Summation Scheme Research Articles

A Beam Summation Representation for 3-D Radiation From a Line Source Distribution

We introduce a Gaussian beams summation (GBS) scheme for radiation from a source distribution along a line in a 3-D configuration. The expansion is based on a discrete phase-space decomposition of the source distribution along the source axis, thus describing the field as a sum of Gaussian beams (GBs) emerging from a discrete set of points along the axis, at a discrete set of directions in a polar coordinate system. Applications to GBS representation of fields diffracted by edges is 3-D configurations are discussed.

Limit theorems for a nonhomogeneous Ornstein-Uhlenbeck process

We describe a construction of a summation scheme with replacements of random variables. As the limit, we obtain a time nonhomogeneous generalization of the Ornstein-Uhlenbeck process. This generalization is described by a transform of Lamperti type in which an arbitrary continuous monotone function is used instead of the exponential one. Bibliography: 16 titles.

Seismic hazard on the French Riviera: observations, interpretations and simulations

SUMMARY We present here a detailed analysis of a seismic data set recorded by a dense seismological network installed over 6 months in the southeast of France. This experiment was set-up at the boundary between the Ligurian basin and the southern subalpine thrust belt (the Nice arc), which is a complex tectonic region that undergoes low to moderate seismicity. We recorded more than 500 microearthquakes, among which 348 occurred exactly in the centre of the network during a very active seismic sequence that lasted mainly over 2 months. We performed an absolute location of all of the events and calculated the magnitudes. Then we applied a cross-correlation technique to gather similar events and to relocate relatively few of them. This method revealed a very clear alignment of 19 events in a direction N120 ◦ oblique to the N20 ◦ general trend of seismicity. Focal mechanisms were determined for the four largest events and composite solutions for 32 smaller ones. Both the alignments of the earthquakes and the focal solutions revealed that two oblique segments of the fault were activated during the crisis. The main segment (8 km long) that was oriented NNE with a left-lateral strike slip movement is called the Blausasc fault. Taking into account the tectonic evolution and the relationships between surface structures and the distribution of earthquakes, and through a paleaoreconstruction of the tectonic evolution, we propose that the Blausasc fault is the hidden root of the Peille-Laghet fault, which has a mapped length of at least 15 km. The smaller segment (0.6 km long) that was activated during the crisis could be interpreted as an antithetic Riedel fracture. The active Blausasc fault is located in a densely populated zone, at only 10 km from the crowded cities of Monaco and Nice. It is thus particularly interesting to analyse it for hazard assessment. In the last section, we present a simulation that is aimed at predicting what the ground motion in the city of Nice would be like if an earthquake of magnitude 5.7 occurs on this fault. For this, we used the recordings of the largest event of the seismic sequence (Ml = 3.2) and an empirical Green’s function summation scheme to simulate the ground motion at two stations situated in urban environments. The values obtained show that especially on soft soil sites, the effects of such an earthquake would be considerable in the city of Nice.

DAMPING OF ULTRASOFT FERMIONS IN FINITE TEMPERATURE QED

We calculate the fermion damping rates to second order in powers of the external momentum in the context of QED at finite temperature using the hard-thermal-loop summation scheme. We find the coefficients of zeroth and first orders finite whereas that of second-order logarithmically infrared sensitive. The calculation is done in covariant gauge and the result is independent of gauge fixing.

Shape optimization of a rigid inclusion in a shear-loaded elastic plane

A rigid inclusion perfectly embedded in a thin plate is considered as a two-dimensional elastostatic composite structure to solve the inverse problem of finding the inclusion shape around which the local maximum of the von Mises equivalent stresses attain the global minimum under shear loading at infinity. Absent optimality preconditions such as the equistress principle for bulk-type loading, a fast and accurate assessment of a given shape is developed by combining complex-valued series expansions with a new infinite summation scheme. This approach to solving the direct problem is then included into a genetic algorithm optimization over the set of shapes obtained from a circle by a finite-term conformal mapping with square symmetry. Compared to a circular inclusion, the stresses may thus be lowered by 15‐25%, depending on the Poisson’s ratio of the plate. The numerical results presented allow us to conjecture that the von Mises stresses around the optimal shape are uniform. The inclusion that minimizes the induced energy increment is also identified by the same approach. Both shapes appear to be very similar, though not identical.

Global Representation of Periodic Solution for Vibro-Impact System (Explicit Form Using Implicit Impact Time Parameters)

This paper gives exact and global representation of all periodic solutions appearing in 1-DOF vibro-impacting system. The previously derived form of periodic solution by authors can be represented as superposition of periodic solution for linearized system and finite number of continuous periodic functions which is evaluated by equivalent impulsive responses caused by nonlinearity. However, it is not practical, because determination of impact velocities characterising periodic solutions includes infinite product and summation procedures in the form. To avoid this difficulties, relation between two types of representation of periodic solutions, traditional and newly one by authors, is reconsidered. As a result, it is clarified that transcendental equations for periodic connecting conditions in these representations can be mutually transformed by a pair of linear transformations. According to apply this fact, the finite product and summation scheme determining impact velocities of periodic solutions is derived.

