´¡²ú²õ³Ù°ù²¹³¦³Ù:Ìý We use a ÌýFeshbach P-Q partitioning technique to derive a closed one- component integro-differential equation. The resultant equationÌýproperly traces the footprint of the target state in quantum control theory. The physical significance of the derived dynamical equation isÌýillustrated by both general analysis and concrete examples. ÌýLeakage from an encoded subspace to the rest of the system space is aÌýparticularly serious problem for which leakage elimination operators (LEOs) were introduced. By using the resultant dynamical equation, weÌýshow that the effectiveness of LEOs depends on the integral of the pulse sequence in the time domain, which has been missing because ofÌýthe idealization of pulse sequences. Our results are applied to a three- level system for the nitrogen-vacancy centers under an externalÌýmagnetic field and are illustrated by the fidelity dynamics of LEO sequences, ranging from regular rectangular pulses, random pulses, andÌýeven disordered (noisy) pulses.

Please contact Prof. Yiqiang Zhao (zhao@math.carleton.ca) if you would like to set up an individual meeting with the speaker.

Abstract: In recent years, a few surprising connections have arisen between information theory, algebra, and topology. This talk is in a similar vein. We will discuss a certain correspondence between Shannon entropy and continuous functions on topological simplices that satisfy an equation akin to the Leibniz rule from Calculus. The correspondence relies heavily on a particular operad, which is an abstract tool with origins in algebraic topology. A broad goal for this talk is to unwind this result and to share why the confluence of these ideas is both unexpected yet intriguing.

Short Bio:

Tai-Danae Bradley is currently a research mathematician at Sandbox AQ <> and a visiting professor of mathematics at The Master’s University <> where she helpsÌý run the Math3ma Institute <>.

She finished her PhD in mathematics in spring 2020 at the CUNY Graduate Center under the supervision of John Terilla <> and spent some timeÌý as a postdoctoral researcher at X, the Moonshot Factory (Google X).Ìý Her research interests lie in the intersection of quantum physics,Ìý machine intelligence, and category theory.

For more information please contact Colin Ingalls.

Abstract: Hypergraphs are generalizations of graphs where each edgeÌý joins an arbitrary number of vertices, and not just two as in the caseÌý of graphs. We define the density of a hypergraph as the ratio of theÌý number of edges to the number of vertices. We solve the problem ofÌý finding in a given large hypergraph a subhypergraph with a maximumÌý possible density. We introduce a notion of a support matrix A of aÌý hypergraph and prove that the maximal density of a subhypergraph isÌý equal to the largest eigenvalue of A*A^T for an optimal support matrixÌý A. The methods that were developed for the proof of this theorem yieldÌý an efficient algorithmic solution of this problem. This topic is rather cross-disciplinary – in addition to graph theory, our results have applications in data science. The proofs, however, are based on linear algebra and linear optimization.

´¡²ú²õ³Ù°ù²¹³¦³Ù:Ìý Item nonresponse in surveys is usually handled through some form of single imputation. Random forests provide flexible tools for obtaining a set of imputed values. Belonging to the class of non-parametric methods, random forests have the ability to capture nonlinear trends in the data and tend to be robust to the non-inclusion of interactions or predictors accounting for curvature. We lay out a set of sufficient conditions needed for establishing the L2-consistency of an imputed estimator based on random forests. We investigate the performance of variance estimators that account for sampling and nonresponse. We present the results from a simulation study to assess the proposed methods in terms of bias, efficiency and coverage rate of normal-based confidence intervals.

´¡²ú²õ³Ù°ù²¹³¦³Ù:Ìý The human gut microbiome is an essential component of our physiology and exploring the relationship between biological/environmental covariates and the resulting taxonomic composition of a given microbial community is an active area of research. Previously, a Dirichlet-multinomial regression framework has been suggested to model this relationship, but it did not account for any underlying latent group structure. An underlying group structure of guts (such as enterotypes) has been observed across gut microbiome samples in which guts in the same group share similar biota compositions. In this talk, a finite mixture of Dirichlet-multinomial regression models will be presented that accounts for this underlying group structure and to allow for a probabilistic investigation of the relationship between bacterial abundance and biological/environmental covariates within each inferred group.

Furthermore, finite mixtures of regression models, which incorporate the concomitant effect of the covariates on the resulting mixing proportions are also proposed and examined within the Dirichlet-multinomial framework.

We utilize the proposed mixture model to gain insight on underlying subgroups in a microbiome dataset comprising of tumor and healthy samples and the relationships between covariates and microbial abundance in those subgroups. The talk will conclude with some current and future research directions involving microbiome data.

For more information please contact Sanjeena Dang –Ìýsanjeena.dang@carleton.ca

´¡²ú²õ³Ù°ù²¹³¦³Ù:Ìý The well-known 1-2-3 Conjecture by Karonski, Luczak and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1, 2 or 3 so that for each edge uv the sums of the weights at u and at v are distinct.

The list version of the 1-2-3 Conjecture by Bartnicki, Grytczuk and Niwczyk states that the same holds if each edge e has the choice of weights not necessarily from {1,2,3}, but from any set {x(e),y(e),z(e)} of three real numbers.

The goal of this talk is to survey developments on the 1-2-3 Conjecture, especially on the list version of the 1-2-3 Conjecture.

Short Bio:Ìý Prof. Zhu received his Ph.D from University of Calgary in 1990.

Before joining Zhejiang Normal University as a chair professor, he worked at National Sun Yat-sen University in Taiwan.

In 2007, he received the Academic Prize from the Taiwan Mathematical Society. He has published over 200 papers in the broad area of discrete mathematics and his work on graph coloring is highly influential. He also serves on the editorial boards of several journals such as Journal of Graph Theory, Discrete Mathematics, SIAM Journal on Discrete Math, European Journal of Combinatorics.

For more information please contact Jason Gao <Zhicheng.Gao@carleton.ca>

Speaker: Vitali Vougalter, University of Toronto

Title: “Solvability of some integro-differential equations with concentrated sources”.

Abstract: “The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used”.

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