arXiv subjects

Alex Kasman

Publications and source records attributed to Alex Kasman.

At least 19 recordsLinked to original sources

arXiv2025

Two Useful Facts About Generating Functions

Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function. The first constructively produces "recursion relations" for the sequence from differential operators in the formal variable having the generating function as an eigenfunction. The second allows one to determine whether the sequence is orthogonal with respect to some inner product by considering the result of taking the inner product of the generating function with itself. Examples presented to demonstrate the use and value of these methods include a sequence of numbers, a family of Exceptional Hermite Polynomials, and an example illustrating the result in a non-commutative setting.

math.CA↗

arXiv2025

Orthogonality with Respect to the Hermite Product, KP Wave Functions, and the Bispectral Involution

It is well known that for any wave function $\psi(x,z)$ of the KP hierarchy, there is another wave function called its ''adjoint'' such that the path integral of their product with respect to $z$ around any sufficiently large closed path is zero. For the wave functions in the adelic Grassmannian ${\rm Gr}^{\rm ad}$, the bispectral involution which exchanges the role of $x$ and $z$ also implies the existence of an ''$x$-adjoint wave function'' $\psi^{\star}(x,z)$ so that the product of the wave function, the $x$-adjoint, and the Hermite weight ${\rm e}^{-x^2/2}$ has no residue. Utilizing this, we show that the sequences of coefficient functions in the power series expansion of any KP wave function in ${\rm Gr}^{\rm ad}$ and its image under the bispectral involution at $t_2=-\frac{1}{2}$ are always ''almost bi-orthogonal'' with respect to the Hermite product. Whether the sequences have the stronger properties of being (almost) orthogonal can easily be determined in terms of KP flows and the bispectral involution. As a special case, the exceptional Hermite orthogonal polynomials can be recovered in this way. This provides both a generalization of and an explanation of the fact that the generating functions of the exceptional Hermites are certain special wave functions of the KP hierarchy. In addition, one new surprise is that the same KP wave function which generates the sequences of functions is also a generating function for the norms when evaluated at $t_1=1$ and $t_2=0$. The main results are proved using Calogero-Moser matrices satisfying a rank one condition. The same results also apply in the case of ''spin-generalized'' Calogero-Moser matrices, which produce instances of matrix orthogonality.

nlin.SI↗

arXiv2025

Exceptional Krall polynomials

In this paper we exhibit and study a novel class of exceptional Krall orthogonal polynomials of Hermite type. This means that the polynomials in question are (i) orthogonal with respect to a Hermite-type weight; (ii) are the eigenfunctions of a higher-order differential operator; (iii) the degree sequence of the polynomial family in question is missing a finite number of degrees. Regarding the second point, unlike the known class of exceptional Hermite polynomials that satisfy a second-order eigenvalue equation, the polynomials we introduce here are not eigenfunctions of any 2nd order differential operator, but are for one of 4th order. Regarding the third point, our family does not include a polynomial of degree zero and consequently satisfies a 5th order recurrence relation instead of the classical 3-term relation.

math.CA↗

arXiv2025

Exceptional Hermite Polynomials and Calogero-Moser Pairs

There are two equivalent descriptions of George Wilson's adelic Grassmannian $Gr^{ad}$, one in terms of differential ``conditions'' and another in terms of Calogero-Moser Pairs. The former approach was used in the 2020 paper by Kasman-Milson which found that each family of Exceptional Hermite Polynomials has a generating function which lives in $Gr^{ad}$. This suggests that Calogero-Moser Pairs should also be useful in the study of Exceptional Hermite Polynomials, but no researchers have pursued that line of inquiry prior to the first author's thesis. The purpose of this note is to summarize highlights from that thesis, including a novel formula for Exceptional Hermite Polynomials in terms of Calogero-Moser Pairs and a theorem utilizing this correspondence to produce explicit finitely-supported distributions which annihilate them.

math.CA↗

arXiv2020

The Adelic Grassmannian and Exceptional Hermite Polynomials

It is shown that when dependence on the second flow of the KP hierarchy is added, the resulting semi-stationary wave function of certain points in George Wilson's adelic Grassmannian are generating functions of the exceptional Hermite orthogonal polynomials. This surprising correspondence between different mathematical objects that were not previously known to be so closely related is interesting in its own right, but also proves useful in two ways: it leads to new algorithms for effectively computing the associated differential and difference operators and it also answers some open questions about them.

math.CA↗

arXiv2020

Did Sequence Dependent Geometry Influence the Evolution of the Genetic Code?

