Iron is an essential transition metal for all eukaryotic cells, and its trafficking throughout th... more Iron is an essential transition metal for all eukaryotic cells, and its trafficking throughout the cell is highly regulated. However, the overall cellular mechanism of regulation is poorly understood despite knowing many of the molecular players involved. Here, an ordinary-differential-equations (ODE) based kinetic model of iron trafficking within a growing yeast cell was developed that included autoregulation. The 9-reaction 8-component in-silico cell model was solved under both steady-state and time-dependent dynamical conditions. The ODE for each component included a dilution term due to cell growth. Conserved rate relationships were obtained from the null space of the stoichiometric matrix, and the reducedrow-echelon-form was used to distinguish independent from dependent rates. Independent rates were determined from experimentally estimated component concentrations, cell growth rates, and the literature. Simple rate-law expressions were assumed, allowing rateconstants for each reaction to be estimated. Continuous Heaviside logistical functions were used to regulate rate-constants. These functions acted like valves, opening or closing depending on component "sensor" concentrations. Two cellular regulatory mechanisms were selected from 134,217,728 possibilities using a novel approach involving 6 mathematically-defined filters. Three cellular states were analyzed including healthy wild-type cells, iron-deficient wild-type cells, and a frataxin-deficient strain of cells characterizing the disease Friedreich's Ataxia. The model was stable toward limited perturbations, as determined by the eigenvalues of Jacobian matrices. Autoregulation allowed healthy cells to transition to the diseased state when triggered by a mutation in frataxin, and to the iron-deficient state when cells are placed in iron-deficient growth medium. The in-silico phenotypes observed during these transitions were similar to those observed experimentally. The model also predicted the observed effects of hypoxia on the diseased condition. A similar approach could PLOS COMPUTATIONAL BIOLOGY
In this article the problem of an interface fracture between two isotropic linear elastic materia... more In this article the problem of an interface fracture between two isotropic linear elastic materials is studied within a continuum modeling framework, which incorporates important nanoscale effects. The proposed model of bi-material crack ascribes curvature-dependent surface tension to both the fracture surfaces and the solid-solid interface. Further, it uses as boundary conditions the jump momentum balance equations, which take into account the excess properties ascribed to the material interfaces. Ultimately, the model leads to a 4 × 4 system of Cauchy-singular integro-differential equations which is equivalent to a system of Fredholm integral equations. It is demonstrated herein, using the Method of Integral Transforms and the Theory of Fredholm Integral equations, that this model leads to bounded stresses and strains, in contrast to the square-root singularities of the stress and strain fields predicted by the classical theory of brittle fracture.
In recent work, force-based canonical approaches have given a unified but different viewpoint on ... more In recent work, force-based canonical approaches have given a unified but different viewpoint on the nature of bonding in pairwise interatomic interactions. The concept of a pairwise canonical potential in these cases was defined for a class of molecules referred to a dimensionless function obtained from each molecule by a readily invertible algebraic transformation. Differing molecular categories (covalent, ionic, van der Waals, hydrogen and halogen bonding) of representative interatomic interactions in which binding energies ranging from 1.01 to 1072.03 kJ/mol have been modeled canonically giving a rigorous semi-empirical verification to high accuracy. However, the fundamental physical basis that is expected to provide the inherent characteristics of these canonical transformations have not yet been elucidated. We now show that canonical force distributions can be formulated from the perspective of the Hellmann-Feynman Theorem and discuss how such canonical approaches can be used to further explain the nature of chemical bonding in pairwise interatomic interactions.
A new approach to the accurate generation of multidimensional intermolecular interaction potentia... more A new approach to the accurate generation of multidimensional intermolecular interaction potentials is demonstrated. The basis for evaluating this application is the available very accurate 3-dimensional morphed potential of ArHBr. Starting from the well-defined potential of the simplest molecule, the diatomic H 2 + , a recently developed canonical approach is applied and used with sufficient additional polyatomic data to generate the adiabatic intermolecular interaction potential in ArHBr, with HBr in the vibrational ground state. This represents the first application of canonical transformations to a higher vibrationally dimensional molecular system, in this case, ArHBr. Results indicate intrinsic bonding characteristics inherent to both systems.
