Analytical Formulas Research Articles

Fully Dynamic G3W2 Self-Energy for Finite Systems: Formulas and Benchmark.

Over the years, Hedin's GW self-energy has been proven to be a rather accurate and simple approximation to evaluate electronic quasiparticle energies in solids and in molecules. Attempts to improve over the simple GW approximation, the so-called vertex corrections, have been constantly proposed in the literature. Here, we derive, analyze, and benchmark the complete second-order term in the screened Coulomb interaction W for finite systems. This self-energy named G3W2 contains all the possible time orderings that combine 3 Green's functions G and 2 dynamic W. We present the analytic formula and its imaginary frequency counterpart, with the latter allowing us to treat larger molecules. The accuracy of the G3W2 self-energy is evaluated on well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. Its link with the simpler static approximation, named SOSEX for static screened second-order exchange, is analyzed, which leads us to propose a more consistent approximation named 2SOSEX. In the end, we find that neither the G3W2 self-energy nor any of the investigated approximations to it improve over one-shot G0W0 with a good starting point. Only quasi-particle self-consistent GW HOMO energies are slightly improved by addition of the G3W2 self-energy correction. We show that this is due to the self-consistent update of the screened Coulomb interaction, leading to an overall sign change of the vertex correction to the frontier quasiparticle energies.

Effect of the degree of an initial mutant in Moran processes in structured populations.

We study effects of the mutant's degree on the fixation probability, extinction, and fixation times in Moran processes on Erdös-Rényi and Barabási-Albert graphs. We performed stochastic simulations and used mean-field-type approximations to obtain analytical formulas. We showed that the initial placement of a mutant has a significant impact on the fixation probability and extinction time, while it has no effect on the fixation time. In both types of graphs, an increase in the degree of an initial mutant results in a decreased fixation probability and a shorter time to extinction. Our results extend previous ones to arbitrary fitness values.

A compressible degree of freedom as a means for improving the performance of heaving wave energy converters

A compressible degree of freedom (CDOF) can provide a means of improving wave power capture by adjusting the system stiffness and providing short-term energy storage, reducing or even eliminating the requirement for two-way electrical power flow as is typical in wave energy control systems. With the overarching goal of investigating and facilitating the application of a CDOF within wave energy converter (WEC) design, this paper focuses primarily on the case of heaving buoys, and on using air volumes to provide the compressible spring effect. Analytical formulae for the natural frequencies are used to gain fundamental insights into the main design considerations. Numerical methods are used to augment this understanding, helping to elucidate the effect of geometric parameters and scale on the required compressible volumes and the efficacy of the CDOF. Beginning with the WaveBot WEC, case studies are then used to demonstrate the design drivers in a more applied context, as well as highlighting the benefit of a self-reacting power take-off (PTO) system. The assimilation of these results with the findings of a detailed literature review culminates in a set of recommendations (or guiding principles) for designing a compressible degree of freedom to improve the performance of a heaving wave energy converter. These recommendations apply both to full-scale WECs, and to the design of scale physical modelling studies for validating the analytical and numerical models.

Excitonic instability towards a Potts-nematic quantum paramagnet

Magnetic frustration can lead to peculiar magnetic orderings that break a discrete symmetry of the lattice in addition to the fundamental magnetic symmetries (i.e., spin rotation invariance and time-reversal symmetry). In this work, we focus on frustrated quantum magnets and study the nature of the quantum phase transition between a paramagnet and a magnetically ordered state with broken threefold (Z3) crystal rotation symmetry. We show that the transition can occur in two stages, giving rise to an intermediate nematic phase in which rotation symmetry is broken but the system remains magnetically disordered. Since the nematic transition is described by the three-state Potts model, the intermediate phase is a Z3 Potts-nematic phase. Our prediction of the existence of a Potts-nematic phase is based on an analysis of bound states formed from two-magnon excitations in the paramagnet, which become gapless while single-magnon excitations remain gapped. By considering three different lattice models, we demonstrate a generic instability towards two-magnon bound state formation in the Potts-nematic channel. We present both numerical results and a general analytical perturbative formula for the bound state binding energy similar to BCS theory. We further discuss a number of different materials that realize key features of the model considered, and thus provide promising venues for possible experimental observation. Published by the American Physical Society 2024

Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

We present in this paper four versions of chaotic and hyperchaotic modified nonlinear Schrödinger equations (MNSEs). These versions are hyperchaotic integer order, hyperchaotic commensurate fractional order, chaotic non-commensurate fractional order, and chaotic distributed order MNSEs. These models are regarded as extensions of previous models found in literature. We also studied their dynamics which include symmetry, stability, chaotic and hyperchaotic solutions. The sufficient condition is stated as a theorem to study the existence and uniqueness of the solutions of hyperchaotic integer order MNSE. We state and prove another theorem to test the dependence of the solution of hyperchaotic integer order MNSE on initial conditions. By similar way, we can introduce the previous two theorems for the other versions of MNSEs. The Runge-Kutta of the order 4, the Predictor-Corrector and the modified spectral numerical methods are used to evaluate the numerical solutions for integer, fractional and distributed orders MNSEs, respectively. We calculate numerically using the Lyapunov exponents the intervals of parameters of the purposed models at which hyperchaotic, chaotic and stable solutions are exist. The MNSEs have an important role in many fields of science and technology, such as nonlinear optics, electromagnetic theory, superconductivity, chemical and biological dynamics, lasers and plasmas. The compound synchronization for these chaotic and hyperchaotic models is investigated. We state its scheme using the tracking control technique among three integer commensurate and non-commensurate orders as the derive models and one distributed order as a slave model. We presented and proved a theorem that provides us with the analytical formula for the control functions which are required to achieve compound synchronization. The analytical results are supported by numerical calculations and agreement is found.

Prediction of Torque Capacity in Circular Concrete-Filled Double-Skin Tubular Members under Pure Torsion via Machine Learning and Shapley Additive Explanations Interpretation

The prediction of torque capacity in circular Concrete-Filled Double-Skin Tubular (CFDST) members under pure torsion is considered vital for structural design and analysis. In this study, torque capacity is predicted using machine learning (ML) algorithms, such as Categorical Boosting (CatBoost), Extreme Gradient Boosting (XGBoost), Gradient Boosting Machine (GBM), Random Forest (RF), and Decision Tree (DT), which are employed. The interpretation of the results is conducted using Shapley Additive Explanations (SHAPs). The performance of these ML models is evaluated against two traditional analytical formulas that have been proposed and are available in the literature. Through comprehensive analysis, it is shown that superior predictive capabilities are possessed by the CatBoost and XGBoost models, characterized by high R2 values and minimal mean errors. Additionally, insights into the influence of input features are provided by SHAP interpretation, with an emphasis on key parameters such as concrete compressive strength and steel tube dimensions. The gap between empirical models and ML techniques is bridged by this study, offering engineers a more accurate and efficient tool for CFDST structural design. Significant implications for optimizing CFDST column designs and advancing structural engineering practices are presented by these findings. Directions for future research include the further refinement of ML models and the integration of probabilistic analyses for enhanced structural resilience. Overall, the transformative potential of ML and SHAP interpretation in advancing the field of structural engineering is showcased by this study.

HUBUNGAN POLA ASUH ORANG TUA DAN KARIES TERHADAP OHRQoL PADA ANAK USIA PRASEKOLAH

Background: Children's dental and oral health problems are caused by many factors including parenting style which has a major influence on children's health and development. In Tanah Bumbu District, children aged over 3 years had a proportion of 50.74% with broken teeth, cavities and pain. Dental caries can cause disruption of children's quality of life such as sleep disturbances, eating disorders, loss of concentration in learning, infections, and other dangerous cases in children. Objective: To determine the relationship between parenting style and caries towards OHRQoL in preschool-aged children in Sungai Loban District. Method: Analytic observational study with a cross sectional approach. The sampling technique of non-probability sampling uses quota sampling. The population is all kindergartens in Sungai Loban District, namely 23 kindergartens with 918 children. The minimum sample size was calculated using the correlational analytic formula and the results obtained were 40 respondents consisting of parents and children taken from 3 kindergartens, namely Harapan Bersama Kindergarten, Taman Sari Kindergarten and Tunas Dewata Kindergarten, Sungai Loban District. Results: Most parents' parenting style has a democratic pattern; the average def-t index is 5.5 and belongs to the high category. The results of the Spearman correlation test showed that there was not a relationship between parenting style and OHRQoL (p>0.05). The results of the Spearman correlation test analysis showed that there was a relationship between caries and OHRQoL (p<0.05). Conclusion: There is a relationship between parenting style and caries to OHRQoL in preschool-aged children in Kindergarten in Sungai Loban District.Keywords: Caries, OHRQoL, Parenting, Preschool children.

