Formulas For Elements Research Articles

Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ invariant boundary conditions

We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses U_q\big(\mathfrak{sl}(2)\big)Uq(𝔰𝔩(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter QQ operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the QQ operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the zz-projection of the total spin operator being equal to zero.

Boundary overlaps from Functional Separation of Variables

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an su3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{su}(3) $$\\end{document} rational spin chain in an alternating fundamental-anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and suN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{su}(N) $$\\end{document} generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates.

Verification analysis for quality control of load-bearing superstructures (Kyzyl)

It is currently relevant to perform the quality control of load-bearing superstructures using verification calculations, which provide strength, rigidity and crack resistance of elements.Purpose: The development of analytical and numerical methods to evaluate bendable reinforced concrete elements of a ribbed floor slab.Methodology/approach: Well-known normative methods for the strength analysis of reinforced concrete structures; comparison of analytical and numerical results obtained in the LIRA software package.Research findings: Well-known formulas for bending elements are used for static calculations. The analysis of the limit state of the floor slab using analytical and numerical methods allows assessing the structural state with sufficient accuracy. The structural analysis shows some discrepancies, which require correction in the future.Originality/value: numerical strength analysis of bent reinforced concrete elements.

Explicitly correlated Gaussians for high-precision variational calculations of S^{e},P^{e}, and D^{e} states of quantum systems: An efficient algorithm.

In this work we consider an efficient algorithm for variational calculations of quantum few-particle systems in S,P, and D states of the even parity using all-particle explicitly correlated Gaussian (ECG) basis sets. We primarily focus on the description of states where the dominant configuration contains either two particles in p states or a single particle in a d state (all other particles are in s states). The basis functions we consider are products of spherically symmetric ECGs and bipolar harmonics. We introduced a scheme for deriving expressions for matrix elements of the overlap, kinetic and potential energy, as well as their derivatives with respect to the nonlinear parameters of the Gaussians. This allowed us to improve the efficiency of numerical calculations of the matrix elements (which is the most critical part of any code that uses ECGs) by one to two orders of magnitude compared to previous implementations. We provide a complete set of formulas for all basic matrix elements using the formalism of matrix differential calculus and discuss some technical details relevant to their efficient implementation. Lastly, we report a few example calculations of the ^{2}D^{e} states of the Li atom, ^{4}P^{e} and ^{2}D^{e} states of the B atom, and ^{3}P^{e} states of the C atom.

Review of Rydberg Spectral Line Formation in Plasmas

The present review is dedicated to the problem of an array of transitions between highly-excited atomic levels. Hydrogen atoms and hydrogen-like ions in plasmas are considered here. The presented methods focus on calculation of spectral line shapes. Fast and simple methods of universal ionic profile calculation for the Hnα (Δn=1) and Hnβ (Δn=2) spectral lines are demonstrated. The universal dipole matrix elements formulas for the Hnα and Hnβ transitions are presented. A fast method for spectral line shape calculations in the presence of an external magnetic field using the formulas for universal dipole matrix elements is proposed. This approach accounts for the Doppler and Stark–Zeeman broadening mechanisms. Ion dynamics effects are treated via the frequency fluctuation model. The accuracy of the presented model is discussed. A comparison of this approach with experimental data and the results of molecular dynamics simulation is demonstrated. The kinetics equation for the populations of highly-excited ionic states is solved in the parabolic representation. The population source associated with dielectronic recombination is considered.

Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with such formulas are investigated in the JB-algebraic setting. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. We also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements.

Analytical solution for rovibrational spectra of diatomic polar molecules in the elliptically-polarized laser field

This article presents an analytical study of the shift/splitting of the rovibrational energy levels of diatomic polar molecules under the elliptically-polarized laser field. First, there are derived general formulas for the matrix elements of the energy shift operator between arbitrary rovibrational states. Then there are provided, as an example, explicit analytical results for the energy shifts/splitting of the rotational level J = 2 and illustrate them. Based on the derived analytical results, there is proposed a new method for the spectroscopic diagnostic of the parameters of the elliptically-polarized laser field inside low-temperature plasmas.