Canonical approaches are applied for investigation of the electronic ground states of H 2 + , H 2... more Canonical approaches are applied for investigation of the electronic ground states of H 2 + , H 2 , HeH + , and LiH using the Virial Theorem. These approaches will be dependent on previous investigations involving the canonical nature of E(R), the Born-Oppenheimer potential, and F(R), the associated force of E(R), that have been demonstrated to be individually canonical to high accuracy in the case of the systems investigated. Now, the canonical nature of the remaining functions in the Virial Theorem [the electronic kinetic energy T(R), the electrostatic potential energy V(R), and the function W(R) =RF(R)] are investigated and applied to H 2 , HeH + , and LiH with H 2 + chosen as reference. The results will be discussed in the context of a different perspective of molecular bonding that goes beyond previous direct applications of the Virial Theorem.
In a previous study, we introduced a generalized formulation for canonical transformations and sp... more In a previous study, we introduced a generalized formulation for canonical transformations and spectra to investigate the concept of canonical potentials strictly within the Born-Oppenheimer approximation. Data for the most accurate available ground electronic state pairwise intramolecular potentials in H 2 + , H 2 , HeH + , and LiH were used to rigorously establish such conclusions. Now, a canonical transformation is derived for the molecular force F(R) from the Hellmann-Feynman Theorem with H 2 + , the simplest molecule as molecular reference. These transformations are demonstrated to be inherently canonical to high accuracy but distinctly different from those corresponding to the respective potentials of H 2 , HeH + , and LiH. Further applications of this methodology to Mg 2 , benzene dimer and to water dimer are also considered within the radial limit. The implications of these results for electrostatic model of chemical bonding will be considered and discussed using this fundamentally force-based canonical approach.
It has been shown that taking into account surface mechanics is extremely important for accurate ... more It has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience, fracture propagation, and contact mechanics. This paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature-dependent surface tension acting on the boundary of the semiplane. Cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered. Using the method of integral transforms, each problem is reduced to a system of singular integro-differential equations, which is further reduced to one or two weakly singular integral equations. It has been shown that the introduction of the curvaturedependent surface tension eliminates the classical singularities of the order 1/2 of the stresses and strains at the end-points of the contact interval. The numerical solution of the problem is obtained by approximation of unknown functions with Taylor polynomials.
The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed... more The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we develop a Markov jump process model for this strategy, and we verify that there is a positive probability for wild-type females going extinct within a finite time. Moreover, when sex-reversed Trojan females are introduced at a constant population size, we formulate a stochastic differential equation (SDE) model as an approximation to the proposed Markov jump process model. Using the SDE model, we investigate the probability distribution and expectation of the extinction time of wild-type females by solving Kolmogorov equations associated with these statistics. The results indicate how the probability distribution and expectation of the extinction time are shaped by the initial conditions and the model parameters.
The concept of chemical bonding is normally presented and simplified through two models: the cova... more The concept of chemical bonding is normally presented and simplified through two models: the covalent bond and the ionic bond. Expansion of the ideal covalent and ionic models leads chemists to the concepts of electronegativity and polarizability, and thus to the classification of polar and non-polar bonds. In addition, the intermolecular interactions are normally viewed as physical phenomena without direct correlation to the chemical bond in any simplistic model. Contrary to these traditional concepts of chemical bonding, recently developed canonical approaches demonstrate a unified perspective on the nature of binding in pairwise interatomic interactions. This new canonical model, which is a force-based approach with a basis in fundamental molecular quantum mechanics, confirms much earlier assertions that in fact there are not fundamental distinctions among covalent bonds, ionic bonds, and intermolecular interactions including the hydrogen bond, the halogen bond, and van der Waals interactions.
Calculus of Variations and Partial Differential Equations, Apr 21, 2015
The main purpose of this study is to establish the existence of a weak solution to the anti-plane... more The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem.
Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using t... more Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using the canonical transformation generated for the Morse potential as a reference, inverse transformations allow the accurate generation of the Born-Oppenheimer potential for H 2 + ion, neutral covalently bound H 2 , van der Waals bound Ar 2 , and the hydrogen bonded one dimensional dissociative coordinate in water dimer. Similar transformations are also generated using the Lennard-Jones and Kratzer potentials as references. Following application of inverse transformations, vibrational eigenvalues generated from the Born-Oppenheimer potentials give significantly improved quantitative comparison with values determined from the original accurately known potentials. In addition, an algorithmic strategy based upon a canonical transformation to dimensionless form applied to the force distribution associated to a potential is presented. The resulting canonical force distribution is employed to construct an algorithm for deriving accurate estimates for the dissociation energy, the maximum attractive force, and the internuclear separations corresponding to the maximum attractive force and the potential well.
