Papers by Patrick Shipman
Biomath Communications, 2017
The mathematics represented in over 50 oral and poster presentations included advances in numeric... more The mathematics represented in over 50 oral and poster presentations included advances in numerical methods, geometric data analysis, dynamical systems and partial differential equations, and surface geometry. The biological topics were similarly diverse: participants learned, for example, about the spread and control of diseases such as malaria, the ubiquity of renewal equations in population dynamics and epidemiology, how magnetic fields interact with DNA, biomechanics and protein folding, and molecular dynamics approaches to biomolecular information processing. In the Roumen Tsanev plenary lecture, Slavka Tcholakova presented on the kinetics of the breakage and coalescence of small droplets in emulsions. As there were no parallel
BMC microbiology, Jan 16, 2003
Bacterial macrofibers twist as they grow, writhe, supercoil and wind up into plectonemic structur... more Bacterial macrofibers twist as they grow, writhe, supercoil and wind up into plectonemic structures (helical forms the individual filaments of which cannot be taken apart without unwinding) that eventually carry loops at both of their ends. Terminal loops rotate about the axis of a fiber's shaft in contrary directions at increasing rate as the shaft elongates. Theory suggests that rotation rates should vary linearly along the length of a fiber ranging from maxima at the loop ends to zero at an intermediate point. Blocking rotation at one end of a fiber should lead to a single gradient: zero at the blocked end to maximum at the free end. We tested this conclusion by measuring directly the rotation at various distances along fiber length from the blocked end. The movement of supercoils over a solid surface was also measured in tethered macrofibers. Macrofibers that hung down from a floating wire inserted through a terminal loop grew vertically and produced small plectonemic struct...
Journal of Chemical Education, 2018
This report summarizes one of the invited papers to the ConfChem online conference on Mathematics... more This report summarizes one of the invited papers to the ConfChem online conference on Mathematics in Undergraduate Chemistry Instruction, held from October 23 to November 27, 2017, and hosted by the ACS DivCHED Committee on Computers in Chemical Education (CCCE). Combining data collected by a team of nine instructors from six Texas institutions for an IRB-approved investigation, this research correlated the arithmetic skills of first-year general chemistry students (n = 2127) to final course grades. The validated, highly reliable (KR-21 = 0.821) instrument, named the Math-Up Skills Test (MUST), evaluated the following topics using 16 questions: multiplication, division, fractions, scientific notation, exponential notation, logarithms, square roots, and balancing chemical equations. The MUST was given twice to each student: first without the use of a calculator followed by a similar, modified version with the use of a calculator. Outcomes suggest that general chemistry students' arithmetic skills are more correlated with course grades when calculators are not used than when calculators are used. Perhaps more emphasis should be placed on improving students' mathematics automaticity.
COMPUTATIONAL MODELING OF THE PHARMACOKINETICS AND PHARMACODYNAMICS OF SELECTED XENOBIOTICS The d... more COMPUTATIONAL MODELING OF THE PHARMACOKINETICS AND PHARMACODYNAMICS OF SELECTED XENOBIOTICS The determination of important endpoints in toxicology and pharmacology continues to involve the acquisition of large amounts of data through resource-intensive experimental studies involving a large number of resources. Because of this, only a small fraction of chemicals in the environment and marketplace can reasonably be evaluated for safety, and many promising drug candidates must be eliminated from consideration based on inadequate evaluation. Promisingly, advances in biologically-based computational models are beginning to allow researchers to estimate these endpoints and make useful extrapolations using a limited set of experimental data. The work described in this dissertation examined how computational models can provide meaningful insight and quantitation of important pharmacological and toxicological endpoints related to toxicity and pharmacological efficacy. To this end, physiolog...
Topological Data Analysis, 2020
We use persistent homology in order to define a family of fractal dimensions, denoted dim i PH (μ... more We use persistent homology in order to define a family of fractal dimensions, denoted dim i PH (μ) for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767-1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space R m for m ≥ 2, then Steele's work implies that dim 0 PH (μ) = m if the absolutely continuous part of μ has positive mass, and otherwise dim 0 PH (μ) < m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i-dimensional persistent homology interval lengths for n random points selected from μ in an i.i.d. fashion. To some measures μ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zerodimensional homology when μ is the uniform distribution over the unit interval, and This work was completed while Elin Farnell was a research scientist in the
Assessment of Muscle Protein Synthesis Rates are Dependent upon Individual Protein Synthetic Rates, Content, and Duration of Label Incorporation
The FASEB Journal, 2015
J. Mach. Learn. Res., 2017
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topologica... more Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological informa...
