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.

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 Drazin-Star and Star-Drazin Inverses of Bounded Finite Potent Operators on Hilbert Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Fernando Pablos Romo
Keyword(s):  
Linear Systems ◽  
Hilbert Spaces ◽  
Explicit Solutions ◽  
Finite Complex ◽  
Complex Matrix ◽  
Infinite Linear Systems ◽  
Infinite Linear

AbstractThe aim of this work is to extend to bounded finite potent endomorphisms on arbitrary Hilbert spaces the notions of the Drazin-Star and the Star-Drazin of matrices that have been recently introduced by D. Mosić. The existence, structure and main properties of these operators are given. In particular, we obtain new properties of the Drazin-Star and the Star-Drazin of a finite complex matrix. Moreover, the explicit solutions of some infinite linear systems on Hilbert spaces from the Drazin-Star inverse of a bounded finite potent endomorphism are studied.

Overdetermined Infinite Linear Systems

2009 ◽  
Author(s):  
Béla Finta ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Linear Systems ◽  
Infinite Linear Systems ◽  
Infinite Linear

Complex Overdetermined Infinite Linear Systems

2011 ◽  
Author(s):  
Béla Finta ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Linear Systems ◽  
Infinite Linear Systems ◽  
Infinite Linear

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.

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