Presented here is a new method for constructing solutions to dynamically accelerating, semi-infin... more Presented here is a new method for constructing solutions to dynamically accelerating, semi-infinite crack problems. The problem of a semi-infinite, anti-plane shear (mode III) crack accelerating dynamically in an infinite, linear, homogeneous, and isotropic elastic body has been solved previously by Freund and Kostrov. However, their methods are based upon the construction of a certain Green's function for the ordinary two-dimensional wave equation and do not generalize to either the opening mode problem in elastic material or viscoelastic material. What is presented here is a new approach based upon integral transforms and complex variable techniques that does, in principle, generalize to both the opening modes of deformation and viscoelastic material. Moreover, the method presented here produces directly, for the mode III crack, a simple closed form expression for the crack-face profile for arbitrary applied crack-face tractions. Generalizations to opening modes of deformation and viscoelastic material produce integral equations for the crack-face displacement profile that in some cases admit closed form solutions and otherwise can be solved numerically. In contrast, the method of Freund and Kostrov yields, in mode III, an expression for the stress in front of the crack, but for opening modes provides only the stress intensity factor.
Public reporting burden for this collection of information is estimated to average 1 hour per res... more Public reporting burden for this collection of information is estimated to average 1 hour per response, including gathering and maintaining the data needed, and completing and reviewing the collection of information. Send c collection of information, including suggestions for reducing this burden, to Washington Headquarters Services,
Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for γ = 0.05 a... more Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for γ = 0.05 and far-field loading σ = 0.01, 0.02, 0.04. 47 3.3 Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for far-field loading σ 0 = 0.02 andγ = 0.
An approach to modeling fracture incorporating interfacial mechanics is applied to the example of... more An approach to modeling fracture incorporating interfacial mechanics is applied to the example of a curvilinear plane strain crack. The classical Neumann boundary condition is augmented with curvaturedependent surface tension. It is shown that the considered model eliminates the integrable crack-tip stress and strain singularities of order 1/2 present in the classical linear fracture mechanics solutions, and also leads to the sharp crack opening that is consistent with empirical observations. Unlike for the case of a straight crack, for a general curvilinear crack some components of the stresses and the derivatives of the displacements may still possess weaker singularities of a logarithmic type. Generalizations of the present study that lead to complete removal of all crack-tip singularities, including logarithmic, are the subject of a future paper.
We consider the problem of the dynamic, transient propagation of a semiinfinite, mode I crack in ... more We consider the problem of the dynamic, transient propagation of a semiinfinite, mode I crack in an infinite elastic body with a nonlinear, viscoelastic cohesize zone. Our problem formulation includes boundary conditions that preclude crack face interpenetration, in contrast to the usual mode I boundary conditions that assume all unloaded crack faces are stress-free. The nonlinear viscoelastic cohesive zone behavior is motivated by dynamic fracture in brittle polymers in which crack propagation is preceeded by significant crazing in a thin region surrounding the crack tip. We present a combined analytical/numerical solution method that involves reducing the problem to a Dirichlet-to-Neumann map along the crack face plane, resulting in a differo-integral equation relating the displacement and stress along the crack faces and within the cohesive zone.
Journal of Mechanics of Materials and Structures, Nov 1, 2008
A careful examination of a dynamic mode I crack problem leads to the conclusion that the commonly... more A careful examination of a dynamic mode I crack problem leads to the conclusion that the commonly used boundary conditions do not always hold in the case of an applied crack face loading, so that a modification is required to satisfy the equations. In particular, a transient compressive stress wave travels along the crack faces, moving outward from the loading region on the crack face. This does not occur in the quasistatic or steady state problems, and is a special feature of the transient dynamic problem that is important during the time interval immediately following the application of crack face loading. We demonstrate why the usual boundary conditions lead to a prediction of crack face interpenetration, and then examine how to modify the boundary condition for a semi-infinite crack with a cohesive zone. Numerical simulations illustrate the resulting approach.
