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Richard Ehrenborg University of Kentuky
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| Title: The descent set polynomial |
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Abstract: One usually encodes the number of permutations beta(S) in the symmetric group having descent set S via the Eulerian polynomial, where the number of such permutations is the coefficient of t^|S|. We instead introduce the descent set polynomial where the statistic beta(S) is the exponent of t. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. Especially we consider factors of the form Phi_2, Phi_{2p} where p is a prime and double cyclotomic factors. As an instance we deduce that the proportion of odd entries in the descent set statistics for the symmetric group on n elements only depends on the number of 1’s in the binary expansion of n. This is joint work first with Denis Chebikin, Pavlo Pylyavskyy and Margaret Readdy, and second with Brad Fox.
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Thurs. October 2 at 12:30pm in the Conference Room
Categories: Spring 2022
Wed. August 13 at 2:00pm in the Conference Room
Categories: Spring 2022
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Yuri Suhov, University of Cambridge, Cambridge, UK
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| Title: The Widom–Rowlinson model and the allelopathy phenomenon |
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Abstract: The Widom–Rowlinson (WR) model was proposed in early 1970s in Chemical Physics, to explain various phenomena at the molecular level. There are several types of particles which repel one another when they belong to different types and have no influence upon each other if they belong to the same type (these assumptions can be made more general). This resembles an allelopathic phenomenon observed in biology where a given species prevents other species from occupying the space nearby by using a variety of means (poisoning soil or water, encouraging parasites harmful to other species but harmless to themselves, etc.). There is also a quantum version of the model.
The WR model became popular in various disciplines. An interesting question is about phase transitions: if the overall particle density is low, there is one equilibrium (Gibbs) distribution resembling Poisson. However, if the density is high, there may be one or several distributions where a particular type will dominate (occupy an overwhelming proportion of the space). This is determined by the collection of hard-core repulsion diameters. The talk will focus on new results on this question and emerging applications.
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Thurs. May 8 at 3:00pm in FRET 402
Categories: Spring 2022
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Mikhail Malioutov, Department of Mathematics, Northeastern University
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| Title: Compressors, Variable memory Length MC and testing homogeneity of time series |
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Abstract: Modeling applications such as literary texts as stationary time series is a firmly established tool in computer linguistics. A popular problem is an authorship attribution of texts which in statistical terms is called homogeneity testing which must be nonparametric with respect to the multivariate distribution of time series since data will never suffice to find their parametric model. An adequate tools turn out to be Variable memory Length MC and Conditional Complexity of Compression. The latter was initially inspired by analogy with non-computable Kolmogorov Complexity. A review of the underlying theory and many applications will be presented.
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Thurs. April 17 at 4:00pm in FRET 406
Categories: Spring 2022
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Michael Klibanov, Department of Mathematics and Statistics, UNC Charlotte
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| Title: Uniqueness of phaseless inverse scattering problems in three dimensions. |
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Abstract: This is the first solution of a long standing problem which was posed in chapter 10 of the book of K. Chadan and P. Sabatier “Inverse Problems in Quantum Scattering Theory”, Springer-Verlag, New York, 1977. In quantum scattering one measures only scattering cross section. This means that the modulus of the complex valued scattering wave field is measured. But phase is not measured. In the meantime the entire theory of inverse quantum scattering is based on the assumption that both modulus and phase are measured. Thus, the question was posed in the above book whether it is possible, at least in principle, to uniquely reconstruct the scattering potential if only the modulus of the scattered field is known for all frequencies. This question was addressed positively for the first time in the paper of the author published in SIAM J. Applied Mathematics, 74, #2, 392-410, 2014. |
Thursday, April 3 at 3:30pm in Room Fretwell 406
Categories: Spring 2022
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Dr. Alexander Bendikov, Department of Mathematics, Cornell University
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Abstract: Let (X,d) be a locally compact separable metric space. We assume that (X,d) is proper and totally disconnected. Given a measure m on X and a choice-function C(B) defined on the set of all non-singleton balls B of X we consider the hierarchical Laplacian L=L_{C}. The operator L acts in L(X,m), is essentially self-adjoint and has a purely point spectrum. Choosing a sequence of i.i.d. random variables we define a perturbated choice-function and a perturbated hierarchical Laplacian. We study asymptotic behaviour of the arithmetic means of the eigenvalues of the perturbated hierarchical Laplacian. We prove that under some mild assumptions the arithmetic means are asymptotically normal whereas without these conditions the limited distribution is not normal. We prove existence of the integrated density of states (i.d.s.) associated with the operator perturbated hierarchical Laplacian – the quantitative characteristics of the phase transition in the Dyson’s model. Assuming that the i.d.s is continuous we prove that the number of eigenvalues which fall in a small interval is approximately Poissonian. As an example we consider random perturbations defined by the i.i.d. Bernoulli random variables of the Vladimirov-Taibleson operator acting on the ring of p-adic numbers.
