Exact Integration Research Articles

Numerical evaluation of orientation averages and its application to molecular physics.

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modeling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also be applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

Analytic forms for the e+e− annihilation cross sections around a resonance including initial state radiation

For the first time, the exact analytic integration of the product between the Kuraev-Fadin radiative function and the Born-level cross section has been achieved for e+e− annihilation near a narrow resonance. The analytic result is in a perfect agreement with the one obtained through direct numerical integration. The extraction of resonance parameters is demonstrated through the unfolding of the initial state radiation effect. The analytic integration significantly accelerates the regression process by more than 160 times compared to numerical integration. Moreover, it facilitates the determination of experimental measurements for physical parameters, such as the two-dimensional confidence region of the branching fraction and the relative phase between the strong and electromagnetic amplitudes, within just one hour instead of one week typically required for numerical integration. Published by the American Physical Society 2024

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

Electrostatic interactions between soft nanoparticles beyond the Derjaguin approximation: Effects of finite size of ions and charges, dielectric decrement and ion correlations

HypothesisElectrostatic interactions between colloids are governed by the overlap of their electric double layers (EDLs) and the ionic screening of the structural charges distributed at their core surface and/or in their peripheral ion-permeable shell, relevant to soft particles like polymer colloids and microorganisms. Whereas ion size-mediated effects on the organization of isolated EDLs have been analysed, their contribution to the electrostatic energy of interacting soft particles has received less attention Theory and simulationsHerein, we elaborate a formalism to evaluate the electrostatic interaction energy profile between spherical core/shell particles, building upon a recent Poisson-Boltzmann theory corrected for the sizes of ions and particle structural charges, for ion correlations and dielectric decrement. Interaction energy is derived from pairwise disjoining pressure and exact Surface Element Integration method, beyond the Derjaguin approximation. The theory is sufficiently flexible to tackle homo- and hetero-interactions that involve weakly to highly charged hard, porous or core/shell nano- to micro-sized particles in asymmetric multivalent electrolytes. FindingsResults illustrate how ion steric effects, ion correlations and dielectric decrement impact the sign, magnitude and range of the interactions depending on the particle size, the Debye length, and the geometric and electrostatic properties of the particle core and shell components.

Proximity ligation-based sequencing for the identification of human papillomavirus genomic integration sites in formalin-fixed paraffin embedded oropharyngeal squamous cell carcinomas.

Human papillomavirus (HPV) infections are an increasing cause of oropharyngeal squamous cell carcinomas (OPSCC). Integration of the viral genome into the host genome is suggested to affect carcinogenesis, however, the correlation with OPSCC patient prognosis is still unclear. Research on HPV integration is hampered by current integration detection technologies and their unsuitability for formalin-fixed paraffin-embedded (FFPE) tissues. This study aims to develop and validate a novel targeted proximity-ligation based sequencing method (targeted locus amplification/capture [TLA/TLC]) for HPV integration detection in cell lines and FFPE OPSCCs. For the identification of HPV integrations, TLA/TLC was applied to 7 cell lines and 27 FFPE OPSCCs. Following preprocessing steps, a polymerase chain reaction (PCR)-based HPV enrichment was performed on the cell lines and a capture-based HPV enrichment was performed on the FFPE tissues before paired-end sequencing. TLA was able to sequence up to hundreds of kb around the target, detecting exact HPV integration loci, structural variants, and chromosomal rearrangements. In all cell lines, one or more integration sites were identified, in accordance with detection of integrated papillomavirus sequences PCR data and the literature. TLC detected integrated HPV in 15/27 FFPE OPSCCs and identified simple and complex integration patterns. In general, TLA/TLC confirmed PCR data and detected additional integration sites. In conclusion TLA/TLC reliably and robustly detects HPV integration in cell lines and FFPE OPSCCs, enabling large, population-based studies on the clinical relevance of HPV integration. Furthermore, this approach might be valuable for clonality assessment of HPV-related tumors in clinical diagnostics.