We reconsider a semi-inverse problem for nonlinearly elastic wedges, versions of which have been ... more We reconsider a semi-inverse problem for nonlinearly elastic wedges, versions of which have been studied by several authors over the past 40 years. An intriguing feature of this problem is the existence of "non-unidirectional" deformations of the wedge, i.e. deformations for which material draws radially inward toward the apex in certain sectors of the wedge while material bulges radially outward in complementary sectors. We observe that such deformations are possible for a much larger class of materials than has been previously noted. We explore more carefully the relation between the displacement boundamy conditions and the "directional" properties of defonnations. Also, we consider the problem under alternative stress boundary conditions that must be satisfied pointwise; most previous treatments of the problem considered either pure displacement boundary conditions or stress boundary conditions satisfied in some average sense. Finally, we investigate a much broader class of generalized plane-strain deformations, consistent with these alternative boundary conditions and with a certain natural physical restriction on the pressure, and we show, surprisingly, that no additional solutions exist within this broader class.
This paper analyzes the stability of a coexistence equilibrium point of a model for competition b... more This paper analyzes the stability of a coexistence equilibrium point of a model for competition between two stage-structured populations. In this model, for each population, competition for resources may affect any one of the following population parameters: reproduction, juvenile survival, maturation rate, or adult survival. The results show that the competitive strength of a population is affected by (1) the ratio of the population parameter influenced by competition under no resource limitation (maximum compensatory capacity) over the same parameter under a resource limitation due to competition (equilibrium rate) and (2) the ratio of interspecific competition over intraspecific competition; this ratio was previously shown to depend on resource-use overlap. The former ratio, which we define as fitness, can be equalized by adjusting organisms' life history strategies, thereby promoting coexistence. We conclude that in addition to niche differentiation among populations, the life history strategies of organisms play an important role in coexistence. T he dynamics of competition between two species have been studied extensively, but few of these prior studies have incorporated variation in life history strategies between competing populations. Here, we present a mechanism for the coexistence of populations with different life history strategies (species). The classic Lotka-Volterra competition model 1 predicts that unless competition between two species is weak, the population that has a competitive advantage will exclude the other; this is also known as the competitive exclusion principle 2,3,4. In order for two species to coexist, their interspecific competition must be weaker than their intraspecific competition (stabilizing mechanism) 5. However, the amount of stabilization required for coexistence can be small if the difference in average fitness between the two species is small (equalizing mechanism) 5. In other words, strong stabilization alone or weak stabilization plus strong equalization is required for species coexistence. Many sympatric populations do not compete for a single resource, which means that stabilization maintains their coexistence. However, one of the interesting questions in ecology is how sympatric populations that do compete strongly with each other (i.e. cases in which interspecific competition is only slightly weaker than intraspecific competition) can coexist. This question may be reduced to the goal of determining how the average fitness associated with two species is equalized. Fitness is a function of an organism's life history parameters within its environment (e.g. ref. 6) assuming no individual heterogeneity within species beyond stage differences. There is clearly a diverse range of life history strategies among species: e.g. some mature early (precocious) while others delay maturation; some adopt semelparous reproductive strategies while others are iteroparous; and some are long-lived and others short-lived. We call this type of diversity demographic biodiversity. Although demographic biodiversity is expected to have profound effects on interactions between species, it has been largely overlooked in theoretical studies containing multiple species (exceptions include refs. 7, 8, 9, 10). A previous study 10 examined a model similar to the one discussed in this paper. Their numerical analysis used two-stage Ricker models 11 , assumed only individuals in the same stage of two populations (i.e. between adults and between juveniles of two populations) compete with each other, and showed the possible asymptotic outcomes, but did not derive the criteria for the outcomes. Here, we use two-stage Beverton-Holt models 12 , which can be derived by assuming competition for available resources, and derive analytical criteria for existence and the stability of coexistence equilibria. This allows us to interpret the criteria in terms of biological processes. Finally, we show how demographic biodiversity can equalize the fitness of two species, thereby promoting their coexistence. To our knowledge, other previous attempts used models that are special cases of our models (e.g. refs 7, 8, 9). We analyze a model that is comprised of two stage-structured populations of different species that experience competition for available resources at some life stage. Although the classic Lotka-Volterra competition model is formulated in continuous-time equations, we use a discrete-time model in order to include stage structured
