Hilbert Formulas Research Articles

ANALYTICAL SOLUTION OF THE HILBERT SINGULAR INTEGRAL EQUATION WITH RESPECT TO THE HILBERT FORMULA

The objective of this paper is to solve the Hilbert singular integral equation of first and second kind by using the Hilbert formula and some known results are obtained as special cases. Applications of these singular integral equations to Boundary value problems of Elasticity and allied subjects are well known.

Integral formulas of the Hilbert, Poincaré-Bertrand, Schwarz and Poisson type for the β–analytic function theory

In the present work we obtain some analogues of the Hilbert formulas on the unit circle and on the upper half-plane for the theory of solutions of a special case of the Beltrami equation in C to be referred as β-analytic functions. Furthermore, a Poincaré–Bertrand formula related to the β-Cauchy singular integral over a closed Jordan curve is derived and it is used to derive the corresponding Schwarz and Poisson formulas.

Correction to: Poincaré–Bertrand and Hilbert formulas for the Cauchy–Cimmino Singular Integrals

Unfortunately, the name of the communicating editor has been published incorrectly. The correct name is Rafal Ablamowicz

On the Hilbert formulas and of change of integration order for some singular integrals in the unit circle

On the Hilbert formulas and of change of integration order for some singular integrals in the unit circle

On the Hilbert formulas on the unit circle for α-hyperholomorphic function theory

ABSTRACTWe obtain some analogues of the Hilbert formulas on the unit circle for -hyperholomorphic function theory in for being a complex quaternionic number. The obtained formulas relate a pair of components of the boundary value of an -hyperholomorphic function in the unit disc with the other pair of components and, hence, being analogous to the case of the theory of functions of one complex variable.

Poincaré–Bertrand and Hilbert formulas for the Cauchy–Cimmino singular integrals

The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincare–Bertrand formula related to the Cauchy–Cimmino singular integrals over piecewise Lyapunov surfaces in $$\mathbb {R}^4$$ is derived with recourse to arguments involving quaternionic analysis. Furthermore, this paper obtains some analogues of the Hilbert formulas on the unit 3-sphere and on the 3-dimensional space for the theory of Cimmino system.

Galois module structure of the square root of the inverse different over maximal orders

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a \mbox{finite group of} odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ be the square root of the inverse different of $K_h/K$, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then $A_h$ is locally free over $\mathcal{O}_{K}G$ by a result of B. Erez, in which case it determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. Such a class in $\mbox{Cl}(\mathcal{O}_KG)$ is said to be $A$-realizable, and tame $A$-realizable if $K_h/K$ is tame. Let $\mathcal{A}(\mathcal{O}_KG)$ and $\mathcal{A}^t(\mathcal{O}_KG)$ denote the sets of all $A$-realizable classes and tame $A$-realizable classes, respectively. For $G$ abelian, we will show that the two sets $\mathcal{A}(\mathcal{O}_KG)$ and $\mathcal{A}^t(\mathcal{O}_KG)$ are equal when extended scalars to the maximal order in $KG$.

On the Galois module structure of the square root of the inverse different in abelian extensions

Let K be a number field with ring of integers OK and let G be a finite group of odd order. Given a G-Galois K-algebra Kh, let Ah denote its square root of the inverse different, which exists by Hilbert's formula. If Kh/K is weakly ramified, then a result of Erez implies that Ah is locally free over OKG and hence defines a class in the locally free class group Cl(OKG) of OKG. In the case that G is abelian, we study the collection of all such classes and show that a subset of them is in fact a subgroup of Cl(OKG).

On the Hilbert operator and the Hilbert formulas on the unit sphere for the time-harmonic Maxwell equations

In this work we establish some analogues of the Hilbert formulas on the unit sphere for the theory of time-harmonic (monochromatic) electromagnetic fields. Our formulas relate one of the components of the limit value of a time-harmonic electromagnetic field in the unit ball to the rest of components. The obtained results are based on the close relation between time-harmonic solutions of the Maxwell equations and the three-dimensional α-hyperholomorphic function theory. Hilbert formulas for α-hyperholomorphic function theory for α being a complex number are also obtained, such formulas relate a pair of components of the boundary value of an α-hyperholomorphic function in the unit ball to the other pair of components, in an analogy with what happens in the case of the theory of functions of one complex variable.