X-Shooting ULLYSES: Massive stars at low metallicity

Context. The winds of massive stars have a significant impact on stellar evolution and on the surrounding medium. The maximum speed reached by these outflows, the terminal wind speed v∞, is a global wind parameter and an essential input for models of stellar atmospheres and feedback. With the arrival of the ULLYSES programme, a legacy UV spectroscopic survey with the Hubble Space Telescope, we have the opportunity to quantify the wind speeds of massive stars at sub-solar metallicity (in the Large and Small Magellanic Clouds, 0.5 Z⊙ and 0.2 Z⊙, respectively) at an unprecedented scale. Aims. We empirically quantify the wind speeds of a large sample of OB stars, including supergiants, giants, and dwarfs at sub-solar metallicity. Using these measurements, we investigate trends of v∞ with a number of fundamental stellar parameters, namely effective temperature (Teff), metallicity (Z), and surface escape velocity vesc. Methods. We empirically determined v∞ for a sample of 149 OB stars in the Magellanic Clouds either by directly measuring the maximum velocity shift of the absorption component of the C IV λλ1548–1550 line profile, or by fitting synthetic spectra produced using the Sobolev with exact integration method. Stellar parameters were either collected from the literature, obtained using spectral-type calibrations, or predicted from evolutionary models. Results. We find strong trends of v∞ with Teff and vesc when the wind is strong enough to cause a saturated P Cygni profile in C IV λλ1548–1550. We find evidence for a metallicity dependence on the terminal wind speed v∞ ∝ Z0.22±0.03 when we compared our results to previous Galactic studies. Conclusions. Our results suggest that Teff rather than vesc should be used as a straightforward empirical prediction of v∞ and that the observed Z dependence is steeper than suggested by earlier works.

Implementation of energy and gradient for the TDDFT-approximate auxiliary function (aas) method.

In this work, we have implemented the time-dependent density functional theory approximate auxiliary s function (TDDFT-aas) method, which is an approximate TDDFT method. Instead of calculating the exact two-center electron integrals in the K coupling matrix when solving the Casida equation, we approximate the integrals, thereby reducing the computational cost. In contrast to the related TDDFT plus tight-binding (TDDFT+TB) method, a new type of gamma function is used in the coupling matrix that does not depend on the tight-binding parameters. The calculated absorption spectra of silver and gold nanoparticles using TDDFT-aas show good agreement with TDDFT and TDDFT+TB results. In addition, we have implemented the analytical excited-state gradients for the TDDFT-aas method, which makes it possible to calculate the emission energy of molecular systems.

Numerical solution of the multi-order fractional differential equation using Legendre wavelets and eigenfunction approach

An eigenfunction approach is implemented in this article to solve the multi-order fractional differential equations (FDEs) with boundary conditions. The approximate unknown solution is expressed as a linear combination of eigenfunctions in the present paper. The proposed approach solved the problem via complex and real eigenfunction. By using the eigenvalue problem Dαϕ(x)=λϕ(x) subjected to u(0)=u(1)=0, first we calculate the eigenvalues and their corresponding eigenfunction and then use the output of this problem in the present scheme to simplify. Exact fractional integration has been utilized to solve the eigenvalue problem. Collocation points have also been used to reduce the problem to a system of algebraic equations. The obtained solutions are compared with the exact solution. In addition, the numerical convergence has also been studied of the proposed technique through three examples. These examples also demonstrate the applicatory and simplicity of the proposed scheme.

Exact Functional Integration of Radial and Complex Slave‐Boson Fields: Thermodynamics and Dynamics of the Two‐Site Extended Hubbard Model

AbstractThe functional integral formulation of the Hubbard model when treated in its Kotliar‐Ruckenstein representation in the radial gauge involves fermionic, as well as complex and radial slave boson fields. In order to improve on the understanding of the interplay of the three types of fields, and on the nature of the latter, a comprehensive investigation of an exactly solvable two‐site cluster is performed, as it entails all pitfalls embodied in this approach. It is first shown that the exact partition function is recovered, even when incorporating in the calculation the square root factors that are at the heart of the representation, when suitably regularized. It is shown that using radial slave boson fields allows to overcome all hurdles following from the normal ordering procedure. It is then demonstrated that this applies to the Green's function as well, and to the correlation functions of physical interest, thereby answering the criticisms raised by Schönhammer [K. Schönhammer, Phys. Rev. B 1990 42, 2591]. In addition, the investigation generalizes the calculations to the Hubbard model extended by a non‐local Coulomb interaction.

