Stable Ergodicity and Frame Flows

In this note we show that all diffeomorphisms close enough to the time-one map of the frame flow on certain negatively curved manifolds are ergodic. As a simple corollary we deduce that the frame flows are ergodic for all compact manifolds with curvature pinched sufficiently close to –1, thus providing results in the case of manifolds of dimension 7 or 8 which were missing from the results of Brin and Karcher.     

Gabor frames and directional time–frequency analysis

We introduce a directionally sensitive time–frequency decomposition and representation of functions. The coefficients of this representation allow us to measure the “amount” of frequency a function (signal, image) contains in a certain time interval, and also in a certain direction. This has been previously achieved using a version of wavelets called ridgelets [E.J. Candès, Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal. 6 (1999) 197–218. [2]; E.J. Candès, D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piesewise-C² singularities, Comm. Pure Appl. Math. 57 (2004) 219–266. [3]] but in this work we discuss an approach based on time–frequency or Gabor elements. For such elements, a Parseval formula and a continuous frame-type representation together with boundedness properties of a semi-discrete frame operator are obtained. Spaces of functions tailored to measure quantitative properties of the time–frequency–direction analysis coefficients are introduced and some of their basic properties are discussed. Applications to image processing and medical imaging are presented.     

On the existence of multiresolution analysis for framelets

We show that a compactly supported tight framelet comes from an MRA if the intersection of all dyadic dilations of the space of negative dilates, which is defined as the shift-invariant space generated by the negative scales of a framelet, is trivial. We also construct examples of (non-tight) framelets, which are arbitrarily close to tight frame framelets, such that the corresponding space of negative dilates is equal to the entire space L ²ℝ.The first author was partially supported by the NSF grant DMS–0441817     

Solar sail orbits at the Earth–Moon libration points

Solar sail technology offer new capabilities for the analysis and design of space missions. This new concept promises to be useful in overcoming the challenges of moving throughout the solar system. In this paper, novel families of highly non-Keplerian orbits for solar sail spacecraft at linear order are investigated in the Earth–Moon circular restricted three-body problem, where the third body is a solar sail. In particular, periodic orbits near the collinear libration points in the Earth–Moon system will be explored along with their applications. The dynamics are completely different from the Earth–Sun system in that the sun line direction constantly changes in the rotating frame but rotates once per synodic lunar month. Using an approximate, first-order analytical solution to the nonlinear nonautonomous ordinary differential equations, periodic orbits can be constructed that are displaced above the plane of the restricted three-body system. This new family of orbits have the property of ensuring visibility of both the lunar far-side and the equatorial regions of the Earth, and can enable new ways of performing lunar telecommunications.     

Numerical Simulation of the Two-Hydrodynamic Film Thickness

We investigate, from a numerical point of view, the coefficients identification problem, for the Elrod–Adams model of cavitation, in the frame of the hydrodynamic lubrication. We relax the control problem, and propose a relaxed augmented Lagrangian algorithm, with a given loop length. Some numerical results are given.     

Some remarks on “On the windowed Fourier transform and wavelet transform of almost periodic functions,” by J.R. Partington and B. Ünalmı

J.R. Partington and B. Ünalmı consider in their 2001 paper [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the windowed Fourier transform and wavelet transform as tools for analyzing almost periodic signals. They establish Parseval-type identities and consider discretized versions of these transforms in order to construct generalized frame decompositions. We have found a gap in the construction of generalized frames in the windowed Fourier transform case; we comment on this gap and give an alternative proof. As for the wavelet transform case, in [J.R. Partington, B. Ünalmı , Appl. Comput. Harmon. Anal. 10 (1) (2001) 45–60] the generalized frame decomposition is done only for the simplest wavelet, the Haar wavelet; we show how to construct generalized frame decompositions for a wide family of wavelets.     

Influence of heat transfer on a peristaltic transport of Herschel–Bulkley fluid in a non-uniform inclined tube

Peristaltic transport in a two-dimensional non-uniform tube filled with Herschel–Bulkley fluid is studied under the assumptions of long wavelength and low Reynold number. The fluid flow is investigated in the wave frame of reference moving with the velocity of the peristaltic wave. Exact solution for the velocity field, the temperature profile, the stream functions and the pressure gradient are obtained. The physical behavior of τ, n, α and on the pressure rise versus flow rate are discussed through graphs. At the end of the article trapping phenomena for Herschel–Bulkley and also for Newtonian, Bingham and power law (which are the special cases of Herschel–Bulkley fluid) fluid are discussed.     

Invariant Euler–Lagrange Equations and the Invariant Variational Bicomplex

In this paper, we derive an explicit group-invariant formula for the Euler–Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.     

Duality Principles in Frame Theory

Duality principles in Gabor theory such as the Ron–Shen duality principle and the Wexler–Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.     

Deficits and Excesses of Frames

The excess of a sequence in a Hilbert space is the greatest number of elements that can be removed yet leave a set with the same closed span. We study the excess and the dual concept of the deficit of Bessel sequences and frames. In particular, we characterize those frames for which there exist infinitely many elements that can be removed from the frame yet still leave a frame, and we show that all overcomplete Weyl–Heisenberg and wavelet frames have this property.     

Quantum mechanics in Riemannian spacetime. II. Operators of observables

The formulation of the generally covariant analog of standard (nonrelativistic) quantum mechanics in a general Riemannian spacetime begun in earlier studies of the author is continued with the introduction of asymptotic (with respect toc ^–2) operators of the spatial position of a spinless particle and of the projection of its momentum onto an arbitrary spacetime direction. The connection between the position operator and the generalization of theV _1,3 Newton—Wigner operator is established. It is shown that the projection of the momentum onto the 4-velocity of the frame of reference (the energy operator) is unitarily equivalent to the Hamiltonian in the Schrödinger equation.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 412–423, March, 1992.     

