Formulas For Elements Research Articles

ZETA ELEMENTS IN DEPTH 3 AND THE FUNDAMENTAL LIE ALGEBRA OF THE INFINITESIMAL TATE CURVE

This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for $\text{SL}_{2}(\mathbb{Z})$. We write down explicit formulae for zeta elements $\unicode[STIX]{x1D70E}_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst–Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.

Construction of R-matrices for symmetric tensor representations related to

In this paper we construct a new factorized representation of the R-matrix related to the affine algebra for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit formulas for the matrix elements of the R-matrix and give a simple proof that a ‘twisted’ R-matrix is stochastic. We also discuss symmetries of the R-matrix, its degenerations and compare our formulas with other results available in the literature.

Somos-4 and elliptic systems of sequences

A general formula for elements of double Somos-4 sequences is obtained. A sufficient integrality condition for such sequences is presented.

Reconstruction of the Hermitian matrix by its spectrum and spectra of some number of its perturbations

Explicit formulas for matrix elements of the Hermitian matrix are found through a spectrum of this matrix and spectra of some number of its perturbations. A dependence of sufficient number of perturbations from the structure of the matrix and the kind of perturbations is established. It is shown that for arbitrary matrix needed number of perturbations is of N2, where N is an order of the matrix. In the case, when the number and locations of zero elements of the matrix is known, needed number of perturbations decreases essentially.

Multiple attribute group decision making based on interval-valued hesitant fuzzy information measures

Under the interval-valued hesitant fuzzy environment, we investigate a multiple attribute group decision making (MAGDM) method on the basis of some information measures. We first introduce three axiomatic definitions of information measures under interval-valued hesitant fuzzy environment, including the entropy, similarity measures and cross-entropy. Several information measure formulas for interval-valued hesitant fuzzy elements (IVHFEs) are further constructed, which is based on the continuous ordered weighted averaging (COWA) operator. Then, the relationship among the entropy, similarity measures and cross-entropy is discussed, from which we find that three information measures can be transformed by each other based on their axiomatic definitions. The programming model is established to determine optimal weight of attribute with the principle of minimum entropy and maximum cross-entropy. Furthermore, an approach to MAGDM is developed, in which the attribute values take the form of IVHFEs. Finally, a numerical example for emergency risk management (ERM) evaluation is provided to illustrate the application of the developed approach.

Entropy measures for hesitant fuzzy sets and their application in multi-criteria decision-making

Entropy is used to measure the uncertain degree of fuzzy sets and has been widely used in many fields. This paper introduces an axiomatic definition of entropy measure and a novel entropy formula for hesitant fuzzy elements (HFEs). Afterwards, a general form of entropy measures for HFEs is proposed, from which a family of concrete entropy formulas for HFEs can be derived. Compared with the existing ones, these formulas can measure both fuzziness and hesitation of HFEs, as a result, the uncertain information can be described in a more appropriate manner. The proposed axiomatic definition and entropy formulas are used to define the entropy measure for hesitant fuzzy sets. A multi-criteria decision making model which uses the proposed entropy measures for HFEs to compute the criteria weights and obtain a ranking of alternatives is introduced.

Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence.

Recently, we introduced a new class of radially polarized cosine-Gaussian correlated Schell-model (CGCSM) beams of rectangular symmetry based on the partially coherent electromagnetic theory [Opt. Express23, 33099 (2015)]. In this paper, we extend the work to study the second-order statistics such as the average intensity, the spectral degree of coherence, the spectral degree of polarization and the state of polarization in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power law α and an effective anisotropic parameter. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in anisotropic turbulence are derived. It is found that the second-order statistics are greatly affected by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic turbulence on the beam parameters mainly appears nearα=3.1, and decreases with the increase of the anisotropic parameter. Furthermore, the polarization state exhibits self-splitting property and each beamlet evolves into a radially polarized structure in the far field. Our work enriches the classical coherence theory and may be important for free-space optical communications.

Vector problem of electromagnetic wave diffraction by a system of inhomogeneous volume bodies, thin screens, and wire antennas

The vector problem of time-harmonic electromagnetic wave diffraction by a system of non-intersecting solid inhomogeneous bodies, infinitely thin perfectly conducting screens and wire antennas in the semi-classical formulation is considered. The original boundary value problem for Maxwell’s equations is reduced to a new system of singular integral equations over the volume domains, the screen surfaces and antennas, that is, over compact domains of dimension 3, 2, and 1. To obtain the equations, the fields scattererd by the bodies and the screens were represented via volume and surface potentials, respectively. To solve the integral equations approximately, the Galerkin method is applied; basis functions on the body, the screens and antennas are introduced as well as formulas for matrix elements in Galerkin method. Combining potential theory and pseudodifferential calculus allowed to obtain results on solvability of new system of integro-differential equations, as well as to theoretically justify the Galerkin method. To solve the problem of diffraction by obstacles of complex shape, novel subhierarchical approach is applied.

