idući (peti) sastanak hrvatsko-slovenskog Seminara za analizu i algebru Alpe-Jadran biti održan ovu subotu 13.5. na PMF-MO, prema sljedećem rasporedu:
- 10:00-10:45 Rudi Mrazović: Tranversals in quasirandom latin squares
- 10:50-11:35 Ganna Kudryavtseva: F-inverse monoids in enriched signature
Coffee break
- 12:00-12:45 Ivica Nakić: Observability inequalities and applications to control problems for partial differential equations
- 12:50-13:35 Aljaž Zalar: A gap between positive maps (resp .copositive matrices) and completely positive ones.
SAŽECI PREDAVANJA
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Ganna Kudryavtseva: F-inverse monoids in enriched signature
Abstract: The class of F-inverse monoids is one of the most studied subclasses of inverse semigroups. They arise naturally and are useful, for instance, in the theories of partial group actions and of C*-algebras. An inverse monoid is F-inverse, if every sigma-class has a maximum element, with respect to the natural partial order, where sigma is the minimum group congruence. Each F-inverse monoid can be equipped with an additional unary operation which assigns to each its element the maximum element of its sigma-class. In this enriched signature, F-inverse monoids form a variety of algebras. We provide a geometric construction of a universal F-inverse monoid assigned to an arbitrary X-generated group and show that this construction contains and unifies the Birget-Rhodes and the Margolis-Meakin expansions of a group. An elegant geometric description of the free F-inverse monoids is a special case of this construction. This is a joint work with K. Auinger (Vienna) and M.B. Szendrei (Szeged).
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Rudi Mrazović: Tranversals in quasirandom latin squares
Abstract: A transversal in a $n \times n$ latin square is a set of $n$ entries not repeating any row, column, or symbol. A famous conjecture of Brualdi, Ryser, and Stein predicts that every latin square has at least one transversal provided $n$ is odd. We will discuss an approach motivated by the circle method from the analytic number theory which enables us to count transversals in latin squares which are quasirandom in an appropriate sense.
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Ivica Nakić: Observability inequalities and applications to control problems for partial differential equations
Abstract: A control system is a dynamical system on which one can act by using suitable controls. If the system is modeled by partial differential equations (PDEs) we talk about PDE control. Observability inequalities are a very useful tool for the control of PDEs as they provide a framework for analyzing control properties of PDE systems, including controllability, observability and stabilizability properties, as well as the existence of time- and norm-optimal controls. There are several techniques that can be used to establish observability inequalities, including Carleman estimates, multiplier methods, spectral inequalities, control-theoretic approaches and moment methods. This talk aims to provide a flavor of these methods and their applications. Introductory material will be combined with several recent results.
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Aljaž Zalar: A gap between positive maps (resp. copositive matrices) and completely positive ones.
Abstract: A ∗-linear map between real symmetric matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its amplifications are positive. A n×n symmetric matrix A is copositive if the quadratic form x^TAx is nonnegative on the nonnegative orthant R^n≥0.The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form BB^t for some n×r matrix B with nonnegative entries. In this talk quantitative bounds on the fractions of positive maps and copositive matrices that are completely positive will be established. The main techniques for both problems come mainly from real algebraic geometry, convex geometry and harmonic analysis. Moreover, algorithms to produce positive but not completely positive examples will be presented; the one for maps based on algebraic geometry, while the one for matrices on free probability.This is joint work with Igor Klep, Scott McCullough, Klemen Sivic and Tea Strekelj.
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