Quantal cumulant dynamics: General theory

The authors have derived coupled equations of motion of cumulants that consist of a symmetric-ordered product of the position and momentum fluctuation operators in one dimension. The key point is the utilization of a position shift operator acting on a potential operator, where the expectation value of the shift operator is evaluated using the cumulant expansion technique. In particular, the equations of motion of the second-order cumulant and the expectation values of the position and momentum operators are given. The resultant equations are expressed by those variables and a quantal potential that consists of an exponential function of the differential operators and the original potential. This procedure enables us to perform quantal (semiclassical) dynamics in one dimension. In contrast to a second-order quantized Hamilton dynamics by Prezhdo and Pereverzev which conserves the total energy only with an odd-order Taylor expansion of the potential [J. Chem. Phys. 116, 4450 (2002); 117, 2995 (2002)], the present quantal cumulant dynamics method exactly conserves the energy, even if a second-order approximation of the cumulants is adopted, because the present scheme does not truncate the given potential. The authors propose three schemes, (i) a truncation, (ii) a summation of derivatives, and (iii) a convolution method, for evaluating the quantal potentials for several types of potentials. The numerical results show that although the truncation method preserves the energy to some degree, the trajectory obtained gradually deviates from that of the summation scheme after 2000 steps. The phase space structure obtained by the truncation scheme is also different from that of the summation scheme in a strongly anharmonic region.

IR renormalization of general effective actions and Hawking flux in 2D gravity theories

The infrared problem of the effective action in 2D is discussed in the framework of the covariant perturbation theory. The divergences are regularized by a mass and the leading term is evaluated up to the third order of perturbation theory. A summation scheme is proposed which isolates the divergences from the finite part of the series and results in a single term. The latter turns out to be equivalent to the coupling to a certain classical external field. This suggests renormalization by factorization.

Computation of methodology-independent ionic solvation free energies from molecular simulations. II. The hydration free energy of the sodium cation

The raw ionic solvation free energies computed from atomistic (explicit-solvent) simulations are extremely sensitive to the boundary conditions (finite or periodic system, system shape, and size) and treatment of electrostatic interactions (Coulombic, lattice sum, or cutoff based) used during these simulations. In the present article, it is shown that correction terms can be derived for the effect of (A) an incorrect solvent polarization around the ion due to the use of an approximate (not strictly Coulombic) electrostatic scheme; (B) the finite size or artificial periodicity of the simulated system; (C) an improper summation scheme to evaluate the potential at the ion site and the possible presence of a liquid-vacuum interface in the simulated system. Taking the hydration free energy of the sodium cation as a test case, it is shown that the raw solvation free energies obtained using seven different types of boundary conditions and electrostatic schemes commonly used in explicit-solvent simulations (for a total of 72 simulations differing in the corresponding simulation parameters) can be corrected so as to obtain a consistent value for this quantity.

Computation of methodology-independent ionic solvation free energies from molecular simulations. I. The electrostatic potential in molecular liquids

The computation of ionic solvation free energies from atomistic simulations is a surprisingly difficult problem that has found no satisfactory solution for more than 15 years. The reason is that the charging free energies evaluated from such simulations are affected by very large errors. One of these is related to the choice of a specific convention for summing up the contributions of solvent charges to the electrostatic potential in the ionic cavity, namely, on the basis of point charges within entire solvent molecules (M scheme) or on the basis of individual point charges (P scheme). The use of an inappropriate convention may lead to a charge-independent offset in the calculated potential, which depends on the details of the summation scheme, on the quadrupole-moment trace of the solvent molecule, and on the approximate form used to represent electrostatic interactions in the system. However, whether the M or P scheme (if any) represents the appropriate convention is still a matter of on-going debate. The goal of the present article is to settle this long-standing controversy by carefully analyzing (both analytically and numerically) the properties of the electrostatic potential in molecular liquids (and inside cavities within them). Restricting the discussion to real liquids of "spherical" solvent molecules (represented by a classical solvent model with a single van der Waals interaction site), it is concluded that (i) for Coulombic (or straight-cutoff truncated) electrostatic interactions, the M scheme is the appropriate way of calculating the electrostatic potential; (ii) for non-Coulombic interactions deriving from a continuously differentiable function, both M and P schemes generally deliver an incorrect result (for which an analytical correction must be applied); and (iii) finite-temperature effects, including intermolecular orientation correlations and a preferential orientational structure in the neighborhood of a liquid-vacuum interface, must be taken into account. Applications of these results to the computation methodology-independent ionic solvation free energies from molecular simulations will be the scope of a forthcoming article.