This is joint work with A. Grigor’yan and S.A. Molchanov.
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Friday, April 4 at 2:00pm in the Conference Room
Categories: Spring 2022
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Dr. Chun Liu, Department of Mathematics, Pennsylvania State University
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| Title: Energetic Variational Approaches: Onsager’s Maximum Dissipation Principle, General Diffusion, Optimal Transport and Stochastic Integrals. |
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Abstract: In the talk, I will explore the underlying mechanism governing various diffusion processes. We will employ a general framework of energetic variational approaches, consisting of, in particular, Onsager’s Maximum Dissipation Principles, and their specific applications to biology and physiology. We will discuss the roles of different stochastic integrals (Ito’s form, Stratonovich’s form and other possible forms), and the procedure of optimal transport in the context of general framework of theories of linear responses.
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Thur. March 27 at 4:00-5:00pm in Fretwell 406
Categories: Spring 2022
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Dr. Marc Ryser, Department of Mathematics, Duke University
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| Title: Of Noise and Cancer |
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Abstract: In the first part of my talk I will discuss the well-posedness of the 2D stochastic Allen-Cahn equation, a commonly used stochastic partial differential equation in physics and material sciences. I will tell the whole tale, from numerical investigations and formulation of a conjecture, to heuristic arguments and the conjecture-turned-theorem. In the second part of my talk I will present ongoing work at the interface of mathematics and cancer biology. I will introduce a stochastic model of carcinogenesis, and show how a combination of analysis and simulations are used to address clinically relevant problems. To finish, I will talk about an ongoing collaboration with cancer surgeons at Duke, where we develop a modeling framework to make patient-specific predictions.
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Fri. March 21 at 2:00-3:00pm in Fretwell 114
Categories: Spring 2022
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Dr. Yichuan Zhao, Department of Mathematics and Statistics, Georgia State University
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Title: Smoothed jackknife empirical likelihood inference for ROC curves with missing data
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Abstract: In this paper, we apply smoothed jackknife empirical likelihood (JEL) method to construct confidence intervals for the receiver operating characteristic (ROC) curve with missing data. After using hot deck imputation, we generate pseudo-jackknife sample to develop jackknife empirical likelihood. Comparing to traditional empirical likelihood method, the smoothed JEL has a great advantage in saving computational cost. Under mild conditions, the smoothed jackknife empirical likelihood ratio converges to a scaled chi-square distribution. Furthermore, simulation studies in terms of coverage probability and average length of confidence intervals demonstrate this proposed method has the good performance in small sample sizes. A real data set is used to illustrate our proposed JEL method. This is joint work with Dr. Hanfang Yang.
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Wed. April 9 at 3:30-4:30pm in Fretwell 306
Categories: Spring 2022
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Dr. Qian, CFA and chief investment officer of the multi asset group at PanAgora Asset Management
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| Title: To Rebalance or Not to Rebalance: A Statistical Comparison of Terminal Wealth of Fixed-Weight and Buy-and-Hold Portfolios |
| Abstract: We carry out statistical analysis under a variety of return assumptions and portfolio settings. For long-only portfolios, we show buy-and-hold approach leads to higher expected terminal wealth but also higher variance of terminal wealth. When there are serial correlations in asset returns, we demonstrate quantitatively that for long-only portfolios mean-reverting returns are relatively more favorable to fixed-weight portfolios whereas trending returns are relatively more favorable to buy-and-hold portfolios. For long-short portfolios, however, the effects of portfolio rebalancing are markedly different from long-only portfolios, mainly due to portfolio leverage. For example, we prove that fixed-weight approach often leads to higher expected value of terminal wealth. But it may also lead to higher variance of terminal wealth although it is not always the case. Furthermore, the effects of serial return correlations on long-short portfolios could be opposite of the effects on long-only portfolios. The overall results suggest that fixed-weight portfolios with portfolio rebalancing are more likely to have better risk-adjusted terminal wealth than buy-and-hold portfolios. |
Wed, Feb 19 at 11:00am in the conference room
Categories: Spring 2022
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Dr. JaEun Ku, Oklahoma State University
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| Title: Solver-friendly hybrid mixed finite element methods |
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A new hybrid mixed finite element method to compute the flux variable
accurately and efficiently will be introduced. The method is a two–step method, based on a system of first-order equations for second-order elliptic partial differential equations. On a coarse mesh, the primary variable is approximated by a standard Galerkin method. Then, on a fine mesh, an H(div) projection is sought as an accurate approximation for the flux variable. The computation on a finer mesh can be carried out very efficiently using well developed preconditioners for the H(div) projection. Also, it will be shown that the mesh size h for the finer mesh can be taken as a square of the coarse meshsize H. This is a joint work with Dr. Young Ju Lee and Dr. Dongwoo Sheen. |