Odd-dimensional Spaces Research Articles

An application of radon inversion to diffraction tomography

Basic equations of linear computer tomography are considered; a connection of the mathermatical formulation of problems of computer tomography with problems of integral geometry is indicated; the properties of the Radon transform for even- and odd-dimensional spaces are analyzed, and applications of Radon inversion to problems of diffraction tomography are presented. Bibliography: 5 titles.

Pointwise Fourier Inversion on Rank One Symmetric Spaces and Related Topics

This paper develops necessary and sufficient conditions for pointwise inversion of Fourier transforms on rank one symmetric spaces of non-compact type for functions in the piecewise smooth category. This extends results of Pinsky for isotropic Riemannian manifolds of constant curvature. Methodologically, a similar result on the Jacobi transform and a transplantation scheme from even dimensional spaces to certain odd dimensional real hyperbolic spaces naturally enter the picture.

Minimal orbits of metrics

The group of diffeomorphisms of a compact manifold acts isometrically on the space of Riemannian metrics with its L 2 metric. Following Arnaudon and Paycha (1995) and Maeda, Rosenberg and Tondeur (1993), we define minimal orbits for this action by a zeta function regularization. We show that odd dimensional isotropy irreducible homogeneous spaces give rise to minimal orbits, the first known examples of minimal submanifolds of infinite dimension and codimension. We also find a flat 2-torus giving a stable minimal orbit. We prove that isolated orbits are minimal, as in finite dimensions.

Slowly decaying solutions for semilinear wave equations in odd space dimensions

Slowly decaying solutions for semilinear wave equations in odd space dimensions

On point to point reflection of harmonic functions across real-analytic hypersurfaces in ℝ n

Let Γ be a non-singular real-analytic hypersurface in some domainU ⊂ ℝ n and let Har0(U, Γ) denote the linear space of harmonic functions inU that vanish on Γ. We seek a condition onx 0,x 1 ∈U/Γ such that the reflection law (RL)u(x 0)+Ku(x 1)=0, ∀u∈Har0(U, Γ) holds for some constantK. This is equivalent to the class Har0 (U, Γ) not separating the pointsx 0,x 1. We find that in odd-dimensional spaces (RL)never holds unless Γ is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR—strong Study reflection—condition), which we described both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P.R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.

THE ζ FUNCTION ANSWER TO PARITY VIOLATION IN THREE-DIMENSIONAL GAUGE THEORIES

We study parity violation in (2+1)-dimensional gauge theories coupled to massive fermions. Using the ζ function regularization approach we evaluate the ground state fermion current in an arbitrary gauge field background, showing that it gets two different contributions which violate parity invariance and induce a Chern–Simons term in the gauge field effective action. One is related to the well-known classical parity breaking produced by a fermion mass term in three dimensions; the other, already present for massless fermions, is related to peculiarities of gauge-invariant regularization in odd-dimensional spaces.

On the critical decay and power for semilinear wave equtions in odd space dimensions

In this paper we study global behaviors of solutions of initial value problem to wave equations with power nonlinearity. We shall derive space-time decay estimates according to decay rates of the initial data with low regularity (in classical sense). Indeed we can control $L^\infty$-norm of a solution in high dimension, provided the initial data are radially symmetric. This enables us to construct a global solution under suitable assumptions and to obtain an optimal estimate for a lifespan of a local solution.

Index Theory, Gerbes, and Hamiltonian Quantization

We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field.

Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces

This work is a continuation of the study of higher-dimensional generalizations of the twistor correspondence in conformally compactified Minkowski space begun in Part I [J. Harnad and S. Shnider, J. Math. Phys. 33, 3197–3209 (1992)]. The case of odd-dimensional spaces is treated, with an equivariant diffeomorphism (Cartan map) defined between Grassmannians (or flag manifolds) of totally isotropic subspaces of a complex vector space with respect to a nondegenerate quadratic form, and minimal orbits in the Grassmannians (or flag manifolds) based upon spinor modules. Then the Cartan maps for the real forms of signature (p,q) are considered. The extensions to super-Grassmannians are also discussed, with emphasis upon the cases of dimensions 4, 6, and 10. It is found that in dimensions greater than 4, the natural choice for the superdouble flag space correspondence does not result in the expected supernull line foliation arising in the application to the supersymmetric Yang–Mills equations although in dimension 6 the model constructed is relevant to chiral supergravity.

Asymptotic behaviors of radial solutions to semilinear wave equations in odd space dimensions

Asymptotic behaviors of radial solutions to semilinear wave equations in odd space dimensions

Asymptotic behaviour of solutions to semilinear wave equations with initial data of slow decay

AbstractSome useful and remarkable property are derived from a representation formula of a radially symmetric solution to the Cauchy problem for a homogeneous wave equation in odd space dimensions. These properties provide us with enough information to consider the semilinear case, namely, the associated integral equation with the problem will be considered on a weighted L∞‐space. This formulation enables us to deal with the problem for slowly decaying initial data.

