Coalescence Point Research Articles

Splitting trees with neutral Poissonian mutations II: Largest and oldest families

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn.We are interested in the sizes and ages at time t of the clonal families carrying the most abundant alleles or the oldest ones, as t→∞, on the survival event. Intuitively, the results must depend on how the mutation rate θ and the Malthusian parameter α>0 compare. Hereafter, N≡Nt is the population size at time t, constants a,c are scaling constants, whereas k,k′ are explicit positive constants which depend on the parameters of the model.When α>θ, the most abundant families are also the oldest ones, they have size cN1−θ/α and age t−a.When α<θ, the oldest families have age (α/θ)t+a and tight sizes; the most abundant families have sizes klog(N)−k′loglog(N)+c and all have age (θ−α)−1log(t).When α=θ, the oldest families have age kt−k′log(t)+a and tight sizes; the most abundant families have sizes (klog(N)−k′loglog(N)+c)2 and all have age t/2.Those informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes (Popovic, 2004, Lambert, 2010), we are also able, when α≤θ, to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes.This is in deep contrast with the largest/oldest families in the standard coalescent with Poissonian mutations, which converge to some point processes after being rescaled by N (Donnelly and Tavaré 1986, Ewens, 2005, Durrett, 2008).

Non-reciprocal photonic bands in a two-dimensional holey metal filled with a magnetoelectric material

We present an electromagnetic tight-binding formalism for the study of the photonic band structure of a two-dimensional lattice of magnetoelectric material infiltrated in a holey metallic host. We find, in particular, that one-way (non-reciprocal) frequency bands may arise in a slab geometry as a result of the explicit time-reversal symmetry breaking induced by the magnetoelectric material. Furthermore, time-reversal symmetry breaking introduces mode coalescence and exceptional points which both manifest the non-trivial topological nature of the frequency bands. For realizing the above lattice in the optical regime, a nematic liquid crystal can be infiltrated within the holes of a noble-metal substrate such as silver or gold.

Antibound poles in cut-off Woods-Saxon and strictly finite-range potentials

The motion of $l=0$ antibound poles of the $S$ matrix with varying potential strength is calculated in a cut-off Woods-Saxon (WS) potential and in a potential that goes to zero smoothly at a finite distance and is exactly zero beyond [P. Salamon and T. Vertse, Phys. Rev. C 77, 037302 (2008)]. The pole positions of the antibound states as well as of the resonances depend on the cutoff radius, especially for higher node numbers. The starting points (at potential zero) of the pole trajectories correlate well with the range of the potential. The normalized antibound radial wave functions on the imaginary $k$ axis below and above the coalescence point have been found to be real and imaginary, respectively.

Relativistic explicit correlation: Coalescence conditions and practical suggestions

To set up the general framework for relativistic explicitly correlated wave function methods, the electron-electron coalescence conditions are derived for the wave functions of the Dirac-Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. The manipulations make full use of the internal symmetries of the reduced two-electron Hamiltonians such that the asymptotic behaviors of the wave functions emerge naturally. The results show that, at the coalescence point of two electrons, the wave functions of the DCG Hamiltonian are regular, while those of the DC and DCB Hamiltonians have weak singularities of the type r(12)(ν) with ν being negative and of O(α(2)). The behaviors of the MDC wave functions are related to the original ones in a simple manner, while the spin-free counterparts are somewhat different due to the complicated electron-electron interaction. The behaviors of the ZORA wave functions depend on the chosen potential in the kinetic energy operator. In the case of the nuclear attraction, the behaviors of the ZORA wave functions are very similar to those of the nonrelativistic ones, just with an additional correction of O(α(2)) to the nonrelativistic cusp condition. However, if the Coulomb interaction is also included, the ZORA wave functions become close to the large-large components of the DC wave functions. Note that such asymptotic expansions of the relativistic wave functions are only valid within an extremely small convergence radius R(c) of O(α(2)). Beyond this radius, the behaviors of the relativistic wave functions are still dominated by the nonrelativistic limit, as can be seen in terms of direct perturbation theory (DPT) of relativity. However, as the two limits α → 0 and r(12) → 0 do not commute, DPT is doomed to fail due to incorrect descriptions of the small-small component Ψ(SS) of the DC wave function for r(12) < R(c). Another deduction from the possible divergence of Ψ(SS) at r(12) = R(c) is that the DC Hamiltonian has no bound electronic states, although the last word cannot be said. These findings enrich our understandings of relativistic wave functions. On the practical side, it is shown that, under the no-pair approximation, relativistic explicitly correlated wave function methods can be made completely parallel to the nonrelativistic counterparts, as demonstrated explicitly for MP2-F12. Yet, this can only be achieved by using an extended no-pair projector.

