Brian Simanek

Exploring the Mycobacteriophage Metaproteome: Phage Genomics as an Educational Platform

Bacteriophages are the most abundant forms of life in the biosphere and carry genomes characterized by high genetic diversity and mosaic architectures. The complete sequences of 30 mycobacteriophage genomes show them collectively to encode 101 tRNAs, three tmRNAs, and 3,357 proteins belonging to 1,536 “phamilies” of related sequences, and a statistical analysis predicts that these represent approximately 50% of the total number of phamilies in the mycobacteriophage population. These phamilies contain 2.19 proteins on average; more than half (774) of them contain just a single protein sequence. Only six phamilies have representatives in more than half of the 30 genomes, and only three—encoding tape-measure proteins, lysins, and minor tail proteins—are present in all 30 phages, although these phamilies are themselves highly modular, such that no single amino acid sequence element is present in all 30 mycobacteriophage genomes. Of the 1,536 phamilies, only 230 (15%) have amino acid sequence similarity to previously reported proteins, reflecting the enormous genetic diversity of the entire phage population. The abundance and diversity of phages, the simplicity of phage isolation, and the relatively small size of phage genomes support bacteriophage isolation and comparative genomic analysis as a highly suitable platform for discovery-based education.

Periodic discrete energy for long-range potentials

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.

Full length article: Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions

Let G be a bounded region with simply connected closure G@? and analytic boundary and let @m be a positive measure carried by G@? together with finitely many pure points outside G. We provide estimates on the norms of the monic polynomials of minimal norm in the space L^q(@m) for q>0. In case the norms converge to 0, we provide estimates on the rate of convergence, generalizing several previous results. Our most powerful result concerns measures @m that are perturbations of measures that are absolutely continuous with respect to the push-forward of a product measure near the boundary of the unit disk. Our results and methods also yield information about the strong asymptotics of the extremal polynomials and some information concerning Christoffel functions.

A new approach to ratio asymptotics for orthogonal polynomials

We use a non-linear characterization of orthonormal polynomials due to Saff in order to prove an equivalence of norm asymptotics and strong asymptotics for polynomials. Several applications of this equivalence are also discussed. One of our main results is that for regular measures on the closed unit disk - including, but not limited to the unit circle - or the interval [-2,2], one has ratio asymptotics along a sequence of asymptotic density 1.

Semilocal formal fibers of principal prime ideals

Let (T,m) be a complete local (Notherian) ring, C a finite set of pairwise incomparable nonmaximal prime ideals of T, and p a nonzero element. We provide necessary and sufficient conditions for T to be the completion of an integral domain A containing the prime ideal pA whose formal fiber is semilocal with maximal ideals the elements of C.

Two universality results for polynomial reproducing kernels

We prove two new universality results for polynomial reproducing kernels of compactly supported measures. The first applies to measures on the unit circle with a jump and a singularity in the weight at

An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

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Universality at an endpoint for orthogonal polynomials with Geronimus-type weights

and the second applies to area-type measures on a certain disconnected polynomial lemniscate. In both cases, we apply methods developed by Lubinsky to obtain our results.

Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions

We show that if m is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial P_n(z;B) solves an explicit second order linear differential equation. We also show that if T and B are distinct, then the pair {P_n(z;B),P_n(z;T)} solves an explicit first order linear system of differential equations. One can use these differential equations to deduce that the zeros of every paraorthogonal polynomial mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.

Full length article: Weak convergence of CD kernels: A new approach on the circle and real line

We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with respect to a Geronimus type weight on an arc of the unit circle. The key tool is a formula of McLaughlin for powers of a two-by-two matrix, which we use to derive convenient formulas for Geronimus polynomials.