Related papers: Jacobians with prescribed eigenvectors
We solve the problem of inversion of an extended Abel-Jacobi map $$ \int_{P_{0}}^{P_{1}}\omega +...+\int_{P_{0}}^{P_{g+n-1}}\omega ={\bf z}, \qquad \int_{P_{0}}^{P_{1}}\Omega_{j1}+... +\int_{P_{0}}^{P_{g+n-1}}\Omega_{j1} =Z_{j},\quad…
The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new…
Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g.,in quasi-Newton methods. Motivated by the latter, we study…
Let $m\geq3$ be an integer. We show that every torsor of the Jacobian of the universal family of degree-$m$ Fermat curve is necessarily a connected component of the Picard scheme. We show that the Jacobian of the generic degree-$m$ Fermat…
In this paper, we study singular systems with complete sets of involutive constraints. The aim is to establish, within the Hamilton-Jacobi theory, the relationship between the Frobenius' theorem, the infinitesimal canonical transformations…
By the Lefschetz embedding theorem, a principally polarized $g$-dimensional abelian variety is embedded into projective space by the linear system of $4^g$ half-characteristic theta functions. Suppose we {\em edit} this linear system by…
The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of…
In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of…
We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present…
The Jacobian Conjecture states that any locally invertible polynomial system in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982) proved a reduction theorem stating that the conjecture is true for any degree of…
It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by…
Some general problems of Jacobian computations in non-full rank matrices are discussed in this work. In particular, the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is revisited. Then the Jacobian in the…
The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In…
A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by…
Our main result describes the relation between the syzygies involving the first order partial derivatives $f_0,...,f_n$ of a homogeneous polynomial $f\in \C[x_0,...x_n]$ and the defect of the linear systems vanishing on the singular locus…
In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some…
Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having…
We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve…
The virial theorem for non-relativistic complex fields in $D$ spatial dimensions and with arbitrary many-body potential is derived, using path-integral methods and scaling arguments recently developed to analyze quantum anomalies in…
Reparameterization from the standard set of orbital elements to Cartesian position-velocity vectors can be computationally advantageous for orbit inference problems, particularly when orbital elements are weakly constrained. Here we present…