95 GeV diphoton and bb¯ excesses in the general next-to-minimal supersymmetric standard model

The CMS and ATLAS Collaborations recently published their results searching for light Higgs bosons, using the complete Run 2 data of the LHC. Both reported an excess in the diphoton invariant mass distribution at mγγ≃95.4 GeV with compatible signal strengths. The combined result corresponded to a local significance of 3.1σ. Besides, the mass of the diphoton signal coincided with that of the bb¯ excess observed at the Large Electron Positron. Given the remarkable theoretical advantages of the general next-to-minimal supersymmetric Standard Model, we interpret these excesses by the resonant productions of the singlet-dominated CP-even Higgs boson predicted by the theory. Using both analytic formulas and numerical results, we show that the idea can interpret the excesses by broad parameter space without contradicting current experimental restrictions, including those from the 125 GeV Higgs data, the dark matter relic abundance and direct detection experiments, and the collider searches for supersymmetry and extra Higgs bosons. Although the explanations are scarcely affected by present Higgs data and the LHC search for supersymmetry, the dark matter physics may leave footprints on them. We also survey the other signals of the light Higgs boson at the LHC. Published by the American Physical Society 2024

Toppling stability of partially buried blocks in granite residual soil slope under extremely heavy rainfall condition

The granite blocks are widely distributed on granite residual soil slopes, particularly in the circumstances of humid granitic terrain. These blocks may be partially buried in the soil slope with their long axis upright, which results in complex toppling mechanisms. The objective of this study is to investigate the mechanical stability of the blocks against toppling failure, considering extremely heavy rainfall. Based on the analysis of limit equilibrium, an analytical formulation taking the influence of the block overhang length and the slope inclination into account is proposed to calculate the factor of safety (FOS) of the blocks partially buried in the soil slope. The results of the model experiments verified the performance of this formulation by showing a consistent behavior regarding the stability of the block with the formulation. The FOS obtained from this analytical formulation first shows a trend of increase and then a decrease according to the overhang length of the blocks. Furthermore, the critical overhang length and the critical slope inclination were found related to the toppling failure of potential blocks. The analytical formula obtained in this study can be used to determine the factor of safety against toppling of the single blocks on granite residual soil slopes, which provide a quantitative method for risk evaluation of toppling disaster in granitic regions.

Theoretical investigation of heavy cluster decay from Z=118 and 120 isotopes: A search for an empirical formula in superheavy region

Various decay modes in superheavy nuclei have been of significant interest among which cluster radioactivity has recently gained sizable attention. The α-decay being a predominant decay mode in the superheavy region, the accurate determination of cluster decay half-lives is also crucial in this region as it has tremendous potential to be explored as one of the major decay channels. The usability of the Royer analytical formula [Nuclear Physics A 683 (2001) 182], which is based on the asymmetric fission model, has been investigated for the cluster and α-decay in superheavy region, by comparing it with several other (semi)empirical/analytical formulas. After fitting the formula on around 100 cluster-decay data and around 423 α-decay data, the refitted Royer formula (RRF) is found to be very robust which is able to estimate the cluster decay and α-decay half-lives with good accuracy. In fact, a comparison of the half-lives of both the decay modes using the same formula points towards a substantial chance of heavy cluster (Kr and Sr) decay from various isotopes of Z=118 and 120. Hence, the formula proposed in this study works fairly well for the estimation of cluster decay half-lives in superheavy regions where most empirical formulas fail to match with the half-lives from the various established theories.

Estimating the reproduction number and transmission heterogeneity from the size distribution of clusters of identical pathogen sequences

Quantifying transmission intensity and heterogeneity is crucial to ascertain the threat posed by infectious diseases and inform the design of interventions. Methods that jointly estimate the reproduction number R and the dispersion parameter k have however mainly remained limited to the analysis of epidemiological clusters or contact tracing data, whose collection often proves difficult. Here, we show that clusters of identical sequences are imprinted by the pathogen offspring distribution, and we derive an analytical formula for the distribution of the size of these clusters. We develop and evaluate an inference framework to jointly estimate the reproduction number and the dispersion parameter from the size distribution of clusters of identical sequences. We then illustrate its application across a range of epidemiological situations. Finally, we develop a hypothesis testing framework relying on clusters of identical sequences to determine whether a given pathogen genetic subpopulation is associated with increased or reduced transmissibility. Our work provides tools to estimate the reproduction number and transmission heterogeneity from pathogen sequences without building a phylogenetic tree, thus making it easily scalable to large pathogen genome datasets.

Comparison of the Prediction of the Temperature Profile in an Ohmic Reactor on Different Modeling Scales

AbstractTo predict the temperature profile in an ohmic reactor, it is essential to know the thermal and electrical conductivity of the reactor geometry. Those properties can be considered on different modeling scales: On the detailed (resolved) scale, the reactor consists of a packed bed structure, where the properties of each substance are considered. On the effective (continuum) scale, the packed bed is considered as effective medium with composite properties. In this contribution, we consider both scales and compare the results. Based on the detailed description with a resolved microstructure, the bed's effective thermal and electrical conductivity are computed. These properties are compared with analytical formulas. The electrical conductivity is the basis for the electrical field and the current density. Finally, these properties are used to evaluate the ohmic heating of the packed bed exposed to flow and to compute the temperature distribution within the reactor.

