bluuebunny/crossref_metadata_2025 huggingface dataset:
Reptation quantum Monte Carlo: A method for unbiased ground-state averages and imaginary-time correlations
Authors: Baroni S , Moroni S
Published: November 2023

This paper introduces a successful new method for extending diffusion QMC with importance sampling to allow the direct calculation of pure ‘¢ 2 distributions in place of mixed ‘¢ 1/Jr distributions. The method removes any bias introduced with a trial function (except that due to node locations) in the calculation of observable quantities which do not commute with the Hamiltonian. It has the advantage, relative to the descendent weighting or forward walking technique,a of a controlled walker population which reduces statistical errors. The authors have named it the reptation quantum Monte Carlo (RQMC) method. Its basic object is a path or sequence of configurations called a “reptile” which may be altered by an action called “reptation,” terms adopted from related work in the polymer area. In conventional diffusion QMC with importance sampling walkers are subjected to diffusion, drift, and a local energy-based multiplication to produce a 1/J’I/Jr distribution. Without multiplication a ‘l/Jr’l/Jr distribution is produced. In RQMC multiplication is replaced by a Metropolis rejection procedure, also based on local energies, which produces a 1/J’!/J distribution of configurations in a reptile. The generation of a new reptile, by removing the tail and adding a new head, is based on diffusion and drift for proposed moves coupled with an acceptance test. Overall, the implementation is quite similar to that for variational QMC. As described, the method was tested first for the hydrogen atom, followed by the successful demonstration of a calculation for a helium fluid, 64 4He atoms in a cubic box with periodic boundary conditions. The advantages of RQMC relative to branching diffusion QMC were pointed out for treatment of clusters, films, and superfluids as well as for estimation of electronic forces.
Relativistic calculations using Monte Carlo methods: One-electron systems
Authors: Alexander S A , Coldwell R L
Published: November 2023

The Schri::idinger equation is not the only fundamental equation of quantum mechanics that can be solved by Monte Carlo methods. In this paper, a solution of the four-component Dirac equation to determine relativistic energies is reported for the one-electron systems U91+, Hf, and Th 79+. The calculations were carried out in variational QMC, with trial functions optimized by minimizing the variance in local energies. For a one-electron system with more than one fixed nucleus, the four-component Dirac equation is required. The equation is similar to the Schri::idinger equation in that H’lj; = E’lj;, but the wave function is a four-component vector with individual components that may be complex, and the Hamiltonian is a 4 x 4 matrix. The exact solution for a one-electron atom is known, and the authors made use of this knowledge to show that variance minimization would produce correct coefficients in a wave function of known functional form. Thus, for U91+, the resulting energy was essentially exact. For the diatomic systems, no exact wave functions are known, and three different carefully chosen series functions of up to eight terms were investigated. Among these a mixed wave function of Slater atomic orbitals and floating Gaussian orbitals was most successful, although all three gave excellent energies. Nevertheless, for both Hf and Th 79+, the energies were slightly higher than the best available estimates from other methods of calculation. Properties other than the energy could be evaluated in a similar fashion using the optimized wave functions, and additional terms could be added to the Hamiltonian.
On the accuracy of the fixed-node diffusion quantum Monte Carlo method
Authors: Manten S , Lüchow A
Published: November 2023

In this work Manten and Liichow assessed the accuracy of allelectron fixed-node diffusion QMC calculations with nodes taken from Hartee-Fock wavefunctions with high-quality basis sets. The electronic energies associated with 17 different reactions involving a total of 20 small molecules were determined and compared with experimental values and with those calculated by Klopper et al.a using the coupled cluster method CCSD(T) with correlation consistent basis sets cc-pVDZ and cc-pVTZ. The molecules involved were limited to those containing the atoms H, C, 0, N, and F. Typical reactions were H2CO + 2H2 --+ CH4 + H2O and H2O + F --+ HOF + H. The geometries for the QMC calculations were optimized with MP2/cc-pVTZ calculations. The QMC total energies for the 20 molecules were lower than CCSD(T) values, but only valence electrons were correlated in the CCSD(T) calculations. Node location error was approximately 0.0015 hartrees for the C, N, 0, and F atoms and their hydrides, but substantial cancellation of node error is to be expected in determining enthalpy changes for reactions.
Role of electronic correlation in the Si(lOO) reconstruction: A quantum Monte Carlo study
Authors: Healy S B , Filippi C , Kratzer, P , Penev E , Scheffler M
Published: November 2023

