arXiv subjects

Michael Shapiro

Publications and source records attributed to Michael Shapiro.

At least 55 records · Page 3Linked to original sources

arXiv2013

Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis

With the goal of providing the foundations for a rigorous study of modules of bicomplex holomorphic functions, we develop a general theory of functional analysis with bicomplex scalars. Even though the basic properties of bicomplex number are well known and widely available, our analysis requires some more delicate discussion of the various structures which are hidden in the ring of bicomplex numbers. We study in particular matrices with bicomplex numbers, bicomplex modules, and inner products and norms in bicomplex modules. We consider two kinds of norms on bicomplex modules: a real-valued norm (as one would expect), and a hyperbolic-valued norm. Interestingly enough, while both norms can be used to build the theory of normed bicomplex modules, the hyperbolic-valued norm appears to be much better compatible with the structure of bicomplex modules. We also consider linear functionals on bicomplex numbers. To conclude we describe a bicomplex version of classical Schur analysis

math.CV↗

arXiv2012

Saturation effects on T-cell activation in a model of a multi-stage pathogen

In previous work, we studied host response to a pathogen which uses a cycle of immunologically distinct stages to establish and maintain infection. We showed that for generic parameter values, the system has a unique biologically meaningful stable fixed point. That paper used a simplified model of T-cell activation, making proliferation depend linearly on antigen-T-cell encounters. Here we generalize the way in which T-cell proliferation depends on the sizes of the antigenic populations. In particular, we allow this response to become saturated at high levels of antigen. As a result, we show that this family of generalized models shares the same steady-state behavior properties with the simpler model contemplated in our previous work.

q-bio.PE↗

arXiv2012

Higher pentagram maps, weighted directed networks, and cluster dynamics

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras. In this paper, we extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems. Our main tool is Poisson geometry of weighted directed networks on surfaces developed by M. Gekhtman, M. Shapiro, and A. Vainshtein. The ingredients necessary for complete integrability -- invariant Poisson brackets, integrals of motion in involution, Lax representation -- are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter $k>1$. The case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. If $k=2$ and the ground field is $\C$, we give a geometric interpretation in terms of circle patterns.

math.QA↗

arXiv2011

Failure Filtrations for Fenced Sensor Networks

In this paper we consider the question of sensor network coverage for a 2-dimensional domain. We seek to compute the probability that a set of sensors fails to cover given only non-metric, local (who is talking to whom) information and a probability distribution of failure of each node. This builds on the work of de Silva and Ghrist who analyzed this problem in the deterministic situation. We first show that a it is part of a slightly larger class of problems which is #P-complete, and thus fast algorithms likely do not exist unless P$=$NP. We then give a deterministic algorithm which is feasible in the case of a small set of sensors, and give a dynamic algorithm for an arbitrary set of sensors failing over time which utilizes a new criterion for coverage based on the one proposed by de Silva and Ghrist. These algorithms build on the theory of topological persistence.

cs.CG↗

arXiv2011

Finding an individual's probability of infection in an SIR network is NP-hard

The celebrated Kermack-McKendric model of epidemics studies the transmission of a disease in a population where each individual is initially susceptible (S), may become infective (I) and then removed or recovered (R) and plays no further epidemiological role. This ODE model arises as the limiting case of a network model where each individual has an equal chance of infecting every other. More recent work gives explicit consideration to the network of social interaction and attendant probability of transmission for each interacting pair. The state of such a network is an assignment of the values {S,I,R} to its members. Given such a network, an initial state and a particular susceptible individual, we would like to compute their probability of becoming infected in the course of an epidemic. It turns out that this problem is NP-hard. In particular, it belongs in a class of problems all of whose known solutions require an exponential amount of computation and for which it is unlikely that there will be more efficient solutions.

q-bio.PE↗

arXiv2011

Cluster structures on simple complex Lie groups and the Belavin-Drinfeld classification

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds to a cluster structure in $Ø(\G)$. We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for $SL_n$, $n<5$, and for any $\G$ in the case of the standard Poisson-Lie structure.

math.QA↗

arXiv2011

Cluster algebras of finite mutation type via unfoldings

We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram characterizing the matrix admits an unfolding which embeds its mutation class to the mutation class of some mutation-finite skew-symmetric matrix. In particular, this establishes a correspondence between a large class of skew-symmetrizable mutation-finite cluster algebras and triangulated marked bordered surfaces.

