REPORT 0809-4:   A Generalized Inverse Censoring Weighted Survival Function Estimator

Sundaraman Subramanian and Peixin Zhang

New Jersey Institute of Technology, Newark, NJ

Abstract:

We investigate an inverse censoring weighted estimator of a survival function when the data are doubly censored but the left censoring is always observed. The estimator reduces to its right censored version when there is no left censoring.

REPORT 0809-12:   Generalized Genomic Matrices, Silver Means, and Pythagorean Triples

Jay Kappraff
Department of Mathematical Science and Center for Applied Mathematics and Statistics New Jersey Institute of Technology, Newark, NJ

Gary W. Adamson,  P.O. Box 124571, San Diego, CA 92112-4571, ntmpkt@yahoo.com  (submitted Nov. 2008) 

Abstract:

Petoukhov has shown that a family of bisymmetric 2n x 2n matrices encode the structure of the four RNA and DNA bases and 64 codons that make up the 20 amino acids in all living structures. He discovered that the elements of the square roots of these matrices are all powers of the golden mean. We have generalized his matrices and shown that the square roots of general bisymmetric matrices are generalizations of the golden mean including a subclass that correspond to the family of silver means. Powers of these matrices are also shown to generate all Pythagorean triples. The integers in these matrices are identical to the set of integers in a table attributed to the second century Syrian mathematician, Nicomachus, who used them to describe the ancient musical scale of Pythagoras. Keywords: dna/rna, golden mean, amino acids, Nichomachus.

REPORT 0809-13:  Influence of Surfactant Solubility on the Deformation and Breakup of a Bubble or Capillary Jet in a Viscous Fluid

Y.-N. Young, M.R. Booty, M. Siegel

New Jersey Institute of Technology

 J. Li

Cambridge University

Abstract:

In a previous study [M. Hameed, M. Siegel, Y.-N. Young, J. Li, M.R. Booty, and D.T. Papageorgiou, J. Fluid Mech. 594, 307 (2008)] the authors investigated the influence of insoluble surfactant on the evolution of a stretched, inviscid bubble surrounded by a viscous fluid via direct numerical simulation of the Navier-Stokes equations, and showed that the presence of surfactant can cause the bubble to contract and form a quasisteady slender thread connecting parent bubbles, instead of proceeding directly toward pinch-off as occurs for a surfactant-free bubble. Insoluble surfactant significantly retards pinch-off and the thread is stabilized by a balance between internal pressure and reduced capillary pressure due to a high concentration of surfactant that develops during the initial stage of contraction. In the present study we investigate the influence of surfactant solubility on thread formation. The adsorption-desorption kinetics for solubility is in the diffusion controlled regime. A long-wave model for the evolution of a capillary jet is also studied in the Stokes flow limit, and shows dynamics that are similar to those of the evolving bubble. With soluble surfactant, depending on parameter values, a slender thread forms but can pinch off later due to exchange of surfactant between the interface and exterior bulk flow.

REPORT 0809-14:  Median Regression using Inverse Censoring Weights

Sundar Subramanian

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark NJ 07102

Gerhard Dikta

Department of Applied Science and Technology Fachhochschule Aachen, Ginsterweg 1, D-52428 Julich, Germany

Abstract:

We implement semiparametric random censorship model aided inference for censored median regression models. This is based on the idea that, when the censoring is specified by a common distribution, a semiparametric survival function estimator acts as an improved weight in the so-called inverse censoring weighted estimating function. We show that the proposed method will always produce estimates of the model parameters that are as good as or better than an existing estimator based on the traditional Kaplan--Meier weights. We also provide an illustration of the method through an analysis of a lung cancer data set.

