Explicit Integral Formula Research Articles

AN EXPLICIT INTEGRATION SCHEME BASED ON RECURSION AND MATRIX MULTIPLICATION FOR THE LINEAR CONVEX QUADRILATERAL ELEMENTS

In this paper we present some analytical integration formulae for computing integrals of rational functions of bivariate polynomial numerator with linear and bilinear denominator over a 2-square |ξ| = 1, |η| = 1 in the local parametric space (ξ, η). These integrals arise in finite element formulations of second order partial differential equations of plane and axisymmetric problems in continuum mechanics to computer the components of element stiffness matrices. In case of a rational integrals of n-th degree bivariate polynomial numerator with a linear of bilinear denominator there are exactly rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these integrals can be computed in two ways, accordingly we have presented an explicit and a recursive scheme. By use of the recursive scheme such integrals can be computed efficiently with less computational effort whenever (n + 1) integrals of order zero to n in one of the variates are known by explicit integration formulae. Integration formulae from zeroth to octic order are, for clarity and reference summarized in tabular forms. Finally to show the application of the derived formulae three application examples to compute the Prandtl stress function values and the torsional constant k are considered. A computer code based on the present integration scheme to obtain the element stiffness matrices for plane problems is also developed.

An Improved Symplectic Integrator for Nosé-Poincaré Thermostat

Simple, completely explicit symplectic integration formula are obtained for the Nose-Poincare thermostat; a Hamiltonian dynamics version of the Nose-Hoover thermostat. The time reversibility holds in each of recursion relations.

Strict Quantization of Solvable Symmetric Spaces

This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic symmetric spaces. Each of these quantizations gives rise to a field of (pre)-C*-algebras whose fibers are function algebras which are closed under the deformed product. The symmetry group of the symmetric space acts on each fiber by C*-algebra automorphisms.

Oscillatory Integral Formulae for Left-invariant Star Products on a Class of Lie Groups

We use star representation geometric methods to obtain explicit oscillatory integral formulae for strongly invariant star products on Iwasawa subgroups AN of SU(1,n)

Generalized variation of mappings with applications to composition operators and multifunctions

We study (set-valued) mappings of bounded Φ-variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the Φ-variation of a metric space valued mapping. We show that the linear span GVΦ(I;X) of the set of all mappings of bounded Φ-variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X → Y is a given mapping and the composition operator ℋ is defined by (ℋf)(t)=h(t,f(t)), where t∈I and f:I → X, we show that ℋ:GVΦ(I;X)→ GVΨ(I;Y) is Lipschitzian if and only if h(t,x)=h0(t)+h1(t)x, t∈I, x∈X. This result is further extended to multivalued composition operators ℋ with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded Φ-variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded Φ-variation.

The Gauss Theorem for Domain Decompositions in Sobolev Spaces

This paper belongs to a broad line of research leaded by Herrera, which encompasses a good number of numerical methods such as Localized Adjoint Method (LAM), Eulerian-Lagrangian LAM (ELAM) and Trefftz-Herrera Method. The results presented in this paper are required in order to incorporate Herrera's general theory in a Sobolev-space setting. In particular, this article introduces a class of partitions (or domain decompositions) whose internal boundaries belong to a category of manifolds with corners, here also presented. Then a version of Gauss (or divergence) theorem, in a wider sense, is established and an explicit integral formula is associated for any given linear partial differential operator L, its adjoint and concomitant. The structure of the bilinear concomitant induced by L is first determined. Then the required formula is given over that class of domain decompositions. Finally, an integral formula well on the way of the Green-Herrera formula is settled.

Fundamental strings in D–D brane systems

We study conformal field theory correlation functions relevant for string diagrams with open strings that stretch between several parallel branes of different dimensions. In the framework of conformal field theory, they involve boundary condition changing twist fields which intertwine between Neumann and Dirichlet conditions. A Knizhnik–Zamolodchikov-like differential equation for correlators of such boundary twist fields and ordinary string vertex operators is derived, and explicit integral formulas for its solutions are provided.

Optimal regularity for $$\bar \partial _b $$ onCR manifoldsonCR manifolds

In this paper a new explicit integral formula is derived for solutions of the tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal estimates in the Lipschitz norms are obtained.