The patterns seen in both the phyllotaxis and surface morphologies in the vicinity of the shoot a... more The patterns seen in both the phyllotaxis and surface morphologies in the vicinity of the shoot apical meristems of plants are discussed. We begin with many pictures and a narrative descriptive of both the universal and anomalous features of desert and other plants. We then briefly outline explanations and open challenges. Although many of the special features of phyllotaxis have been known for over four centuries, only now are mechanistic explanations beginning to emerge.
The Journal of Chemical Physics, 2019
Vapor-phase ammonia, NH 3(g) , and hydrochloric acid, HCl (g) , undergo a series of complex react... more Vapor-phase ammonia, NH 3(g) , and hydrochloric acid, HCl (g) , undergo a series of complex reactions, including nucleation and growth, to form solid ammonium chloride, NH 4 Cl (s). The counterdiffusional experiment, whereby HCl (g) and NH 3(g) diffuse from opposite ends of a tube and react to form spatiotemporally complex patterns, has a rich history of study. In this paper, we combine experimental data, molecular simulations, and analysis and simulations of a partial differential equation model to address the questions of where the first unobserved vapor product NH 4 Cl (g) and visually observable precipitate NH 4 Cl (s) form and how these positions depend on experimental parameters. These analyses yield a consistent picture which involves a moving reaction front as well as previously unobserved heterogeneous nucleation, wall nucleation, and homogeneous nucleation. The experiments combined with modeling allow for an estimate of the heterogeneous and homogeneous nucleation thresholds for the vapor-to-solid phase transition. The results, synthesized with literature on this vapor-to-particle reaction, inform a discussion of the details of the reaction mechanism, including the role of water, which concludes the paper.
Discrete & Continuous Dynamical Systems - S, 2018
The defect of a complex Hadamard matrix H is an upper bound for the dimension of a continuous Had... more The defect of a complex Hadamard matrix H is an upper bound for the dimension of a continuous Hadamard orbit stemming from H. We provide a new interpretation of the defect as the dimension of the center subspace of a gradient flow and apply the Center Manifold Theorem of dynamical systems theory to study local structure in spaces of complex Hadamard matrices. Through examples, we provide several applications of our methodology including the construction of affine families of Hadamard matrices.
The Journal of physiology, 2018
Muscle fibre cross sectional area is enhanced with massage in the form of cyclic compressive load... more Muscle fibre cross sectional area is enhanced with massage in the form of cyclic compressive loading during regrowth after atrophy. Massage enhances protein synthesis of the myofibrillar and cytosolic, but not the mitochondrial fraction, in muscle during regrowth. Focal adhesion kinase activation and satellite cell number are elevated in muscles undergoing massage during regrowth. Muscle fibre cross sectional area and protein synthesis of the myofibrillar fraction, but not DNA synthesis, are elevated in muscle of the contralateral non-massaged limb. Massage in the form of cyclic compressive loading is a potential anabolic intervention during muscle regrowth after atrophy. Massage, in the form of cyclic compressive loading (CCL), is associated with multiple health benefits, but its potential anabolic effect on atrophied muscle has not been investigated. We hypothesized that the mechanical activity associated with CCL induces an anabolic effect in skeletal muscle undergoing regrowth a...
Graphs and Combinatorics, 2017
For positive integers k, n, a de Bruijn sequence B(k, n) is a finite sequence of elements drawn f... more For positive integers k, n, a de Bruijn sequence B(k, n) is a finite sequence of elements drawn from k characters whose subwords of length n are exactly the k n words of length n on k characters. This paper introduces the unoriented de Bruijn sequence uB(k, n), an analog to de Bruijn sequences, but for which the sequence is read both forwards and backwards to determine the set of subwords of length n. We show that nontrivial unoriented de Bruijn sequences of optimal length exist if and only if k is two or odd and n is less than or equal to 3. Unoriented de Bruijn sequences for any k, n may be constructed from certain Eulerian paths in Eulerizations of unoriented de Bruijn graphs.