A note on fully commutative elements in complex reflection groups

In a Coxeter group, an element is fully commutative if all its reduced expressions can be obtained each from the other by interchanges of adjacent commuting generators. These elements were extensively studied by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving an enumeration of such elements in G(m,1,n). In this note, we prove a connection between fully commutative elements in Bn and in G(m,1,n), which allows us to characterize fully commutative elements in G(m,1,n) by pattern avoidance. Further, we present a counting formula for such elements in G(m,1,n).

A high-precision formula for mixed-order polygon elements based on SBFEM

A satisfactory balance between solution precision and computational efficiency can be achieved via mixed-order elements, however, the generality and robustness of commonly used methods can be further optimized. In this paper, a flexible and versatile mixed-order analysis method is developed based on Scaled Boundary Finite Element Method (SBFEM) theory, wherein one-dimensional linear and quadratic functions are combined to interpolate circular boundaries. The applicability and accuracy of the proposed method for different problems are explored and compared, and good accuracy and tolerance of element distortion performance are revealed. Moreover, compared with the quadratic element, computational efforts can be reduced considerably, enabling efficiency to be enhanced. Additionally, a polygon mixed-order formula can be directly derived, benefiting from the advantages of the SBFEM, wherein stronger universalities are rendered. In summary, the proposed method improves the flexibility and robustness of the mixed-order cells, and distinctive superiorities are revealed, providing an alternative option for the efficient and high-precision numerical analysis of vital structures.

NONRECURSIVE CANONICAL BASIS COMPUTATIONS FOR LOW RANK KASHIWARA CRYSTALS OF TYPE A

We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type A on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type A and rank e=2, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with Λ=aΛ0+aΛ1, we give formulae for the canonical basis elements for all the e-regular multipartitions with defects either k(a−k) or k(a−k)+2a, for 0≤k≤a.

The Method of Modal Parameters for the Wire Segments With Symmetrical Geometry and Singularity Expansion Method

The previously developed Method of Modal Parameters (MoMP) was applied to calculate the Singularity Expansion Method (SEM) poles of the single and double wire segments with symmetrical geometry: straight wire, circular wire and helix wire, which are important for EMC practice. The symmetry leads to the simplification of the corresponding formulas for matrix elements and allows to obtain SEM poles of the first, second and third layers, which correspond both to the known analytical results and numerical data. Moreover, for the first layer of the poles one can use a simplified variant of the method taking into account only diagonal elements of the p.u.l. impedance matrix.

On Elliptic Modular Foliations, II

We give an example of a one dimensional foliation $\cal F$ of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves $X_0(d), d\in\Bbb B$ minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on introducing faster algorithms to compute equations for small values of $d$.

On the wavefront sets associated with theta representations

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over p-adic fields. In particular, it is shown that the formula agrees with the existing work in the literature for various families of groups. We also recapitulate the results of an analogous formula in the archimedean case, which motivated the conjectural formula in the p-adic setting.

Central elements in affine modpHecke algebras via perverse -sheaves

AbstractLet$G$be a split connected reductive group over a finite field of characteristic$p > 2$such that$G_\text {der}$is absolutely almost simple. We give a geometric construction of perverse$\mathbb {F}_p$-sheaves on the Iwahori affine flag variety of$G$which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod$p$Hecke algebra to the center of the Iwahori mod$p$Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod$p$Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse$\mathbb {F}_p$-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly$F$-regular, and hence they are$F$-rational and have pseudo-rational singularities.

Ценообразование и бюджетирование в производстве товарного выпуска. Учет добавленной стоимости

This article discusses the organization of cost accounting in the context of items and elements of a manufacturing enterprise. For this, a general cost matrix is analyzed and a price formula for cost items and cost elements is developed. Particular attention is paid to the pricing and budgeting procedures, as well as the justification of the produced value added of the commodity output.

The R-Matrix of the Quantum Toroidal Algebra $$U_{q,t}(\overset{..}{gl}_1)$$ in the Fock Module

We propose a method to compute the $R$-matrix $R$ on a tensor product of Fock modules from coproduct relations in a Hopf algebra. We apply this method to the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ for which $R$ is currently not explicitly known. We show that the coproduct relations of $U_{q,t}(\overset{..}{gl}_1)$ reduce to a single elegant equation for $R$. Using the theory of symmetric Macdonald polynomials we show that this equation provides a recursive formula for the matrix elements of $R$.

