Hyperbolic Rotations Research Articles

Oblique Airy wave packets in bidispersive optical media

Using hyperbolic rotations, we show that a new class of skewed, nonspreading, accelerating Airy wave packets is possible in optical bidispersive systems. Their obliquity factor is found to have a profound effect on their spatiotemporal acceleration dynamics. Pertinent examples are provided.

Braided vector fields on a quantum hyperboloid via the quantum group Uq(sl(2))

We introduce a new method of defining a space of ‘braided vector fields’ on a quantum hyperboloid. Our method is based on the Lyubashenko–Sudbery construction (see Lyubashenko and Sudbery 1998 J. Math. Phys. 39 3487 and references therein) and it consists in realizing its generators (braided analogs of hyperbolic infinitesimal rotations) via those of the quantum group Uq(sl(2)). Our main result consists in showing that the space of braided vector fields is a projective module over the coordinate algebra of the quantum hyperboloid.

Nonorthogonal Joint Diagonalization by Combining Givens and Hyperbolic Rotations

A new algorithm for computing the nonorthogonal joint diagonalization of a set of matrices is proposed for independent component analysis and blind source separation applications. This algorithm is an extension of the Jacobi-like algorithm first proposed in the joint approximate diagonalization of eigenmatrices (JADE) method for orthogonal joint diagonalization. The improvement consists mainly in computing a mixing matrix of determinant one and columns of equal norm instead of an orthogonal mixing matrix. This target matrix is constructed iteratively by successive multiplications of not only Givens rotations but also hyperbolic rotations and diagonal matrices. The algorithm performance, evaluated on synthetic data, compares favorably with existing methods in terms of speed of convergence and complexity.

Shrinking microwave filters

An adaptive predistortion technique has been presented and verified through the design and fabrication of practical filters in both the C and Ku bands. The method allows the realization of microwave filters at a lower cost, lighter mass, smaller volume, and better performance with minimum insertion loss penalties.The concept of lossy filters has been presented from a practical perspective. A simple lossy synthesis technique using any synthesized lossless (nontransversal) filter was shown, which can be used with hyperbolic rotations for loss distribution. Moreover, the limitation on the minimum Q of lossy resonators has been studied using a one-pole filter as a fundamental building block. Lossy four-pole Chebyshev and quasi-elliptic synthesis examples were presented. A four- pole Chebyshev lossy filter in the Ku band has been synthesized, modeled, and fabricated successfully using mixed combline and microstrip technologies. The design has the advantage of having all input-output paths going through more than one resonator, which minimizes unwanted source-to-load coupling, especially at high frequencies. The lossy approach is still at its early stages of development and needs more research and development effort to become as mature as the predistorted filters.

Unicyclic graphs with strong reciprocal eigenvalue property

A graph G is bipartite if and only if the negative of each eigenvalue of G is also an eigenvalue of G. It is said that a graph has property (R), if G is nonsingular and the reciprocal of each of its eigenvalues is also an eigenvalue. Further, if the multiplicity of an eigenvalue equals that of its reciprocal, the graph is said to have property (SR). The trees with property (SR) have been recently characterized by Barik, Pati and Sarma. Barik, Neumann and Pati have shown that for trees the two properties are, in fact, equivalent. In this paper, the structure of a unicyclic graph with property (SR) is studied. It has been shown that such a graph is bipartite and is a corona (unless it has girth four). In the case it is not a corona, it is shown that the graph can have one of the three specified structures. Families of unicyclic graphs with property (SR) having each of these specific structures are provided.

Thermal symmetry of the Markovian master equation

The quantum Markovian master equation of the reduced dynamics of a harmonic oscillator coupled to a thermal reservoir is shown to possess thermal symmetry. This symmetry is revealed by a Bogoliubov transformation that can be represented by a hyperbolic rotation acting on the Liouville space of the reduced dynamics. The Liouville space is obtained as an extension of the Hilbert space through the introduction of tilde variables used in the thermofield dynamics formalism. The angle of rotation depends on the temperature of the reservoir, as well as the value of Planck's constant. This symmetry relates the thermal states of the system at any two temperatures. This includes absolute zero, at which purely quantum effects are revealed. The Caldeira-Leggett equation and the classical Fokker-Planck equation also possess thermal symmetry. We compare the thermal symmetry obtained from the Bogoliubov transformation in related fields and discuss the effects of the symmetry on the shape of a Gaussian wave packet.