Synthetic division based integration of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0⩽ ξ, η⩽1, ξ+ η⩽1} in the local parametric space ( ξ, η)

The domain of real problems in mechanics often contains curved boundaries. Curved boundaries are often more accurately modelled by curved finite elements than by straight edged elements, as straight sides are perfectly satisfactory if the domain has a polygonal boundary. Because fewer curved elements are required, the effort needed to obtain a solution is usually reduced. If some parts of the boundary are curved, however, elements with at least one curved side are desirable. Our aim in this paper is to consider the triangular element with two straight sides and one curved side. This paper is concerned with explicit formulae for evaluating integrals of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0⩽ξ, η⩽1, ξ+η⩽1} in the local parametric two dimensional space ( ξ, η). These integrals arise in finite element formulations of second order linear partial differential equations by use of triangular element with two straight sides and one curved side of quadratic variation which often require relatively large numerical effort to integrate. The curved elements considered here are the four node, six node and ten node triangular elements with one curved side of quadratic variation and the other two sides have straight edges under the isoparametric and subparametric transformations, respectively. We have shown that by use of a method similar to synthetic division, the rational integrals of nth order bivariate polynomial numerator with a linear denominator having ( n+1)( n+2)/2 integrals can be reduced to rational integrals of nth order polynomial numerator in one variate with the same linear denominator having ( n+1) integrals and a simple integral of bivariate polynomial expression containing n( n+1)/2 terms (which is free from denominator), and this amounts to a substantial reduction in the numerical effort for such integrals. The explicit analytical integration formulae (obtained) up to sextic polynomial numerator in bivariates ξ, η due to different element geometry are, for clarity and reference, summarised in tables. Finally three application examples are also considered. For the first example we have used integration formulae derived in all theorems of this paper and explained the detailed computational scheme. For the other two examples computational scheme follows in a similar way. It is observed in the solution of all problems that the displacements and torsional constants are satisfactory when components of all element matrices are calculated by analytical integration formulae derived in this paper. It is also observed that in the calculation of components of element matrices by using numerical integration formulae (e.g., 7-point and 13-point rule) much discrepancy occurs if the element geometry is coarse with a concave curve side. But it is found that the analytical integration technique is always consistent. Therefore the symbolic integration formulae presented in this paper are reliable and may lead to an easy and systematic incorporation of element matrices required in the finite element solution procedure.

A Decoupling Solution Method for Implicit Numerical Integration of Constrained Mechanical Systems*

A decoupling solution method is presented for implicit numerical integration of constrained multibody systems. Compared to implicit numerical integration methods, explicit numerical integration methods usually spend less computation time per iteration, since the equations of motion and an explicit integration formula coupled with the constraint manifolds are solved separately. However, the integration step size can be excessively small for highly nonlinear or stiff problems, due to the small stability region of the explicit method. In contrast, implicit numerical integration methods do not have the problem of excessive small step size. Implicit integration methods, however, generally require one to solve a large system equations simultaneously. The solution method proposed in this paper eliminates the problem of large system equations, while maintaining the advantage of the implicit method. Furthermore, the proposed method provides a well-conditioned iteration matrix for the implicit method, which is important to obtain accurate and stable solutions.

Inversion for the Radon line transform in higher dimensions

Twistor theory is used to develop a new procedure for inverting the Radon line transform in three or more dimensions. First, the inversion problem is reduced to a ∂–problem on the complement of real projective space in complex projective space. The ∂–problem in turn is solved by means of an explicit integral formula.

Boundary integration of polynomials over an arbitrary linear hexahedron in euclidean three-dimensional space

This paper is concerned with explicit integration formulas and algorithms for computing volume integrals of trivariate polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space. Three different approaches are discussed. The first algorithm is obtained by transforming a volume integral into a sum of surface integrals and then into convenient and computationally efficient line integrals. The second algorithm is obtained by transforming a volume integral into a sum of surface integrals over the boundary quadrilaterals. The third algorithm is obtained by transforming a volume integral into a sum of surface integrals over the triangulation of boundary. These algorithms and finite integration formulas are then followed by an application example, for which we have explained the detailed computational scheme. The symbolic finite integration formulas presented in this paper may lead to efficient and easy incorporation of integral properties of arbitrary linear polyhedra required in the engineering design process.

Integration of rational functions of bivariate polynomial numerators with linear denominators over a (−1,1) square in a local parametric two-dimensional space

This paper is concerned with explicit formulas and algorithms for computing integral of rational function of bivariate polynomial numerators with linear denominators over a (−1,1) square in the local parametric space. These integrals arise in finite element formulations of second-order partial differential equations. Explicit evaluation of these integrals produce analytical finite element relations, provided that the original element geometry is restricted to a linear convex quadrilateral. We have also presented two algorithms, either of these can be used to compute n(n + 1) 2 integrals of 0th to nth order bivariate polynomial numerator whenever ( n + 1) such integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulas. The analytical quadrature formulas from cubic to quintic monomial numerator in one of the variâtes due to different element geometry are, for clarity and reference, summarized in tabular forms. All the explicit formulas are constructed with three simple functions of element nodal values. They can be easily coded and save much computation time besides their inherent merit on numerical accuracy associated with analytical integration. We have also derived some integration formulas for the product of global derivatives which has applications to second-order linear partial differential equations in a variety of disciplines. Finally, an application example to compute the torsional constant K for an equilateral triangular cross-section is also considered for which we have explained the detailed computational scheme.

