Matrix Elements Research Articles

A user selection algorithm for downlink MU-MIMO systems using product of singular values of effective channels

User selection plays a crucial role when aiming to improve channel throughput in multiuser multiple- input multiple-output downlink systems with block diagonalization linear precoding. While an optimal group of users can be found by exhaustively searching all possible combinations of users, the process becomes prohibitively complex when there is a large number of users. In this paper, we propose a greedy suboptimal user selection algorithm, which iteratively selects a user to maximize the product of the singular values of the effective channels from a set of unselected users. The proposed algorithm is based on the iterative precoder design method, whereby interuser interference is perfectly cancelled with lower complexity than with the singular value decomposition-based precoding technique. Moreover, to avoid frequent use of singular value decomposition, the proposed algorithm applies QR decomposition to the selected channel matrices in order to obtain singular values, which are equivalent to the product of diagonal elements in the upper triangular matrix. Simulation results show that, under a high SNR regime, the proposed algorithm outperforms several other greedy algorithms and can provide the same performance as a capacity-based algorithm for both correlated and uncorrelated channels with significantly reduced complexity.

Fluorescence by incoherently pumped polar quantum dot placed in a low-frequency monochromatic field

Abstract We studied spectral properties of high-frequency fluorescent radiation from a two-level quantum system with broken inversion spatial symmetry, which can be implemented as a model of a one-electron two-level asymmetric polar semiconductor quantum dot whose electric dipole moment operator has permanent unequal diagonal matrix elements. The case of the excitation of this system by incoherent pumping of some sort supplemented with an additional driving monochromatic field, which frequency is much lower than the optical transition frequency of the quantum dot, was considered. An analytical expression for the fluorescence spectrum as a function of the amplitude and frequency of the monochromatic driving field, as well as of the pumping intensity, was derived. We also discussed feasible practical applications of the effects under study in the field of quantum optics and optoelectronics for the generation of high-frequency radiation, possessing stationary equidistant spectrum, and for the control of the radiative properties of such nanodevices as spasers (dipole nanolasers).

Odd-even mass differences of well and rigidly deformed nuclei in the rare earth region: A test of a newly proposed fit of average pairing matrix elements

Abstract We discuss a test of a recently proposed approach to determine average pairing matrix elements within a given interval of single-particle states (sp) around the Fermi level $\lambda$ as obtained in the so-called uniform gap method (UGM). It takes stock of the crucial role played by the averaged sp level density $\tilde{\rho}(e)$. These matrix elements are deduced within the UGM approach, from microscopically calculated $\tilde{\rho}(e)$ and gaps obtained from analytical formulae of a semi-classical nature. Two effects generally ignored in similar fits have been taken care of. They are: (a) the correction for a systematic bias in choosing to fit pairing gaps corresponding to equilibrium deformation solutions as discussed by M"{o}ller and Nix [Nucl. Phys. A 476, 1 (1992)] and (b) the correction for a systematic spurious enhancement of $\tilde{\rho}(e)$ for protons in the vicinity of $\lambda$, because of the local Slater approximation used for the treatment of the Coulomb exchange terms in most calculations (see e.g. [Phys. Rev C 84, 014310 (2011)]). This approach has been deemed to be very efficient upon performing Hartree-Fock + BCS (with seniority force and self-consistent blocking when dealing with odd nuclei) calculations of a large sample of well and rigidly deformed even-even rare-earth nuclei. The reproduction of their experimental moments of inertia has been found to be at least of the same quality as what has been obtained in a direct fit of these data [Phys. Rev C 99, 064306 (2019)]. We extend here the test of our approach to the reproduction, in the same region, of three-point odd-even mass differences centered on odd-$N$ or odd-$Z$ nuclei. The agreement with the data is again roughly of the same quality as what has been obtained in a direct fit, as performed in [Phys. Rev C 99, 064306 (2019)].

Mixed quantum/classical theory for rotationally inelastic scattering of identical collision partners revised.