The 100 surface of a silicon crystal is believed to reconstruct to form rows of silicon dimers in either a “symmetric” pattern ( 11 I 111 I I) or a somewhat deformed “buckled” pattern (/\/\/\/\) in which the dimers are twisted alternately within each row. At room temperature the dimers may flip back and forth between configurations and they appear symmetric in scanning tunneling microscopy. Below 120 K the buckled configuration appears, but below 20 K the symmetric configuration has been indicated but might be due to flipping, and arguments in favor of each of the configurations have been made. Theoretical work is equivocal. Density functional (DFT) calculations favor the buckled configuration. Multiconfiguration self-consistent field (MCSCF) and configuration interaction (CI) calculations favor the symmetric. The indicated differences in energies for the two structures are small, and electron correlation (note the title) is thus expected to be important in determining the minimum-energy structure. Diffusion QMC provides the required accurate description of correlation for large systems. As in the earlier studies with other methods the authors treated the model systems Si9H12, Sii5H16, and Si21H20 - clusters which have one, two, and three dimers in configurations like those on the Si(lO0) surface. The calculations were made using silicon pseudopotentials tested in calculations for Si2. For the intermediate cluster the energies of the symmetric and the buckled configurations were nearly identical. For the larger cluster, with three dimers, the buckled configuration was favored by 0.34(6) eV or 0.11(2) eV per dimer. This energy difference is similar to that for DFT calculations (LDA to B3LYP). The agreement suggests that the dynamic correlation effects, described better in DQMC and DFT than in MCSCF and CI, are important. The buckled configuration is thus to be favored, but many questions in both theory and experiment remain unresolved for this system.
Quantum Monte Carlo study of the three- to one-dimensional crossover for a trapped Bose gas
Authors: Astrakharchik G E , Giorgini S
Published: November 2023

These two papers, published back to back, provide superb examples of the use of a quantum Monte Carlo method to obtain exact solutions for a problem difficult to treat by other methods. The problem is that of a bosonic atomic gas confined in a cigarshaped trap. As the cigar is narrowed the atoms are less able to pass each other, their behavior approaches that of fermions, and they are said to exhibit “fermionization.” Using diffusion QMC the authors were able to determine the ground-state energies, examine structural properties, confirm that fermionization can occur, and predict conditions under which it might be observed. The calculations by Blume were carried out for 2-20 atoms confined in axially symmetric traps of varied aspect ratios. The confining potential for each atom was given by Vk = CpPl + Czzl (radial and axial positions squared). The atom-atom interactions were simple hard-sphere potentials. The (nodeless) guided diffusion calculations used a trial wave function of the Jastrow type, a product of atom-trap terms and atom-atom terms. The calculations by Astrakharchik and Giorgini were carried out similarly but included both hard and soft spheres with numbers in the range of 5 to 100 atoms. Comparisons were made with predictions of the Lieb Liniger equation of state.
Quantum Monte Carlo calculations of neutron matter
Authors: Carlson J , Morales J , Pandharipande V R , >Ravenhall D E
Published: November 2023

Pure neutron matter is not likely to be found in any of the Earth’s laboratories, but it has been found to exist as the material of neutron stars. It presents an interesting and difficult manybody problem which may be attacked by QMC methods. In this paper the authors report variational, approximate fixed-node, and released-node QMC calculations for uniform neutron matter in a three-dimensional box with periodic boundary conditions. These allow the determination of the energy E(p) as a function of the neutron density p for a realistic two-neutron interaction expression. The results are used to assess the accuracy of earlier variational approaches. The system treated was that of 14 neutrons in a cube at densities in the range of 1 / 4 to 3/2 the density p0 typical for nuclear matter. Most of the calculations were done for the Argonne v8’ two-body interaction potential. The trial wavefunction was a correlated Slater function, identical for the variational and diffusion calculations, specifying the nodes and used in importance sampling. Corrections were made for the effects of finite box dimensions. The released-node calculations were limited to very short time intervals especially for the higher densities. The energies E(p) found in the diffusion QMC calculations were 5 to 10% lower than those of the variational QMC. The releasednode results, with relatively large statistical errors, were not significantly different from the fixed-node results. The energies determined for the lower densities were estimated to be accurate within 2% and those for the higher densities somewhat less accurate. The results showed earlier predictions to be generally correct. Comparisons of results from variational chain summations (VCS) indicated an overall accuracy of about 10% for that method.
Rotational dynamics of CO solvated in small He clusters: A quantum Monte Carlo study
Authors: Cazzato P , Paolini S , Moroni S , Baroni S
Published: November 2023