math.CO↗

arXiv2010

Poisson Geometry of Directed Networks in an Annulus

As a generalization of Postnikov's construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in arXiv: 0805.3541 induce a family of Poisson structures on rational-valued matrix functions and on the space of loops in the Grassmannian. In the former case, this family includes, for a particular kind of networks, the Poisson bracket associated with the trigonometric R-matrix.

math.QA↗

arXiv2010

Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N_{u,v} in the disk that correspond to the choice of a pair (u,v) of Coxeter elements in the symmetric group and the corresponding networks N_{u,v}^\circ in the annulus. Boundary measurements for N_{u,v} represent elements of the Coxeter double Bruhat cell G^{u,v} in GL_n. The Cartan subgroup acts on G^{u,v} by conjugation. The standard Poisson structure on the space of weights of N_{u,v} induces a Poisson structure on G^{u,v}, and hence on its quotient by the Cartan subgroup, which makes the latter into the phase space for an appropriate Coxeter--Toda lattice. The boundary measurement for N_{u,v}^\circ is a rational function that coincides up to a nonzero factor with the Weyl function for the boundary measurement for N_{u,v}. The corresponding Poisson bracket on the space of weights of N_{u,v}^\circ induces a Poisson bracket on the certain space of rational functions, which appeared previously in the context of Toda flows. Following the ideas developed in our previous papers, we introduce a cluster algebra A on this space, compatible with the obtained Poisson bracket. Generalized Bäcklund--Darboux transformations map solutions of one Coxeter--Toda lattice to solutions of another preserving the corresponding Weyl function. Using network representation, we construct generalized Bäcklund-Darboux transformations as appropriate sequences of cluster transformations in A.

math.QA↗

arXiv2010

A model of host response to a multi-stage pathogen

We model the immune surveillance of a pathogen which passes through $n$ immunologically distinct stages. The biological parameters of this system induce a partial order on the stages, and this, in turn, determines which stages will be subject to immune regulation. This corresponds to the system's unique asymptotically stable fixed point.

q-bio.PE↗

arXiv2009

Skew-symmetric cluster algebras of finite mutation type

In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems $E_6$, $E_7$, $E_8$, $\widetilde E_6$, $\widetilde E_7$, $\widetilde E_8$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$; two remaining were recently found by Derksen and Owen. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.

math.CO↗

arXiv2009

Poisson structures compatible with the cluster algebra structure in Grassmannians

We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian $G_k(n)$ and show that any such bracket endows $G_k(n)$ with a structure of a Poisson homogeneous space with respect to the natural action of $SL_n$ equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Drinfeld classification. Moreover, every compatible Poisson structure can be obtained this way.

math.QA↗

arXiv2009

Poisson Geometry of Directed Networks in a Disk

We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriately chosen R-matrix Poisson-Lie structure. We also prove that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.

math.QA↗

arXiv2008

Groups that do and do not have context-sensitive word problem

We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and Shapiro). We generalise their results to find many examples of groups not admitting non-deterministic Cannon's algorithms. This adds to the examples of Kambites and Otto of groups separating context-sensitive and growing context-sensitive word problems, and provides a new language-theoretic separation result.

math.GR↗

arXiv2008

On a generalization of Dehn's algorithm

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

math.GR↗

arXiv2007

On eigenvalues of rectangular matrices

Given a $(k+1)$-tuple $A, B_1,...,B_k$ of $(m\times n)$-matrices with $m\le n$ we call the set of all $k$-tuples of complex numbers $\{\la_1,...,\la_k\}$ such that the linear combination $A+\la_1B_1+\la_2B_2+...+\la_kB_k$ has rank smaller than $m$ the {\it eigenvalue locus} of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see \cite{He} and \cite{Vol}, we study a number of properties of the eigenvalue locus in the most important case $k=n-m+1$.

math.AG↗

arXiv2007

Cluster algebras and triangulated surfaces. Part I: Cluster complexes

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of "tagged triangulations" of the surface, and determine its homotopy type and its growth rate.

math.RA↗

arXiv2006

On double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers $\CP^1\to\CP^1$ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.

math.AG↗