Mixed quantum/classical theory (MQCT) for the treatment of rotationally inelastic transitions during collisions of two identical molecules, described either as indistinguishable or distinguishable partners, is reviewed. The treatment of two molecules as indistinguishable includes symmetrization of rotational wavefunctions, introduces exchange parity, and leads to state-to-state transition matrix elements different from those in the straightforward treatment of molecules as distinguishable. Moreover, the treatment of collision partners as indistinguishable is eight times faster. Numerical results presented herein for H2 + H2, CO + CO and H2O + H2O systems indicate good agreement of MQCT calculations with full-quantum calculations from the literature and show that an a posteriori correction, applied after treatment of the collision partners as distinguishable, generally produces good results that agree well with the rigorous treatment of collision partners as indistinguishable. This correction for the cross section includes either multiplication by 2 or a summation over physically indistinguishable processes, depending on the transition type. After this correction, the results of the two treatments agree within 5% for most but may reach 10-20% for some transitions. At low collision energies dominated by scattering resonances, these differences can be larger, but they tend to decrease as collision energy is increased. It is also shown that if the system is artificially forced to follow the same collision path in the indistinguishable and distinguishable treatments, then all differences between the results of the two treatments disappear. This interesting finding gives new insight into the collision process and indicates that the indistinguishability of identical collision partners comes into play through the collision path itself, rather than through matrix elements of inelastic transitions.

Electron work function guided tailoring of (W4-x, Mx)C4 /doped Ni matrix interfacial bonding: Insights from first-principles calculations

Heavy tungsten carbide (WC) may cause inhomogeneous distributions in WC-metal matrix composite hardfacing overlays, thus negatively affecting its performance as reinforcement. WC can be lightened by partially substituting W with lighter metals, e.g., Mo and Cr, while maintaining its strength. However, the bonding between modified WC and matrix metals such as nickel (a typical metal-matrix for overlays) is uncertain. This article reports a study on the interfacial bonding between (W4-x, Mx)C4 (M=Mo or Cr) and Ni matrix via first-principle calculations. Different atomic interactions i.e., metal-metal and metal-carbon interactions, at the interface were studied to understand the interfacial bonding through analyses of electron work function (EWF), electron localization function, electronic density of states, bond order, and net charge. It was demonstrated that the lighter (W4-x, Mx)C4 carbides exhibit strong bonding with Ni, comparable to or even stronger than that of WC/Ni interface, and the interfacial bonding includes covalent, ionic and metallic bond components. It is demonstrated that the interfacial bonding can be tuned by doping elements in the Ni matrix with different EWFs, e.g., Mn, Cu, Au, Pt, and Se. Efforts have been made to verify a hypothesis that EWF is an indicator, which can be used to guide selecting effective dopants for tailoring the interfacial bond strength.

Moshinsky brackets for a wide range of quantum numbers using generating functions

We used a new Python code to reproduce the brackets for the Moshinsky harmonic oscillator, which was based on the generating function. We made these brackets by transforming the wave functions of two groups of coupled particle harmonic oscillators, Φn1l1,n2l2,Λm1,m2,λ(r→1,r→2) and Φnala,nblb,Λma,mb,λ(r→a,r→b). To convert between the supplied position and momentum coordinates in both frames, we performed orthogonal transformations on nuclei with both low and high angular momentum.In our derivation, we have used the expansion of the generating functions e2p→.r→−p2 and e2cpi.pj in spherical coordinates in terms of harmonic oscillator wave functions. When we modified the Moshinsky brackets for two-coupled oscillator states, we used generating functions with two variables. The number of indices has significantly decreased compared to the oscillator brackets in previous references; this reduction in the program code's iterative process has yielded influential results. Compared to the previous version of the Moshinsky brackets code, the new Python code is easier to use. Our approach utilizes this code to assess Moshinsky brackets across a broad spectrum of quantum numbers. According to the revelation, adding more variables to the generating function makes the number of Moshinsky brackets that work for the higher body interactions increase. Program summaryProgram Title: GFBRACKETSCPC Library link to program files:https://doi.org/10.17632/jvbnwp35rm.1Licensing provisions: MITProgramming language: PythonSupplementary material: AppendixNature of problem: The generating functions were used to obtain a concise representation of the oscillator states for single particles. This was done to formulate and compute the generalized version of the Moshinsky brackets and matrix elements of two-body operators. By enlarging sets of basis oscillator states, it becomes possible to cover interactions involving more than two bodies. This can be easily achieved using the expanded generating function to compute the Moshinsky brackets for high-quantum numbers.Solution method: The Python code requires fewer iterations than similar codes generated by the method of the generating functions.Additional comments including restrictions and unusual features: When calculating Moshinsky brackets for large quantum numbers, restrictions arise. There is no data for quantum numbers greater than 10 in the formation of reaction potential. Additionally, a transformation matrix can be used to switch from single-particle to centre-of-mass coordinates on a two-particle harmonic oscillator basis. This transformation is applicable even when the masses of the two particles are unequal.