The reptation QMC method introduced by Baroni and Moroni0 a few years earlier is demonstrated in this paper to be the method of choice for predicting the rotational-vibrational features of the infrared spectrum of CO in small helium droplets. The calculations gave good agreement with experimental measurements for these relatively complex systems and provided insights into some of their characteristics not observable in experiments. The calculations were carried out for clusters containing one CO molecule and up to 30 He atoms. The CO molecule was treated as rigid, and HeHe and He-CO interactions were treated as pairwise additive, with pair terms from ab initio calculations. The trial wavefunction for this nodeless bosonic system was a Jastrow function made up of exponential functions dependent on the CO distance and orientation with respect to He atom and He-He distances. The reptation QMC calculations were carried out as diffusion QMC calculations with importance-sampling drift, but without multiplication so that the energy obtained was the expectation value of the energy for the trial wavefunction. The reptation part of the calculation produced exact (or nearly exact) expectation values and, more important in these calculations, unbiased quantum correlation functions from which the absorption spectra of the CO molecule could be determined. The results exhibit a variety of interesting phenomena. The binding energy for an additional atom first increases and then decreases before leveling out with increasing cluster size. The calculated rotational absorption spectra show two types of lines, one increasing in strength with cluster size, the other decreasing with cluster size. Much more detail, especially in regard to the droplet shape, revealed in the calculations.
Interpretation of Hund’s multiplicity rule for the carbon atom
Authors: Hongo K , Maezono R , Kawazoe Y , Towler M D , Needs R J
Published: November 2023

Originating from experimental observations and published in 1925, Hund’s rulea states that the electronic state with the lowest energy is the one with the highest spin multiplicity S. Obeyed by most atoms and molecules, although exceptions are known, the rule has been explained (incorrectly) as due to the lower average electron-electron repulsion flee for electrons of the same spin in higher S states, and (correctly)b as due to lowering of the electron nucleus energy Ven aselectrons of the same spin avoid each other and can move closer to the nucleus. The authors call this Boyd’s less screeningc mechanism. A completely satisfactory analysis of the phenomenon requires very accurate wavefunctions, and in this paper the authors used VQMC and DQMC calculations to provide those wavefunctions for singlet and triplet states of the carbon atom.
Monte Carlo solution of Schrödinger’s equation for the hydrogen atom in a magnetic field
Authors: Tomashima Y , Ozaki J
Published: November 2023

The problem of the hydrogen atom in a magnetic field can be treated satisfactorily by approximate methods which are quite accurate for very low or very high magnetic field strengths. Before this paper the only useful treatments for intermediate field strengths were variational methods giving upper bounds to the energies. As described in the paper the diffusion QMC method was applied in the same way as in earlier solutions for the hydrogen atom, but with additional magnetic terms added to the Hamiltonian expression. These include complex expressions and the required wavefunction includes a corresponding factor. With the magnetic field aligned along one axis, integration over the angle ϕ in (r, θ, ϕ) coordinates reduces the additional terms to real functions of r and z as well as the magnetic quantum number m. In the diffusion/multiplication processes of a random walk procedure these may be simply included with the potential energy term. In this study the energy at each step was determined from the average multiplication factor for the walkers. Importance sampling was not used. Results were obtained for quantum numbers m = 0, 1 at each of three magnetic field strengths. A comparison of results showed good agreement with the lowest values obtained in several of the earlier analytic variational calculations.
Interdimensional degeneracies in van der Waals clusters and quantum Monte Carlo computation of rovibrational states
Authors: Nightingale M P , Moodley M
Published: November 2023

In the course of an investigation of n-body correlations in trial wavefunctions for noble gas clusters the authors varied the spatial dimensionality of some of the clusters and discovered a surprising phenomenon: an interdimensional degeneracy for rotationally invariant states. For Lennard-Jones clusters of N particles in D dimensions they found identical energies for the ground states in dimensions D = N - l and D = N + l. They were able to explain this behavior by considering a D-dimensional Schrodinger equation for all S-states in dimensions D 2’. N - l. The minimum energy was observed and proven to occur for D = N. The calculations were carried out in dimensions D = l to 6 for systems corresponding to clusters of Kr, Ar, Ne, and a hypothetical ½-Ne with half the mass of Ne. The interactions were taken as pairwise-additive L-J 6-12. Very high accuracies were obtained with the aid of trial wavefunctions which were linear combinations of basis functions with nonlinear variational parameters optimized with an iteration procedure.
Quantum Monte Carlo and the Hydride Ion
Authors: Mcdowell K , Doll J D
Published: November 2023

These two papers introduced the term “Quantum Monte Carlo” to the scientific literature. Earlier in the same year McDowell0 and Berneb had used the term in abstracts for an American Chemical Society meeting, and later in the year Gordon, Rothstein, and Proctorc used it in the title of a paper submitted to the Journal of Computational Physics. The papers report investigations of variational QMC methods with Metropolis sampling using the hydride ion H- and the helium atom He as examples. As had been found by earlier workers it is shown that relatively simple trial functions explicitly incorporating the interelectron distance rij can produce a high level of accuracy comparable to those of configuration interaction (CI) calculations based on many-term trial functions without rij
Muon-Alpha-Particle Sticking Probability In Muon-Catalyzed Fusion
Authors: Ceperley D , Alder B J
Published: November 2023