Dieudonné Points of Holomorphic Self-Maps of Regions

Schwarz’s Lemma says that if a holomorphic self-map f of the unit disc fixes the origin then ¦f′(0)¦ ≤ 1. In 1931, Dieudonne extended this by showing that ¦f′(z)¦ ≤ 1 when \(|z|< {\sqrt 2}-1\). We show that there is no analogue of Dieudonne’s Lemma for a simply connected hyperbolic region unless the region is a disc and the fixed point is the centre. We also discuss the analogue of Dieudonne’s result for multiply connected regions. In order to do this, we establish a local version of the Aumann-Caratheodory Rigidity Theorem, and we determine the maximal number of points fixed by some non-trivial conformal automorphism of a region in terms of its connectivity. More generally we investigate the topological relationship between the group of conformal automorphisms and its complement as subsets of the space of holomorphic self-maps of a hyperbolic region. For a simply connected hyperbolic region Ω we give sharp conformally invariant bounds on the hyperbolic distance between a holomorphic self-map f with a fixed point a and the hyperbolic rotation about a through angle arg f ′(a).

The Robin Function and Its Eigenvalues

Abstract The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Green's function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.

On nonsingular trees and a reciprocal eigenvalue property

Let G be a simple connected graph. Let A(G) be the adjacency matrix of G. We give a combinatorial description of A(G)−1 of a bipartite graph G with a unique perfect matching. As a corollary, we obtain the combinatorial description of the inverse of a nonsingular tree. A graph is said to have property (R) if is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and have the same multiplicity, for each eigenvalue λ then it is said to have the property (SR). We characterize all trees with property (R) and show that it is exactly the class of all trees with property (SR). The class of trees with property (R) is also identical to the class of corona trees, namely, the trees which can be obtained from other trees by adding pendants to all vertices. Other equivalent conditions for a tree to have property (R) are also given.

On Extended RLS Lattice Adaptive Variants: Error-Feedback, Normalized, and Array-Based Recursions

Error-feedback, normalized, and array-based recursions represent equivalent RLS lattice adaptive filters which are known to offer better numerical properties under finite-precision implementations. This is the case when the underlying data structure arises from a tapped-delay-line model for the input signal. On the other hand, in the context of a more general orthonormality-based input model, these variants have not yet been derived and their behavior under finite precision is unknown. This paper develops several lattice structures for the exponentially weighted RLS problem under orthonormality-based data structures, including error-feedback, normalized, and array-based forms. As a result, besides nonminimality of the new recursions, they present unstable modes as well as hyperbolic rotations, so that the well-known good numerical properties observed in the case of FIR models no longer exist. We verify via simulations that, compared to the standard extended lattice equations, these variants do not improve the robustness to quantization, unlike what is normally expected for FIR models.

On the Factorization of Hyperbolic and Unitary Transformations into Rotations

This paper presents a $\Sigma$-unitary analogue to the CS decomposition of a partitioned unitary matrix. The hyperbolic rotations revealed by the decomposition are shown to be optimal in that, among a broader class of decompositions of $\Sigma$-unitary matrices into elementary hyperbolic rotations, they are the smallest possible in a sum-of-squares sense. A similar optimality property is shown to hold for the sines in the CS decomposition of a unitary matrix.

Solving the block-Toeplitz least-squares problem in parallel

In this paper we present two versions of a parallel algorithm to solve the block–Toeplitz least-squares problem on distributed-memory architectures. We derive a parallel algorithm based on the seminormal equations arising from the triangular decomposition of the product T TT . Our parallel algorithm exploits the displacement structure of the Toeplitz-likematrices using theGeneralized SchurAlgorithm to obtain the solution in O(mn) flops instead of O(mn2) flops of the algorithms for non-structured matrices. The strong regularity of the previous product of matrices and an appropriate computation of the hyperbolic rotations improve the stability of the algorithms. We have reduced the communication cost of previous versions, and have also reduced the memory access cost by appropriately arranging the elements of the matrices. Furthermore, the second version of the algorithm has a very low spatial cost, because it does not store the triangular factor of the decomposition. The experimental results show a good scalability of the parallel algorithm on two different clusters of personal computers. Copyright c © 2005 John Wiley & Sons, Ltd.

Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs

We consider the reduction of a symmetric indefinite matrix pair (A,B), with B nonsingular, to tridiagonal-diagonal form by congruence transformations. This is an important reduction in solving polynomial eigenvalue problems with symmetric coefficient matrices and in frequency response computations. The pair is first reduced to symmetric-diagonal form. We describe three methods for reducing the symmetric-diagonal pair to tridiagonal-diagonal form. Two of them employ more stable versions of Brebner and Grad's pseudosymmetric Givens and pseudosymmetric Householder reductions, while the third is new and based on a combination of Householder reflectors and hyperbolic rotations. We prove an optimality condition for the transformations used in the third reduction. We present numerical experiments that compare the different approaches and show improvements over Brebner and Grad's reductions.

A new Method for Determining the Interplanetary Current-Sheet Local Orientation

In this work we have developed a new method for determining the interplanetary current sheet local parameters. The method, called `HYTARO' (from Hyperbolic Tangent Rotation), is based on a modified Harris magnetic field. This method has been applied to a pool of 57 events, all of them recorded during solar minimum conditions. The model performance has been tested by comparing both, its outputs and noise response, with these of the `classic MVM' (from Minimum Variance Method). The results suggest that, despite the fact that in many cases they behave in a similar way, there are specific crossing conditions that produce an erroneous MVM response. Moreover, our method shows a lower noise level sensitivity than that of MVM.

Computation of Signature Symmetric Balanced Realizations

A new numerical scheme for computing balancing coordinate transformations for signature symmetric realizations in linear systems theory is presented. The method is closely related to the Jacobi method for diagonalizing symmetric matrices. Here the minimization of the sum of traces of the Gramians by orthogonal and hyperbolic Jacobi-type rotations is considered. Local quadratic convergence of the algorithm is shown.

Sums of Reciprocal Eigenvalues of the Laplacian

The purpose of this paper is to present formulas for the sum of the squares of all reciprocal eigenvalues λ ofthe fixed membrane and the free membrane μ for plane simply connected domains. As a consequence we obtain that among all simply connected plane domains with the maximal conformal radius 1 and the area A only the unit disk yields the minimum of .

Hyperbolic representation of light propagation in a multilayer medium.

By analogy with the representation of the polarization of light on the Poincaré sphere, we describe the propagation and the reflection/transmission of light in a multilayer on a hyperbolic surface. We show that the propagation of light corresponds to a classical rotation on this surface and that its reflection/transmission corresponds to a hyperbolic rotation.

Fresnelcoefficients as hyperbolic rotations

We describe the action of a plane interface between twosemi-infinite media in terms of a transfer matrix. We find aremarkably simple factorization of this matrix, which enables usto express the Fresnel coefficients as hyperbolic rotations.

Unifying unitary and hyperbolic transformations

In this paper, we describe unified formulas for unitary and hyperbolic reflections and rotations, and show how these unified transformations can be used to compute a Hermitian triangular decomposition R ̂ H D R ̂ of a strongly nonsingular indefinite matrix A ̂ given in the form A ̂ =X 1 H X 1+αX 2 H X 2, α=±1 . The unification is achieved by the introduction of signature matrices which determine whether the applicable transformations are unitary, hyperbolic, or their generalizations. We derive formulas for the condition numbers of the unified transformations, propose pivoting strategies for lowering the condition number of the transformations, and present a unified stability analysis for applying the transformations to a matrix.

Fresnel formulas as Lorentz transformations

From a matrix formulation of the boundary conditions we obtain the fundamental invariant for an interface and a remarkably simple factorization of the interface matrix, which enables us to express the Fresnel coefficients in a new and compact form. This factorization allows us to recast the action of an interface between transparent media as a hyperbolic rotation. By exploiting the local isomorphism between SL(2, C) and the (3 + 1)-dimensional restricted Lorentz group SO(3, 1), we construct the equivalent Lorentz transformation that describes any interface.