Alex Kasman

Overcoming the Phage Replication Threshold: a Mathematical Model with Implications for Phage Therapy

ABSTRACT Prior observations of phage-host systems in vitro have led to the conclusion that susceptible host cell populations must reach a critical density before phage replication can occur. Such a replication threshold density would have broad implications for the therapeutic use of phage. In this report, we demonstrate experimentally that no such replication threshold exists and explain the previous data used to support the existence of the threshold in terms of a classical model of the kinetics of colloidal particle interactions in solution. This result leads us to conclude that the frequently used measure of multiplicity of infection (MOI), computed as the ratio of the number of phage to the number of cells, is generally inappropriate for situations in which cell concentrations are less than 107/ml. In its place, we propose an alternative measure, MOIactual, that takes into account the cell concentration and adsorption time. Properties of this function are elucidated that explain the demonstrated usefulness of MOI at high cell densities, as well as some unexpected consequences at low concentrations. In addition, the concept of MOIactual allows us to write simple formulas for computing practical quantities, such as the number of phage sufficient to infect 99.99% of host cells at arbitrary concentrations.

Bispectral Darboux transformations: the generalized Airy case

Abstract This paper considers Darboux transformations of a bispectral operator which preserve its bispectrality. A sufficient condition for this to occur is given, and applied to the case of generalized Airy operators of arbitrary order r > 1. As a result, the bispectrality of a large family of algebras of rank r is demonstrated. An involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters, generalizing Wilsons rank one bispectral involution. Spectral geometry and the relationship to the Sato grassmannian are discussed.

Bispectral KP solutions and linearization of Calogero-Moser particle systems

Rational and soliton solutions of the KP hierarchy in the subgrassmannianGr1 are studied within the context of finite dimensional dual grassmannians. In the rational case, properties of the tau function, τ, which are equivalent to bispectrality of the associated wave function, ψ, are identified. In particular, it is shown that there exists a bound on the degree of all time, variables in τ if and only if ψ is a rank one bispectral wave function. The action of the bispectral involution, β, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions, and particle systems, it is demonstrated that β is a linearizing map of the Calogero-Moser particle system and is essentially the map σ introduced by Airault, McKean and Moser in 1977 [2].

D-Modules and Darboux Transformations

A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.

Darboux Transformations from n-KdV to KP

The iterated Darboux transformations of an ordinary differential operator are constructively parametrized by an infinite-dimensional Grassmannian of finitely supported distributions. In the case that the operator depends on time parameters so that it is a solution to the n-KdV hierarchy, it is shown that the transformation produces a solution of the KP hierarchy. The standard definitions of the theory ofτ -functions are applied to this Grassmannian and it is shown that these new τ-functions are quotients of KPτ -functions. The application of this procedure for the construction of ‘higher rank’ KP solutions is discussed.

On Decompositions of the KdV 2-Soliton

AbstractThe KdV equation is the canonical example of an integrable nonlinear partial differential equation supporting multisoliton solutions. Seeking to understand the nature of this interaction, we investigate different ways to write the KdV 2-soliton solution as a sum of two or more functions. The paper reviews previous work of this nature and introduces new decompositions with unique features, putting it all in context and in a common notation for ease of comparison.

On KP generators and the geometry of the HBDE

Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the “shift” operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota bilinear difference equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a “rank one condition”, tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

Solitons and almost-intertwining matrices

We define the algebraic variety of almost intertwining matrices to be the set of triples (X,Y,Z) of n×n matrices for which XZ=YX+T for a rank one matrix T. A surprisingly simple formula is given for tau functions of the KP hierarchy in terms of such triples. The tau functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars–Schneider particle system.

Recurrence relations for exceptional Hermite polynomials

The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in x.

When is negativity not a problem for the ultradiscrete limit

The “ultradiscrete limit” has provided a link between integrable difference equations and cellular automata displaying soliton-like solutions. In particular, this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the “Max” operator. Although it certainly is the case that dropping these positivity conditions creates potential difficulties, it is still possible for solutions to persist under the ultradiscrete limit, even in their absence. To recognize when this will occur, one must consider whether a certain expression, involving a measure of the rates of convergence of different terms in the difference equation and their coefficients, is equal to zero. Applications discussed include the solution of elementary ordinary difference equations, a discretization of the Hirota Bilinear Difference Equation and the identification of integrals of motion for ultradiscrete equations.