Iron is an essential transition metal for all eukaryotic cells, and its trafficking throughout th... more Iron is an essential transition metal for all eukaryotic cells, and its trafficking throughout the cell is highly regulated. However, the overall cellular mechanism of regulation is poorly understood despite knowing many of the molecular players involved. Here, an ordinary-differential-equations (ODE) based kinetic model of iron trafficking within a growing yeast cell was developed that included autoregulation. The 9-reaction 8-component in-silico cell model was solved under both steady-state and time-dependent dynamical conditions. The ODE for each component included a dilution term due to cell growth. Conserved rate relationships were obtained from the null space of the stoichiometric matrix, and the reducedrow-echelon-form was used to distinguish independent from dependent rates. Independent rates were determined from experimentally estimated component concentrations, cell growth rates, and the literature. Simple rate-law expressions were assumed, allowing rateconstants for each reaction to be estimated. Continuous Heaviside logistical functions were used to regulate rate-constants. These functions acted like valves, opening or closing depending on component "sensor" concentrations. Two cellular regulatory mechanisms were selected from 134,217,728 possibilities using a novel approach involving 6 mathematically-defined filters. Three cellular states were analyzed including healthy wild-type cells, iron-deficient wild-type cells, and a frataxin-deficient strain of cells characterizing the disease Friedreich's Ataxia. The model was stable toward limited perturbations, as determined by the eigenvalues of Jacobian matrices. Autoregulation allowed healthy cells to transition to the diseased state when triggered by a mutation in frataxin, and to the iron-deficient state when cells are placed in iron-deficient growth medium. The in-silico phenotypes observed during these transitions were similar to those observed experimentally. The model also predicted the observed effects of hypoxia on the diseased condition. A similar approach could PLOS COMPUTATIONAL BIOLOGY
In this article the problem of an interface fracture between two isotropic linear elastic materia... more In this article the problem of an interface fracture between two isotropic linear elastic materials is studied within a continuum modeling framework, which incorporates important nanoscale effects. The proposed model of bi-material crack ascribes curvature-dependent surface tension to both the fracture surfaces and the solid-solid interface. Further, it uses as boundary conditions the jump momentum balance equations, which take into account the excess properties ascribed to the material interfaces. Ultimately, the model leads to a 4 × 4 system of Cauchy-singular integro-differential equations which is equivalent to a system of Fredholm integral equations. It is demonstrated herein, using the Method of Integral Transforms and the Theory of Fredholm Integral equations, that this model leads to bounded stresses and strains, in contrast to the square-root singularities of the stress and strain fields predicted by the classical theory of brittle fracture.
In recent work, force-based canonical approaches have given a unified but different viewpoint on ... more In recent work, force-based canonical approaches have given a unified but different viewpoint on the nature of bonding in pairwise interatomic interactions. The concept of a pairwise canonical potential in these cases was defined for a class of molecules referred to a dimensionless function obtained from each molecule by a readily invertible algebraic transformation. Differing molecular categories (covalent, ionic, van der Waals, hydrogen and halogen bonding) of representative interatomic interactions in which binding energies ranging from 1.01 to 1072.03 kJ/mol have been modeled canonically giving a rigorous semi-empirical verification to high accuracy. However, the fundamental physical basis that is expected to provide the inherent characteristics of these canonical transformations have not yet been elucidated. We now show that canonical force distributions can be formulated from the perspective of the Hellmann-Feynman Theorem and discuss how such canonical approaches can be used to further explain the nature of chemical bonding in pairwise interatomic interactions.
A new approach to the accurate generation of multidimensional intermolecular interaction potentia... more A new approach to the accurate generation of multidimensional intermolecular interaction potentials is demonstrated. The basis for evaluating this application is the available very accurate 3-dimensional morphed potential of ArHBr. Starting from the well-defined potential of the simplest molecule, the diatomic H 2 + , a recently developed canonical approach is applied and used with sufficient additional polyatomic data to generate the adiabatic intermolecular interaction potential in ArHBr, with HBr in the vibrational ground state. This represents the first application of canonical transformations to a higher vibrationally dimensional molecular system, in this case, ArHBr. Results indicate intrinsic bonding characteristics inherent to both systems.