BIOMATH, 2017
A wide variety of natural and labo-ratory systems can produce patterns of ripples, hexagons, or s... more A wide variety of natural and labo-ratory systems can produce patterns of ripples, hexagons, or squares. The formation of stable square patterns from partial differential equation models requires specific cubic nonlinearities involving higher-order derivatives. Motivated by plant phyllotaxis, we demonstrate that the coupling of more than one pattern-forming system can produce square patterns without these special nonlinearities.
Optimally Topologically Transitive Orbits in Discrete Dynamical Systems
The American Mathematical Monthly, 2016
Abstract Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. Howeve... more Abstract Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. However, the apparent uniformity of the distribution of iterates after a finite number of iterations appears strikingly different for various choices of a rotation angle. Motivated by this observation, we introduce a scalar function on the orbits of a discrete dynamical system defined on a bounded metric space, called the linear limit density, which we interpret as a measure of an orbit's approach to density. Utilizing the three-distance theorem, we compute the exact value of the linear limit density of orbits of rigid rotations by irrational rotation angles with period-1 continued fraction expansions. We further show that any discrete dynamical system defined by an orientation-preserving diffeomorphism of the circle has an orbit with a larger linear limit density than any orbit of the rigid rotation by the golden number. Bernoulli shift maps acting on sequences over a finite alphabet provide another illustrative class of dynamical systems with dense orbits. Our study of the efficiency of an orbit's approach to density leads us to demonstrate the existence of a class of infinite sequences with finite linear limit density constructed by recursively extending finite de Bruijn sequences.
Expositiones Mathematicae, 2017
While conformal transformations of the plane preserve Laplace's equation, Lorentzconformal mappin... more While conformal transformations of the plane preserve Laplace's equation, Lorentzconformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvilinear quadrilaterals where three sides determine the fourth by a geometric "rectangle rule," which can be expressed also by functional formulas. There is an explicit functional degree of freedom in choosing the mapping taking the square to a given quadrilateral. We characterize classes of Lorentz-conformal maps by their symmetries under subgroups of the dihedral group of order eight. Unfoldings of non-invertible mappings into invertible ones are reflected in a change of the symmetry group. The questions are simple; but the answers are not obvious, yet have beautiful geometric, algebraic, and functional descriptions and proofs. This is due to the very simple form of nonlinear Lorentz-conformal transformations in dimension 1+1, provided by characteristic coordinates.
Biomath Communications, Apr 5, 2015
Physical review. E, 2016
A theory is developed for the nanoscale patterns formed when the (001) surface of a crystalline b... more A theory is developed for the nanoscale patterns formed when the (001) surface of a crystalline binary material with fourfold rotational symmetry is subjected to normal-incidence ion bombardment. The deterministic nonlinear continuum equations account for the Ehrlich-Schwoebel barrier, which produces uphill atomic currents on the crystal surface. We demonstrate that highly ordered square arrays of nanopyramids can form in a certain region of parameter space. An Ehrlich-Schwoebel barrier is required for patterns of this kind to develop. For another range of parameters, a disordered square array of nanodots forms and the pattern coarsens over time.
Physical Review E, 2015
When the surface of a nominally flat binary material is bombarded with a broad, normallyincident ... more When the surface of a nominally flat binary material is bombarded with a broad, normallyincident ion beam, disordered hexagonal arrays of nanodots can form. Shipman and Bradley have derived equations of motion that govern the coupled dynamics of the height and composition of such a surface [P. D. Shipman and R. M. Bradley, Phys. Rev. B 84, 085420 (2011)]. We investigate the influence of initial conditions on the hexagonal order yielded by integration of those equations of motion. The initial conditions studied are hexagonal and sinusoidal templates, straight scratches and nominally flat surfaces. Our simulations indicate that both kinds of template lead to marked improvements in the hexagonal order if the initial wavelength is approximately equal to or double the linearly selected wavelength. Scratches enhance the hexagonal order in their vicinity if their width is close to or less than the linearly selected wavelength. Our results suggest that prepatterning a binary material can dramatically increase the hexagonal order achieved at large ion fluences.
Topochemistry in Starch-Iodine Diffusion Reaction Systems
A Lie-algebraic formulation for triply orthogonal and general coordinate systems in three-dimensional Euclidean and Lorentz spaces
Journal of Lie Theory, 2014
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Papers by Patrick Shipman