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# Algebra

**DMS Linear Algebra/Algebra Seminar**

Feb 20, 2018 04:00 PM

Parker Hall 352

Speaker: **Frank Uhlig**

Title: The Eight Epochs of Math as regards past and future Matrix Computations

Abstract: This survey paper gives a personal assessment of Epoch making advances in Matrix Computations, from antiquity and with an eye towards tomorrow. It traces the development of number systems and elementary algebra and the uses of Gaussian Elimination methods from around 4000 BC on to current real-time Neural Network computations to solve time-varying matrix equations. The paper includes relevant advances from China from the 3rd century AD on and from India and Persia in the 9th and later centuries. Then it discusses the conceptual genesis of vectors and matrices in central Europe and in Japan in the 14th through 17th centuries AD. Followed by the 150 year cul-de-sac of polynomial root finder research for matrix eigenvalues, as well as the superbly useful matrix iterative methods and Francis’s matrix eigenvalue algorithm from the last century. Finally we explain the recent use of initial value problem solvers and high order 1-step ahead discretization formulas to master time-varying linear and nonlinear matrix equations via Zhang Neural Networks. This paper ends with a short outlook upon new hardware schemes with multilevel processors that go beyond the 0-1 base 2 framework which all of our past and current electronic computers have been using.

**DMS Linear Algebra/Algebra Seminar**

Feb 13, 2018 04:00 PM

Parker Hall 352

Speaker: **Luke Oeding**

Title: Higher Order Partial Least Squares and an Application to Neuroscience.

Abstract: Partial least squares (PLS) is a method to discover a functional dependence between two sets of variables X and Y. PLS attempts to maximize the covariance between X and Y by projecting both onto new subspaces. Higher order partial least squares (HOPLS) comes into play when the sets of variables have additional tensorial structure. Simultaneous optimization of subspace projections may be obtained by a multilinear singular value decomposition (MSVD). I'll review PLS and SVD, and explain their higher order counterparts. Finally I'll describe recent work with G. Deshpande, A. Cichocki, D. Rangaprakash, and X.P. Hu where we propose to use HOPLS and Tensor Decompositions to discover latent linkages between EEG and fMRI signals from the brain, and ultimately use this to drive Brain Computer Interfaces (BCI)'s with the low size, weight and power of EEG, but with the accuracy of fMRI.

**DMS Linear Algebra/Algebra Seminar**

Feb 06, 2018 04:00 PM

Parker Hall 352

Speaker: **Samir Raouafi**

Title: A survey of Crouzeix's Conjecture

Abstract: Please click here

**DMS Linear Algebra/Algebra Seminar**

Jan 30, 2018 04:00 PM

Parker Hall 352

Speaker: **Mehmet Gumus**

Title: On the Lyapunov-type diagonal stability

Abstract: In this talk, we present several recent developments regarding Lyapunov diagonal stability. This type of matrix stability plays an important role in various applied areas such as population dynamics, systems theory, complex networks, and mathematical economics. First, we examine a result of Redheer that reduces Lyapunov diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. An enhanced statement of this result based on the Schur complement formulation is presented here along with a shorter and purely matrix-theoretic proof. We develop a number of extensions to this result and formulate the range of feasible common diagonal Lyapunov solutions. In particular, we derive explicit algebraic conditions for a set of 2x2 matrices to share a common diagonal Lyapunov solution. Second, we present a new characterization involving Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. This extends a useful characterization, due to Kraaijevanger, of Lyapunov diagonal stability in terms of the P-matrix property under similar Hadamard multiplications. Our development mainly relies on a new notion called P-sets, which is a generalization of P-matrices, and a recent result of Berman, Goldberg, and Shorten.

**DMS Linear Algebra/Algebra Seminar**

Jan 23, 2018 04:00 PM

Parker Hall 352

Speaker: **Huajun Huang **

Title: A survey of the Marcus-de Oliveira determinantal conjecture

Abstract: The Marcus-de Oliveira determinantal conjecture (MOC) says that the complex number det(A+B), where A and B are normal matrices, is located in the convex hull of the determinants of sum of diagonal matrices similar to A and B, respectively. I will survey in this talk the history, progress, and challenges of this conjecture, and introduce the related geometric properties and extended results to the determinants of sums of matrices.

