The number of vertices missed by a maximum matching in a graph G is the multiplicity of zero as a root of the matchings polynomial μ(G,x) of G, and hence many results in matching theory can be expressed in terms of this multiplicity. Thus, if mult(θ,G) denotes the multiplicity of θ as a zero of μ(G,x), then Gallai’s lemma is equivalent to the assertion that if mult(θ,G\u) < mult(θ,G) for each v...

Algebraic complexity theory investigates the computational cost of solving problems with an algebraic flavor. Several cost measures are of interest. We consider arithmetic circuits, which can perform the (exact) arithmetic operations +, -, *, / at unit cost, and take their size ( = sequential time) or their depth (=parallel time) as cost functions. This is a natural "structured" model of comput...

For a long time people have been trying to develop probability theory starting from ‘finite’ events rather than collections of infinite events. In this way one can find natural replacements for measurable sets and integrable functions, but measurable functions seemed to be more difficult. We present a solution. Moreover, our results are constructive (in the sense of Bishop).

We present the mathematical setting of an algebraic approach to quantum theory, including the Tomita–Takesaki theory, liouvilleans and quantum dynamical systems, and non-commutative integration up to the Falcone–Takesaki theory.

This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a linear signal model defined as a triple (A,M,Φ), where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively, and Φ generalizes the concept of the z-transform to bijective linear mappings from a vector sp...

Simeon Ball, Universitat Politécnica de Catalunya On subsets of a finite vector space in which every subset of basis size is a basis In this talk we consider sets of vectors S of the vector space Fq with the property that every subset of S of size k is a basis. The classical example of such a set is the following. Example (Normal Rational Curve) The set S = {(1, t, t, . . . , tk−1) | t ∈ Fq} ∪ ...