Pion generalized parton distributions within a fully covariant constituent quark model

We extend the investigation of the Generalized Parton Distribution for a charged pion within a fully covariant constituent quark model, in two respects: (i) calculating the tensor distribution and (ii) adding the treatment of the evolution, needed for achieving a meaningful comparison with both the experimental parton distribution and the lattice evaluation of the so-called generalized form factors. Distinct features of our phenomenological covariant quark model are: (i) a 4D Ansatz for the pion Bethe-Salpeter amplitude, to be used in the Mandelstam formula for matrix elements of the relevant current operators, and (ii) only two parameters, namely a quark mass assumed to hold $m_q=~220$ MeV and a free parameter fixed through the value of the pion decay constant. The possibility of increasing the dynamical content of our covariant constituent quark model is briefly discussed in the context of the Nakanishi integral representation of the Bethe-Salpeter amplitude.

The Voronoi inverse mapping

Given an arbitrary set T in the Euclidean space whose elements are called sites, and a particular site s, the Voronoi cell of s, denoted by VT(s), consists of all points closer to s than to any other site. The Voronoi mapping of s, denoted by ψs, associates to each set T∋s the Voronoi cell VT(s) of s w.r.t. T. These Voronoi cells are solution sets of linear inequality systems, so they are closed convex sets. In this paper we study the Voronoi inverse problem consisting in computing, for a given closed convex set F∋s, the family of sets T∋s such that ψs(T)=F. More in detail, the paper analyzes relationships between the elements of this family, ψs−1(F), and the linear representations of F, provides explicit formulas for maximal and minimal elements of ψs−1(F), and studies the closure operator that assigns, to each closed set T containing s, the largest element of ψs−1(F), where F=VT(s).

Geometric elements and classification of quadrics in rational Bézier form

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational Bézier triangular form (such as the center, the conic at infinity, the vertex and the axis of paraboloids and the principal planes), using just the control vertices and the weights for the quadric patch. The results are extended also to quadric tensor product patches. Our results rely on using techniques from projective algebraic geometry to find suitable bilinear forms for the quadric in a coordinate-free fashion, considering a pencil of quadrics that are tangent to the given quadric along a conic. Most of the information about the quadric is encoded in one coefficient, involving the weights of the patch, which allows us to tell apart oval from ruled quadrics. This coefficient is also relevant to determine the affine type of the quadric. Spheres and quadrics of revolution are characterized within this framework.

Fibonacci, and Lucas Pascal Triangles

In this paper, we give explicit formulas for elements of the Fibonacci, and Lucas Pascal triangles. The structure of these objects and Pascal's original triangle coincide. Keeping the rule of addition, we replace both legs of the Pascal triangle by the Fibonacci sequence, and the Lucas sequence, respectively. At the end of the study we describe how to determine such a formula for any binary recurrence $\{G_n\}^{\infty}_{n=0}$ satisfying $G_n=G_{n−1}+G_{n−2}$. Other scattered results are also presented.

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non-acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra.

Intertwiners of $U^{\prime }_q\bigl (\widehat{sl}(2)\bigr )$Uq′sl̂(2)-representations and the vector-valued big q-Jacobi transform

Linear operators R are introduced on tensor products of evaluation modules of \documentclass[12pt]{minimal}\begin{document}$U^{\prime }_q\bigl (\widehat{sl}(2)\bigr )$\end{document}Uq′sl̂(2) obtained from the complementary and strange series representations. The operators R satisfy the intertwining condition on finite linear combinations of the canonical basis elements of the tensor products. Infinite sums associated with the action of R on six pairs of tensor products are evaluated. For two pairs, the sums are related to the vector-valued big q-Jacobi transform of the matrix elements defining the operator R. In one case, the sums specify the action of R on the irreducible representations present in the decomposition of the underlying indivisible sum of \documentclass[12pt]{minimal}\begin{document}$U_q\bigl (sl(2)\bigr )$\end{document}Uqsl(2)-tensor products. In both cases, bilinear summation formulae for the matrix elements of R provide a generalization of the unitarity property. Corresponding results are given for the remaining pairs.

Analysing the behaviour of partially coherent divergent Gaussian beams propagating through oceanic turbulence

Theoretical study of propagation behaviour of partially coherent divergent Gaussian beams through oceanic turbulence has been performed. Based on the previously developed knowledge of propagation of a partially coherent beam in atmosphere, the spatial power spectrum of the refractive index of ocean water, extended Huygens–Fresnel principle and the unified theory of coherence and polarization, analytical formulas for cross-spectral density matrix elements are derived. The analytical formulas for intensity distribution, beam width and spectral degree of coherence are determined by using cross-spectral density matrix elements. Then, the effects of some source factors and turbulent ocean parameters on statistical properties of divergent Gaussian beam propagating through turbulent water are analysed. It is found that beam’s statistical propagation behaviour is affected by both environmental and source parameters variations.