The genetic code is the function from the set of codons to the set of amino acids by which a DNA sequence encodes proteins. Since the codons also influence the shape of the DNA molecule itself, the same sequence that encodes a protein also has a separate geometric interpretation. A question then arises: How well-duplexed are these two "codes"? In other words, in choosing a genetic sequence to encode a particular protein, how much freedom does one still have to vary the geometry (or vice versa). A recent paper by the first author addressed this question using two different methods. After reviewing those results, this paper addresses the same question with a third method: the use of Monte Carlo and Gaussian sampling methods to approximate a multi-integral representing the mutual information of a variety of possible genetic codes. Once again, it is found that the genetic code used in nuclear DNA has a slightly lower than average duplexing efficiency as compared with other hypothetical genetic codes. A concluding section discusses the significance of these surprising results.

q-bio.OT↗

arXiv2019

Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions

Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these three basic types. A complete characterization of the parameters that lead to non-singular 1-soliton and periodic solutions is given. Surprisingly, it is shown that such solutions are never singular when the solution is essentially non-commutative. When a 1-soliton solution is combined with another solution through an iterated Darboux transformation, the result behaves asymptotically like a combination of different solutions. This ``non-linear superposition principle'' is used to find a formula for the phase shift in the general 2-soliton interaction. A concluding section compares these results with other research on non-commutative soliton equations and lists some open questions.

nlin.SI↗

arXiv2015

On Factoring an Operator Using Elements of its Kernel

A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation $\partial$ acting on some ring of functions, this paper considers the more general situation of an endomorphism $\mathfrak{D}$ acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of $\mathfrak{D}$ followed by left multiplication by elements of the algebra. Assume that the set of such operators is closed under multiplication and that a Wronski-like matrix produced from some finite list of elements of the algebra is invertible (analogous to the linear independence condition). Then, it is shown that the set of operators whose kernels contain all of those elements is the left ideal generated by an explicitly given operator. In other words, an operator has those elements in its kernel if and only if it has that generator as a right factor. Three examples demonstrate the application of this result in different contexts, including one in which $\mathfrak{D}$ is an automorphism of finite order.

math.RA↗

arXiv2015

Bispectrality of $N$-Component KP Wave Functions: A Study in Non-Commutativity

A wave function of the $N$-component KP Hierarchy with continuous flows determined by an invertible matrix $H$ is constructed from the choice of an $MN$-dimensional space of finitely-supported vector distributions. This wave function is shown to be an eigenfunction for a ring of matrix differential operators in $x$ having eigenvalues that are matrix functions of the spectral parameter $z$. If the space of distributions is invariant under left multiplication by $H$, then a matrix coefficient differential-translation operator in $z$ is shown to share this eigenfunction and have an eigenvalue that is a matrix function of $x$. This paper not only generates new examples of bispectral operators, it also explores the consequences of non-commutativity for techniques and objects used in previous investigations.

nlin.SI↗

arXiv2015

Factorization of a Matrix Differential Operator Using Functions in its Kernel

Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a straight-forward generalization to the case of matrix coefficient differential operators that applies even in the case that the leading coefficient is singular.

math.RA↗

arXiv2008

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case $r=1$, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs $(L, Λ)$ of bispectral matrix differential operators is different than those previously studied in that $L$ acts from the left, but $Λ$ from the right on a common $r\times r$ eigenmatrix.

nlin.SI↗

arXiv2007

Universality of Rank 6 Plucker Relations and Grassmann Cone Preserving Maps

The Plucker relations define a projective embedding of the Grassmann variety Gr(k,n). We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a certain finite set of linear maps from the pth exterior power of n-dimensional space to the 2nd exterior power of 4-dimensional space and pulling back the unique Plucker relation on the latter. We also give a quadratic equation depending on (p+2) parameters having the same properties.

math.AG↗

arXiv2006

When is negativity not a problem for the ultra-discrete limit?

The `ultra-discrete 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 ultra-discrete 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 ultra-discrete equations.

nlin.SI↗

arXiv2006

On Decompositions of the KdV 2-Soliton

The KdV equation is the canonical example of an integrable non-linear partial differential equation supporting multi-soliton 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.

nlin.PS↗

arXiv2005

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.

math-ph↗

arXiv2004

Integrable Systems and Rank One Conditions for Rectangular Matrices

We provide a determinantal formula for tau-functions of the KP hierarchy in terms of rectangular, constant matrices $A$, $B$ and $C$ satisfying a rank one condition. This result is shown to generalize and unify many previous results of different authors on constructions of tau-functions for differential and difference integrable systems from square matrices satisfying rank one conditions. In particular, it contains as explicit special cases the formula of Wilson for tau-functions of rational KP solutions in terms of Calogero-Moser Lax matrices as well as our previous formula for KP tau functions in terms of almost-intertwining matrices.

math-ph↗

arXiv2004

On the Quantization of a Self-Dual Integrable System

In this note, we apply canonical quantization to the self-dual particle system describing the motion of poles to a higher rank solution of the KP hierarchy, explicitly determining both the quantum Hamiltonian and the wave function. It is verified that the quantum Hamiltonian is trivially bispectral (that is, that the wave function can be taken to be symmetric) as predicted by a widely held hypothesis of mathematical physics.

math-ph↗

arXiv2000

Solitons and Almost-Intertwining Matrices

We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some 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.

math-ph↗