Physical Sciences and Mathematics Commons^™

Entropy Analysis Of Boolean Network Reduction According To The Determinative Power Of Nodes, Matthew J. Pelz, Mihaela T. Velcsov

Mathematics Faculty Publications

Boolean networks are utilized to model systems in a variety of disciplines. The complexity of the systems under exploration often necessitates the construction of model networks with large numbers of nodes and unwieldy state spaces. A recently developed, entropy-based method for measuring the determinative power of each node offers a new method for identifying the most relevant nodes to include in subnetworks that may facilitate analysis of the parent network. We develop a determinative-power-based reduction algorithm and deploy it on 36 network types constructed through various combinations of settings with regards to the connectivity, topology, and functionality of networks. We …

A Single-Scale Fractal Feature For Classification Of Color Images: A Virus Case Study, Walker Arce, James E. Pierce, Mihaela T. Velcsov

Mathematics Faculty Publications

Current methods of fractal analysis rely on capturing approximations of an images’ fractal dimension by distributing iteratively smaller boxes over the image, counting the set of box and fractal, and using linear regression estimators to estimate the slope of the set count line. To minimize the estimation error in those methods, our aim in this study was to derive a generalized fractal feature that operates without iterative box sizes or any linear regression estimators. To do this, we adapted the Minkowski-Bouligand box counting dimension to a generalized form by fixing the box size to the smallest fundamental unit (the individual …

Analytical And Numerical Convexity Results For Discrete Fractional Sequential Differences With Negative Lower Bound, Christopher S. Goodrich, Benjamin Lyons, Andrea Scapellato, Mihaela T. Velcsov

Mathematics Faculty Publications

We investigate relationships between the sign of the discrete fractional sequential difference (Δ^v _1+a-μ Δ^μ_af)(t) and the convexity of the function t→f(t). In particular, we consider the case in which the bound (Δ^v _1+a-μ Δ^μ_af)(t) ≥εf(a), for some ε > 0 and where f(a) < 0 is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation.

Analytical And Numerical Monotonicity Results For Discrete Fractional Sequential Differences With Negative Lower Bound, Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov

Mathematics Faculty Publications

We investigate the relationship between the sign of the discrete fractional sequential difference(Δ^v_1+a-μ Δ_a^μf)(t) and the monotonicity of the function t→f(t). More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound (Δ^v_1+a-μ Δ_a^μf)(t) ≥ -εf(a), for some ε >0. We prove that even though the fractional difference can be negative, the monotonicity of the function f, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges …

Boolean Network Topologies And The Determinative Power Of Nodes, Bronson W. Wacker, Mihaela T. Velcsov, Jim A. Rogers

Mathematics Faculty Publications

Boolean networks have been used extensively for modeling networks whose node activity could be simplified to a binary outcome, such as on-off. Each node is influenced by the states of the other nodes via a logical Boolean function. The network is described by its topological properties which refer to the links between nodes, and its dynamical properties which refer to the way each node uses the information obtained from other nodes to update its state. This work explores the correlation between the information stored in the Boolean functions for each node in a property known as the determinative power and …

Approximate And Exact Merging Of Knapsack Constraints With Cover Inequalities, Fabio Vitor, Todd Easton

Mathematics Faculty Publications

This paper presents both approximate and exact merged knapsack cover inequalities, a class of cutting planes for knapsack and multiple knapsack integer programs. These inequalities combine the information from knapsack constraints and cover inequalities. Approximate merged knapsack cover inequalities can be generated through a O(n log n) algorithm, where n is the number of variables. This class of inequalities can be strengthened to an exact version with a pseudo-polynomial time algorithm. Computational experiments demonstrate an average improvement of approximately 8% in solution time and 5% in the number of ticks from CPLEX when approximate merged knapsack cover …

Two Dimensional Search Algorithms For Linear Programming, Fabio Torres Vitor

Mathematics Faculty Publications

Linear programming is one of the most important classes of optimization problems. These mathematical models have been used by academics and practitioners to solve numerous real world applications. Quickly solving linear programs impacts decision makers from both the public and private sectors. Substantial research has been performed to solve this class of problems faster, and the vast majority of the solution techniques can be categorized as one dimensional search algorithms. That is, these methods successively move from one solution to another solution by solving a one dimensional subspace linear program at each iteration. This dissertation proposes novel algorithms that move …

Weighted Composition Operators On The Hilbert Hardy Space Of A Half-Plane, Valentin Matache

Mathematics Faculty Publications

Operators of type f→ψf∘ϕ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on the Hilbert Hardy space of a half-plane and study compactness, boundedness, invertibility, normality and spectral properties of such operators.

Analysis Of Optimal Superconvergence Of A Local Discontinuous Galerkin Method For Nonlinear Second-Order Two-Point Boundary-Value Problems, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1toward the derivatives of Gauss-Radau projections of …

Not So Many Non-Disjoint Translations, Andrzej Roslanowski, Vyacheslav V. Rykov

Mathematics Faculty Publications

We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.

