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These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…

This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global…

In this paper, we consider the existence and uniqueness of weak solutions of a nonlinear elliptic equation with a variable exponent, a monotonic type operator and a convection term. With the topological degree theory, we prove the existence…

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

We establish the unique solvability of a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two-dimensional first order systems and is also of relevance…

In this paper we study singularities of third secant varieties of Veronese embedding $v_d(\mathbb{P}^n)$, which corresponds to the variety of symmetric tensors of border rank at most three in $(\mathbb{C}^{n+1})^{\otimes d}$.

Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.

A monotone iterative method is proposed to solve nonlinear discrete boundary value problems with the support of upper and lower solutions. We establish some new existence results. Under some sufficient conditions, we establish maximum…

All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…

We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems.

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…

We extend existence and uniqueness results of [4] for nonlinear integro-differential equations of Volterra type between real locally complete vector spaces

For a complex tensor A, Minimal Gersgorin tensor eigenvalue inclusion set of A is presented, and its sufficient and necessary condition is given. Furthermore, we study its boundary by the spectrums of the equimodular set and the extended…

A complete characterization of the complexity of the reachability problem for vector addition system has been open for a long time. The problem is shown to be Tower complete.

We investigate the existence and multiplicity of solutions for higher order discrete boundary value problems via critical point theory.

The concepts of P- and P$_0$-matrices are generalized to P- and P$_0$-tensors of even and odd orders via homogeneous formulae. Analog to the matrix case, our P-tensor definition encompasses many important classes of tensors such as the…

Gravity theories with non-minimally coupled scalar fields are used as characteristic examples in order to demonstrate the challenges, pitfalls and future perspectives of considering alternatives to general relativity. These lecture notes…

We prove existence and uniqueness of solutions for an entropic version of the semi-geostrophic equations. We also establish convergence to a weak solution of the semi-geostrophic equations as the entropic parameter vanishes. Convergence is…

A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…