Matrix elements of one-body and two-body operators between arbitrary HFB multi-quasiparticle states

We present new formulae for the matrix elements of one-body and two-body physical operators, which are applicable to arbitrary Hartree–Fock–Bogoliubov wave functions, including those for multi-quasiparticle excitations. The testing calculations show that our formulae may substantially reduce the computational time by several orders of magnitude when applied to many-body quantum system in a large Fock space.

The complex variable fast multipole boundary element method for the analysis of strongly inhomogeneous media

The paper aims to develop the efficient method tailored for accurate, robust and stable calculations of 2D local fields in strongly inhomogeneous materials with arbitrary interaction conditions on multiple contacts of structural blocks. The method is also to be of immediate use for solving homogenisation problems. To reach the goal we employ: (i) special forms of the complex variable singular and hypersingular integral equations with the densities representing those physical quantities, which enter the contact conditions; (ii) circular-arc boundary elements (in addition to straight elements) for smooth approximation of smooth parts of the external boundaries and contacts; (iii) higher order approximations of densities, which account for arbitrary power asymptotics of physical fields near singular points (crack tips, corner points, common apexes of structural blocks); (iv) analytical recurrent evaluation of all influence coefficients; (v) analytical recurrent evaluation of all moments; (vi) the complex variable fast multipole method (CV FMM), for solving the resulting system of the complex variable boundary element method (CV BEM), with large number (up to million) of unknowns. As a result, we obtain free of numerical integration, higher-order CV fast multipole boundary element method (CV FM-BEM) for a medium with multiple structural elements and multiple singular points. In the due course, we suggest the simplified starting quadrature formulae for singular boundary elements, the adjustment of the procedure for building the hierarchical tree and the proper choice of the key parameters of the developed CV FM-BEM: the number of elements in a leaf; the number of moments in the truncated Taylor expansions; the reasonable tolerance, when iteratively solving the system by the FMM. Numerical examples illustrate the abilities of the method developed, as regard to local fields in strongly inhomogeneous structures with multiple singular points. The study of local fields shows application of the method to finding extreme distributions of stress intensity factors in a medium with many cracks, which may intersect. The homogenisation problem is solved, as well.

Lie-series for orbital elements: I. The planar case

Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor-coefficients of the solution as a set of recurrence relations. In this paper we present these recurrence formulae for orbital elements and other integrals of motion for the planar $N$-body problem. We show that if the reference frame is fixed to one of the bodies -- for instance to the Sun in the case of the Solar System --, the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies.

SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra

The analysis of the transfer matrices associated with the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method, which generalizes to these integrable quantum models the method first introduced by Sklyanin. For representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras, for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of the transfer matrix spectrum (eigenvalues and eigenstates) and its non-degeneracy. Moreover, we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states, which particularly apply to transfer matrix eigenstates. A comparison of our analysis of the 8-vertex reflection algebra with that of (Niccoli G 2012 J. Stat. Mech. P10025, Faldella S et al 2014 J. Stat. Mech. P01011) for the 6-vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products, which exists for these two different realizations of the reflection algebra once they are described by the SOV method. As will be shown in a future publication, this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8-vertex and 6-vertex type.

Numerical study on fatigue crack growth at a web-stiffener of ship structural details by an objected-oriented approach in conjunction with ABAQUS

It is necessary to manage the fatigue crack growth (FCG) once those cracks are detected during in-service inspections. This is particular critical as high strength steels are being used increasingly in ship and offshore structures. In this paper, a simulation program (FCG-System) is developed utilizing the commercial software ABAQUS with its object-oriented programming interface to simulate the fatigue crack path and to compute the corresponding fatigue life. In order to apply FCG-System in large-scale marine structures, the substructure modeling technique is integrated in the system under the consideration of structural details and load shedding during crack growth. Based on the nodal forces and nodal displacements obtained from finite element analysis, a formula for shell elements to compute stress intensity factors is proposed in the view of virtual crack closure technique. Neither special singular elements nor the collapsed element technique is used at the crack tip. The established FCG-System cannot only treat problems with a single crack, but also handle problems with multiple cracks in case of simultaneous but uneven growth. The accuracy and the robustness of FCG-System are demonstrated by two illustrative examples. No stability and convergence difficulties have been encountered in these cases and meanwhile, insensitivity to the mesh size is confirmed. Therefore, the FCG-System developed by authors could be an efficient tool to perform fatigue crack growth analysis on marine structures.