Microwave Electrothermal Thruster Chamber TemperatureMeasurements and Performance Calculations

The microwave electrothermal thruster (MET) uses microwave frequency energy to create and sustain a resonant cavity plasma to heat a propellant. A 2.45-GHz aluminum cylindrical thruster with converging copper-alloy nozzles was used for this study. A spectroscopic system was used to collect light emitted through a window in the plasma chamber. A Schumann-Runge oxygen-emission model was developed assuming an anharmonically vibrating, nonrigid rotating oxygen molecule. The commercially available LIFBASE software was used to model the ionized molecular nitrogen first negative system emission from the nitrogen plasmas. Experimental data were compared to the temperature-dependent models using least-squared difference summation schemes. Oxygen rotational temperatures of 2000 K and ionized nitrogen rotational temperatures of 5500 K were measured. These measurements were nearly constant for all chamber pressures and investigated absorbed specific powers. CEA2 code equilibrium thermochemical calculations show the relationship among enthalpy addition, temperature, and specific impulse for realistic operating conditions. Nitrogen was found to be an excellent choice as a propellant or propellant component, whereas oxygen was found to be a poor choice because of the temperatures achieved for the respective gases in the MET chamber.

An Efficient Methodology for Fatigue Reliability Analysis for Mechanical Components

This paper presents an efficient methodology to solve a fatigue reliability problem. The fatigue failure mechanism and its reliability assessment must be treated as a rate process since, in general, the capacity of the component and material itself changes irreversibly with time. However, when fatigue life is predicted using the S-N curve and a damage summation scheme, the time dependent stress can be represented as several time-independent stress levels using the cycle counting approach. Since, in each counted stress cycle, the stress amplitude is constant, it becomes a random variable problem. The purpose of this study is to develop a methodology and algorithm to solve this converted random variable problem by combining the accumulated damage analysis with the first-order reliability analysis (FORM) to evaluate fatigue reliability. This task was tackled by determining a reliability factor using an inverse reliability analysis. The theoretical background and algorithm for the proposed approach to reliability analysis will be introduced based on fatigue failure modes of mechanical components. This paper will draw on an exploration of the ability to predict spectral fatigue life and to assess the corresponding reliability under a given dynamic environment. Next, the process for carrying out this integrated method of analysis will be explained. Use of the proposed methodology will allow for the prediction of mechanical component fatigue reliability according to different mission requirements.

A Two-Stage Method for Ground-Motion Simulation Using Stochastic Summation of Small Earthquakes

We present a two-stage method to simulate the ground motions produced by an earthquake by using stochastic summation of small earthquakes. In this method, identical small earthquakes are multiplied by a scaling factor and summed together with time delays randomly distributed, during the two stages, over the source duration. The summation scheme is characterized by four fundamental parameters: the number of summed small earthquakes, the scaling factor, and both probability densities of time delays used in the first and second stages. By a proper choice of these parameters, this method generates a large number of synthetic time histories that, on average, agree exactly with the ω −2 model in the whole frequency band. The produced time histories are sufficiently realistic and different from each other to be associated with a multitude of rupture processes that could happen during an earthquake. However, because the extended target fault is approximated by a point source, this method does not take into account possible directivity effects and is not appropriate to simulate ground motions for near-source sites. We test this method on the Oaxaca earthquake (1999, M w 7.5, Mexico) at regional distances and on the two mainshocks of the Umbria Marche crisis (1997, M w 5.7 and M w 6.0, Italy) at local distances. We found that the simulated ground motions fit the observed data well, both in time and in frequency domains. Within simulation context, only specification of seismic moment and stress drop is required for the target event. Because the magnitude and then the seismic moment are necessarily specified, the stress drop plays a major role in ground-motion simulation.

Robust and accurate method for free-energy calculation of charged molecular systems

A new approach is presented to eliminate the problem of creation and/or annihilation of atoms in free-energy calculations of charged molecular systems. The method employs a damping potential in the Ewald summation scheme, which is an exact solution of the electrostatics for three-dimensional periodic systems. The proposed method enables entire molecules to be mutated from a noninteracting (ideal) state in an efficient and robust way, thus providing a means by which accurate absolute free energies of structurally complex molecules can be determined. This methodology will enable chemical and phase equilibria to be determined for large molecular species with significant charge distributions, e.g., biomolecules and drugs.