Canonical approaches are applied for investigation of the electronic ground states of H 2 + , H 2... more Canonical approaches are applied for investigation of the electronic ground states of H 2 + , H 2 , HeH + , and LiH using the Virial Theorem. These approaches will be dependent on previous investigations involving the canonical nature of E(R), the Born-Oppenheimer potential, and F(R), the associated force of E(R), that have been demonstrated to be individually canonical to high accuracy in the case of the systems investigated. Now, the canonical nature of the remaining functions in the Virial Theorem [the electronic kinetic energy T(R), the electrostatic potential energy V(R), and the function W(R) =RF(R)] are investigated and applied to H 2 , HeH + , and LiH with H 2 + chosen as reference. The results will be discussed in the context of a different perspective of molecular bonding that goes beyond previous direct applications of the Virial Theorem.
In a previous study, we introduced a generalized formulation for canonical transformations and sp... more In a previous study, we introduced a generalized formulation for canonical transformations and spectra to investigate the concept of canonical potentials strictly within the Born-Oppenheimer approximation. Data for the most accurate available ground electronic state pairwise intramolecular potentials in H 2 + , H 2 , HeH + , and LiH were used to rigorously establish such conclusions. Now, a canonical transformation is derived for the molecular force F(R) from the Hellmann-Feynman Theorem with H 2 + , the simplest molecule as molecular reference. These transformations are demonstrated to be inherently canonical to high accuracy but distinctly different from those corresponding to the respective potentials of H 2 , HeH + , and LiH. Further applications of this methodology to Mg 2 , benzene dimer and to water dimer are also considered within the radial limit. The implications of these results for electrostatic model of chemical bonding will be considered and discussed using this fundamentally force-based canonical approach.
It has been shown that taking into account surface mechanics is extremely important for accurate ... more It has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience, fracture propagation, and contact mechanics. This paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature-dependent surface tension acting on the boundary of the semiplane. Cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered. Using the method of integral transforms, each problem is reduced to a system of singular integro-differential equations, which is further reduced to one or two weakly singular integral equations. It has been shown that the introduction of the curvaturedependent surface tension eliminates the classical singularities of the order 1/2 of the stresses and strains at the end-points of the contact interval. The numerical solution of the problem is obtained by approximation of unknown functions with Taylor polynomials.
The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed... more The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we develop a Markov jump process model for this strategy, and we verify that there is a positive probability for wild-type females going extinct within a finite time. Moreover, when sex-reversed Trojan females are introduced at a constant population size, we formulate a stochastic differential equation (SDE) model as an approximation to the proposed Markov jump process model. Using the SDE model, we investigate the probability distribution and expectation of the extinction time of wild-type females by solving Kolmogorov equations associated with these statistics. The results indicate how the probability distribution and expectation of the extinction time are shaped by the initial conditions and the model parameters.
The concept of chemical bonding is normally presented and simplified through two models: the cova... more The concept of chemical bonding is normally presented and simplified through two models: the covalent bond and the ionic bond. Expansion of the ideal covalent and ionic models leads chemists to the concepts of electronegativity and polarizability, and thus to the classification of polar and non-polar bonds. In addition, the intermolecular interactions are normally viewed as physical phenomena without direct correlation to the chemical bond in any simplistic model. Contrary to these traditional concepts of chemical bonding, recently developed canonical approaches demonstrate a unified perspective on the nature of binding in pairwise interatomic interactions. This new canonical model, which is a force-based approach with a basis in fundamental molecular quantum mechanics, confirms much earlier assertions that in fact there are not fundamental distinctions among covalent bonds, ionic bonds, and intermolecular interactions including the hydrogen bond, the halogen bond, and van der Waals interactions.
Calculus of Variations and Partial Differential Equations, Apr 21, 2015
The main purpose of this study is to establish the existence of a weak solution to the anti-plane... more The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem.
Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using t... more Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using the canonical transformation generated for the Morse potential as a reference, inverse transformations allow the accurate generation of the Born-Oppenheimer potential for H 2 + ion, neutral covalently bound H 2 , van der Waals bound Ar 2 , and the hydrogen bonded one dimensional dissociative coordinate in water dimer. Similar transformations are also generated using the Lennard-Jones and Kratzer potentials as references. Following application of inverse transformations, vibrational eigenvalues generated from the Born-Oppenheimer potentials give significantly improved quantitative comparison with values determined from the original accurately known potentials. In addition, an algorithmic strategy based upon a canonical transformation to dimensionless form applied to the force distribution associated to a potential is presented. The resulting canonical force distribution is employed to construct an algorithm for deriving accurate estimates for the dissociation energy, the maximum attractive force, and the internuclear separations corresponding to the maximum attractive force and the potential well.