Pluricanonical cycles and tropical covers

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

Computational Analysis of the Micromechanical Stress Field in Undamaged and Damaged Unidirectional Fiber-Reinforced Plastics Using a Modified Principal Component Analysis

This study investigated the internal stress distribution of unidirectional fiber-reinforced plastics (UD-FRP) at the micro level using principal component analysis (PCA). The composite material was simulated using a representative volume element model together with the embedded cell approach. Two fundamental quasi-static load cases, transverse and longitudinal tensile deformation, were considered. The experimental results show that mechanical failure occurred at 2.15 ± 0.06% transverse tensile strain and at 1.52 ± 0.07% longitudinal tensile strain. Furthermore, the undamaged state and a combination of matrix and interface damage, as well as fiber breakage, were simulated. From the simulations, the octahedral shear stress and octahedral normal stress were computed at the integration points of the matrix elements, constituting what is known as the octahedral stress field. A modification on the PCA to obtain mesh-independent eigenvalues is presented and was used to investigate the effects of damage events on the octahedral stress field. The results indicate that each damage mechanism had a distinct signature in the redistribution of the stress field, characterized by specific changes in the eigenvalues and orientation of the principal component (θ1). Furthermore, the PCA suggests that the accumulation of matrix damage began to be relevant at the 1% strain, while fiber breakage began at an average longitudinal strain of 0.98 ± 0.12%. Additionally, it is shown that the first principal component served as an indicator of the predominant stress state of the stress field. This investigation suggests that the PCA can provide valuable insights regarding the complex damage mechanisms of UD-FRP that may not be captured by conventional mechanical analysis.

Some notes on the consequences of pretreatment of multivariate data

With the advent of data technologies, we have various types of data, such as structured, unstructured and semi-structured. Performing certain statistical or machine learning techniques may require careful preprocessing or pretreatment of the data to make them suitable for analysis. For example, given a data matrix X, which represents n multivariate observations or cases on p variables or features, the columns/rows of X may be pretreated before applying statistical or machine learning techniques to the data. While centering and/or scaling the variables do not alter the correlation structure nor the graphical representation of the data, centering/scaling the observations do. We investigate various row pretreatment methods more closely and show with theoretical proofs and numerical examples that centering/scaling the rows of X changes both the graphical structure of the observations in the multi-dimensional space and the correlation structure among the variables. There may be good reasons for performing row centering/scaling on the data and we are not against it, but analysts who use such row operations should be aware of the geometrical and correlation structures one has performed on the data and should also demonstrate that the process results in a new, more appropriate structure for their questions.