Radon transform and the Cavalieri condition: a cohomological approach

We apply the theory of integral transforms for sheaves andD-modules to the study of the real Radon transform. By identifying abstract adjunction formulas with the explicit integral formulas appearing in the literature, we give, in particular, a cohomological interpretation of Gelfand’s Cavalieri condition, and of Helgason’s support theorem.

Boundary integration of polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space

This paper is concerned with explicit integration formulas and algorithms for computing integrals of trivariate polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space. This basic three-dimensional integral governing the problem is transformed to surface integrals by use of the divergence theorem. The resulting two-dimensional integrals are then transformed into convenient and computationally efficient line integrals. These algorithms and explicit finite integration formulas are followed by an application—example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with previous works. Further, it is shown that the present algorithms are much simpler and more economical as well, in terms of arithmetic operations. The symbolic finite integration formulas presented in this paper may lead to an easy incorporation of geometric properties of solid objects, for example, the centre of mass, moment of inertia, etc. required in the engineering design process as well as several applications of numerical analysis where integration is required, for example in the finite element and boundary integral equation methods.

Integration of polynomials over n-dimensional linear polyhedra

This paper is concerned with explicit integration formulae for computing integrals of n-variate polynomials over linear polyhedra in n-dimensional space R n . Two different approaches are discussed; the first set of formulae is obtained by mapping the polyhedron in n-dimensional space R n into a standard n-simplex in R n , while the second set of formulae is obtained by reducing the n-dimensional integral to a sum of n − 1 dimensional integrals which are n + 1 in number. These formulae are followed by an application example for which we have explained the detailed computational scheme. The symbolic integration formulae presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, such as, for example, volume, centre of mass, moments of inertia etc., required in engineering design problems.

The green function for the weighted biharmonic operator Δ(1−∣z∣2)−αΔ, and factorization of analytic functions, and factorization of analytic functions

An explicit integral formula is obtained for the Green function of the weighted biharmonic operator Δ(1−∣z∣2)−αΔ in the unit disk of the complex plane for the case α ∈ (−1, 0). The formula shows the positivity of the Green function. This is a basis for a theorem on factorization of analytic functions in the weighted Bergman spaces with the weights ω(z)=(1−∣z∣2)α as products of a nonvanishing function and a function of special form responsible for the zeros. Bibliography: 16 titles.

Cubic correspondences arising from G 2

We use the automorphic realization of the minimal representation (the theta representation) of the three fold cover of G 2 to construct a correspondence of automorphic forms between members of dual pairs inside G 2 . The dual pairs under consideration are (SL 2 , SL 2 ) and (SL 3 , Z 3 ). In the first case, we obtain the Shimura correspondence between the three fold cover of SL 2 and SL 2 , through explicit integral formulae. In the second case, we decompose theta over the three fold cover of SL 3 , both locally and globally. The constituents are parametrized by their central characters and are cuspidal, except (essentially) when the central character is trivial. We thus obtain "cuspidal theta representations" and "the unramified theta representation" of the three fold cover of SL 3 .

Determination of the global and regional gravitational fields from satellite and balloon gradiometry missions

Future satellite gradiometry missions (e. g. GOCE, STEP) and projects for low altitude freefalling probes, with gravity gradiometers on board, will be the most promising experiments for refining the Earth's gravitational field. An optimal approach is developed for solving the spatial gradiometry problems. The full magnitude Δ Γ of the disturbance of the gravity gradient vector, with respect to its normal value, is chosen as the observable. Correspondingly, a unique spatial boundary value relation (A) is derived, Δ Γ = ΔΓ N , where ΔΓ N is the truncated to the degree N spherical harmonic series with the unknown global potential coefficients C n,m . Based on (A), from satellite gradiometry missions the potential coefficients can be evaluated by the least squares technique. Besides, an explicit integral formula is derived for C n,m . Low altitude balloon missions with free falling gradiometers will allow to evaluate the regional surface gravity anomaly Δg. For solving this problem from (A) another basic relation (B) is derived by means of downward continuation of Δ γ from a spatial observation domain to its projection onto a terrain region of interest. Formula (B) represents Δg in form of a sum of a Stokes integral, depending on Δ Γ , and correction terms due to downward continuation and the Earth's nonsphericity. In addition to (B) corresponding to a point-wise approach, a mean-value relation is derived by averaging Δ Γ over observation altitudes of a free falling gradiometer.

Integration of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space

In this paper, we present explicit integration formulas and algorithms for computing integrals of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space. Two different approaches are discussed: the first algorithm/formula is obtained by mapping the arbitrary tetrahedron into a unit orthogonal tetrahedron, while the second algorithm/formula computes the required integral as a sum of four integrals over the unit triangle. These algorithms/formulas are followed by an example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with the previous work. Further, it is shown that the present algorithms are much simpler and more economical as well in terms of arithmetic operations.