相关论文: Continuation Methods for Computing Z-/H-eigenpairs…
We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical…
A nonconforming linear element method is developed for a three-dimensional generalized tensor-valued Stokes equation associated with the Hessian complex in this paper. A discrete Helmholtz decomposition for the piecewise constant space of…
Extending our method for investigating Real cobordism (which was recently used by Hill, Hopkins and Ravenel in their solution of the Kervaire invariant 1 problem), we investigate the $RO(G)$-graded homotopy groups of a (non-complete)…
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a…
We present a continuation method to compute all zeros of a harmonic mapping $f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solution of…
The tensor complementarity problem is a specially structured nonlinear complementarity problem, then it has its particular and nice properties other than ones of the classical nonlinear complementarity problem. In this paper, it is proved…
A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it…
We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in "near-colliding" situations. We consider a case when the structure of the…
We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant…
Positively graded algebras are fairly natural objects which are arduous to be studied. In this article we query quotients of non-standard graded polynomial rings with combinatorial and commutative algebra methods.
In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schr\"odinger equations. A discretization of the saturable nonlinear Schr\"odinger equation leads to a nonlinear algebraic eigenvalue…
We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical…
We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The…
The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We…
We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients…
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$,…
This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…
The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has…
We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…