Related papers: Discrete Conley Index for Zero-dimensional Basic S…

For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials in several variables, a finite dimensionality criterion for the Lie algebra generated by these derivations is known. Also the structure of the…

To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…

We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend…

We give a complete classification of analytic equivalence of germs of parametric families of systems of complex linear differential equations unfolding a generic resonant singularity of Poincare rank 1 in dimension $n = 2$ whose leading…

We compute the arithmetic rank as well as the local/\'etale cohomological dimension of nullcone ideals arising from the classical actions of the symplectic group, the general linear group, and the orthogonal group. We use these calculations…

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing…

In this paper we investigate Deep Learning Models using topological dynamical systems, index theory, and computational homology. These mathematical machinery was invented initially by Henri Poincare around 1900 and developed over time to…

A novel basis of discrete analytic polynomials on a rhombic lattice is introduced and the associated convolution product is studied. A class of discrete analytic functions that are rational with respect to this product is also described.

An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…

Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article…

We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an…

In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the…

The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core…

An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist…

We introduce middle convolution for systems of linear differential equations with irregular singular points, and we presend a tentative definition of the index of rigidity for them. Under some assumption, we show a list of terminal patterns…

Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most…

In this work we complement the description of the extreme rays of the $6 \times 6$ copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of pieces…

We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalizing classical results of Raynaud in two different ways. The first is a higher dimensional generalization of Raynaud's…

The structural invariant subspaces of the discrete-time singular Hamiltonian system are used in 1] to give an analytic nonrecursive expression of all the admissible trajectories. A deeper insight into the features of these subspaces,…

Using homogeneous dynamical approach, we obtain a complete description of the forms with discrete set of values at the integer points and not representing zero non-trivially over the rational numbers. As a consequence, we obtain a general…