Joins in the Frame of Nuclei

Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixed-point theorem due to Pataraia, we deduce an induction principle for joins of nuclei. As an illustration of the technique, we offer a simple (and also intuitionistic) proof of the localic Hofmann–Mislove Theorem.     

Problems of the mechanics of winding thick-walled composite structures

Conclusions Dividing the winding process into individual stages, examination of the interply compliance of the composites, and the introduction of the circular metals made it possible to link the parameters of this process and, in particular, the tensioning force, with the properties of the completed component. It was thus possible to propose engineering methods of calculating thick-walled wound components with an allowance made for technological prior history. Structures for specific applications were constructed and a new technological method proposed and applied. These methods include the methods of producing spatial reinforced driveshafts whose load-carrying frame is produced on the basis of the system of the single filament. The studies examined in this work were used as a starting point for further investigations and development of the mechanics of winding in a number of scientific centers of the country.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 618–626, September–October, 1992.     

Space curve evolution,geometric phase,and solitons

Certain moving space curves are endowed with a geometric phase. This phase arises due to the path dependence of the rotation of an orthonormal triad (frame) defined at every point on the curve. In the present work we use the connection between moving curves and soliton dynamics to find the geometric phase associated with a class of soliton-supporting equations.The Institute of Mathematical Sciences, C.I.T. Campus, Madras 600 113, India. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 172–176, May, 1994.     

The Keldysh-Sedov problem for functions holomorphic in a bi-halfplane

We pose and solve the Keldysh-Sedov problem: determine a function holomorphic in a bi-halfplane where its real and imaginary parts are given on mutually nonintersecting rectangles which exhaust the plane and which are the frame of the boundary of the bi-halfplane. Necessary and sufficient conditions for this problem to be solvable are given.Translated from Matematicheskii Zametki, Vol. 19, No. 5, pp. 681–690, May, 1976.     

Singular Values of Compact Pseudodifferential Operators

This paper investigates the asymptotic decay of the singular values of compact operators arising from the Weyl correspondence. The motivating problem is to find sufficient conditions on a symbol which ensure that the corresponding operator has singular values with a prescribed rate of decay. The problem is approached by using a Gabor frame expansion of the symbol to construct an approximating finite rank operator. This establishes a variety of sufficient conditions for the associated operator to be in a particular Schatten class. In particular, an improvement of a sufficient condition of Daubechies for an operator to be trace-class is obtained. In addition, a new development and improvement of the Calderón–Vaillancourt theorem in the context of the Weyl correspondence is given. Additional results of this type are then obtained by interpolation.     

Hypothesis ofP-odd contribution to the gravitational interactions and its consequences

A generalized Lagrangian of spinor particles in a gravitational field containingP-odd terms is proposed. It is used to obtain a quasirelativistic equation of spinors in the field of a gravitational source with nonvanishing intrinsic angular momentum in both an inertial frame and in a frame rotating with the source. For these cases the quasirelativistic Hamiltonian of a test body with nonzero intrinsic angular momentum is obtained in the classical limit; it is shown that the presence in the Hamiltonian ofP-odd terms leads to the appearance of forces that, depending on the orientation of the intrinsic angular momentum of the test body, are attractive or repulsive. It is noted that theC noninvariance of the theory can in principle result in an evolution of the universe in which particles predominate over antiparticles. Other macroscopic consequences whose experimental verification would yield information about the values of the constants introduced in the theory are noted.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 309–319, May, 1992.     

The Matrix-Valued Riesz Lemma and Local Orthonormal Bases in Shift-Invariant Spaces

We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrix-valued Fejér–Riesz lemma for Laurent polynomials.     

General-covariant quantum mechanics in Riemannian space-time III. The Dirac particle

A general covariant analog of standard nonrelativistic quantum mechanics with relativistic corrections is constructed for the Dirac particle in a normal geodesic frame in general Riemannian space-time. Not only the Pauli equation with a Hermitian Hamiltonian and the pre-Hilbert structure of the space of its solutions, but also matrix elements of the Hermitian operators of momentum, (curvilinear) spatial coordinates, and spin of the particle, are deduced, as a general-covariant asymptotic approximation in c^–2 (c is the velocity of light), to their naturally determined general-relativistic pre-images. It is shown that the Pauli equation Hamiltonian, generated by the Dirac equation, is unitary-equivalent to the energy operator generated by the metric energymomentum tensor of the spinor field. Commutation and other properties of the observables associated with variation in the geometrical background of quantum mechanics are briefly discussed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 122–132, January, 1996.     

Reconstruction of Wideband Reflectivity Densities by Wavelet Transforms

This paper is concerned with reconstruction problems arising in the context of radar signal analysis. The goal in radar is to obtain information about objects by emitting certain signals and analyzing the reflected echoes. In this paper, we shall focus on the general wideband model for radar echoes and on the case of continuously distributed objects D (reflectivity density). In this case, the echo is given by an inverse wavelet transform of the density D where the role of the analyzing wavelet is played by the transmitted signal. However, the null space of an inverse wavelet transform is nontrivial, it is described by the corresponding reproducing kernel. Following the approach of Naparst [14] and Rebolla-Neira et al. [16], we suggest to treat this problem by transmitting not just one signal but a family of signals. Indeed, a reconstruction formula for one- and 2-dimensional reflectivity densities can be derived, provided that the set of outgoing signals forms an orthogonal basis or – more general – a frame. We also present some rigorous error estimates for these reconstruction formulas. The theoretical results are confirmed by some numerical examples.