Splitting trees with neutral Poissonian mutations I: Small families

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree [9], and the population counting process (Nt;t≥0) is a homogeneous, binary Crump–Mode–Jagers process.We assume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called an allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,…,Nt.We mainly use two classes of tools: coalescent point processes, as defined in [15], and branching processes counted by random characteristics, as defined in [11,13]. We provide explicit formulae for the expectation of A(k,t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/Nt and of A(t)/Nt thanks to random characteristics, in the same vein as in [19].Last, we separately compute the expected homozygosity by applying a method introduced in [14], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.

Improved Large‐Scale Liquid‐Phase Synthesis and High‐Temperature NMR Characterization of Short ­(F‐)PNAs

AbstractWe report on a large‐scale synthesis of F‐PNA trimer 10 and PNA trimer 11. The key improvement is the facile two‐step synthesis of (2,4‐difluoro‐5‐methylphenyl)acetic acid (2). Water solubility of the corresponding F‐PNA oligomer 10 was achieved by synthesizing solubility enhancer 5a, which is twofold positively charged and only consists of inherent structural elements of PNA. Protected and unpaired PNA n‐mers exist in a mixture of 2n conformers undergoing slow exchange and leading to complicated NMR spectra. Structure analysis was improved by recording 1H‐ and 13C‐NMR spectra at elevated temperatures above the coalescence point. Fully protected backbone derivatives show sharp resonances where expected, and spectra of protected PNAs are remarkably simplified, thereby allowing an interpretation for the first time. Both trimers 10 and 11 are considered as building blocks for a self‐replicating system based on PNA.

Characterization of creep and creep damage by in-situ microtomography

Application of in-situ microtomography to characterization of power law creep and creep damage in structural materials is presented. It is shown first that the successively reconstructed volumes are adequately monitoring the macroscopic sample shape and that microtomography is an optimal tool to characterize inhomogeneous specimen deformation. Based on a two-step image correlation technique the evolution of single voids is revealed and the basis of a pioneering approach to creep damage studies is presented. The method allows the unequivocal separation of three concurrent damage mechanisms: nucleation, growth, and coalescence of voids. The results indicate that growth rate of voids with equivalent diameters in the range of 2–5 mm is of about one order of magnitude higher than the prediction of continuum solid mechanics. Analysis of void coalescence points out the presence of two stable growth regimes related to coalescence between primary and secondary voids, respectively.

Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.

Correlated n 1,3 S states for coulomb three-body systems

n1,3S (n = 1 − 4) states for atomic three-body systems are studied with the Angular Correlated Configuration Interaction method. A recently proposed angularly correlated basis set is used to construct, simultaneously and with a single diagonalization, ground and excited states wave functions which: (i) satisfy exactly Kato cusp conditions at the two-body coalescence points; (ii) involve only linear parameters; (iii) show a fast convergency rate for the energy; and (iv) form an orthogonal set. The efficiency of the method is illustrated by the study a variety of three-body atomic systems [mmm] with two negatively charged light particles, with diverse masses m and m, and a heavy positively charged nucleus m. The calculated ground 11S and excited n1,3S (n = 2 − 4) state energies are compared with those given in the literature, when available. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Preparation of nanostructured ultrathin silver layer

Silver is widely used for a fabrication of plasmonic devices due to its unique optical constants. Nanostructured Ag layer can exhibit strong localized surface plasmon resonance, which mainly affects its optical behavior in visible and near infrared spectra. The nanostructure of the Ag layer is mainly influenced during the initial stage of the silver nucleation. Therefore we focused our attention on the study of this stage of the silver growth. The nanostructured ultra-thin silver layers were prepared by means of the magnetron sputtering. The nucleation mode and the resulting nanostructure was controlled by the deposition conditions. The initial stage of the nucleation and the layer growth was studied by means of an optical monitoring, which is based on a principle of spectrophotometric measurement of sample reflectivity. The measured data were fitted to a model of layered structure. The non-continual (Volmer-Weber) mode of the layer nucleation was clearly distinguished in the monitored data. Thus we were able to estimate the point of the non-continual layer coalescence as well as the subsequent evolution of the surface roughness. The prepared nanostructured Ag layers were analyzed by Atomic Force Microscopy (AFM) and Scanning Electron Microscopy (SEM). Optical properties were studied by spectroscopic ellipsometry and spectrophotometry. The aim of this work was to study the formation of nanostructured Ag layer and its correlation to the optical behavior.