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

Extraction of exact soliton solutions of (2+1)-dimensional Chaffee-Infante equation using two exact integration techniques

In this research paper, (2 + 1)-dimensional Chaffee-Infante equation, with applications in fluid dynamics, nonlinear fiber optics, coastal engineering, ion-acoustic modeling, sound waves, electromagnetic field waves, and plasma physics, is explored and analyzed to calculate is analytic exact solutions utilizing two effective exact methods named as, extended (G′G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\frac{G'}{G^2})$$\\end{document}-expansion method and new (F/G)-expansion method. These techniques work wonderfully to produce exact solutions for many nonlinear problems that arise in various disciplines of scientific fields. The proposed methods are employed to extract the soliton solutions of the proposed equation. These exact methods are implemented successfully and provide many soliton solutions. The extracted soliton solutions contain hyperbolic, trigonometric and rational functions. Graphical analysis is provided using 3D surface plots and 2D contour plots in order to comprehend the physical structure of the obtained solutions.

Microscopic contributions to the entropy production at all times: from nonequilibrium steady states to global thermalization

Based on exact integration of the Schrödinger equation, we numerically study microscopic contributions to the entropy production for the single electron transistor, a paradigmatic model describing a single Fermi level tunnel coupled to two baths of free fermions. To this end, we decompose the entropy production into a sum of information theoretic terms and study them across all relevant time scales, including the nonequilibrium steady state regime and the final stage of global thermalization. We find that the entropy production is dominated for most times by microscopic deviations from thermality in the baths and the correlation between (but not inside) the baths. Despite these microscopic deviations from thermality, the temperatures and chemical potentials of the baths thermalize as expected, even though our model is integrable. Importantly, this observation is confirmed for both initially mixed and pure states. We further observe that the bath-bath correlations are quite insensitive to the system-bath coupling strength contrary to intuition. Finally, the system-bath correlation, small in an absolute sense, dominates in a relative sense and displays pure quantum correlations for all studied parameter regimes.

Lagrangian stochastic integrals of motion in isotropic random flows

A set of exact integrals of motion is found for systems driven by homogenous isotropic stochastic flow. The integrals of motion describe the evolution of (hyper-)surfaces of different dimensions transported by the flow and can be expressed in terms of local surface densities. The expression for the integrals is universal: it represents general geometric properties and does not depend on the statistics of the specific flow.

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for 1Σ States

AbstractEnergetics of two‐center two‐electron (2c–2e) systems carry challenges in theoretical understanding of Schrödinger equation (SE) for well‐known divergence of Coulomb interactions and nuclear separation (R) in modified H‐like AOs, Slater type orbitals (STOs), Gaussian type orbitals (GTOs), B‐spline, Sturmian function and etc. employed to VBT and MOT. Certain elegant computational and analytical techniques were developed for STO, GTO and other square integrable basis set within Born‐Oppenheimer (BO) approximation. STOs and GTOs have an essential limitation of absence of radial nodes. Thus, analytical treatment has become an urge for H‐like AOs. We have considered the diatomic molecules only for the sake of simplicity. Employing Sheffer identity in associated Laguerre polynomial/Whittaker‐M function forms of H‐like AOs and transforming integrals into elliptic coordinates with two nuclei on two foci furnishes exact, analytical and simple Coulomb integrals (Js) in terms of R. Lah number originated from for nuclear coordinates only due to Sheffer identity shows that energetics of diatomic molecules can be anticipated as extremum function of R. Therefore, the optimization of potential energy surface (PES) of electrons as gradient of R may lead to σ‐bond formation. In this paper, we have developed diagonal Js for bound states of H2 molecule.

