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Yi Ming Zou

Publications and source records attributed to Yi Ming Zou.

At least 19 recordsLinked to original sources
arXiv2023

Isomorphism Classes of Idempotent Evolution Algebras

We showed that isomorphism classes of idempotent evolution algebras are in bijection with the orbits of the semidirect product group of the symmetric group and the torus, considered the combinatoric problem of enumeration of isomorphism classes for these algebras over arbitrary finite fields, derived a general counting formula, and obtained explicit formulas for the numbers of isomorphism classes in dimensions 2, 3, and 4 over any finite field.

math.RA↗
arXiv2022

On Automorphism Groups of Idempotent Evolution Algebras

We study the automorphism group of an idempotent evolution algebra, show that any finite group can be the automorphism group of an evolution algebra, and describe certain evolution algebras with given automorphism groups. In particular, we classify $n$-dimensional idempotent evolution algebras whose automorphism group is isomorphic to the symmetric group $S_n$, and classify idempotent evolution algebras with maximal diagonal automorphism subgroups.

math.RA↗
arXiv2017

Dynamics of Linear Systems over Finite Commutative Rings

The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous publication, the last two authors developed an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system or not. The algorithm can also be used to reduce the problem of finding the cycles of such a system to the case where the system is given by an automorphism. Here, we further analyze the cycle structure of such a system and develop a method to determine its cycles.

math.RA↗
arXiv2014

Newton's Method Backpropagation for Complex-Valued Holomorphic Multilayer Perceptrons

The study of Newton's method in complex-valued neural networks faces many difficulties. In this paper, we derive Newton's method backpropagation algorithms for complex-valued holomorphic multilayer perceptrons, and investigate the convergence of the one-step Newton steplength algorithm for the minimization of real-valued complex functions via Newton's method. To provide experimental support for the use of holomorphic activation functions, we perform a comparison of using sigmoidal functions versus their Taylor polynomial approximations as activation functions by using the algorithms developed in this paper and the known gradient descent backpropagation algorithm. Our experiments indicate that the Newton's method based algorithms, combined with the use of polynomial activation functions, provide significant improvement in the number of training iterations required over the existing algorithms.

math.CV↗
arXiv2014

An Algorithm for Detecting Fixed Points of Boolean Networks

In the applications of Boolean networks to modeling biological systems, an important computational problem is the detection of the fixed points of these networks. This is an NP-complete problem in general. There have been various attempts to develop algorithms to address the computation need for large size Boolean networks. The existing methods are usually based on known algorithms and thus limited to the situations where these known algorithms can apply. In this paper, we propose a novel approach to this problem. We show that any system of Boolean equations is equivalent to one Boolean equation, and thus it is possible to divide the polynomial equation system which defines the fixed points of a Boolean network into subsystems that can be solved easily. After solving these subsystems and thus reducing the number of states, we can combine the solutions to obtain all fixed points of the given network. This approach does not depend on other algorithms and it is straightforward and easy to implement. We show that our method can handle large size Boolean networks, and demonstrate its effectiveness by using MAPLE to compute the fixed points of Boolean networks with hundreds of nodes and thousands of interactions.

q-bio.QM↗
arXiv2014

Boolean Networks with Multi-Expressions and Parameters

To model biological systems using networks, it is desirable to allow more than two levels of expression for the nodes and to allow the introduction of parameters. Various modeling and simulation methods addressing these needs using Boolean models, both synchronous and asynchronous, have been proposed in the literature. However, analytical study of these more general Boolean networks models is lagging. This paper aims to develop a concise theory for these different Boolean logic based modeling methods. Boolean models for networks where each node can have more than two levels of expression and Boolean models with parameters are defined algebraically with examples provided. Certain classes of random asynchronous Boolean networks and deterministic moduli asynchronous Boolean networks are investigated in detail using the setting introduced in this paper. The derived theorems provide a clear picture for the attractor structures of these asynchronous Boolean networks.

q-bio.MN↗
arXiv2014

Finitely Generated Nil but Not Nilpotent Evolution Algebra

To use evolution algebras to model population dynamics that both allow extinction and introduction of certain gametes in finite generations, nilpotency must be built into the algebraic structures of these algebras with the entire algebras not to be nilpotent if the populations are assumed to evolve for a long period of time. To adequately address this need, evolution algebras over rings with nilpotent elements must be considered instead of evolution algebras over fields. This paper develops some criteria, which are computational in nature, about the nilpotency of these algebras, and shows how to construct finitely generated evolution algebras which are nil but not nilpotent.