Presented here is a new method for constructing solutions to dynamically accelerating, semi-infin... more Presented here is a new method for constructing solutions to dynamically accelerating, semi-infinite crack problems. The problem of a semi-infinite, anti-plane shear (mode III) crack accelerating dynamically in an infinite, linear, homogeneous, and isotropic elastic body has been solved previously by Freund and Kostrov. However, their methods are based upon the construction of a certain Green's function for the ordinary two-dimensional wave equation and do not generalize to either the opening mode problem in elastic material or viscoelastic material. What is presented here is a new approach based upon integral transforms and complex variable techniques that does, in principle, generalize to both the opening modes of deformation and viscoelastic material. Moreover, the method presented here produces directly, for the mode III crack, a simple closed form expression for the crack-face profile for arbitrary applied crack-face tractions. Generalizations to opening modes of deformation and viscoelastic material produce integral equations for the crack-face displacement profile that in some cases admit closed form solutions and otherwise can be solved numerically. In contrast, the method of Freund and Kostrov yields, in mode III, an expression for the stress in front of the crack, but for opening modes provides only the stress intensity factor.
Public reporting burden for this collection of information is estimated to average 1 hour per res... more Public reporting burden for this collection of information is estimated to average 1 hour per response, including gathering and maintaining the data needed, and completing and reviewing the collection of information. Send c collection of information, including suggestions for reducing this burden, to Washington Headquarters Services,
Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for γ = 0.05 a... more Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for γ = 0.05 and far-field loading σ = 0.01, 0.02, 0.04. 47 3.3 Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for far-field loading σ 0 = 0.02 andγ = 0.
An approach to modeling fracture incorporating interfacial mechanics is applied to the example of... more An approach to modeling fracture incorporating interfacial mechanics is applied to the example of a curvilinear plane strain crack. The classical Neumann boundary condition is augmented with curvaturedependent surface tension. It is shown that the considered model eliminates the integrable crack-tip stress and strain singularities of order 1/2 present in the classical linear fracture mechanics solutions, and also leads to the sharp crack opening that is consistent with empirical observations. Unlike for the case of a straight crack, for a general curvilinear crack some components of the stresses and the derivatives of the displacements may still possess weaker singularities of a logarithmic type. Generalizations of the present study that lead to complete removal of all crack-tip singularities, including logarithmic, are the subject of a future paper.
We consider the problem of the dynamic, transient propagation of a semiinfinite, mode I crack in ... more We consider the problem of the dynamic, transient propagation of a semiinfinite, mode I crack in an infinite elastic body with a nonlinear, viscoelastic cohesize zone. Our problem formulation includes boundary conditions that preclude crack face interpenetration, in contrast to the usual mode I boundary conditions that assume all unloaded crack faces are stress-free. The nonlinear viscoelastic cohesive zone behavior is motivated by dynamic fracture in brittle polymers in which crack propagation is preceeded by significant crazing in a thin region surrounding the crack tip. We present a combined analytical/numerical solution method that involves reducing the problem to a Dirichlet-to-Neumann map along the crack face plane, resulting in a differo-integral equation relating the displacement and stress along the crack faces and within the cohesive zone.
Journal of Mechanics of Materials and Structures, Nov 1, 2008
A careful examination of a dynamic mode I crack problem leads to the conclusion that the commonly... more A careful examination of a dynamic mode I crack problem leads to the conclusion that the commonly used boundary conditions do not always hold in the case of an applied crack face loading, so that a modification is required to satisfy the equations. In particular, a transient compressive stress wave travels along the crack faces, moving outward from the loading region on the crack face. This does not occur in the quasistatic or steady state problems, and is a special feature of the transient dynamic problem that is important during the time interval immediately following the application of crack face loading. We demonstrate why the usual boundary conditions lead to a prediction of crack face interpenetration, and then examine how to modify the boundary condition for a semi-infinite crack with a cohesive zone. Numerical simulations illustrate the resulting approach.
