Abstract:  Classical limit theorems are discussed for multiple sums of independent random variables. Among those results are weak and strong law of large numbers, almost sure convergence of multiple series, and law of the iterated logarithm.

 

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Abstract:  The evolution of differential $k$-forms in dynamical systems has proven to be a versatile tool. It facilitates theinvestigation of local and global properties of the dynamical systems and topics such as existence and stability of periodic orbits. The main tools in finite dimensional dynamical systems have been multiplicative and additive $k$-th compound matrices whose algebraic and spectral properties have provided useful insights. In this talk, I will give a report on the finite dimensional theory developed by Prof. Jim Muldowney and Prof. Michael Li. Applications including a higher dimensional generalization of the Poincar'{e} stability condition of periodic orbits and the Bendixson conditions for the non-existence of

periodic orbits will be discussed. For the second half of the talk, I will discuss a joint work with Prof. Jim Muldowney on how to extend the compound theory to infinite dimensional dynamical systems.

 

 Abstract: We will formulate an inessential generalization to Goodstein's Theorem, 1944, (see Wikipedia on Goodstein Sequences) which is an arithmetical truth not provable in arithmetic.  No proofs, but examples to show how things work.

 

Abstract:  The direct sum decomposition of tensor products for su(3) has many applications to physics, and the problem has been studied extensively.  This has resulted in many decomposition methods, each with its advantages and disadvantages.  The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities.  In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and helping to draw general conclusions.  After a description and proof of the method, several consequences are discussed and proved.  In particular, questions regarding multiplicities are considered.

 

Please invite anyone you think might be interested.

 

 

 

ABSTRACT: It sometimes happens in mathematics - for instance, several famous examples coming from number theory are very meaningful - that the answer to questions which appear to be easy and accessible can actually be extremely difficult. With some disappointment, one finds out very soon that the question of the title has exactly the above characteristics.  Moreover, by knowing how many (linearly independent) partial derivatives of any order a (homogeneous) polynomial in $n$ variables may have, one can deduce important consequences in a number of areas of mathematics, ranging from invariant theory to commutative algebra and algebraic geometry, from combinatorics to complexity theory.  I will describe some important and some curious properties of the derivatives of a polynomial, and will discuss what those properties imply in my field, commutative algebra. Indeed, they are at the core of what we call "the theory of Macaulay's inverse systems", which has been a very powerful tool in some of my recent research. Most of the the talk will be accessible to a general audience.

 

Please invite anyone who might be interested.

Abstract:  We study the relationship between the no free lunches condition and relevant coherent measures of risk. In particular, we provide a solution to the hedging price problem: the asset price process is a relevant convergent martingale. Two applications are given: worst conditional mean and value-at-risk.

 

Please invite anyone who might be interested.

 

Abstract: In this talk, we explore the Gross-Pitaevskii equation describing superfluidity (nonlinear Schrodinger equation of Ginzburg Landau type).  We will establish linearized instability of +/- 2 vortices and derive and effective dynamical law of the motion of (the centres of) single vortices splitting from the double vortex.

 

This is joint work with I.M. Sigal, Y. Ovchinnikov and L. Jonsson.

 

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Abstract:  An odd induced cycle is an induced subgraph of a graph G that is cycle of odd length.   The recent proof of the Strong Perfect Graph Theorem has shown that a graph is perfect if and only if a graph or its complement does not have an odd induced cycle of length 5 or more.  It is therefore of interest to determine when a graph has an odd induced cycle.  In this talk I will describe how the odd induced

cycle structure of the graph is encoded into the associated primes of an ideal constructed from the graph, and how to use tools from commutative algebra to determine if a graph contains such cycles.

 

This is joint work with Chris Francisco and Tai Ha.

 

Please invite your upper year undergraduate students or any one you feel might be interested.

 

 

 

Abstract:  Time series models for serial data are characterized by certain parameters.  A standard statistical problem is to decide whether a statistically significant change has occurred in the parameters at a specified time point.  However, the problem becomes non-standard if one is asked to decide a statistically significant change has occurred at an unspecified point in the time series.  Identification of whether a change of parameters has occurred and, if so, estimation of the location of the change is referred to as the changepoint problem.

Changepoint statistics are discussed for a variety of models, and distributional results are provided under the assumption the observations that make up the time series are independent. Optimality properties are discussed.  Methodology is presented for the case of serially correlated observations.  The problem of multiple changepoints is considered and an elegant solution is presented.  The use of change detection for monitoring purposes is discussed.  The changeboundary problem, which is the spatial analogue of the changepoint problem, is also discussed.

Several applications are presented.   

 

 

 

Abstract:  The study of geometrical and topological properties of arithmetic sums of Cantor sets appears naturally in distinct fields as dynamical systems (particularly, in the study of homoclinic bifurcations related to non-trivial hyperbolic sets) and number theory (particularly, in the study of geometrical properties of the Markov and Lagrange spectra, related to Diophantine approximations).  We discuss the topological structure of the arithmetic sums of Cantor sets of a particular type.  This is a joint work with Dr. Razvan Anisca.