**DMS Linear Algebra/Algebra Seminar**

Jan 16, 2018 04:00 PM

Parker Hall 352

Organizational Meeting

**DMS Linear Algebra/Algebra Seminar**

Dec 05, 2017 04:00 PM

Parker Hall 356

Speaker: **Sima Ahsani**

Title: Metrics on the manifold of positive definite matrices and matrix means

Abstract: I will talk about the set of positive definite matrices as a Riemannian manifold. Then I will introduce two metrics on this set and explain their basic properties. After describing their connection through the geometric mean of two positive definite matrices, similarities and differences between two metrics will be discussed.

**DMS Linear Algebra/Algebra Seminar**

Nov 28, 2017 04:00 PM

Parker Hall 356

Speaker: **Samir Raouafi**

Title: On the extension of the Kreiss Matrix Theorem

Abstract: Let A be a matrix with spectrum δ(A). The Kreiss Matrix Theorem (KMT), a well-known fact in applied matrix analysis, gives estimates of upper bounds for ∥A^{n}∥ if δ(A) is in the unit disc, or for ∥e^{tA}∥ if δ(A) is in the left-half plane based on the resolvent norm. In this talk, we shall discuss some extensions of this celebrated theorem to simply connected compact sets in the complex plane.

**DMS Linear Algebra/Algebra Seminar**

Nov 14, 2017 04:00 PM

Parker Hall 356

Speaker: **Wei Gao**

Title: Sign patterns that require H_n and its generalization to zero-nonzero patterns

Abstract: The refined inertia of a square real matrix is the ordered 4-tuple (n_+, n_-, n_z, 2n_p), where n_+ (resp., n_-) is the number of eigenvalues with positive (resp., negative) real part, n_z is the number of zero eigenvalues and 2n_p is the number of pure imaginary eigenvalues.

The set of refined inertias H_n=(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0) is important for the onset of Hopf bifurcation in dynamical systems. In this talk, I will introduce some results about sign patterns, i.e., matrices whose entries are from the set {+, -, 0}, that require H_n.

Recently, Berliner et al. expend H_n to H_n* for zero-nonzero patterns, i.e., matrices whose entries are from the set {0, *}. In this talk, I will show that there is no zero-nonzero pattern that requires H_n*.

**DMS Linear Algebra/Algebra Seminar**

Nov 07, 2017 04:00 PM

Parker Hall 356

Speaker:

**Luke Oeding**

Title: Higher Order Partial Least Squares and an Application to Neuroscience.

Abstract: Partial least squares (PLS) is a method to discover a functional dependence between two sets of variables X and Y. PLS attempts to maximize the covariance between X and Y by projecting both onto new subspaces. Higher order partial least squares (HOPLS) comes into play when the sets of variables have additional tensorial structure. Simultaneous optimization of subspace projections may be obtained by a multilinear singular value decomposition (MSVD). I'll review PLS and SVD, and explain their higher order counterparts. Finally I'll describe recent work with G. Deshpande, A. Cichocki, D. Rangaprakash, and X.P. Hu where we propose to use HOPLS and Tensor Decompositions to discover latent linkages between EEG and fMRI signals from the brain, and ultimately use this to drive Brain Computer Interfaces (BCI)'s with the low size, weight and power of EEG, but with the accuracy of fMRI.

**DMS Linear Algebra/Algebra Seminar**

Oct 31, 2017 04:00 PM

Parker Hall 356

Speaker: **Reimbay Reimbayev**

Title: Synchronization in Neuronal Networks, Part II

Abstract: In the second part of my talk I will concentrate on proving the stability of global synchronization manifold for the network of square-wave bursting neurons using semi-numerical tools and graph theoretical considerations.

Last Updated: 01/20/2017