the next extension of dimension theorems of H. Martens and D. Mumford on the variety of special linear systems with the fullest possible generality. We then proceed to show that, under the assumption g≥11, [Formula: see text] has dimension g- 7 if and only if C is either a trigonal curve or a double covering of a curve of genus 2.

Solving Symmetric Arrowhead and Special Tridiagonal Linear Systems by Fast Approximate Inverse Preconditioning

A new class of approximate inverses for arrowhead and special tridiagonal linear systems, based on the concept of sparse approximate Choleski-type factorization procedures, are introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of symmetric linear systems. A theorem on the rate of convergence of the explicit preconditioned conjugate gradient scheme is given and estimates of the computational complexity are presented. Applications of the proposed method on linear and nonlinear systems are discussed and numerical results are given.

Singular bielliptic curves and special linear systems

Here we study the Brill–Noether theory of special linear systems on a singular Gorenstein bielliptic curve Y. Any such linear system is associated to a spanned rank 1 torsion-free sheaf, L. We obtain the same results as in the case of a smooth curve if either L∈Pic( Y) or Y has only ordinary nodes or ordinary cusps as singularities.

Trigonal gorenstein curves and special linear systems

LetY be a Gorenstein trigonal curve withg:=pa(Y)≥0. Here we study the theory of special linear systems onY, extending the classical case of a smoothY given by Maroni in 1946. As in the classical case, to study it we use the minimal degree surface scroll containing the canonical model ofY. The answer is different if the degree 3 pencil onY is associated to a line bundle or not. We also give the easier case of special linear series on hyperelliptic curves. The unique hyperelliptic curve of genusg which is not Gorenstein has no special spanned line bundle.

Introduction to fractional linear systems. Part 2: Discrete-time case

In the paper, the class of discrete linear systems is enlarged with the inclusion of discrete-time fractional linear systems. These are systems described by fractional difference equations and fractional frequency responses. It is shown how io compute the impulse response and transfer function. Fractal signals are introduced as output of special linear systems: fractional differaccumulators, systems that can be considered as having fractional poles or zeros. The concept of fractional differaccumulation is discussed, gencralising the notions of fractal and lif noise, and introducing two kinds of fractional differaccumulated stochastic proccss: hyperbolic, resulting from fractional accumulation (similar to the continuous-time casc), and parabolic noise, resulting from fractional differencing.