Identification Of Biologically Essential Nodes Via Determinative Power In Logical Models Of Cellular Processes, Trevor Pentzien, Bhanwar L. Puniya, Tomáš Helikar, Mihaela Teodora Matache

Mathematics Faculty Publications

A variety of biological networks can bemodeled as logical or Boolean networks. However, a simplification of the reality to binary states of the nodes does not ease the difficulty of analyzing the dynamics of large, complex networks, such as signal transduction networks, due to the exponential dependence of the state space on the number of nodes. This paper considers a recently introduced method for finding a fairly small subnetwork, representing a collection of nodes that determine the states of most other nodes with a reasonable level of entropy. The subnetwork contains the most determinative nodes that yield the highest information …

The Double Pivot Simplex Method, Fabio Vitor, Todd Easton

Mathematics Faculty Publications

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time …

A Flexible Distribution Class For Count Data, Kimberly F. Sellers, Andrew W. Swift, Kimberly S. Weems

Mathematics Faculty Publications

The Poisson, geometric and Bernoulli distributions are special cases of a flexible count distribution, namely the Conway-Maxwell-Poisson (CMP) distribution – a two-parameter generalization of the Poisson distribution that can accommodate data over- or under-dispersion. This work further generalizes the ideas of the CMP distribution by considering sums of CMP random variables to establish a flexible class of distributions that encompasses the Poisson, negative binomial, and binomial distributions as special cases. This sum-of-Conway-Maxwell-Poissons (sCMP) class captures the CMP and its special cases, as well as the classical negative binomial and binomial distributions. Through simulated and real data examples, we demonstrate this …

An Optimal A Posteriori Error Estimates Of The Local Discontinuous Galerkin Method For The Second-Order Wave Equation In One Space Dimension, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L 2 -norm to (p + 1)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p + …

Nonminimal Cyclic Invariant Subspaces Of Hyperbolic Composition Operators, Valentin Matache

Mathematics Faculty Publications

Operators on function spaces acting by composition to the right with a fixed self-map ϕ are called composition operators. We denote them Cϕ. Given ϕ, a hyperbolic disc automorphism, the composition operator Cϕ on the Hilbert Hardy space H2 is considered. The bilateral cyclic invariant subspaces Kf, f ∈ H2, of Cϕ are studied, given their connection with the invariant subspace problem, which is still open for Hilbert space operators. We prove that nonconstant inner functions u induce non–minimal cyclic subspaces Ku if they have unimodular, orbital, cluster points. Other results about Ku when u is inner are obtained. If …

Logical Reduction Of Biological Networks To Their Most Determinative Components, Mihaela Teodora Matache, Valentin Matache

Mathematics Faculty Publications

Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. …

Systems Perturbation Analysis Of A Large-Scale Signal Transduction Model Reveals Potentially Influential Candidates For Cancer Therapeutics, Bhanwar L. Puniya, Laura Allen, Colleen Hochfelder, Mahbubul Majumder, Tomáš Helikar

Mathematics Faculty Publications

Dysregulation in signal transduction pathways can lead to a variety of complex disorders, including cancer. Computational approaches such as network analysis are important tools to understand system dynamics as well as to identify critical components that could be further explored as therapeutic targets. Here, we performed perturbation analysis of a large-scale signal transduction model in extracellular environments that stimulate cell death, growth, motility, and quiescence. Each of the model’s components was perturbed under both loss-of-function and gain-of-function mutations. Using 1,300 simulations under both types of perturbations across various extracellular conditions, we identified the most and least influential components based on …

Some New Refinements Of The Arithmetic, Geometric And Harmonic Mean Inequalities With Applications, Steven G. From, R. Suthakaran

Mathematics Faculty Publications

No abstract provided.

Analysis Of A Posteriori Error Estimates Of The Discontinuous Galerkin Method For Nonlinear Ordinary Differential Equations, Mahboub Baccouch

Mathematics Faculty Publications

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O(h^p+1) convergence rate in the L²-norm when p-degree piece-wise polynomials with p ≥1 are used. We further prove that the DG solution is O(h^2p+1) superconvergent at the downwind points. We …

Analysis Of Optimal Error Estimates And Superconvergence Of The Discontinuous Galerkin Method For Convection-Diffusion Problems In One Space Dimension, Mahboub Baccouch, Helmi Temimi

Mathematics Faculty Publications

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(h^p+1) and O(h^p) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h^2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG …

The Discontinuous Galerkin Finite Element Method For Ordinary Differential Equations, Mahboub Baccouch

Mathematics Faculty Publications

We present an analysis of the discontinuous Galerkin (DG) finite element method for nonlinear ordinary differential equations (ODEs). We prove that the DG solution is $(p + 1) $th order convergent in the $L^2$-norm, when the space of piecewise polynomials of degree $p$ is used. A $ (2p+1) $th order superconvergence rate of the DG approximation at the downwind point of each element is obtained under quasi-uniform meshes. Moreover, we prove that the DG solution is superconvergent with order $p+2$ to a particular projection of the exact solution. The superconvergence results are used to show that the leading term of …

Invertible And Normal Composition Operators On The Hilbert Hardy Space Of A Half–Plane, Valentin Matache

Mathematics Faculty Publications

Operators on function spaces of form... is a fixed map are called composition operators with symbol φ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Improving The Solution Time Of Integer Programs By Merging Knapsack Constraints With Cover Inequalities, Fabio Vitor

Mathematics Faculty Publications

Integer Programming is used to solve numerous optimization problems. This class of mathematical models aims to maximize or minimize a cost function restricted to some constraints and the solution must be integer. One class of widely studied Integer Program (IP) is the Multiple Knapsack Problem (MKP). Unfortunately, both IPs and MKPs are NP-hard, potentially requiring an exponential time to solve these problems.