Singlet oxygen: there is still something new under the sun, and it is better than ever

Singlet oxygen, O(2)(a(1)Δ(g)), the lowest excited electronic state of molecular oxygen, has a characteristic chemistry that sets it apart from the triplet ground state of oxygen, O(2)(X(3)Σ). Although singlet oxygen can be produced in a variety of ways, it is arguably best known as a product of the interaction between a light absorbing chromophore (i.e., a photosensitizer) and O(2)(X(3)Σ). In this way, it plays a key role in the many disciplines that comprise the "photosciences". As outlined in this Perspective, which is part of the program "Photosciences: a look to the future", the study of singlet oxygen provides a coalescence point for many topics currently at the cutting edge of modern science ranging from materials science to biology and medicine. As such, although singlet oxygen is a "mature citizen" and has been examined for many years in a variety of contexts, there is still something new under the sun, and the future looks exceedingly bright for this photo-related discipline.

Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components

Let A be a matrix with distinct eigenvalues and let w ( A ) be the distance from A to the set of defective matrices (using either the 2-norm or the Frobenius norm). Define Λ ϵ , the ϵ -pseudospectrum of A, to be the set of points in the complex plane which are eigenvalues of matrices A + E with ‖ E ‖ < ϵ , and let c ( A ) be the supremum of all ϵ with the property that Λ ϵ has n distinct components. Demmel and Wilkinson independently observed in the 1980s that w ( A ) ⩾ c ( A ) , and equality was established for the 2-norm by Alam and Bora (2005). We give new results on the geometry of the pseudospectrum near points where first coalescence of the components occurs, characterizing such points as the lowest generalized saddle point of the smallest singular value of A - zI over z ∈ C . One consequence is that w ( A ) = c ( A ) for the Frobenius norm too, and another is the perhaps surprising result that the minimal distance is attained by a defective matrix in all cases. Our results suggest a new computational approach to approximating the nearest defective matrix by a variant of Newton’s method that is applicable to both generic and nongeneric cases. Construction of the nearest defective matrix involves some subtle numerical issues which we explain, and we present a simple backward error analysis showing that a certain singular vector residual measures how close the computed matrix is to a truly defective matrix. Finally, we present a result giving lower bounds on the angles of wedges contained in the pseudospectrum and emanating from generic coalescence points. Several conjectures and questions remain open.

A molecular dynamics study of void interaction in copper

Molecular dynamics simulations in three-dimensional single crystal copper under remote uniaxial tension at high strain rate (109/s) were performed to analyze the microscopic mechanism of dislocation emission and void interaction. Two different cases, characterized by whether the line joining centers of two embedded cylindrical voids was perpendicular to or paralleled the tension direction, were studied using embedded atom method (EAM) potentials. The mean-squared displacements of atoms around the voids marking both yielding and coalescence points were presented. The critical position of the first slide was predicted where shear stress at slip plane was maximum.

Surface morphology of c-plane sapphire (α-alumina) produced by high temperature anneal

A comparative study of the morphological surface evolution of c-plane (0001) α-Al 2O 3 upon annealing was investigated for non-miscut (i.e. substrates with 0° nominal miscut) and vicinal substrates. The samples were annealed in air at 1100 °C for different durations of time. Although non-miscut samples do not show any step bunching at this temperature, miscut substrates show a regular and ordered stepped morphology with clearly defined terraces as revealed by Atomic Force Microscopy (AFM) and Transmission Electron Microscopy (TEM) image analysis. The surface morphology presents a number of coalescence points, i.e. locations where two steps merge and form a multiple step. Close to the coalescence points, parallel steps change direction to different low index direction.

Microstructural properties and dislocation evolution on a GaN grown on patterned sapphire substrate: A transmission electron microscopy study

The microstructural properties of a GaN layer grown on a patterned sapphire substrate (PSS) were studied in detail using transmission electron microscope techniques to determine dislocation and growth behaviors. Regular and uniform recrystallized GaN islands were observed on the protruding pattern. On a flat sapphire surface, the crystallographic orientation relationship of ⟨1¯21¯0⟩GaN on FS//⟨11¯00⟩sapphire and {11¯01}GaN on FS//{12¯13}sapphire existed between the GaN and the substrate. On the other hand, the orientation relationship of ⟨1¯21¯0⟩GaN layer//⟨1¯21¯0⟩GaN island on IS//⟨11¯00⟩sapphire and {11¯01}GaN layer//{0002}GaN island on IS//{12¯13}sapphire was confirmed among the GaN layer, the recrystallized GaN islands on an inclined sapphire surface and the PSS. The flat surface among the protruding patterns began to fill rapidly with GaN. Then, the GaN gradually overgrew the protruding pattern and coalesced near the summit as the growth time increased. The generation of threading dislocations was observed in the vicinity of the coalescence points near the top of the protruding patterns.