RADIATIVE TRANSFER: ASYMPTOTIC SOLUTION OF THE KINETIC EQUATION OF RADIATION PROPAGATION, <i>N</i>-th ORDER ASYMPTOTIC APPROXIMATION AND IMPROVED BOUNDARY CONDITIONS

In the article, new asymptotic approximation of the \(n\)-th order is obtained and proposed to be used in calculations of radiation propagation in optically thick media without scattering; the asymptotic approximation is simpler and more precise than the known diffusion approximation. It is shown, that for optically thick media the asymptotic solution of the kinetic equation of radiation propagation without scattering is asymptotic expansion of the exact integral solution of that kinetic equation. The rigorous derivation of the diffusion approximation equation is obtained. Improved boundary conditions, which are essential for practical application in calculations of radiation propagation, are derived.

An In Vivo PiggyBac Insertional Mutagenesis Screen Reveals Oncogenic Lesions Cooperating with Myd88 L265P

Diffuse large B cell lymphoma (DLBCL) is the most common Non-Hodgkin lymphoma and originates from transformed germinal center-experienced B cells. Traditionally, DLBCL has been divided into two subtypes, depending on whether the transcriptional profile of the tumor relates to an activated B cell or a germinal center B cell (ABC and GCB DLBCL, Alizadeh et al., 2000). More recent efforts classified DLBCL cases based on their genetic aberrations and identified several clusters with distinct mutational profiles (Chapuy et al., 2018, Schmitz et al., 2018, Wright et al., 2020). The MCD/C5 cluster is characterized by recurrent mutations in MYD88, PRDM1 and frequent amplifications of BCL2, amongst others. We recently showed that mice harboring a B cell-specific Myd88 L252P mutation (orthologous position of the human p.L265P mutation) develop B cell proliferation and occasional transformation into DLBCL (Knittel et al., 2016). Lymphomagenesis is further increased when Myd88 L252P is combined with BCL2 overexpression and a genetically engineered block in plasmacytic differentiation by loss of Prdm1 or overexpression of Spib (Flümann et al., 2021). To ask which further genes and pathways cooperate with Myd88 L252P in lymphomagenesis, we performed an in vivo piggyBac insertional mutagenesis screen. In this system, a conditionally expressed transposase mobilizes transposable elements. These elements can then reintegrate into the genome and either silence or drive the expression of genes, depending on the exact integration site and orientation of the transposon cassette (Rad et al., 2015). We crossed this piggyBac system onto a Myd88 L252P background to identify genes that cooperate with Myd88 mutations in lymphomagenesis. Mice harboring both the Myd88 mutant allele and the piggyBac system lived significantly shorter than controls harboring only the Myd88 mutation or just the piggyBac system. Myd88/piggyBac animals developed B220 +/CD138 - lymphomas and DNA isolated from these lesions allowed the detection of common transposon insertion sites. Among the genes significantly enriched for integrations, we observed known genetic drivers of human MCD/C5 DLBCL, such as PIM1 and ETV6. We also identified TBL1XR1 and SPIB as common insertion sites, as well as BCL2, BIM1 and BCL-XL, further validating our approach of engineering a plasma cell differentiation block as well as anti-apoptotic BCL2 overexpression on a Myd88-mutant background to model MCD DLBCL. Of note, the hits identified in this screen were significantly distinct from hits identified to drive Myc-driven B cell lymphomagenesis (Weber et al., 2019). Additionally, we identified several candidate genes that are not reported to be frequently genetically altered in DLBCL, however associated with processes relevant to B cell biology, particularly to B cell receptor signaling. As ‘B cell receptor signaling’ was a prominent term in our screen hits and CD79B ITAM mutations are an additional hallmark of MCD/C5 DLBCL (Chapuy et al., 2018, Schmitz et al., 2018, Wright et al., 2020), we introduced a conditional Cd79b p.Y195H allele to our Cd19 Cre/wt;Myd88 cond.p.L252P/wt; Rosa26 LSL.BCL2-IRES-GFP/wt; Prdm1 fl/fl MCD/C5 DLBCL model. While we did not observe significant differences in the immunohistochemial and transcriptional phenotype of Cd79b WT and mutant lymphomas, Cd79b p.Y195H tumors showed increased B cell receptor (BCR) signaling activity, indicated by increased levels of phosphorylated SYK and PLCg2. Cd79b mutant lymphomas also showed an increased formation of cytoplasmic signaling complexes comprised of MYD88 and several components of the BCR pathway, including MALT1 and BTK. The formation of these complexes depended on active BTK, as treatment with the BTK inhibitor ibrutinib reduced complex formation to levels found in Cd79b WT lymphomas. Consequently, we investigated the effects of ibrutinib treatment in Cd79b-mutant and wildtype MCD DLBCL mouse models and found Cd79b-mutant lymphomas to be significantly more sensitive to ibrutinib treatment than their Cd79b WT counterparts.