Nonlinear static solution of suspension bridge formulated from a surrogate model with secant stiffness of suspension cables

Explicit analytical solution may offer great convenience for the design and parametric study of suspension bridges. However, the nonlinear behavior of suspension cable exacerbates the difficulty of such theoretical derivation. Adopting secant stiffness is an effective way to deal with the nonlinearity of the cable, but the crux is how to find its deformed state under external loading. The strategy of this study is primarily targeted at a typically arranged three-span suspension bridge. It starts from formulating the basic form of nonlinear stiffness of suspension cables by setting the elastic stiffness and geometric stiffness in series, then a surrogate model is accordingly established with the nonlinear springs to replace the mid-span and side-span cables. The deformed state of the surrogate model subjected to the specified external load cases will be found by restricting and releasing its tower tops, and thus the analytical formula of the secant stiffness can be obtained. Consequently, the nonlinear static solution of suspension bridge, including force and deformation changes in the cables and towers, can be explicitly expressed with the secant stiffness. The general solution procedure can also be extended to multi-span suspension bridges. Finally, the nonlinear static solution is verified by numerical examples of a three-span suspension bridge and a four-span suspension bridge. The parametric analysis reveals that the nonlinearity of suspension bridge will become significant as the side-span to mid-span ratio increases, or when the side-span cables are hung with hangers.

The far-field diffraction of a plane wave on a system of randomly and in average periodically located point scatterers: the Debye-Waller factor

The description of the superposition field of a system of point sources with a random and, on average, periodic structure is considered. The description is given in the far field and the point sources are considered as centers of secondary sphere waves generated by an external plane wave. In contrast to the traditional approach, where description of the diffraction pattern is given on the base of wave field averaging, here the intensity is averaged.In the framework of the suggested approach the analytic formula of dependence of the average intensity on the direction observation is found. The influence of disorder on the values of intensity of main maximums of a periodic structure is investigated. It is shown that this influence depends on the ordinal numbers of the maximum and in the case of central main maximum this influence vanishes. The question of applicability of the Debye-Weller factor for describing of the statistic of diffracted field is discussed.

Polarized signatures of orbiting hot spots: Special relativity impact and probe of spacetime curvature

Context. The Galactic Center supermassive black hole is well known to exhibit transient peaks of flux density on a daily basis across the spectrum. Recent infrared and millimeter observations have strengthened the case for the association between these flares and circular orbital motion in the vicinity of the event horizon. The strongly polarized synchrotron radiation associated with these events leads to specific observables called QU loops, that is, looping motion in the Stokes QU plane of linear polarization. These patterns have been observed by the Submillimeter Array, VLTI/GRAVITY, and ALMA. Aims. We want to deepen the understanding of the QU loops associated with orbiting hot spots. To this end, we computed such loops in Minkowski and Schwarzschild spacetimes in order to determine which aspects of the observed patterns are due to special- or general-relativistic phenomena. Methods. We considered a parcel of energized plasma in circular motion in Minkowski spacetime and in Keplerian orbit in the Schwarzschild spacetime. We computed, using the GYOTO ray-tracing code, the polarized radiative transfer associated with this orbiting hot spot and derived the evolution of the flux density, astrometry, and Stokes Q and U parameters. Results. We show that QU loops in Minkowski spacetime at low or moderate inclination i ≲ 45° (where i = 0° is a face-on view) share all the qualitative features of Schwarzschild QU loops. There exist QU loops for all setups considered (including for the face-on view and vertical magnetic field), there may be one or two QU loops per orbital period for a vertical magnetic field configuration, and there are always two QU loops in case of a toroidal magnetic field. The simplicity of Minkowski spacetime is a key asset for allowing us to provide analytical formulas that explain the details of this behavior. Moreover, we analyzed the flux variation of the hot spot and show that it is dictated either by the angular dependence of the radiative transfer coefficients or by relativistic beaming. In the former case, this can lead to extreme flux ratios, even at a moderate inclination. Finally, we highlight the increasing mirror asymmetry of the Schwarzschild QU track with increasing inclination and show that this behavior is a specific Schwarzschild feature caused by light bending. Conclusions. Although special-relativistic effects have not been extensively discussed in this context, they are a crucial part in generating the observed QU loops. However, general-relativistic light bending leads to a specific observable feature encoded in the asymmetry of the observed loops, and this feature might allow the spacetime curvature to be quantified.