Statistical properties of a return wave generated from an arbitrary rough target in atmospheric turbulence by a partially coherent beam with radial polarization

The interaction of a radially polarized partially coherent beam with an arbitrary rough target embedded in a bi-static LIDAR system in the presence of atmospheric turbulence was illustrated. The target could modelled as a combination of a Goodman-like thin random phase screen and a Gaussian lens. Within the framework of paraxial approximation, the analytical formulas for the cross-spectral density matrix elements of the beam at the source plane, at the target plane, just after the target and at the receiver plane were derived. It was found that the statistical properties of the return wave were dependent not only upon the transverse correlation width of the target, but also on the turbulence. In addition, it was possible to predict the target type by measuring the spectrum and polarization properties of the return beam in the receiver plane.

Stark–Zeeman and Blokhintsev Spectra of Rydberg Atoms

The problem of Rydberg atomic spectra in crossed electric F and magnetic B fields (combined Stark–Zeeman effects), as well as an oscillating electric field are under consideration. The main problem is the great array of radiative transitions between the Rydberg atomic states. The different versions of semiclassical approaches are applied for the solution of the problem. New approximate selection rules are established to simplify the matrix elements of radiative transitions, making it possible to reduce the complex expressions of Gordon’s hypergeometric series to the trivial trigonometric functions. Simple analytical formulas for the dipole matrix elements of a Rydberg hydrogen-like atom in external crossed electric and magnetic fields for Hnα are obtained. The specific calculations are presented for n = 10 to 9 transition both for parallel and perpendicular orientations of F–B fields. The result of Blokhintsev for a single Stark component in oscillating field is generalized for the array of radiative transitions between Rydberg atomic states. Such spectra with large values of modulation index are compared with static ones.

Inclusive probability of particle creation on classical backgrounds

The quantum theories of boson and fermion fields with quadratic nonstationary Hamiltoanians are rigorously constructed. The representation of the algebra of observables is given by the Hamiltonian diagonalization procedure. The sufficient conditions for the existence of unitary dynamics at finite times are formulated and the explicit formula for the matrix elements of the evolution operator is derived. In particular, this gives the well-defined expression for the one-loop effective action. The ultraviolet and infrared divergencies are regularized by the energy cutoff in the Hamiltonian of the theory. The possible infinite particle production is regulated by the corresponding counterdiabatic terms. The explicit formulas for the average number of particles N_D recorded by the detector and for the probability w(D) to record a particle by the detector are derived. It is proved that these quantities allow for no-regularization limit and, in this limit, N_D is finite and w(D)in [0,1). As an example, the theory of a neutral boson field with stationary quadratic part of the Hamiltonian and nonstationary source is considered. The average number of particles produced by this source from the vacuum during a finite time evolution and the inclusive probability to record a created particle are obtained. The infrared and ultraviolet asymptotics of the average density of created particles are derived. As a particular case, quantum electrodynamics with a classical current is considered. The ultraviolet and infrared asymptotics of the average number of photons are derived. The asymptotics of the average number of photons produced by the adiabatically driven current is found.

Far-zone behaviors of scattering-induced statistical properties of partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam**Project supported by the National Natural Science Foundation of China (Grant No. 11504286), the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2019JM-470), the Fund from the International Technology Collaborative Center for Advanced Optical Manufacturing and Optoelectronic Measurement, and the Science

In this study, we explore the far-zero behaviors of a scattered partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam irradiating on a deterministic medium. The analytical formula for the cross-spectral density matrix elements of this beam in the spherical coordinate system is derived. Within the framework of the first-order Born approximation, the effects of the scattering angle θ, the source parameters (i.e., the pulse duration T0 and the temporal coherence length Tcxx), and the scatterer parameter (i.e., the effective width of the medium σR) on the spectral density, the spectral shift, the spectral degree of polarization, and the degree of spectral coherence of the scattered source in the far-zero field are studied numerically and comparatively. Our work improves the scattering theory of stochastic electromagnetic beams and it may be useful for the applications involving the interaction between incident light waves and scattering media.