REPORT 0809-15:  Hydrodynamic Interactions between Two Semi-Flexible In-Extensible Filaments in Stokes Flow

Y.-N. Young

Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, NJIT, Newark, NJ, 07102

Abstract:

Hydrodynamic interactions between two semi-flexible in-extensible filaments are shown to have a significant impact on filament buckling and their subsequent motion in Stokesian fluids. In linear shear flow, hydrodynamic interactions lead to filament shear dispersion that depends on the filament aspect ratio and the initial filament separation. In linear extensional flow, hydrodynamic interactions lead to complex filament dynamics around the stagnation point. These results suggest that hydrodynamic interactions need to be taken into account to determine the self-diffusion of non-Brownian semi-flexible filaments in a cellular flow [Phys.Rev. Lett., 99, 058303, 2007].

REPORT 0809-16:  Nonlinear Hydrodynamic Phenomena in Stokes Flow Regime

R. H. Goodman¹, Y.-N. Young¹, N. Khurana², J. Blawzdziewicz² and E. Wajnryb³

¹ Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, NJIT, Newark, NJ, 07102
² Department of Mechanical Engineering, Yale University, P. O. Box 208286, New Haven, CT 06520-8286
³ Institute of Fundamental Technological Research, Swietokrzyska 21, 00-049 Warsaw, Poland

Abstract:

We investigate nonlinear phenomena in dispersed two-phase systems under creeping-flow conditions. We consider nonlinear evolution of a single deformed drop and collective dynamics of arrays of hydrodynamically coupled particles. To explore physical mechanisms of system instabilities, chaotic drop evolution, and structural transitions in particle arrays we use simple models, such as small deformation equations and effective medium theory. We find numerical and analytical solutions of the simplified governing equations. The small-deformation equations for drop dynamics are analyzed using results of dynamical-systems theory. Our investigations shed new light on the dynamics of complex fluids, where the nonlinearity often stems from the evolving boundary conditions in Stokes flow.

REPORT 0809-17:  Generalized DNA matrices, Silver Means, and Pythagorean Triples

Jay Kappraff¹ and Gary W. Adamson
¹Mathematics Department, NJIT, University Heights, Newark, NJ 07102, kappraff@verizon.net

Abstract:

Petoukhov [1], [2], [3] has studied a family of bisymmetric 2n x 2n matrices that code the structure of the four DNA bases, the 64 codons that make up the 20 amino acids in all living structures, and beyond that, the proteins assembled from the amino acids as building blocks [1], [2], [3]. As the result of his studies he has found that the amino acids express certain degeneracies, 8 with high degeneracy containing 4 or more codons, and 12 with low degeneracy, containing less than 4 codons. He suggests that these values may be the result of 24 hour chronocycles. These degeneracies are propagated through 17 different classes of DNA.  Although different groups of codons correspond to the same amino acid in different DNA types, the quality of the degeneracy (high or low) is preserved. The first matrix of the family expresses the fact that two of the DNA bases have 3 hydrogen bonds while the other two have 2 hydrogen bonds. The rows and columns of his family of matrices reproduce the sequences of musical fifths, i.e., integer ratios of 3:2, found in a table attributed to the Roman mathematician of the first century AD, Nicomachus [4]. The integer values in this table have multiplicities given by the columns of Pascal’s triangle. The square roots of this family of matrices have entries that are all powers of the golden mean.

REPORT 0809-18:  SYMMETRIES, GENERALIZED NUMBERS AND HARMONIC LAWS IN MATRIX GENETICS

J. Kappraff * and S.Petoukhov**
* Mathematical Sciences Department, New Jersey Institute of Technology, USA. Email: kappraff@verizon.net
** Biophysics, Department of Biomechanics, Mechanical Engineering Research Institute, Russian Academy of Sciences, d.4, Malyi Kharitonievskiy pereulok, Moscow, 101990, Russia, Email: petoukhov@hotmail.com

Abstract:

This paper is devoted to the presentation and analysis of matrix representations of the genetic code. Principal attention is paid to a family of the genetic
matrices which are constructed on the basis of Gray code ordering of their rows and columns. This Gray code ordering reveals new connections of the genetic code to: 8-dimensional bipolar algebras; Hadamard matrices; golden matrices; Pythagorean musical scale, and an integer triangle attributed to Nicomachus, a Syrian
mathematician from second century A.D. All of these mathematical entities possess symmetrical features. Some questions about silver means and Pythagorean triples are also described by these genetic matrices and their generalizations.