A negative muon µ has the negative charge of an electron but a mass 207 times that of an electron. Like an electron e binding two protons pin the molecular ion Ht or (epp)+, it can bind two protons in a molecular ion (µpp)+, and because of its higher mass, it binds more strongly. Similarly, a negative muon can bind a deuteron d and a triton t in a molecular ion (µdt)+. With the deuteron and triton held tightly together, they may fuse together, ejecting a neutron, forming an alpha particle, and releasing substantial energy. In about 1% of the events, the muon sticks to the positively charged alpha particle and is lost as a catalyst to additional fusions. This paper describes the use of Green’s function QMC to calculate the wavefunction for the (µdt)+ molecular ion for the purpose of estimating the sticking probability of the muon with the alpha particle. The essential functions are the initial and final wavefunctions at the instant of fusion with the deuteron and triton at the same position (coalescence point), and these functions along with the sudden approximation can be used to estimate the sticking probability. Earlier analytic variational treatments based on the Born-Oppenheimer approximation were not satisfactory. A modified version of Green’s function QMC with the use of a trial density matrix, along with importance sampling, was used to determine energies and sample the wavefunction. Calculations of relative wavefunction values, at the coalescence point relative to others, were made using the evolution procedure described by Liu, Kalos, and Chestera. Determination of normalization factors required special consideration. The calculations were clearly successful and gave a sticking probability of 0.90%, a value about 25% smaller than given by a Born-Oppenheimer wavefunction and closer to estimates derived from experimental measurements.
Quantum Monte Carlo method with the model potential
Authors: Yoshida T , Iguchi K
Published: November 2023

This third paper on the subject of effective potentials in 1987 and 1988 reports a somewhat simpler approach to combining effective core potentials with QMC than other papers of the same era. In this work the effective potential is based on an improved version of one introduced by Bonifacic and Huzinaga,a but it is simplified by elimination of core-valence exchange interaction and by use of an approximate core-valence orthogonality. All that remains of the original Hamiltonian are the kinetic energy terms for valence electrons, the potential energy terms for valence electrons interacting with each other and a reduced-charge nucleus, and a simpler radius-dependent potential energy term for each valence electron. FixeD nodes are introduced with a Slater-determinant trial function also used for importance sampling. Tests were made in calculations of the valence energies of the atoms Mg, Ca, and Sr, along with their cations, so that ionization potentials for these atoms could be calculated and compared with experimental values. The statistical uncertainties in the energies were about 0.03 eV, and the deviations in ionization potentials from experiment were about 0.1 eV. The results were significantly better than those of SCF calculations exhibited for comparison. Additional calculations were reported in a later paper.b
Damped-core quantum Monte Carlo method: Effective treatment for large-Z systems
Authors: Haimmond B L , Reynolds P J , Lester W A
Published: November 2023

Heavy atoms are problematic in QMC calculations for several reasons. Their energies are large, they have many electrons, and the time scales for core and valence electron motions are widely different. Larger local energy fluctuations are associated with higher energies. More electrons require longer computational efforts. Different time scales require short steps in one region and long equilibration times in others. The result is an estimated scaling of computational effort with zB-7, the sixth to seventh power of the nuclear charge Z. The Z-dependence can be reduced with the use of pseudopotentials, and this approach has been pursued with success, but not without introducing uncertainties. This paper describes an entirely different method which retains all the electrons, but treats core and valence electrons in different ways. The basic idea is that of treating core electrons by Metropolis sampling and valence electrons by diffusion with drift. Separate trial functions t/JT,core and were chosen orthogonal such that the valence electrons were prevented from collapsing into the core. For the core electrons the Metropolis walks were taken without branching and produced a distribution of walkers. For the valence electrons normal diffusion walks with drift produced a distribution. A smooth transition between regions was obtained with a damping function to eliminate branching as valence electrons approached the core.
Simulation of the electronic structure of an atom dissolved in a hard-sphere liquid
Authors: Dobrosavljević V , Henebry C W , Stratt R M
Published: November 2023

For this paper the authors used diffusion Monte Carlo methods to obtain a first solution to the problem of electronic structure of an atom in a structured liquid. Although models of several types using the assumption of no spatial variation of solvent effects have had some success, they miss essential features which are included naturally in a QMC calculation. The authors treated the problem of a hydrogenic atom in a hard-sphere liquid, specifically a proton and an electron in the presence of a hard-sphere helium-atom solvent. A separate classical simulation of 31 helium atoms and 1 hydrogen atom was used to generate equilibrium configurations of the solvent for 300 K. These were followed by diffusion QMC calculations for the electron in the field of the fixed proton, along with the boundary conditions of a zero wavefunction at the surfaces of helium spheres. A highly effective importance sampling function, incorporating factors of the Jastrow form, reduced the uncertainties in calculated energies and simplified the analysis to determine the characteristics of the electronic structure. Results were obtained for thousands of configurations for each of a number of conditions for liquid structures as well as for bee structures. The major observations were a smooth increase in ground-state energy with liquid density and a solvent-induced hybridization of the p orbital into the ground state at intermediate densities.
Cohesive energy of silicon by the Green’s-function :Monte Carlo method
Authors: Li X -p. , Ceperley D M , Martin R M
Published: November 2023

