Department of Mathematics

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Abacus proofs of Schur function identities
Loehr, N. A. (Siam Publications, 2010)
This article uses combinatorial objects called labeled abaci to give direct combinatorial proofs of many familiar facts about Schur polynomials. We use abaci to prove the Pieri rules, the Littlewood-Richardson rule, the equivalence of the tableau definition and the determinant definition of Schur polynomials, and the combinatorial interpretation of the inverse Kostka matrix (first given by Egecioglu and Remmel). The basic idea is to regard formulas involving Schur polynomials as encoding bead motions on abaci. The proofs of the results just mentioned all turn out to be manifestations of a single underlying theme: when beads bump, objects cancel.
Acceleration of tensor-product operations for high-order finite element methods
Świrydowicz, K.; Chalmers, N.; Karakus, A.; Warburton, T. (2017-09)
This paper is devoted to GPU kernel optimization and performance analysis of three tensor-product operators arising in finite element methods. We provide a mathematical background to these operations and implementation details. Achieving close-to-the-peak performance for these operators requires extensive optimization because of the operators' properties: low arithmetic intensity, tiered structure, and the need to store intermediate results inside the kernel. We give a guided overview of optimization strategies and we present a performance model that allows us to compare the efficacy of these optimizations against an empirically calibrated roofline.
ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra
Hinkelmann, Franziska; Brandon, Madison; Guang, Bonny; McNeill, Rustin; Blekherman, Grigoriy; Veliz-Cuba, Alan; Laubenbacher, Reinhard C. (2011-07-20)
Background Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed. Results We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness and robustness. For a large set of published complex discrete models, ADAM identified the attractors in less than one second. Conclusions Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools. ADAM provides analysis methods based on mathematical algorithms as a web-based tool for several different input formats, and it makes analysis of complex models accessible to a larger community, as it is platform independent as a web-service and does not require understanding of the underlying mathematics.
An Adaptive Zolotarev Upper-Bound for the Singular Values of Loewner Matrices
Garcia Hilares, Nilton; Embree, Mark P. (2021-11-12)
Additive averages of multiplicative correlation sequences and applications
Donoso, Sebastian; Le, Ahn N.; Moreira, Joel; Sun, Wenbo (Springer, 2023-04-01)
We study sets of recurrence, in both measurable and topological settings, for actions of (ℕ, ×) and (ℚ>0, ×). In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form {(an+b)1/(cn+d) ℓ: n ∈ ℕ} are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.
An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice
Mare, A.-L.; Mihalcea, L. C. (2016-06-23)
Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $\Lambda_0, \ldots, \Lambda_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for $G= \mathrm{SL}_n(\mathbb{C})$. The first quantum ring is a deformation of the subalgebra of $\mathrm{H}^*(\mathcal{Fl}_G)$ generated by divisors. The second ring, denoted $\mathrm{QH}^*_{\mathrm{af}}(G/B)$, deforms the ordinary quantum cohomology ring $\mathrm{QH}^*(G/B)$ by adding an affine quantum parameter $q_0$. We prove that $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of $G$.
Algebras and Varieties
Green, Edward L.; Hille, Lutz; Schroll, Sibylle (2020-03)
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The cases of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.
Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control
Gunzburger, Max D.; Manservisi, S. (Siam Publications, 2000-05)
We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier-Stokes ow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.
Analysis of Entropy Generation Rate in an Unsteady Porous Channel Flow with Navier Slip and Convective Cooling
Chinyoka, Tirivanhu; Makinde, Oluwole Daniel (MDPI, 2013-05-28)
This study deals with the combined effects of Navier Slip, Convective cooling, variable viscosity, and suction/injection on the entropy generation rate in an unsteady flow of an incompressible viscous fluid flowing through a channel with permeable walls. The model equations for momentum and energy balance are solved numerically using semi-discretization finite difference techniques. Both the velocity and temperature profiles are obtained and utilized to compute the entropy generation number. The effects of key parameters on the fluid velocity, temperature, entropy generation rate and Bejan number are depicted graphically and analyzed in detail.
