Complex fuzzy linear systems

In paper the complex fuzzy linear equation nbspnbspin which nbspis a crisp complex matrix and nbspis an arbitrary complex fuzzy numbers vector, is investigated. The complex fuzzy linear system is converted to a equivalent high order fuzzy linear system . Numerical procedure for calculating the complex fuzzy solution is designed and thenbsp sufficient condition for the existence of strong complex fuzzy solution is derived. A example is given to illustrate the proposed method.nbs

Linear systems with adiabatic fluctuations

We consider a dynamical system subjected to weak but adiabatically slow fluctuations of external origin. Based on the ``adiabatic following'' approximation we carry out an expansion in \alpha/|\mu|, where \alpha is the strength of fluctuations and 1/|\mu| refers to the time scale of evolution of the unperturbed system to obtain a linear differential equation for the average solution. The theory is applied to the problems of a damped harmonic oscillator and diffusion in a turbulent fluid. The result is the realization of `renormalized' diffusion constant or damping constant for the respective problems. The applicability of the method has been critically analyzed.Comment: Plain Latex, no figure, 21 page

Linear systems on irregular varieties

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety, $a\colon X\rightarrow A$ a morphism to an abelian variety such that $\rm{Pic}^0(A)$ injects into $\rm{Pic}^0(T)$ and let $L$ be a line bundle on $X$. Denote by $X^{(d)}\to X$ the connected \'etale cover induced by the $d$-th multiplication map of $A$, by $T^{(d)} \subseteq X^{(d)}$ the preimage of $T$ and by $L^{(d)}$ the pull-back of $L$ to $X^{(d)}$. For $\alpha\in \rm{Pic}^0(A)$ general, we study the restricted linear system $|L^{(d)}\otimes a^*\alpha|_{|T^{(d)}}$: if for some $d$ this gives a generically finite map $\varphi^{(d)}$, we show that f $\varphi^{(d)}$ is independent of $\alpha$ or $d$ sufficiently large and divisible, and is induced by the {\em eventual map} $\varphi\colon T\to Z$ such that $a_{|T}$ factorizes through $\varphi$. The generic value $h^0_a(X_{|T}, L)$ of $h^0(X_{|T}, L\otimes\alpha)$ is called the {\em (restricted) continuous rank.} We prove that if $M$ is the pull back of an ample divisor of $A$, then $x\mapsto h^0_a(X_{|T}, L+xM)$ 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. In the case when $X$ and $T$ are smooth, combining the above results we prove Clifford-Severi type inequalities, i.e., geographical bounds of the form $$\rm{vol}_{X|T}(L)\geq C(m) h^0_a(X_{|T},L),$$ where $C(m)={\mathcal O}(m!)$.Comment: Revised version, 37 pages. The final section has been remove