After variational QMC calculations with pseudopotentials for diamond-structure carbon and silicon, the next step higher is fixednode diffusion QMC for these materials. This paper reports that step for silicon with use of a pseudo Hamiltonian° to eliminate core electrons. In this case, the pseudo-Hamiltonian was tested and found accurate in fixed-node diffusion calculations for atomic and diatomic silicon. The trial wavefunction specifying the node locations for valence electrons was a Slater Jastrow combination based on a plane-wave basis set with constants determined in LDA density functional calculations. Both VQMC and fixed-node diffusion QMC calculations were performed for supercells of 64 atoms (216 electrons) with a range of lattice constants. Corrections of results to infinite systems were made with the assumption of size dependence the same as for LDA calculations. The predicted minimumenergy structures for VQMC and FN-DQMC were approximately the same, with lattice constants within 1% of the experimental value. The cohesive energy for FN-DQMC was approximately 0.4 e flower than for VQMC and, at 4.51(3) eV per atom, was within the uncertainties in agreement with experimental values of 4.63(8) eV per atom. Questions regarding the transferability of the LDA-based pseudo-Hamiltonian led the authors to recommend the use of pseudo-Hamiltonians or pseudopotentials based on many body theory in future calculations. As noted by the authors, this work demonstrated that QMC calculations of solids can provide high accuracies.
Quantum chemistry by random walk: Higher accuracy for Ht
Authors: Anderson J B
Published: November 2023

Although the ground-state Born-Oppenheimer wavefunction of the molecular ion Ht is nodeless and cancellation of positive and negative walkers is not required to treat a node structure, the larger step sizes of “exact” cancellation calculationsa in Green’s function QMC made it the choice for the calculations reported in this paper. In this case the higher efficiencies allowed, for the first time ever, the determination of potential energies for a polyatomic system with microhartree accuracy. The calculations were essentially identical to those described earliera for H2 1 E;t” and linear symmetric H-H-H. The calculated energy for Ht in its equilibrium equilateral triangle configuration of side length 1.6500 bohr was found to be -1.343 835 ± 0.000 001 hartrees.
Quantum :Monte Carlo calculation of the Fe atom
Authors: S L Mitáš
Published: November 2023

A variety of earlier calculations for the third-row transition element iron using argon-core pseudopotentials had failed to give good agreement with experimental measurements of the ionization potential, the electron affinity, and excitation energies. This paper reports QMC calculations which give excellent agreement and which are likely to be the most accurate of any type of calculation for the Fe atom in the 1990s. For these the author used neon-core pseudopotentials (leaving 16 electrons, including 3s and 3p electrons, in the valence space) for both variational and fixednode diffusion QMC calculations. The pseudopotential used was an ab initio pseudopotential tested against all-electron results in limited configuration interaction calculations. Relativistic effects were included. The complete Hamiltonian was modified slightly to facilitate treatment of the nonlocal part of the pseudopotential expression. The trial functions for importance sampling and specifying nodes were linear combinations of Slater determinants along with Jastrow functions. Six low-lying neutral states were examined in addition to the ground states of the anion and cation. Recovery of valence correlation energies was 0.6 to 0.8 hartrees for all of these, typically 0.1 to 0.2 hartrees greater that of coupled cluster and configuration interaction calculations. Most impressive is the agreement with experimental measurements: ionization potential, DQMC 7.67(6) eV vs. expt. 7.87 eV; 5  D  3  F transition, DQMC 4.24(9) eV vs. expt. 4.07 eV; 5  D  5  F transition, DQMC 0.84(6) eV vs. expt.
Quantum Monte Carlo of nitrogen: Atom, dimer, atomic and molecular solids
Authors: Mitáš L , Martin R M
Published: November 2023

This paper reports the first electronic structure calculations of any type to treat nitrogen solids with very high accuracies accuracies similar to those obtained for atomic and molecular nitrogen. It also reports the first estimate of a band gap for a solid using QMC methods. The calculations were carried out as valence-only calculations with abinitio nonlocal pseudopotentials using diffusion QMC with importance sampling. For the atomic and molecular solids, compressed /213 and Pa3 crystals, cubic structures with eight atoms in a simulation cell and periodic boundary conditions were used. The trial functions for importance sampling were composed of Slater determinants from Gaussian92, Garness, and Crystal92 calculations along with correlation terms optimized in variational QMC calculations. For the atom three different forms for the pseudopotential were found to give identical correlation energies. The pseudopotential of Stevens, Basch, and Kraussa was subsequently used for all systems. For the dimer the DQMC calculations gave a binding energy within 2% of the experimental binding energy . . . a result “comparable with the accuracy of the most extensive quantum chemistry calculations.” For the two compressed solids the binding energies were found to be 3.3(1) and 3.4(1) eV, values much higher than those of Hartree-Fock calculations, 0.01 and 0.25 eV, and lower than those of LDA density functional calculations at 4.45 and 4.75 eV. No experimental energies were available.
Improved quantum Monte Carlo calculation of the ground-state energy of the hydrogen molecule
Authors: Chen B , Anderson J B
Published: November 2023