D-Brane effective Lagrangian in spacetimes with boundaries

AbstractIn this study, we explore the transformation of $$D_p$$ D p -branes to $$D_{p-1}$$ D p - 1 -branes under T-duality when the D-brane is embedded in a spacetime with a boundary. Our goal is to derive the higher-derivative corrections to the Dirac–Born–Infeld (DBI) Lagrangian for both the bulk and boundary terms. For the bulk terms, we calculate the $$\alpha '$$ α ′ corrections for the massless open string fields, up to the 8th order in the dimensionless Maxwell field strength. We demonstrate that the bulk Lagrangian can satisfy the T-duality constraint without residual total derivative terms in the base space. This determines the most general independent couplings of the massless open string fields in the bulk Lagrangian, encompassing 145 coupling constants, up to three parameters. Two of these parameters are physical and are determined by disk-level S-matrix elements, while the third is unphysical and can be eliminated by field redefinitions and integration by parts. The final result for the bulk Lagrangian consists of 49 couplings. For the boundary terms, the application of T-duality symmetry to massless open and closed string fields enables us to extend the DBI Lagrangian to include the boundary unit normal vector field $$n_\mu $$ n μ and its first derivative.

Finite-order method to calculate approximate density matrices in the Fock-space multireference coupled cluster theory

An efficient approach to calculate approximate pure-state and transition reduced density matrices in the framework of the multireference relativistic Fock-space coupled cluster (FS CC) theory is proposed. The method is based on the effective operator formalism and consists of the direct substitution of the FS CC Ansatz for a wave operator into the effective operator expression with the subsequent truncation of expansion at the terms quadratic in cluster amplitudes. The final density matrix is defined by active-space density matrices of different ranks ‘dressed’ with contributions from cluster operators. The method gives a connected expression for pure-state density matrices, provided that the intermediate normalisation condition is fulfilled. Moreover, under some additional assumptions, the connectivity can also be ensured for calculated transition property matrix elements and natural transition spinors. The developed technique allows for fast and accurate calculations of one-particle reduced density matrices for a wide range of electronic states. A pilot application of the new technique to construct averaged atomic natural orbital (ANO) basis sets for fully relativistic electronic structure calculations is presented.

Algorithms for numerically stable scattering amplitudes

The numerically stable evaluation of scattering matrix elements near the infrared limit of gauge theories is of great importance for the success of collider physics experiments. We present a novel algorithm that utilizes double-precision arithmetic and reaches higher precision than a naive quadruple-precision implementation at smaller computational cost. The method is based on physics-driven modifications to propagators, vertices, and external polarizations. Published by the American Physical Society 2024

Moderate Deviation Principle for the Determinant of Sample Correlation Matrix

ABSTRACTLet be a sequence of independence random vectors from with population correlation matrix , and its sample correlation matrix is denoted as , where the matrix element represents the Pearson correlation coefficient between the and columns of the data matrix . is an interesting part in multivariate analysis, and the likelihood ratio test (LRT) statistic used to test the complete independence of all components of can be expressed as a function of . On this basis, under the condition that the sample size and the dimension satisfy as , we derive the moderate deviation principle (MDP) of the LRT statistic for the hypothesis testing problem vs .

Systematic Uncertainties from Gribov Copies in Lattice Calculation of Parton Distributions in the Coulomb gauge

Abstract Recently, it has been proposed to compute parton distributions from boosted correlators fixed in the Coulomb gauge within the framework of Large-Momentum Effective Theory. This method does not involve Wilson lines and could greatly improve the efficiency and precision of lattice QCD calculations. However, there are concerns about whether the systematic uncertainties from Gribov copies, which correspond to the ambiguity in lattice gauge-fixing, are under control. This work gives an assessment of the Gribov copies’ effect in the Coulomb-gauge-fixed quark correlators. We utilize different strategies for the Coulomb-gauge fixing, selecting two different groups of Gribov copies based on the lattice gauge configurations. We test the difference in the resulted spatial quark correlators in the vacuum and a pion state. Our findings indicate that the statistical errors of the matrix elements from both Gribov copies, regardless of the correlation range, decrease proportionally to the square root of the number of gauge configurations. The difference between the strategies does not show statistical significance compared to the gauge noise. This demonstrates that the effect of the Gribov copies can be neglected in the practical lattice calculation of the quark parton distributions.