On the Hilbert formulas on the unit sphere for the time-harmonic relativistic Dirac bispinors theory

In this paper there are established some analogues of the Hilbert formulas on the unit sphere for the theory of time-harmonic (monochromatic) relativistic Dirac bispinors. The formulas relate a pair of the components of the limit value of a time-harmonic Dirac bispinor in the unit ball to the other pair of components. The obtained results are based on the intimate connection between time-harmonic solutions of the relativistic Dirac equation and the three-dimensional α-hyperholomorphic function theory. Hilbert formulas for α-hyperholomorphic function theory for α being a complex quaternionic number are also presented, such formulas relate a pair of components of the boundary value of an α-hyperholomorphic function in the unit ball to the other pair of components, in an analogy with what is known for the case of the theory of functions of one complex variable.

Hilbert formulas for 𝑟-analytic functions and the Stokes flow about a biconvex lens

The so-called r r -analytic functions are a subclass of p p -analytic functions and are defined by the generalized Cauchy-Riemann system with p ( r , z ) = r p(r,z)=r . In the system of toroidal coordinates, the real and imaginary parts of an r r -analytic function are represented by Mehler-Fock integrals with densities, which are assumed to be meromorphic functions. Hilbert formulas, establishing relationships between those functions, are derived for the domain exterior to the contour of a biconvex lens in the meridional cross-section plane. The derivation extends the framework of the theory of Riemann boundary-value problems, suggested in our previous work, to solving the three-contour problem for the case of meromorphic functions with a finite number of simple poles. For numerical calculations, Mehler-Fock integrals with Hilbert formulas reduce to the form of regular integrals. The 3D problem of the axially symmetric steady motion of a rigid biconvex lens-shaped body in a Stokes fluid is solved, and the Hilbert formula for the real part of an r r -analytic function is used to express the pressure in the fluid via the vorticity analytically. As an illustration, streamlines and isobars about the body, the vorticity and pressure at the contour of the body and the drag force exerted on the body by the fluid are calculated.

Hilbert Formulas for r-Analytic Functions in the Domain Exterior to Spindle

Hilbert formulas for an r-analytic function, defined by a generalized Cauchy--Riemann system in the domain exterior to the contour of a spindle in the meridional cross-section plane, have been derived. The derivation is based on the theory of Riemann boundary-value problems for analytic functions. For numerical calculations, Fourier integrals with Hilbert formulas representing the real and imaginary parts of the r-analytic function have been reduced to the form of regular integrals. The problem of the axially symmetric steady motion of a rigid spindle-shaped body in a Stokes fluid has been solved, and the pressure in the fluid has been expressed analytically based on a Hilbert formula. As an illustration, streamlines about the body, vortex and pressure functions at the contour of the body, and the drag force, exerted on the body by the fluid, have been calculated.

On the hilbert operator for α-hyperholomorphic function theory in

The purpose of this article is to obtain some generalizations of the usual Hilbert formulas and Hilbert operator, for two-dimensional α-hyperholomorphic function theory. Our Hilbert operator relates a pair of components of the limit function of the α-hyperholomorphic function with the other pair of components, in an exact anology with what the usual Hilbert operator does for the limit function of a function holomorphic in the half-plane.

Stress–energy–momentum tensors in higher order variational calculus

Given a variational problem defined by a natural Lagrangian density Lω on the k-jet extension J k ( Y/ X) of a natural bundle p: Y→ X over an n-dimensional manifold X, oriented by a volume element ω, a stress–energy–momentum tensor T( s) is constructed for each section s∈ Γ( X, Y) from the multimomentum map μ Θ:Γ(X,Y)→ Hom R ( X(X),Ω n−1(X)) associated to any Poincaré–Cartan form Θ and to the natural lifting of vector fields X(X) to the bundle Y→ X. The characterization made for T( s) gives an intrinsic expression of this tensor as well as a generalization of the classical Belinfante–Rosenfeld formula. This tensor satisfies the typical properties of a stress–energy–momentum tensor: Diff( X)-covariance, Hilbert formula, conservation law, etc. Furthermore, it plays the expected role in the theory of minimal gravitational interactions.

The different and differentials of local fields with imperfect residue fields

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

The Hilbert theory of transport phenomena in multiple gas mixtures

The Hilbert solution of the Boltzmann equation for a multiple gas mixture is presented. It is shown that the first approximation of the Hilbert theory yields the Euler equations for ideal flow. The Hilbert field equations of the second approximation are derived and the transport processes that appear in this approximation are discussed in a manner similar to that given in Chapman and Cowling. The Hilbert formulas for the transport coefficients of a multiple gas mixture are obtained and are seen to be precisely the same formulas of Chapman and Cowling obtained by the Chapman–Enskog method.