Since the ground state of the hydrogen molecule is nodeless, it is easily treated by “exact” QMC cancellation methods in a full 12-dimensional, four-particle coordinate space to yield the groundstate energy without the use of the Born-Oppenheimer approximation. Following the discovery of quantum mechanics, the hydrogen molecule has been the frequent target of theoretical predictions of increasing accuracy, in parallel with experimental measurements of its ionization potential and its dissociation energy, also with increasing accuracy. The calculations reported in this paper took advantage of importance sampling and efficient cancellation to yield an accuracy of better than 1 microhartree in the total energy, with the result of -1.164 0237 ± 0.000 0009 hartrees, a value slightly less accurate than analytic variational calculations at the time. Expressed as the dissociation energy and corrected for relativistic and radiative effects, the QMC result in 36117.84 ± 0.20 cm-1, which may be compared with the experimental valuea of 36118.11 cm-1 determined in 1992.
Introduction
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

The subject of this work is the description of physical systems that possess symmetry exhibit chaotic behavior These systems are described by sets of nonlinear ordinary differential equations. The existence of a symmetry has a profound and far-reaching impact on the properties of the solutions of such equations. The presence of symmetry sometimes complicates and sometimes simplifies our understanding of the physical processes that generate chaotic behavior. In the following sections we describe a number of physical systems with symmetry and present a number of mathematical reasons for studying the relation between symmetries and dynamical systems. We also outline the structure and contents of this book.
Image Dynamical Systems
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

Dynamical systems with symmetry can be mapped to closely related systems without symmetry by introducing transformations that are diffeomorphisms almost everywhere. For such mappings the Jacobian of the transformation is nonsingular almost everywhere. It vanishes on a set of measure zero. In this chapter and the next we introduce the relation between equivariant dynam­ical systems and their images, invariant dynamical systems. The coordinates for the equivariant dynamical system are capitalized, those for the invariant dynamical sys­tem are in lowercase letters.
Three-Fold and Four-Fold Covers
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

In this chapter we will describe three-fold and four-fold covers of image dynamical systems. It is possible to numerically generate strange attractors equivariant under the groups C3 , C4 , and V4 . This is done by lifting image dynamical systems without symmetry. In this section we present three canonical image dynamical systems. One is the Rossler attractor, the other two are images of equivariant dynamical systems. The strange attractors for all three dynamical systems without symmetry are shown in Fig. 6.1.
Covering Dynamical Systems
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

The image dynamical system is the foundation upon which all its Q-equivariant covers are built. It is therefore useful-even imperative-to understand in detail all the most important properties of the image dynamical system. The variables that describe the image dynamical system are (u1, u2, ... , UN). These represent points in an N -dimensional phase space. The phase space is often taken to be ]RN . However, the phase space is more generally an N-dimensional manifold MN. We have already seen that the phase spaces for typical image dynamical systems are N dimensional algebraic submanifolds of IRK defined by K - N algebraic constraints called syzygies.
Symmetries Due to Symmetry
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

Schur’s lemmas play a fundamental role in the analysis of the representations of groups and algebras. They are powerful tools for the construction of a complete set of irreducible representations of any group. We exploit the first of Schur ‘s lemmas to detem1ine the most general form that a dynamical system in JR.N can have if it is equivariant under a group g. The structure is determined by the representations rDef (Q) and rRing (Q) . Although only one of the two lemmas of Schur is needed for our purposes, the lemmas are so powerful, so pretty, and so similar, we provide the proofs of both. The proofs are summarized in Figs. 14.1 and 14.2. The essential idea is that groups act on linear vector spaces through linear trans formations. These are determined by matrices. The matrices are the group representations. Definition: A subspace is invariant if it is mapped into itself under all linear transformations associated with the matrix representation of the group.
Ancient Greek Scholarship
Authors: Dickey Eleanor
Published: November 2023