REPORT 0809-19:  Ancient Harmonic Law (version 2)

Jay Kappraff
New Jersey Institute of Technology, Newark, NJ 07102, kappraff@verizon.net

Abstract:

The matrix arithmetic for ancient harmonic theory is presented here for two tuning systems with opposite defects: “Spiral fifths” as presented by Nicomachus, a Syrian Neo-Pythagorean of the second century A.D., and Plato’s “Just tuning” as reconstructed by the ethnomusicologist, Ernest McClain, from clues preserved by Nicomachus and Boethius (6th c. AD). These tables lie behind the system of architectural proportions used during the Renaissance, and their basic ratios now pervade modern science as the foundation of a “string theory” formally presented first in Euclid. Calculation employs an early form of log table governed by vectors of 2-3-4 in the first, and by 3-4-5 in the second. The square root of 2 plays a central role in integrating these systems governing 12-tone theory from the perspective of four primes--2, 3, and 5 generate all ratios under the overview of 7—as disciplined “self-limitation” within a “balance of perfect opposites. “

REPORT 0809-20:  The influence of the A-current on the dynamics of an oscillator-follower inhibitory network

Yu Zhang
yu.zhang@njit.edu
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102
Current Address: Department of Pharmacology and Systems Therapeutics, Mount Sinai School of Medicine, New York, NY 10029

Amitabha Bose
bose@njit.edu
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Farzan Nadim
farzan@njit.edu
Department of Mathematical Sciences, New Jersey Institute of Technology
Department of Biological Sciences, Rutgers University, Newark, NJ 07102

Abstract:

The transient potassium A-current is present in almost all neurons and plays an essential role in determining the timing and frequency of action potential generation. We use a three-variable mathematical model to examine the role of the A-current in a rhythmic inhibitory network, as is common in central pattern generation. We focus on a feedforward architecture consisting of an oscillator neuron inhibiting a follower neuron. We use separation of time scales to demonstrate that the trajectory of the follower neuron within each cycle can be tracked by analyzing the dynamics on a 2-dimensional slow manifold that as determined by the two slow model variables: the recovery variable and the inactivation of the A-current. The steady-state trajectory, however, requires tracking the slow variables across multiple cycles. We show that tracking the slow variables, under simplifying assumptions, leads to a one-dimensional map of the unit interval with at most a single discontinuity depending on gA, the maximal conductance of the A-current, or other model parameters. We demonstrate that, as the value of gA is varied, the trajectory of the follower neuron goes through a set of bifurcations to produce n:m periodic solutions where the follower neuron becomes active m times for each n cycles of the oscillator. Using a generalized Pascal triangle, each n:m trajectory can be constructed as a combination of solutions from a higher level of the triangle.

REPORT 0809-21:  Regulation of motor patterns by the central spike initiation zone of a sensory neuron

Nelly Daur
Institute of Neurobiology, Ulm University, D-89069 Ulm, Germany

Farzan Nadim
Dept. Math Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Wolfgang Stein
Institute of Neurobiology, Ulm University, D-89069 Ulm, Germany

Abstract:

Sensory feedback from muscles and peripheral sensors acts to initiate, tune or reshape motor activity according to the state of the body. Yet, sensory neurons often show low levels of activity even in the absence of sensory input. Here we examine the functional role of spontaneous low-frequency activity of such a sensory neuron. The anterior gastric receptor (AGR) is a muscle tendon organ in the crab stomatogastric nervous system whose phasic activity shapes the well-characterized gastric mill (chewing) and pyloric (filtering) motor rhythms. Phasic activity is driven by a spike initiation zone near the innervated muscle. We here demonstrate that AGR possesses a second spike initiation zone, which is located spatially distant from the innervated muscle in a central section of the axon. This initiation zone generates tonic activity and is responsible for the spontaneous activity of AGR in vivo, but does not code sensory information. Rather, it is sensitive to the neuromodulator octopamine. A computational model indicates that the activity at this initiation zone is not caused by excitatory input from another neuron, but generated intrinsically. This tonic activity is functionally relevant, because it modifies the activity state of the gastric mill motor circuit and changes the pyloric rhythm. The sensory function of AGR is not impaired since phasic activity suppresses spiking at the central initiation zone. Our results thus demonstrate that sensory neurons are not mere reporters of sensory signals. Neuromodulators can elicit non-sensory coding activity in these neurons that shapes the state of the motor system.

REPORT 0809-22:  Calculation of complex singular solutions to the 3D incompressible Euler equations

M. Siegel

Department of Mathematical Sciences, New Jersey Institute of Technology

R. Caflisch

Department of Mathematics, UCLA and Institute for Pure and Applied Mathematics, Los Angeles, CA 90095

Abstract:

This paper presents numerical computations of complex singular solutions to the 3D incompressible Euler equations. The Euler solutions found here consist of a complex valued velocity field $\bf u_+$ that contains all positive wavenumbers; ${\bf u_+}$ satisfies the usual Euler equations but with complex initial data. The real valued velocity ${\bf u}={\bf u}_+ + {\bf u}_-$ (where ${\bf u}_-= \overline{\bf u}_+$) is an approximate singular solution to the Euler equations under Moore's approximation. The method for computing singular solutions is an extension of that in Caflisch (1993) for axisymmetric flow with swirl, but with several improvements that prevent the extreme magnification of round-off error which affected previous computations.  This enables the first clean analysis of the singular surface in three-dimensional complex space. We find singularities in the velocity field of the form ${\bf u}_+ \sim \xi^{\alpha-1}$ for $\alpha$ near $3/2$ and ${\bf u}_+ \sim \log \xi$, where $\xi=0$ denotes the singularity surface. The logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations.

REPORT 0809-23:  Stokes-Darcy Boundary Integral Solutions Using Preconditioners

Yassine Boubendir and Svetlana Tlupova
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, 323 Dr. Martin Luther King, Jr. Boulevard, University Heights, Newark, NJ 07102

Abstract:

In a system where a free fluid flow is coupled to flow in a porous medium, diffrent PDEs are solved simultaneously in two subdomains. We consider steady Stokes equations in the free region, coupled across a fixed interface to Darcy equations in the porous substrate. In this paper, the numerical solution is obtained using the boundary integral formulation with regularized Green's function. Higher accuracy is achieved by applying a correction process, which also results in the improvement of the condition number of the linear system. In this work, an appropriate preconditioner based on the singular part of corrections is introduced to improve the convergence of a Krylov subspace method applied to solve the integral formulation. Key words: Preconditioner, Boundary integral method, Stokes flow, Darcy equations

REPORT 0809-24:  Discovery and Assessment of New Target Sites for Anti-HIV Therapies

C. Breward, University of Oxford, Oxford, UK
J. Heffernan, York University, Toronto, ON, Canada
N. Madras, York University, Toronto, ON, Canada
R.M. Miura, New Jersey Institute of Technology, Newark, NJ, USA
M.P. Sorensen, Technical University of Denmark, Lyngby, Denmark

Abstract:

Human immunodeficiency virus (HIV) infects cells by endocytosis and takes over parts of the cell's reaction pathways in order to reproduce itself and spread the infection. One such pathway taken over by HIV becomes the inflammatory pathway which uses Nuclear Factor $\kappa$B (NF-$\kappa$B) as the principal transcription factor. Therefore, knocking out the NF-$\kappa$B pathway would prevent HIV from reproducing itself. In this report, our goal is to produce a simple model for this pathway with which we can identify potential targets for anti-HIV therapies and test out various hypotheses.