Related papers: Measures and Jacobians of singular random matrices

This article presents a unified approach to simultaneously compute the Jacobians of several singular matrix transformations in the real, complex, quaternion and octonion cases. Formally, these Jacobians are obtained for real normed division…

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 Jacobian algebras are introduced and their various properties are studied.

A particular case of the Jacobian conjecture is considered and for small dimensional cases a computational approach is offered

We state two recent results concerning the linearization of integrable systems on generalised Jacobians. Then we apply this to the (complexified) spherical pendulum.

This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric…

We survey the theory of the compactified Jacobian associated to a singular curve. We focus on describing low genus examples using the Abel map.

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…

We look at the decomposition of the compactified jacobian of a singular curve into components and discuss some examples.

We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.

Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class…

Numerical examples demonstrated that a prescribed positive Jacobian determinant alone can not uniquely determine a diffeomorphism. It is conjectured that the uniqueness of a transformation can be assured by its Jacobian determinant and the…

We propose a new starting point for comparing dose distributions in therapeutic radiation physics using a Jacobian-based measure. The measure is normalization independent, free of tunable parameters, bounded and converges to a unique value…

In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete-continuous Sobolev-type inner product defined in terms of the Jacobi measure.

The purpose of this review paper is the collection, systematization and discussion of recent results concerning the quantization approach to the Jacobian conjecture, as well as certain related topics.

The paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent rectangular…

When $k$ is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated $k$-algebra as the closed subset of an ideal generated by appropriate minors of the so-called Jacobian matrix.…

To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian…

We study the proportion of metric matroids whose Jacobians have nontrivial $p$-torsion. We establish a correspondence between these Jacobians and the $\mathbb{F}_p$-rational points on configuration hypersurfaces, thereby relating their…

We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by…