Jacobians of linear systems on an algebraic variety

1956 ◽  
Vol 52 (2) ◽  
pp. 198-201 ◽  
Author(s):  
D. Monk
Keyword(s):  
Linear Systems ◽  
Algebraic Variety ◽  
Image Position

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and letThe purpose of this note is to prove that the Jacobian of these systems is given by the equivalencewhere Kh is Eger's operator, defined as follows:

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

scholarly journals Remarks on the Asymptotic Behaviour of Perturbed Linear Systems

1970 ◽  
Vol 11 (1) ◽  
pp. 84-84 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Asymptotic Behaviour ◽  
Linear Systems ◽  
Image Position ◽  
Perturbed Linear Systems

Remarks 1, 3 and 5 are incorrect as stated. They should be supplemented by the following observations:(i) In case the perturbing term is linear in y, i.e. f(t, y) = B(t)y, the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6),The hypothesis given in Trench's theorem is sufficient to imply (*) but not (6). A similar comment applies to Remark 5.

scholarly journals Infinite linear systems with homogeneous kernel of degree –1

1965 ◽  
Vol 5 (2) ◽  
pp. 129-168
Author(s):  
T. M. Cherry
Keyword(s):  
Continuous Function ◽  
Linear Systems ◽  
Algebraic Function ◽  
Additional Restriction ◽  
Main Concern ◽  
Homogeneous Kernel ◽  
Infinite Linear Systems ◽  
Image Position ◽  
Infinite Linear

The main concern of this paper is with the solution of infinite linear systems in which the kernel k is a continuous function of real positive variables m, n which is homogeneous with degree –1, so that If k is a rational algebraic function it is supposed further that the continuity extends up to the axes m = 0, n > 0 and n = 0, m > 0; the possibly additional restriction when k is not rational is discussed in § 1,2.

Global attraction for systems with almost C1 vector fields

2005 ◽  
Vol 135 (4) ◽  
pp. 815-832
Author(s):  
Zhanyuan Hou
Keyword(s):  
Linear Systems ◽  
Global Attractor ◽  
Differential System ◽  
Vector Fields ◽  
Single Point ◽  
Sufficient Conditions ◽  
Global Attraction ◽  
Almost Everywhere ◽  
Image Position ◽  
Special Case

Sufficient conditions are given for an autonomous differential system to have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, ℝN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in ℝN.

Bounds for the characteristic exponents of linear systems

1965 ◽  
Vol 61 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. A. Smith
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Characteristic Exponent ◽  
Zero Solution ◽  
System Of Differential Equations ◽  
Characteristic Exponents ◽  
Integrable Functions ◽  
Image Position ◽  
Complex Valued

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equationscan be defined, following Perron ((12)), aswhere ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

Some Theorems on Abelian Integrals Associated with an Algebraic Variety

1935 ◽  
Vol 31 (1) ◽  
pp. 18-25 ◽  
Author(s):  
W. V. D. Hodge
Keyword(s):  
Algebraic Variety ◽  
Abelian Integrals ◽  
Image Position

1. In some recent papers I have developed the theory of what I have called harmonic integrals associated with an algebraic variety. My object has been to provide an apparatus with which to investigate the properties of the classical integralsof total differentials which are finite everywhere on an algebraic variety Vm of m dimensionsThese integrals are referred to in what follows as “Abelian integrals”, as it is desirable to have a single name to denote all integrals of this type. The usual qualification “of the first kind” is omitted, and is always to be understood. In the present paper it is shown how the theory of harmonic integrals can be applied to deduce certain results concerning Abelian integrals, and a number of interesting problems are suggested by these results.

scholarly journals Linear systems of nonoscillatory differential equations with delayed arguments

1984 ◽  
Vol 30 (2) ◽  
pp. 307-314
Author(s):  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Delayed Arguments ◽  
Image Position

Sufficient conditions are obtained for a not necessarily scalar system of the formto be nonoscillatory.

scholarly journals Remarks on the asymptotic behaviour of perturbed linear systems

1969 ◽  
Vol 10 (2) ◽  
pp. 116-120 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Continuous Function ◽  
Asymptotic Behaviour ◽  
Linear Systems ◽  
Linearly Independent ◽  
Image Position ◽  
Perturbed Linear Systems

We are here concerned with following result of Trench:Theorem. (Trench [5]). Let v1 and v2 be two linearly independent solutions of the differential equationwhere a(t) is continuous on [0, ∞), and let b(t) be a continuous function of t for t ≧ 0 satisfyingwhere m(t) = max {|v1(t)|2, |v2(t)|2}. Then, if α1and α2are two arbitrary constants, there exists a solution u ofwhich can be written in the form,withfori = 1, 2.

The Behaviour of Damped Linear Systems in Steady Oscillation

1956 ◽  
Vol 7 (4) ◽  
pp. 353-354
Author(s):  
R. E. D. Bishop
Keyword(s):  
Linear Systems ◽  
Harmonic Excitation ◽  
Steady Oscillation ◽  
Image Position ◽  
Limiting Condition ◽  
Partial Fractions ◽  
Generalised Coordinates

In this paper (published in The Aeronautical Quarterly, Vol. VII, Part 2, p. 156, May 1956), a set of equations is derived giving the responses at the generalised coordinates q1, q2, … qn which are caused by a harmonic generalised force Φseiωt corresponding to qs. They are equations (26), namelyThis note is concerned with a sentence which follows these equations which says (of them) that “ … there is no longer any assurance that the ratios between the constants mBrs in any column of partial fractions are the same for all co-ordinates qs at which excitation is applied, since the previous limiting condition cannot now be applied. Thus the shapes of the ‘ modes ’ which are associated with the various columns depend upon the nature of the applied harmonic excitation.”

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