We reconsider a semi-inverse problem for nonlinearly elastic wedges, versions of which have been ... more We reconsider a semi-inverse problem for nonlinearly elastic wedges, versions of which have been studied by several authors over the past 40 years. An intriguing feature of this problem is the existence of "non-unidirectional" deformations of the wedge, i.e. deformations for which material draws radially inward toward the apex in certain sectors of the wedge while material bulges radially outward in complementary sectors. We observe that such deformations are possible for a much larger class of materials than has been previously noted. We explore more carefully the relation between the displacement boundamy conditions and the "directional" properties of defonnations. Also, we consider the problem under alternative stress boundary conditions that must be satisfied pointwise; most previous treatments of the problem considered either pure displacement boundary conditions or stress boundary conditions satisfied in some average sense. Finally, we investigate a much broader class of generalized plane-strain deformations, consistent with these alternative boundary conditions and with a certain natural physical restriction on the pressure, and we show, surprisingly, that no additional solutions exist within this broader class.
This paper analyzes the stability of a coexistence equilibrium point of a model for competition b... more This paper analyzes the stability of a coexistence equilibrium point of a model for competition between two stage-structured populations. In this model, for each population, competition for resources may affect any one of the following population parameters: reproduction, juvenile survival, maturation rate, or adult survival. The results show that the competitive strength of a population is affected by (1) the ratio of the population parameter influenced by competition under no resource limitation (maximum compensatory capacity) over the same parameter under a resource limitation due to competition (equilibrium rate) and (2) the ratio of interspecific competition over intraspecific competition; this ratio was previously shown to depend on resource-use overlap. The former ratio, which we define as fitness, can be equalized by adjusting organisms' life history strategies, thereby promoting coexistence. We conclude that in addition to niche differentiation among populations, the life history strategies of organisms play an important role in coexistence. T he dynamics of competition between two species have been studied extensively, but few of these prior studies have incorporated variation in life history strategies between competing populations. Here, we present a mechanism for the coexistence of populations with different life history strategies (species). The classic Lotka-Volterra competition model 1 predicts that unless competition between two species is weak, the population that has a competitive advantage will exclude the other; this is also known as the competitive exclusion principle 2,3,4. In order for two species to coexist, their interspecific competition must be weaker than their intraspecific competition (stabilizing mechanism) 5. However, the amount of stabilization required for coexistence can be small if the difference in average fitness between the two species is small (equalizing mechanism) 5. In other words, strong stabilization alone or weak stabilization plus strong equalization is required for species coexistence. Many sympatric populations do not compete for a single resource, which means that stabilization maintains their coexistence. However, one of the interesting questions in ecology is how sympatric populations that do compete strongly with each other (i.e. cases in which interspecific competition is only slightly weaker than intraspecific competition) can coexist. This question may be reduced to the goal of determining how the average fitness associated with two species is equalized. Fitness is a function of an organism's life history parameters within its environment (e.g. ref. 6) assuming no individual heterogeneity within species beyond stage differences. There is clearly a diverse range of life history strategies among species: e.g. some mature early (precocious) while others delay maturation; some adopt semelparous reproductive strategies while others are iteroparous; and some are long-lived and others short-lived. We call this type of diversity demographic biodiversity. Although demographic biodiversity is expected to have profound effects on interactions between species, it has been largely overlooked in theoretical studies containing multiple species (exceptions include refs. 7, 8, 9, 10). A previous study 10 examined a model similar to the one discussed in this paper. Their numerical analysis used two-stage Ricker models 11 , assumed only individuals in the same stage of two populations (i.e. between adults and between juveniles of two populations) compete with each other, and showed the possible asymptotic outcomes, but did not derive the criteria for the outcomes. Here, we use two-stage Beverton-Holt models 12 , which can be derived by assuming competition for available resources, and derive analytical criteria for existence and the stability of coexistence equilibria. This allows us to interpret the criteria in terms of biological processes. Finally, we show how demographic biodiversity can equalize the fitness of two species, thereby promoting their coexistence. To our knowledge, other previous attempts used models that are special cases of our models (e.g. refs 7, 8, 9). We analyze a model that is comprised of two stage-structured populations of different species that experience competition for available resources at some life stage. Although the classic Lotka-Volterra competition model is formulated in continuous-time equations, we use a discrete-time model in order to include stage structured
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Papers by Jay Walton