12,750 research outputs found

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

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

### Indices of Coincidence Isometries of the Hyper Cubic Lattice $Z^n$

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

### 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

### Quantum super spheres and their transformation groups, representations, and little $t$-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

### 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

### 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

### 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

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