Experimental and analytical investigation on eccentric performance of circular CFST columns jacketed by concrete-filled square tube

This paper presents an experimental and analytical investigation of the eccentric behaviour of circular CFST columns strengthened by a square steel tube and sandwiched concrete jackets. The main parameters include the eccentricity, strength of sandwiched concrete, and thickness of the outer steel tube. The failure modes, load-axial displacement curves, load-lateral deflection curves, and longitudinal strain distributions are presented. The effects of these parameters on the ultimate strength, ductility index, and lateral deflection are discussed in detail. Based on the longitudinal strain results obtained from the experiment and basic hypotheses, the possible positions of the neutral axis in the column section at the ultimate limit state were analysed, and three failure patterns of the specimens were determined. Analytical formulas corresponding to each failure pattern were derived to calculate the eccentric load-bearing capacity of the strengthened columns based on the limit equilibrium theory. Finally, a simplified formula was proposed to serve as the basis for future designs and applications.

Option valuation via nonaffine dynamics with realized volatility

This paper evaluates the improvement in option pricing brought about by realized volatility (RV) through nonaffine dynamics as advocated by Christoffersen et al. (2014). We complement their studies by developing a closed-form approximation of option pricing for the nonaffine models with RV, and then study the trade-off between the degradation in data fitting and the computational convenience offered by the analytical formula. Our studies confirm the literature that the nonaffine dynamics consistently outperform the affine in option pricing. In particular, we find that RV can significantly improve return fitting and option pricing through both affine and nonaffine models. For the affine models, we find strong evidence in favor of the RV information for both returns and options; for the nonaffine models, the evidence is less convincing for option pricing. We also provide additional new evidence that RV and nonaffine structures are equally competent at improving option pricing; moreover, these two features are complements rather than substitutes for GARCH option pricing, and the importance of one feature for option pricing is further enhanced when the other is present. All of these results are robust across moneyness, maturity, and volatility levels, and point to the necessity of including RV in nonaffine option pricing models.

On the elongational viscosity of viscoelastic slip flows in hyperbolic confined geometries

We study theoretically the elongational viscosity (or Trouton ratio, in dimensionless form) for steady viscoelastic flows in confined and symmetric hyperbolic tubes considering Navier-type slip along the wall(s). Both the planar and the cylindrical axisymmetric geometrical configurations are addressed. Under the classic lubrication approximation, and for a variety of constitutive models such as Phan-Thien and Tanner, Giesekus, and Finite Extensibility Nonlinear Elastic with the Peterlin approximation models, the same general analytical formula for the Trouton ratio is derived as for the Oldroyd-B model, in terms of the velocity at the midplane/axis of symmetry and the Deborah number only. Assuming that the velocity field is approximated by the Newtonian lubrication profile, based on our previous study in the absence of slip, we show that a constant extensional strain rate can be achieved in the limits of zero or infinite slip. For finite slip, a slight modification of the geometry is required to achieve a constant strain rate. In these cases, the formula for the steady state Trouton ratio reduces to that for transient homogeneous elongation. We also provide analytical formulae for the modification (decrease) for both the extensional strain rate and the Hencky strain achieved in the confined geometries because of introducing wall slip.

Using Monte Carlo Simulation to Forecast the Scientific Utility of Psychological App Studies: A Tutorial

Mobile applications offer a wide range of opportunities for psychological data collection, such as increased ecological validity and greater acceptance by participants compared to traditional laboratory studies. However, app-based psychological data also pose data-analytic challenges because of the complexities introduced by missingness and interdependence of observations. Consequently, researchers must weigh the advantages and disadvantages of app-based data collection to decide on the scientific utility of their proposed app study. For instance, some studies might only be worthwhile if they provide adequate statistical power. However, the complexity of app data forestalls the use of simple analytic formulas to estimate properties such as power. In this paper, we demonstrate how Monte Carlo simulations can be used to investigate the impact of app usage behavior on the utility of app-based psychological data. We introduce a set of questions to guide simulation implementation and showcase how we answered them for the simulation in the context of the guessing game app Who Knows (Rau et al., 2023). Finally, we give a brief overview of the simulation results and the conclusions we have drawn from them for real-world data generation. Our results can serve as an example of how to use a simulation approach for planning real-world app-based data collection.

On the diagonal product of special unitary matrices

We study the image of SU(n) under the diagonal product map. Using only elementary tools of linear algebra, analysis and general topology, we find the analytical formula for the boundary of this image and find all special unitary matrices whose diagonal product lies on that boundary.