Utilization of cutting planes is one common method to improve the solution time of IPs. A cutting plane is a valid inequality that cuts off a portion of the linear relaxation space. This thesis presents a new class …

Difference Equation For Tracking Perturbations In Systems Of Boolean Nested Canalyzing Functions, Elena S. Dimitrova, Oleg I. Yordanov, Mihaela Teodora Matache

Mathematics Faculty Publications

This paper studies the spread of perturbations through networks composed of Boolean functions with special canalyzing properties. Canalyzing functions have the property that at least for one value of one of the inputs the output is fixed, irrespective of the values of the other inputs. In this paper the focus is on partially nested canalyzing functions, in which multiple, but not all inputs have this property in a cascading fashion. They naturally describe many relationships in real networks. For example, in a gene regulatory network, the statement “if gene A is expressed, then gene B is not expressed regardless of …

On Spectra Of Composition Operators, Valentin Matache

Mathematics Faculty Publications

In this paper we consider composition operators Cφ on the Hilbert Hardy space over the unit disc, induced by analytic selfmaps φ. We use the fact that the operator C∗φCφ is asymptotically Toeplitz to obtain information on the essential spectrum and spectrum of Cϕ, which we are able to describe in select cases (including the case of some hypercyclic composition operators or that of composition operators with the property that the asymptotic symbol of C∗φCφ is constant a.e.). One of our tools is the Nikodym derivative of the pull-back measure induced by φ. An alternative formula for the essential norm …

Asymptotically Exact Local Discontinuous Galerkin Error Estimates For The Linearized Korteweg-De Vries Equation In One Space Dimension, Mahboub Baccouch

Mathematics Faculty Publications

We present and analyze a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the linearized Korteweg-de Vries (KdV) equation in one space dimension. These estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We extend the work of Hufford and Xing [J. Comput. Appl. Math., 255 (2014), pp. 441-455] to prove new superconvergence results towards particular projections of the exact solutions for the two auxiliary variables in the LDG method that approximate the first and second derivatives of the solution. The order of convergence …

Sensitivity Analysis Of Biological Boolean Networks Using Information Fusion Based On Nonadditive Set Functions, Naomi Kochi, Tomáš Helikar, Laura Allen, Jim A. Rogers, Zhenyuan Wang, Mihaela Teodora Matache

Mathematics Faculty Publications

Background: An algebraic method for information fusion based on nonadditive set functions is used to assess the joint contribution of Boolean network attributes to the sensitivity of the network to individual node mutations. The node attributes or characteristics under consideration are: in-degree, out-degree, minimum and average path lengths, bias, average sensitivity of Boolean functions, and canalizing degrees. The impact of node mutations is assessed using as target measure the average Hamming distance between a non-mutated/wild-type network and a mutated network.

Results: We find that for a biochemical signal transduction network consisting of several main signaling pathways whose nodes represent signaling …

Isometric Weighted Composition Operators, Valentin Matache

Mathematics Faculty Publications

A composition operator is an operator on a space of functions defined on the same set. Its action is by composition to the right with a fixed selfmap of that set. A composition operator followed by a multiplication operator is called a weighted composition operator. In this paper, we study when weighted composition operators on the Hilbert Hardy space of the open unit disc are isometric. We find their Wold decomposition in select cases and apply it to the computation of numerical ranges.

Global Convergence Of A Posteriori Error Estimates For A Discontinuous Galerkin Method For One-Dimensional Linear Hyperbolic Problems, Mahboub Baccouch

Mathematics Faculty Publications

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L² -norm under mesh refinement. The order of convergence is proved to be k + 2, when k-degree piecewise polynomials with k ≥ 1 are used. As a consequence, we …

Small-World Properties Of Facebook Group Networks, Jason Wohlgemuth, Mihaela Teodora Matache

Mathematics Faculty Publications

Small-world networks permeate modern society. In this paper we present a methodology for creating and analyzing a practically limitless number of networks exhibiting small-world network properties. More precisely, we analyze networks whose nodes are Facebook groups sharing a common word in the group name and whose links are mutual members in any two groups. By analyzing several numerical characteristics of single networks and network aggregations, we investigate how the small-world properties scale with a coarsening of the network. We show that Facebook group networks have small average path lengths and large clustering coefficients that do not vanish with increased network …