Invariants and orthogonal qq-polynomials associated with Cq(osp(1,2))\Bbb{C}_{q}(osp(1,2))

The spaces of invariants and the zonal spherical functions associated with quantum super 2-shpheres defined by $\Bbb{C}_{q}(osp(1,2))$ are discussed. Connection between the zonal spherical functions and orthogonal $q$-polynomials from the Askey-Wilson scheme is investigated.Comment: AMS-Tex, preprint 18 page

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 $\beta$-galactosidase regulatory pathway when glucose is absent based on a delay differential equation modelComment: Main results of this article appeared as a 4 page abstract in the ICBBE 2012 Conference Proceeding, pp. 517-52

Indices of Coincidence Isometries of the Hyper Cubic Lattice ZnZ^n

The problem of computing the index of a coincidence isometry of the hyper cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational orthogonal matrix is analyzed in detail, and explicit formulas for the index of certain coincidence isometries of $\mathbb{Z}^{n}$ are obtained. These formulas generalize the known results for $n\le 4$.Comment: AMS-Latex, preprint 10 page

Quantum super spheres and their transformation groups, representations, and little tt-Jacobi polynomials

Quantum super 2-shpheres and the corresponding quantum super transformation group are introduced in analogy to the well-known quantum 2-shpheres and quantum SL(2), connection between little $t$-Jacobi polynomials and the finite dimensional representations of the quantum super group is formulated, and the Peter-Weyl theorem is obtained.Comment: AMS-Tex, preprint 18 page

Structures of Coincidence Symmetry Groups

The structure of the coincidence symmetry group of an arbitrary $n$-dimensional lattice in the $n$-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries which are elements of the orthogonal group). Conditions under which the coincidence isometry group can be generated by reflections defined by vectors of the lattice will be discussed, and an algorithm to decompose an arbitrary element of the coincidence isometry group in terms of reflections defined by vectors of the lattice will be given.Comment: AMS-Latex, preprint 13 page

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.Comment: A shorter version of this paper appeared in the conference proceeding of ICME 2013 (Beijing), pp. 670 - 67

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.Comment: A version of this paper appeared in IEEE Transactions on Computational Biology and Bioinformatic

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

Gaussian binomials and the number of sublattices

The purpose of this short communication is to make some observations on the connections between various existing formulas of counting the number of sublattices of a fixed index in an $n$-dimensional lattice and their connection with the Gaussian binomials.Comment: AMS-Latex, preprint 3 page

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.Comment: A shorter version of this article appeared in LNCS 6677, 201