A direct proof of the non-existence of anomalies in odd dimension

By the use of a Mellin-transform technique for the heat kernel expansion we have introduced in A. P. C. Malbouissonet al, J. Math. Phys.,30, 2016 (1989), concurrently with the results contained in G. Cognola and S. Zerbini,Phys. Lett. B,195, 435 (1988),Mod. Phys. Lett. A,3, 599 (1988) for the study of anomalies, we have been able to show directly the non-existence of a large class of anomalies (including the axial anomaly) in odd-dimensional compact spaces.

Maxwell-Chern-Simons Casimir effect. II. Circular boundary conditions.

In odd-dimensional spaces, gauge invariance permits a Chern-Simons mass term for the gauge fields in addition to the usual Maxwell-Yang-Mills kinetic energy term. We study the Casimir effect in such a (2 + 1)-dimensional Abelian theory. The case of parallel conducting lines was considered by us in a previous paper. Here we discuss the Casimir effect for a circle and examine the effect of finite temperature. The Casimir stress is found to be attractive at both low and high temperatures.

On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa

AbstractLet π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

Spin factors and geometric phases in arbitrary dimensions

The Polyakov spin factor is discussed using Strominger's intuitive analysis of the fermion propagator in loop space. Modulo a “helicity” projection, the Polyakov-Strominger spin factor is shown to be a cumulant of infinitesimal “boosts” or Thomas precessions along the particle orbit. Alternative forms for the spin factor and the fermion propagator are also discussed. The results are extended to higher euclidean space dimensions with a natural interpretation in terms of abelian/non-abelian Berry phases. In even (2 n) and odd (2 n+1) space dimensions with n > 1 the induced Berry potential is U (2 n−1 )-valued with an implicit monopole structure.

Expressions for the zeta-function regularized Casimir energy

The expansion of the Casimir energy for a scalar field with mass m, in a space where one dimension has been compactified into a circle of length a, leads to a double-infinite series that can be regularized by analytic continuation in the space dimension. The dimensionally regularized sum is then expressed as a power series in am by means of zeta-function expansions. The two possibilities of odd and even space dimensions are distinguished. In the odd space dimension we give a power expansion for small am, in addition to the asymptotic behavior. For the even space dimension, an expansion valid for any value of am is obtained. The contribution of higher-order terms is studied and, for the three-dimensional space, results for different values of the compactification length are shown.

𝜂-invariants and their adiabatic limits

0. Introduction 1. Algebraic preliminaries 2. The finite dimensional case; superconnections (a) Odd dimensional base spaces (b) Even dimensional base spaces Appendix 1. Coupling to general superconnections 3. Auxiliary Grassman variables and auxiliary adiabatic limits 4. The infinite dimensional case; fibrations (a) Elementary geometry of fibrations (b) Odd dimensional base spaces (c) Even dimensional base spaces (d) The case in which the Dirac operator on the fiber is not invertible Appendix 2. Multifibrations

Spherical waves in odd-dimensional space

The general solution is given of the ( 2 N + 1 ) \left ( {2N + 1} \right ) -dimensional wave equation with spherical symmetry, u t t − u x x − 2 N x u x = 0 {u_{tt}} - {u_{xx}} - \frac {{2N}}{x}{u_x} = 0 , in terms of two arbitrary functions and their first N N derivatives. Simple transformations then yield the general solutions to the Euler-Poisson-Darboux equation, u x y + N ( x + y ) ( u x + u y ) = 0 {u_{xy}} + \frac {N}{{\left ( {x + y} \right )}}\left ( {{u_x} + {u_y}} \right ) = 0 , for integer N N , and also the one-dimensional wave equation, u t t − c 2 u x x = 0 {u_{tt}} - {c^2}{u_{xx}} = 0 , for certain variable wave speeds c ( x ) c\left ( x \right ) .

Anomalies in spaces of even and odd dimensions in the scheme of stochastic quantization

Axial anomalies in spaces of even and odd dimensions are studied by the method of stochastic quantization with a generalized scheme of stochastic regularization. Although the stochastic regularization formally preserves the axial and gauge symmetries, the standard even-dimensional axial anomalies of Dirac fermions are correctly reproduced in the limit in which the regularization is lifted, while the anomalies of the chiral fermions are reproduced in covariant form. An analysis is also made of the general conditions for the existence and canceling of anomalies of massless fermions that violate parity in odd-dimensional spaces. The stochastic scheme works in odd-dimensional spaces when the parity-violating anomalies are absent. The P-anomalous part of the effect action of infinitely heavy fermions in odd-dimensional spaces is calculated explicitly.

Uniform approximation by algebraic polynomials on the sphere of an odd-dimensional space

Uniform approximation by algebraic polynomials on the sphere of an odd-dimensional space