math.RA↗
arXiv2013

Characterization of Boolean Networks with Single or Bistable States

Many biological systems, such as metabolic pathways, exhibit bistability behavior: these biological systems exhibit two distinct stable states with switching between the two stable states controlled by certain conditions. Since understanding bistability is key for understanding these biological systems, mathematical modeling of the bistability phenomenon has been at the focus of researches in quantitative and system biology. Recent study shows that Boolean networks offer relative simple mathematical models that are capable of capturing these essential information. Thus a better understanding of the Boolean networks with bistability property is desirable for both theoretical and application purposes. In this paper, we describe an algebraic condition for the number of stable states (fixed points) of a Boolean network based on its polynomial representation, and derive algorithms for a Boolean network to have a single stable state or two stable states. As an example, we also construct a Boolean network with exactly two stable states for the lac operon's $β$-galactosidase regulatory pathway when glucose is absent based on a delay differential equation model

q-bio.QM↗
arXiv2013

Representing Boolean Functions Using Polynomials: More Can Offer Less

Polynomial threshold gates are basic processing units of an artificial neural network. When the input vectors are binary vectors, these gates correspond to Boolean functions and can be analyzed via their polynomial representations. In practical applications, it is desirable to find a polynomial representation with the smallest number of terms possible, in order to use the least possible number of input lines to the unit under consideration. For this purpose, instead of an exact polynomial representation, usually the sign representation of a Boolean function is considered. The non-uniqueness of the sign representation allows the possibility for using a smaller number of monomials by solving a minimization problem. This minimization problem is combinatorial in nature, and so far the best known deterministic algorithm claims the use of at most $0.75\times 2^n$ of the $2^n$ total possible monomials. In this paper, the basic methods of representing a Boolean function by polynomials are examined, and an alternative approach to this problem is proposed. It is shown that it is possible to use at most $0.5\times 2^n = 2^{n-1}$ monomials based on the $\{0, 1\}$ binary inputs by introducing extra variables, and at the same time keeping the degree upper bound at $n$. An algorithm for further reduction of the number of terms that used in a polynomial representation is provided. Examples show that in certain applications, the improvement achieved by the proposed method over the existing methods is significant.

cs.CC↗
arXiv2013

Dynamics of Boolean Networks

Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in studying these dynamical systems is to link their long term behaviors to the structures of the functions that define them. In this paper, a method for deriving a Boolean network's dynamical information via its disjunctive normal form is explained. For a given Boolean network, a matrix with entries 0 and 1 is associated with the polynomial function that represents the network, then the information on the fixed points and the limit cycles is derived by analyzing the matrix. The described method provides an algorithm for the determination of the fixed points from the polynomial expression of a Boolean network. The method can also be used to construct Boolean networks with prescribed limit cycles and fixed points. Examples are provided to explain the algorithm.

q-bio.QM↗
arXiv2011

Factoring Pseudoidentity Matrix Pairs

The problem of factorization and parametrization of compactly supported biorthogonal wavelets was reduced to that of pseudoidentity matrix pairs by Resnikoff, Tian, and Wells in their 2001 paper. Based on a conjecture on the pseudoidentity matrix pairs of rank 2 stated in the same paper, they proved a theorem which gives a complete factorization result for rank 2 compactly supported biorthogonal wavelets. In this paper, we first provide examples to show that the conjecture is not true, then we prove a factorization theorem for pseudoidentity matrix pairs of rank $m\ge 2$. As a consequence, our result shows that a slightly modified version of the factorization theorem in the rank 2 case given by Resnikoff, Tian, and Wells holds. We also provide a concrete constructive method for the rank 2 case which is determined by applying the Euclidean algorithm to two polynomials.

math.FA↗
arXiv2009

Ideal Structure of Clifford Algebras

The structures of the ideals of Clifford algebras which can be both infinite dimensional and degenerate over the real numbers are investigated.

math.RA↗
arXiv2009

Spherical functions on homogeneous superspaces

Homogeneous superspaces arising from the general linear supergroup are studied within a Hopf algebraic framework. Spherical functions on homogeneous superspaces are introduced, and the structures of the superalgebras of the spherical functions on classes of homogeneous superspaces are described explicitly.

math.RT↗
arXiv2009

Linear Transformations and Restricted Isometry Property

The Restricted Isometry Property (RIP) introduced by Candés and Tao is a fundamental property in compressed sensing theory. It says that if a sampling matrix satisfies the RIP of certain order proportional to the sparsity of the signal, then the original signal can be reconstructed even if the sampling matrix provides a sample vector which is much smaller in size than the original signal. This short note addresses the problem of how a linear transformation will affect the RIP. This problem arises from the consideration of extending the sensing matrix and the use of compressed sensing in different bases. As an application, the result is applied to the redundant dictionary setting in compressed sensing.

cs.IT↗
arXiv2008

Linear Dynamical Systems over Finite Rings

The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring $R$ such that $|R|=q$, where $q$ is a positive integer, the algorithm determines whether a given linear system over $R^n$ is a fixed point system or not in time $O(n^3\log(n\log(q)))$.

math.DS↗
arXiv2008

Toeplitz Block Matrices in Compressed Sensing

Recent work in compressed sensing theory shows that $n\times N$ independent and identically distributed (IID) sensing matrices whose entries are drawn independently from certain probability distributions guarantee exact recovery of a sparse signal with high probability even if $n\ll N$. Motivated by signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. This paper considers Toeplitz block matrices as sensing matrices. They naturally arise in multichannel and multidimensional filtering applications and include Toeplitz matrices as special cases. It is shown that the probability of exact reconstruction is also high. Their performance is validated using simulations.

cs.IT↗