Equation-of-motion coupled-cluster study on exciton states of polyethylene with periodic boundary condition

Equation-of-motion coupled cluster with singles and doubles (EOM-CCSD) method has been applied to exciton states of polyethylene using ab initio crystal Hartree-Fock method with one-dimensional periodic boundary condition. Full transformation of two-electron integrals from atomic-orbital basis to crystal-orbital basis has been performed for EOM-CCSD calculations. In order to make transformed integrals to have correct properties of translational symmetry, a lattice summation scheme has been proposed. The EOM-CCSD excitation energies have been obtained for the lowest singlet and triplet exciton states of polyethylene. The excitation energies converge with system size much faster than oligomer calculations using n-alkanes. Quasiparticle energy-level calculations by second-order many-body perturbation theory and by solving the inverse Dyson equation have also been performed to obtain exciton binding energies. Basis set dependencies on excitation energy, quasiparticle band gap, and exciton binding energy have been investigated. At the 6-31+G level, the excitation energy of the lowest singlet-exciton state and its binding energy are calculated to be 8.1 and 3.2 eV, respectively. The calculated excitation energy is well comparable with the corresponding experimental value, 7.6 eV.

A Low-Power Systolic Array Architecture for Block-Matching Motion Estimation

SUMMARY This paper proposes a low-power systolic array architecture for a block-matching motion estimation processor IP for portable and high-resolution video applications. The architecture features a ringconnected processing element (PE) array to reduce both computation cycles and memory access cycles at the same time, allowing lower power characteristics. The feature of low memory access cycles allows concurrent operation of a half-pel processing unit with no extra cache. Furthermore, the architecture allows various summation schemes for absolute difference values. For that reason, it is applicable to various video coding modes such as the adaptive field/frame mode in MPEG2 and multiple macroblock mode in H.264. When the architecture is introduced to a design of a MPEG2 MP@HL motion estimation processor VLSI, the power consumption of the VLSI is reduced by 45–73% in comparison to cases with conventional

Nonconvergence of the Solute Potential in an Infinite Solvent and Its Implications in Continuum Models

The conventional way of computing the electrostatic potential at a solute charge in water, Φ ∞ (Q), includes contributions from an infinite number of water molecules. Φ ∞ (Q) varies by as large as 20 kcal/mol depending onthe summation scheme or dielectric boundary. Here we show that Φ ∞ (Q) is nonconvergent. By computing Φ(Q) as a sum of contributions from an interior, explicit water region, an intermediate, continuum water region, and an exterior, explicit water region using the same summation scheme, the two implicit water/ explicit water surface effects cancel and Φ(Q) converges to a single value, independent of the summation scheme.

Fourth-order perturbative extension of the single-double excitation coupled-cluster method

Fourth-order many-body corrections to matrix elements for atoms with one valence electron are derived. The obtained diagrams are classified using coupled-cluster-inspired separation into contributions from n-particle excitations from the lowest-order wavefunction. The complete set of fourth-order diagrams involves only connected single, double, and triple excitations and disconnected quadruple excitations. Approximately half of the fourth-order diagrams are not accounted for by the popular coupled-cluster method truncated at single and double excitations (CCSD). Explicit formulae are tabulated for the entire set of fourth-order diagrams missed by the CCSD method and its linearized version, i.e. contributions from connected triple and disconnected quadruple excitations. A partial summation scheme of the derived fourth-order contributions to all orders of perturbation theory is proposed.

Asymmetric laplace laws and modeling financial data

Asymmetric Laplace laws form a subclass of geometric stable distributions, the limiting laws in the random summation scheme with a geometric number of terms. Among geometric stable laws, they play a role analogous to that of normal distribution among stable Paretian laws. However, with steeper peaks and heavier tails than normal distribution, asymmetric Laplace laws reflect properties of empirical financial data sets much better than the normal model. Despite heavier than normal tails, they have finite moments of any order. In addition, explicit analytical forms of their one-dimensional densities and convenient computational forms of their multivariate densities make estimation procedures practical and relatively easy to implement. Thus, asymmetric Laplace laws provide and interesting, efficient, and user friendly alternative to normal and stable Paretian distributions for modeling financial data. We present an overview of the theory of asymmetric Laplace laws and their applications in modeling currency exchange rates.

LEVY-STABLE DISTRIBUTIONS REVISITED: TAIL INDEX > 2 DOES NOT EXCLUDE THE LEVY-STABLE REGIME

Power-law tail behavior and the summation scheme of Levy-stable distributions is the basis for their frequent use as models when fat tails above a Gaussian distribution are observed. However, recent studies suggest that financial asset returns exhibit tail exponents well above the Levy-stable regime (0 < α ≤ 2). In this paper, we illustrate that widely used tail index estimates (log–log linear regression and Hill) can give exponents well above the asymptotic limit for α close to 2, resulting in overestimation of the tail exponent in finite samples. The reported value of the tail exponent α around 3 may very well indicate a Levy-stable distribution with α ≈ 1.8.