We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R{modules. In this note, we describe so...

We prove that if (a; b) is an R-diagonal pair in some non-commutative probability space (A; ') then (a p ; b p) is R-diagonal too and we compute the determining series f (a p ;b p) in terms of the distribution of ab. We give estimates of the upper and lower bounds of the support of free multiplicative convolution of probability measures compactly supported on 0; 1, and use the results to give n...

Ideals generated by prescribed powers of linear forms have attracted a great deal of attention recently. In this paper we study properties that hold when the linear forms are general, in a sense that we make precise. Analogously, one could study so-called “general forms” of the same prescribed degrees. One goal of this paper is to highlight how the differences between these two settings are rel...

In this paper we prove a special case of the transversal conjecture of Tverberg and Vrećica. We consider the case when the numbers of parts ri in this conjecture are powers of the same prime. We also prove some results on common transversals that generalize the classical nonembeddability theorems. We also prove an analogue of colored Tverberg’s theorem by Živaljević and Vrećica. Instead of mult...

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski’s theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R–modules. In this note, we describe so...

The k-th power D of a directed graph D is defined to be the directed graph on the vertices of D with an arc from a to b in D iff one can get from a to b in D with exactly k steps. This notion is equivalent to the k-fold composition of binary relations or k-th powers of Boolean matrices. A k-th root of a directed graph D is another directed graph R with R = D. We show that for each k ≥ 2, comput...

the porcine eye is often used as an ex vivo animal model in ophthalmological research. it is well-suited for investigations concerning refractive surgery; however, corneal topography data are scarce. this study investigated the corneal topography and pachymetry of the porcine eye to provide further reproducible data. we evaluated freshly enucleated porcine eyes (n = 16) by performing computeriz...

We generalize an example, due to Sylvester, and prove that any monomial of degree d in R[x0, x1], which is not a power of a variable, cannot be written as a linear combination of fewer than d powers of linear forms.

We are concerned with fractional powers of the so-called hyponormal operators of Putnam type. Under some suitable assumptions it is shown that if A, B are closed hyponormal linear operators of Putnam type acting on a complex Hilbert space H, then D((A+B)α) = D(Aα)∩D(Bα) = D((A+B)∗α) for each α ∈ (0,1). As an application, a large class of the Schrödinger’s operator with a complex potential Q ∈ L...

We consider sequences of generalized Bell numbers B(n), n = 0, 1,. .. which can be represented by Dobi´nski-type summation formulas, i.e. B(n) = 1 C ∞ k=0 [P (k)] n D(k) , with P (k) a polynomial, D(k) a function of k and C = const. They include the standard Bell numbers (P (k) = k, D(k) = k!, C = e), their generalizations B r,r (n), r = 2, 3,. .. appearing in the normal ordering of powers of b...