The triton lifetime from nuclear lattice effective field theory

In this work, we present a calculation of the triton β-decay lifetime using Nuclear Lattice Effective Field Theory (NLEFT) at next-to-next-to-next-to-leading order in the chiral expansion. By incorporating a non-perturbative treatment of the higher-order corrections, we achieve consistent predictions for the Fermi and Gamow-Teller matrix elements, which are crucial for determining the triton lifetime. Our results are consistent with earlier theoretical calculations, confirming the robustness of our approach. This study marks a significant advancement in the systematic application of NLEFT to nuclear β-decay processes, paving the way for future high-precision calculations in more complex nuclear systems. Additionally, we discuss potential improvements to our approach, including the explicit inclusion of two-pion exchange mechanisms and the refinement of three-nucleon forces. These developments are essential for extending the applicability of NLEFT to a broader range of nuclear phenomena, including neutrinoless double-β decay.

Numerical simulation of shale gas reservoir based on ILU-GMRES method

ABSTRACT The numerical simulation of shale gas reservoirs requires transforming mathematical models into large linear equation systems, and solving large linear equation systems requires a lot of time. A novel ILU-GMRES method for solving large linear equations is presented, which treats the sub matrix discretized at nodes as the basic elements of the coefficient matrix. By combining the incomplete LU decomposition method (ILU) with the generalized minimum residue method (GMRES), the discretized large linear equations are iteratively solved. A comparison was made between the new method and other methods for calculating the actual shale gas reservoir model. The results reveal that the new method has a faster calculation speed in solving linear equation systems. The calculation time of the novel ILU-GMRES method is 1029s, the calculation time of the generalized minimum residual method (GMRES) method is 2709s, the calculation time of the Gauss–Seidel method (GS) is 3786s, and the calculation time of the successive over relaxation method (SOR) method is 3386s, the calculation time of the preprocessing conjugate gradient method (PCG) is 3115s. The calculation speed of the novel method is almost 2.6 times faster than the GMRES method, the novel method is more effective in the numerical simulation of shale gas.

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Complex system fault diagnosis based on interpretive structural model

To realize the fault diagnosis of complex systems, a fault diagnosis method for complex systems based on the explanatory structural model is proposed. According to the fault mechanism analysis, the fault association matrix of system elements is established. The explanatory structural model is applied to transform the fault association relationship of complex systems into an intuitive hierarchical structural model through matrix transformation, so as to realize the structuring and hierarchical propagation of system faults. The PageRank algorithm is introduced to evaluate the influence and impact of system element faults. According to the influence and impact of the same-layer components and the fault transmission logic, the main cause of system fault transmission is clarified, which provides a basis for fault diagnosis. Finally, a specific application is carried out with a certain device as an example to verify the effectiveness of the method.

Hydrogen energy levels from the anomalous energy-momentum QED trace

Energy levels of hydrogen are calculated as one-loop matrix elements of the QED energy-momentum tensor trace in the external field approximation. An explicit connection established between the one-loop trace diagrams and the standard Lamb shift one-loop diagrams. Our calculations provide an argument against inclusion of the anomalous trace contribution as a separate term in the decomposition of the QED quantum field Hamiltonian and serve as an illustration how the trace anomaly is realized in the bound state QED. Published by the American Physical Society 2024

Measurement of single top-quark production in association with a W boson in pp collisions at s=13 TeV with the ATLAS detector

The inclusive cross section for the production of a single top quark in association with a W boson is measured using 140 fb−1 of proton-proton collision data collected with the ATLAS detector at s=13 TeV. Events containing two charged leptons and at least one jet identified as originating from a b-quark are selected. A multivariate discriminant is constructed to separate the tW signal from the tt¯ background. The cross section is extracted using a profile likelihood fit to the signal and control regions and it is measured to be σtW=75−14+15 pb, in good agreement with the Standard Model prediction. The measured cross section is used to extract a value for the left-handed form factor at the Wtb vertex times the Cabibbo-Kobayashi-Maskawa matrix element |fLVVtb| of 0.97±0.10. © 2024 CERN, for the ATLAS Collaboration 2024 CERN