Ancient greek sholarship constitutes a precious resource for classicists, but one that is underutilized because graduate students and even mature scholars lack familiarity with its conventions. The peculiarities of scholarly Greek and the lack of translations or scholarly aids often discourages readers from exploiting the large body of commentaries, scholia, lexica, and grammatical treatises that have been preserved on papyrus and via the manuscript tradition. Now, for the first time, there is an introduction to such scholarship that will enable students and scholars unfamiliar with this material to use it in their work. Ancient Greek Scholarship includes detailed discussion of the individual ancient authors on whose works scholia, commentaries, or single-author lexica exist, together with explanations of the probable sources of that scholarship and the ways it is now used, as well as descriptions of extant grammatical works and general lexica. These discussions, and the annotated bibliography of more than 1200 works, also include evaluations of the different texts of each work and of a variety of electronic resources. This book not only introduces readers to ancient scholarship, but also teaches them how to read it. Here readers will find a detailed, step-by-step introduction to the language, a glossary of over 1500 grammatical terms, and a set of more than 200 passages for translation, each accompanied by commentary. The commentaries offer enough help to enable undergraduates with as little as two years of Greek to translate most passages with confidence; in addition, readers are given aids to handling the ancient numerical systems, understanding the references found in works of ancient scholarship, and using an apparatus criticus (including an extensive key to the abbreviations used in an apparatus). Half the passages are accompanied by a key, so that the book is equally suitable for those studying on their own and for classes with graded homework.
Energy Characteristics and Control Strategies
Authors: Robertson Leon S
Published: November 2023

Injury epidemiologists should have at least a rudimentary understanding of the forms of energy involved in injury and the tolerance of human beings to exposure to energy. Since incidence and severity of injury are usually classified by vehicles or circumstances rather than by agent, the precise numbers attributable to a given agent are not available in many data sets. At the turn of the century in the United States, about 58 percent of fatal injuries and 62 percent of hospitalized injuries were attributed to motor vehicles, falls, and firearms (Finkelstein et al., 2006).
Local Injury Surveillance
Authors: Robertson Leon S
Published: November 2023

Some states and other entities have established systems of surveillance for particular types of injury outcome, such as spinal cord or brain injuries. For example, in response to congressional mandate, the Centers for Disease Control and Prevention (CDC) funds traumatic brain injury surveillance in 12 U.S. states (Langlois et al., 2005). Focus on such injuries may be useful because of the effects on mortality (e.g., Salassie et al., 2005) and the lives of the injured and those who must care for them, as well as the costs. Surveillance of injury from activities such as skiing is obviously not applicable to areas where the activity is not done but provides useful insights into prevention in areas where it is relevant (Xiang et al., 2004).
The Use and Abuse of Causal Analysis
Authors: Robertson Leon S
Published: November 2023

On an exam in a course on the use of epidemiology for injury control, the author asked the students, “What is wrong with this statement: ‘Alcohol is consistently associated with unintentional injury deaths; therefore, alcohol abuse must be reduced to lower rates of death and disability due to injury’?” Only 1 in 20 came close to writing a correct answer. Yet in the course, numerous instances to the contrary were noted. Motor vehicle injuries have been reduced, for example, by improved vehicle crashworthiness, lighting dark sections of road, and channeling traffic, without reducing alcohol or otherwise changing drivers.
Research Designs and Data Analysis
Authors: Robertson Leon S
Published: November 2023

The choice of study designs to investigate a given set of factors is affected by numerous considerations. What is the unit of analysis (people, vehicles, environments)? In what population should the study be conducted? To what population of people, vehicles, or environments will the results be generalized? What measurements of the factors are available or could be obtained? How reliable and valid are the measurements? Can the data be collected without violating ethical guidelines? How can the study isolate the effects of given factors independent of, or in combination with, other relevant factors? How much time will be needed to complete the study? How much will the study cost? No single study will specify degree of effect of all the factors in a causal model. The goal of a prevention-oriented research project is to specify the extent to which injury or injury severity would be reduced by changing a given factor hypothesized to contribute to the injury or severity, other things being equal. A study design should be chosen that eliminates or minimizes the probability that factors other than the changeable factor of interest somehow bias that estimate. See appendix 9-1 for an example of elimination of alternative explanations.
Evaluation of Programs to Change Human Factors Voluntarily
Authors: Robertson Leon S
Published: November 2023

Programs aimed at changing human factors without resort to law are of three types: (1) attempts to change abilities, such as driving skills among drivers and strength and gait among the elderly; (2) attempts to change behaviors that are presumed or known to contribute to risk, such as storage of materials hazardous to children, violent responses to peer challenges, speeding, and alcohol use; and (3) attempts to increase use of protection, such as child restraints, seat belts, motorcycle and bicycle helmets, and certain sports equipment. Various techniques are used to promote behavioral change-information and motivation by education in schools and media campaigns, outreach programs for children and the elderly, behavior modification using incentives or punishments, and enhancement of perception.
Evaluation of Laws and Rules Directed at Individual Behavior
Authors: Robertson Leon S
Published: November 2023

Governments sometimes impose laws applicable to the public or segments of the public in an attempt to change behavior to reduce injuries. Also, in organizations such as corporations, sports leagues, and summer camps, there are often formal rules that proscribe or require particular behaviors thought to be related to injury risk. For example, changing certain rules for sports events has been found to strongly reduce injury rates (Roberts et al., 1996). This chapter focuses primarily on the use of epidemiological data in evaluating the effects of laws and their enforcement on injury incidence or severity.
Evaluation of Postinjury Treatment and Rehabilitation
Authors: Robertson Leon S
Published: November 2023