Analysis of GMRES for Low‐Rank and Small‐Norm Perturbations of the Identity Matrix
Carr, Arielle K.; de Sturler, Eric; Embree, Mark P. (Wiley, 2023-03-24)
Analytic theory for the selection of a two-dimensional needle crystal at arbitrary Péclet number
Tanveer, S. (American Physical Society, 1989-10)
An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Péclet number for small values of the surface-tension parameter. The velocity selection is caused by the effect of transcendentally small terms that are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small-Péclet-number analytical results of other investigators, although there are some discrepancies in details. It also addresses questions raised by a recent investigator on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.
Analytical solutions of model equations for two phase gas mixtures: Transverse velocity perturbations
Cavalier, J. F.; Greenberg, William (AIP Publishing, 1984)
Model equations for a dilute binary gas system are derived, using a linear BGK scheme. Complete analytical solutions for the stationary half_space problem are obtained for transverse velocity perturbations. The method of solution relies on the resolvent integration technique.
Annual Report Mathematics Department 2005
(Department of Mathematics, 2006)
Annual Report Mathematics Department 2006
(Department of Mathematics, 2007)
Annual Report Mathematics Department 2007-2008
(Department of Mathematics, 2008)
Annual Report Mathematics Department 2008-2009
(Department of Mathematics, 2009)
Annual Report Mathematics Department 2009-2010
(Department of Mathematics, 2010)
Anomalous decay of a prepared state due to non-Ohmic coupling to the continuum
Sela, I.; Aisenberg, J.; Kottos, T.; Elgart, A.; Cohen, D. (American Physical Society, 2010-03)
We study the decay of a prepared state E(0) into a continuum {E(k)} in the case of non-Ohmic models. This means that the coupling is |V(k),(0)| proportional to |E(k)-E(0)|(s-1) with s not equal 1. We find that irrespective of model details there is a universal generalized Wigner time t(0) that characterizes the decay of the survival probability P(0)(t). The generic decay behavior which is implied by rate equation phenomenology is a slowing down stretched exponential, reflecting the gradual resolution of the band profile. But depending on nonuniversal features of the model a power-law decay might take over: it is only for an Ohmic coupling to the continuum that we get a robust exponential decay that is insensitive to the nature of the intracontinuum couplings. The analysis highlights the coexistence of perturbative and nonperturbative features in the dynamics. It turns out that there are special circumstances in which t(0) is reflected in the spreading process and not only in the survival probability, contrary to the naive linear-response theory expectation.
Anomaly meltdown
Gajdzinski, C.; Streater, R. F. (AIP Publishing, 1991-08)
It is shown that at nonzero temperature it is possible that anomalies in representations of symmetry groups and gauge groups, present at zero temperature, disappear. Several examples are given. Thus the idea that anomalies in baryon currents might have caused the baryon imbalance in the early hot universe needs reconsideration.
Antibiotics ameliorate lupus-like symptoms in mice
Mu, Qinghui; Tavella, Vincent J.; Kirby, Jay L.; Cecere, Thomas E.; Chung, Matthias; Lee, Jiyoung; Li, Song; Ahmed, Sattar Ansar; Eden, Kristin; Allen, Irving C. (Nature, 2017-10-20)
Gut microbiota and the immune system interact to maintain tissue homeostasis, but whether this interaction is involved in the pathogenesis of systemic lupus erythematosus (SLE) is unclear. Here we report that oral antibiotics given during active disease removed harmful bacteria from the gut microbiota and attenuated SLE-like disease in lupus-prone mice. Using MRL/lpr mice, we showed that antibiotics given after disease onset ameliorated systemic autoimmunity and kidney histopathology. They decreased IL-17-producing cells and increased the level of circulating IL-10. In addition, antibiotics removed Lachnospiraceae and increased the relative abundance of Lactobacillus spp., two groups of bacteria previously shown to be associated with deteriorated or improved symptoms in MRL/lpr mice, respectively. Moreover, we showed that the attenuated disease phenotype could be recapitulated with a single antibiotic vancomycin, which reshaped the gut microbiota and changed microbial functional pathways in a time-dependent manner. Furthermore, vancomycin treatment increased the barrier function of the intestinal epithelium, thus preventing the translocation of lipopolysaccharide, a cell wall component of Gram-negative Proteobacteria and known inducer of lupus in mice, into the circulation. These results suggest that mixed antibiotics or a single antibiotic vancomycin ameliorate SLE-like disease in MRL/lpr mice by changing the composition of gut microbiota.

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