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

We study the numerical solution of the nonrelativistic Schr\"odinger equation for two-electron atoms in ground and excited $S$ states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wave function, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wave function's cusplike behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence and is more computationally efficient than a single subdomain. A comparison of several different treatments of the cusps suggests that the simplest prescription is sufficient. We investigate two alternate methods for handling the semi-infinite domain, one which involves a sequence of truncated versions of the domain and the other which employs an algebraic mapping of the semi-infinite domain to a finite one and imposes no explicit cutoffs on the wave function. The latter prescription proves superior. For many purposes it proves unnecessary to handle the three-particle coalescence in a special way. The presence of logarithmic terms in the exact solution is expected to limit the convergence to being nonexponential but the only clear evidence of that is the rate of convergence of derivatives near the three-particle coalescence point. Higher resolution than achieved in this work will ultimately be needed to see its limiting effect on other measures of error. As developed and applied here the PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wave function near cusps or at large distances, the local energy ($\mathcal{H}\ensuremath{\psi}/\ensuremath{\psi}$) is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. As the number of collocation points grows, the method achieves exponential convergence up to the resolution tested. This study serves as a proof-of-principle of the method for more general two- and possibly three-electron applications.

Study of hot tearing of A206 aluminum alloy using Instrumented Constrained T-shaped Casting method

The hot tearing susceptibility of A206 aluminum alloy was investigated using Instrumented Constrained T-shaped Casting method and the effect of the casting temperature on hot tearing was studied. The Instrumented Constrained T-shaped Casting apparatus enabled real-time measurements of the contraction load developed in the casting and the temperature variations during solidification as a function of time. Critical temperatures and points during solidification of the castings were extracted from these data. The contraction load developed at the coalescence point of the castings was identified as a comparative criterion for predicting the hot tearing susceptibility of the alloys which could be utilized even when no visual tearing had occurred. The results showed that hot tearing susceptibility increased with the casting temperature. This was associated with reduced cooling rate, increased solute segregation and more localized hot spot formation at the T-junction area. Increase in the casting temperature also increased the grain size which may in turn have affected the initiation of the hot tears. The visual observations were further validated with radiographic tests.

The contour of splitting trees is a Lévy process

Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {∞}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point (v, τ) of some individual v (vertex) in a discrete tree where τ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping φ from the tree into the real line which preserves this order. The inverse of φ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level τ, we prove that a thus defined contour process is a Lévy process reflected below τ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall–Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

Ground and excited states for exotic three-body atomic systems

An Angular Correlated Configuration Interaction method is extended and applied to exotic threebody atomic systems with general masses. A recently proposed angularly correlated basis set is used to construct, simultaneously and with a single diagonalization, ground and excited states wave functions which: (i) satisfy exactly Kato cusp conditions at the two-body coalescence points; (ii) have only linear parameters; (iii) show a fast convergency rate for the energy; (iv) form an orthogonal set. The efficiency of the construction is illustrated by the study a variety of three-body atomic systems [m1− m2− m3z3+ ] with two negatively charged light particles, with 123 diverse masses m1− and m2−, and a heavy positively charged nucleus m3z3+. The calculated ground 11S and several excited n1,3S state energies are compared with those given in the literature, when available. We also present a short discussion on the critical charge necessary to get a stable three-body system supporting two electrons, an electron and a muon, or two muons.

Ground state for two‐electron and electron‐muon three‐body atomic systems

AbstractIn this article, the angular correlated configuration interaction method previously introduced by some of the authors is extended to three‐body atomic systems with general masses. A recently proposed angularly correlated basis set is used to construct ground state wave functions which: (i) satisfy exactly Kato cusp conditions at the two‐body coalescence points; (ii) have only linear coefficients; and (iii) show a fast convergency rate for the energy. The efficiency of the construction is illustrated by the study of the negatively charged hydrogen‐like systems (∞H−, T−, D−, 1H−, and Mu−), neutral helium‐like systems (e−e− ∞He+2,e−e− 4He+2, e−e− 3He+2, e−μ− ∞He+2, e−μ −4He+2, and e−μ− 3He+2), and positively charged lithium‐like systems (e−e− ∞Li+3, e−e− 7Li+3, e−e− 6Li+3, e−μ− ∞Li+3, e−μ− 7Li+3, and e−μ− 6Li+3). The ground state energies and other mean values are compared with those given in the literature, when available. Wave functions with a moderate number of (20 maximum) linear coefficients are given explicitly; they are sufficiently simple and accurate to be used in practical calculations of atomic collision in which multidimensional integrations are involved. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010