We consider the problem of low canonical polyadic (CP) rank tensor completion. A completion is a tensor whose entries agree with the observed entries and its rank matches the given CP rank. We analyze the manifold structure corresponding to the tensors with the given rank and define a set of polynomials based on the sampling pattern and CP decomposition. Then, we show that finite completability...

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Generic finite dimensional (not necessarily associative) kalgebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over k rational functions in the structure constants. (2) There exists an “algebraic normal form”, to wh...

One of the techniques used to prove the algebraic independence of numbers is Mahler’s method, which deals with the values of so-called Mahler functions satisfying a certain type of functional equation. In order to apply the method, one must confirm the algebraic independence of the Mahler functions themselves. This can be reduced, in many cases, to their linear independence modulo the rational ...

It is well-known that the associated analytic space of an affine variety defined over complex number field is Stein but the converse is not true, that is, an algebraic Stein variety is not necessarily affine. In this talk, we will give sufficient and necessary conditions for an algebraic Stein variety to be affine. One of our results is that a quasi-projective variety Y defined over complex num...

A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this condition for Coxeter groups in certain classes, possibly of infinite ranks. We also determine the finite irreducible components of another Coxeter group, t...

we find an explicit expression of the tutte polynomial of an $n$-fan. we also find a formula of the tutte polynomial of an $n$-wheel in terms of the tutte polynomial of $n$-fans. finally, we give an alternative expression of the tutte polynomial of an $n$-wheel and then prove the explicit formula for the tutte polynomial of an $n$-wheel.

For an ancestral graph with four vertices, we show that the associated binary model exhibits different properties than the previously studied Gaussian analog. The Gaussian likelihood function may be multimodal, whereas the likelihood function of the binary model is guaranteed to be unimodal. Canonical hidden variable models associated with ancestral graph models impose additional polynomial con...

Given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. In particular, we do not assume independence or equidistribution of the coefficients of the polynomial. We apply this result to various problems in both random and deterministic sequences of polyno-mials, including ...

If sk denotes the number of independent sets of cardinality k and α(G) is the size of a maximum independent set in graph G, then I(G;x) = s0 + s1x+ ...+ sα(G)x α(G) is the independence polynomial of G [8]. In this paper we provide an elementary proof of the inequality |I(G;−1)| ≤ 2, where φ(G) is the decycling number of G.

The well-known greedy algorithm MIN for nding a maximal independent set in a graph G is based on recursively removing the closed neighborhood of a vertex which has (in the currently existing graph) minimum degree. We give a forbidden induced subgraph condition under which algorithm MIN always results in nding a maximum independent set of G, and hence yields the exact value of the independence n...