Finite element 3D model of a double quantum ring: effects of electric and laser fields on the interband transition

In this work, the changes in the energy of electrons and holes, oscillator strength and interband transition time when external fields are applied to a GaAs/AlGaAs semiconductor double ring grown by the droplet epitaxy technique are theoretically analyzed. We consider a static electric field and an intense laser field nonresonant with the quantum structure, with variable intensities and orientations with respect to the symmetry axis of the quantum ring (QR). In the formalism of the effective mass approximation for electrons and holes, the energies and wavefunctions were numerically computed using the finite element method implemented with an accurate three-dimensional model of the real QR. Laser dressing of the confining potential was performed using the exact integration formula at each point. Our results show major differences between the effects of the two types of applied fields, caused mainly by the static electric-field-induced strong polarizability of the confined electron-hole pair. In addition, the effects of both fields exhibit strong anisotropy in the electronic properties as a result of the particular flattened geometry of the QR. Proper combinations of field strengths and orientations are helpful in designing accurate tools for the sensitive manipulation of interband radiative properties.

CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models

We propose a method for finding optimal quadrature/cubature rules with positive weights for parameterized functions in 1D, 2D or 3D spatial domains. The method takes as starting point the values of the functions at the Gauss points of a finite element (FE) mesh of the spatial domain for a representative sample of input parameters, and then construct an elementwise continuous orthogonal basis for such functions using the truncated Singular Value Decomposition (SVD) along with element polynomial fitting. To avoid possible memory bottlenecks in computing the SVD, we propose a Sequential Randomized SVD (SRSVD) in which the matrix is provided in a column-partitioned format, and which uses randomization to accelerate the processing of each individual block. After computing the basis functions, the method determines an exact integration rule for such functions, featuring as many points as functions, and in which the points are selected among the Gauss points of the FE mesh. Finally, the desired optimal rule is obtained by an sparsification process in which the algorithm zeroes one weight at a time while readjusting the positions and weights of the remaining points so that the constraints of the problem are satisfied. We apply this methodology to multivariate polynomials in cartesian domains to demonstrate that the method is indeed able to produce optimal rules – i.e., Gauss product rules –, and to a 2-parameters, 3D sinusoidal–exponential function to illustrate the use of the SRSVD in scenarios in which the standard SVD cannot handle the operation because of memory limitations. Lastly, the fact that the method does not require the analytical expression of the integrand functions – just their values at the FE Gauss points – makes it suitable for dealing with the so-called hyperreduction of parameterized finite element models. We exemplify this by showing its performance in the derivation of low-dimensional surrogate models in the context of the multiscale FE method. The Matlab source codes of both the CECM and the SRSVD, along with the scripts for launching the numerical tests, are openly accessible in the public repository https://github.com/Rbravo555/CECM-continuous-empirical-cubature-method.

A closed-form expression of electric polarizability for elliptical apertures excited by nonuniform electric field

It is well known that electric field penetration through a small aperture in a perfectly conducting plane behaviors like an electric dipole. When excited by uniform electric field, the dipole moment has analytical solution for elliptical apertures and numerical/experimental values for other shapes. In this paper, for the case of nonuniform field excitation, based on the electrostatic reciprocity theorem, the closed-form expression of electric dipole moment is derived for elliptical apertures. This expression is in integral form, and its integrand is the product of the nonuniform electric field distribution and the potential distribution when excited by uniform field in the aperture plane. Taking a point charge located on the aperture plane as an example of nonuniform field excitation, the dipole moment is calculated using the proposed closed-form expression. The validation is verified by comparisons with exact integral solution for circular apertures and with finite element simulations for elliptical apertures.