Research questions regarding the treatment of injuries and rehabilitation range from the profound issue of who survives trauma to the more mundane estimates of costs and payment. Only a small minority of physicians and other professionals delivering care to traumatized patients conduct research, but because of their familiarity of the treatment and rehabilitation scene, these persons are usually in a better position to ask good research questions than are academic epidemiologists (Findley, 1989).
Injury Epidemiology and Economics
Authors: Robertson Leon S
Published: November 2023

In addition to the humane reasons for injury control, there are huge cost savings to be realized. The issue in cost of injury studies is not just cost per se but also the extent of cost savings realizable by reduction of incidence and severity (Currie et al., 2000). Cost is usually measured in “human capital cost” and does not include the intangible toll in pain, grief, social disruption, and disorganization. It also does not include substitution of economic productivity with care giving to the injured by family members, which greatly reduces public costs (Rice and MacKenzie, 1989; Leigh et al., 2000; Finkelstein et al. 2006).
On Second (and Third and Fourth) Thought: Obsessions and Compulsions
Authors: Kant Jared Douglas , Franklin Martin , Andrews Linda Wasmer
Published: November 2023

When I was much younger, long before the symptoms manifested, I was a happy kid. At least, most of my memories from being younger seem to center around a knowledge that I was somehow special.

I was one of those kids who would study an anthill instead of playing soccer. Rather than longing for the prefabricated action figures and toys that were advertised on television, I would construct cities out of building blocks. I could be entertained for whole days this way. As I got older, the things I built began to reflect the trouble that was to come. Like a horse that can smell rain, I could smell the storm.
Adventures in Daily Living: OCD at Home and School
Authors: Kant Jared Douglas , Franklin Martin , Andrews Linda Wasmer
Published: November 2023

Locker rooms are full of disease, or at least they feel that way. So it’s no surprise that one of the first times my OCD manifested itself at school involved a locker room. My classmates seemed to like it there. They ran around the room, squirting bottles of cologne at each other, telling tasteless jokes, and pelting each other with dirty clothes like a cotton snowball fight. But I hated that room. The smell of feet, sweat, and body odor was enough to make me gag.

Some days, the soccer team brought a sizable chunk of the field inside on their cleats, and the dank smell of earth added to the nauseating odors. I would look down and think about how many people had traveled that dirty floor. No matter how much the janitor cleaned, I was still anxious. To protect myself, I usually wore boots, which became my foot armor.
Introduction: Continually Creating New Inner Psychological Space: An Ongoing Process of Mindfulness
Authors: Wicks Robert J
Published: November 2023

Sometimes a brief interchange with a patient may leave us speechless. One such instance for me happened at the end of a long course of psychotherapy with an adult client who was sexually abused as a child. As I began to ask a series of wrap-up questions, my goal was to elicit summative answers or paradigms she had learned that would provide internal guidance for her in the future. Then, when she entered darkness again–for darkness comes again and again for all who are self-aware and compassionate, as she was–she would know how to better handle it. I also wanted her to take credit for all the hard work she had done in therapy. At least those were my intentions. Her response to my intervention, though, would leave me with a lesson worth remembering.
Daily Debrieﬁng: Mindfulness and Positive Psychology as an Integral Part of the Clinician’s Ongoing Self-Reflective Process
Authors: Wicks Robert J
Published: November 2023

If acute and chronic secondary stress are to be limited and one’s personal and professional well-being are to be enhanced, then self-knowledge and the enlightened behavior that it should give rise to are not a nicety in clinical practice; they must be a given. Personal discipline and self-management are essential parts of clinical practice. This is referred to as “self-regulation.”
Minding the Gap: Race, Ethnicity, Achievement, and Cultural Meaning
Authors: Carter Prudence L
Published: November 2023

A few years before I embarked on the study discussed in these pages, I traveled across eight states to recruit academically talented students for admission to Brown University. I visited an array of schools, from those in Detroit’s inner city to the upper-middle-class suburban enclaves of Shaker Heights, Ohio, to the de facto segregated schools of Memphis, Tennessee, to the vast campuses of wealthy prep schools along the eastern seaboard in Massachusetts. The students I encountered varied, too, from the top achievers to the students who loitered in the hallways and cut classes. Often those in the latter group pointed me in the direction of either the principal’s office or the counselor’s office or would even escort me to the door.
New “Heads” and Multicultural Navigators
Authors: Carter Prudence L
Published: November 2023

Increasing the school attachment and achievement of low-income African American and Latino students requires both improved academic skills and resources—economic, cultural, and social. Most will have some difficulty making their way out of poverty if they lack resourceful social connections— or social capital. “Capital”—there is that metaphor again, but in this instance it refers to the resources that come with personal connections and social networks (Bourdieu 1986; Coleman 1988; Loury 1976; Portes 1998).