Hyperbolic Rotations Research Articles

On non-bipartite graphs with strong reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A(G) is nonsingular and X=SA(G)−1S−1 is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by G+. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A(G) is nonsingular, and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) T+ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G+ exists and G+ is isomorphic to G. It follows that each graph G in this class has property (SR).

Reciprocal eigenvalue properties using the zeta and Möbius functions

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.

Singular graphs and the reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A(G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3], the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: “Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?” Suppose that G is singular and the characteristic polynomial of G is xn−k(xk+a1xk−1+⋯+ak). Assume that A(G) has rank k, so that ak≠0. Can we ever have |ak|=1? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, “Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?” We provide a class of such graphs. Furthermore, we extend our results to weighted graphs.

From Wigner hyperbolic rotation to fractional squeezing transformation

Based on the usual Wigner-Weyl transformation theory we find that the Wigner hyperbolic rotation in phase space will map onto fractional squeezing operator in Hilbert space. The merit of Weyl ordering and the coherent state representation of Fresnel operator is used in our derivation.

Radix-4 CORDIC algorithm based low-latency and hardware efficient VLSI architecture for Nth root and Nth power computations

In this article, a low-complexity VLSI architecture based on a radix-4 hyperbolic COordinate Rotion DIgital Computer (CORDIC) is proposed to compute the N{{rm th}} root and N{{rm th}} power of a fixed-point number. The most recent techniques use the radix-2 CORDIC algorithm to compute the root and power. The high computation latency of radix-2 CORDIC is the primary concern for the designers. N{{rm th}} root and N{{rm th}} power computations are divided into three phases, and each phase is performed by a different class of the proposed modified radix-4 CORDIC algorithms in the proposed architecture. Although radix-4 CORDIC can converge faster with fewer recurrences, it demands more hardware resources and computational steps due to its intricate angle selection logic and variable scale factor. We have employed the modified radix-4 hyperbolic vectoring (R4HV) CORDIC to compute logarithms, radix-4 linear vectoring (R4LV) to perform division, and the modified scaling-free radix-4 hyperbolic rotation (R4HR) CORDIC to compute exponential. The criteria to select the amount of rotation in R4HV CORDIC is complicated and depends on the coordinates X^j and Y^j of the rotating vector. In the proposed modified R4HV CORDIC, we have derived the simple selection criteria based on the fact that the inputs to R4HV CORDIC are related. The proposed criteria only depend on the coordinate Y^j that reduces the hardware complexity of the R4HV CORDIC. The R4HR CORDIC shows the complex scale factor, and compensation of such scale factor necessitates the complex hardware. The complexity of R4HR CORDIC is reduced by pre-computing the scale factor for initial iterations and by employing scaling-free rotations for later iterations. Quantitative hardware analysis suggests better hardware utilization than the recent approaches. The proposed architecture is implemented on a Virtex-6 FPGA, and FPGA implementation demonstrates 19% less hardware utilization with better error performance than the approach with the radix-2 CORDIC algorithm.

Hyperbolic Rotation with Euclidean Angle Illuminates Space Time Spinors

In this article it will be shown that by introducing a hyperbolic rotation as function of a Euclidean rotation angle, all possible Lorentz group spacetime rotations can be performed by Euclidean rotation parameters. This unification in Lorentz group rotations parameters (all Euclidean) allows for the calculation of the volume of a spacetime region bounded by the light cone of a past event, i.e., a causality-volume with the shape of a spacetime three-sphere. As will be calculated, the surface area of the spacetime three-sphere is the volume of our 3D space. Additionally, the unification in Lorentz group rotation parameters allows for the derivation of a Dirac spinor purely from the spacetime three-sphere symmetries. This derivation from the spacetime symmetries of a geometrical object yields the same result as solving the Dirac equation with quantum mechanical eigenvalue eigenvector complex matrix calculations.

Signed graphs with strong anti-reciprocal eigenvalue property

The ordered pair , where G is an underlying graph and is a signature function, is called a signed graph. A nonsingular signed graph is said to satisfy strong reciprocal (or strong anti-reciprocal) eigenvalue property if for each eigenvalue there exists (or ) in the spectrum of having same multiplicities, if we remove this multiplicity constraint then the signed graph is said to satisfy reciprocal (respectively anti-reciprocal) eigenvalue property. In this article, we investigate strong anti-reciprocal eigenvalue property in some families of signed graphs.

Reframing the general theory of relativity

The general theory of relativity extends the special theory of relativity to account for gravity (acceleration) and results in the postulation that matter curves spacetime, where gravitation is the embodiment of the spacetime curvature. The curvature of spacetime is modeled using Riemannian geometry. In this paper, it is proposed that matter rotates spacetime hyperbolically, rather than curves spacetime. As a result, gravitation is the embodiment of the hyperbolic rotation of spacetime. Regions in spacetime that have a larger magnitude of rotation relative to other regions of spacetime, ceteris paribus, experience stronger gravity, gravitational time dilation, gravitational length contraction, and gravitational frequency shift toward the red (blue) in the electromagnetic waves moving away (toward).

On weighted noncorona graphs with property R and −SR

Let Gw be a simple weighted graph with adjacency matrix A(Gw). The set of all eigenvalues of A(Gw) is called the spectrum of weighted graph Gw denoted by σ(Gw). The reciprocal eigenvalue property (or property R) for a connected weighted nonsingular graph Gw is defined as, if η ∈ σ(Gw) then 1 η ∈ σ(Gw). Further, if η and 1 η have the same multiplicities for each η ∈ σ(Gw) then this graph is said to have strong reciprocal eigenvalue property (or property SR). Similarly, a connected weighted nonsingular graph Gw is said to have anti-reciprocal eigenvalue property (or property −R) if η ∈ σ(Gw) then −1 η ∈ σ(Gw). Furthermore, if η and −1 η have the same multiplicities for each η ∈ σ(Gw) then strong anti-reciprocal eigenvalue property (or property −SR) holds for the weighted graph Gw. In this article, classes of weighted noncorona graphs satisfying property R and property −SR are studied.

Analysis of the heliospheric current sheet’s local structure based on a magnetic model

Context.An approach to the problem of estimating several physical properties of the heliospheric current sheet (HCS) local structure from a single spacecraft observation point of view was published in 2003. This method, HYperbolic TAngent ROtation (HYTARO), is based on the following two main assumptions: (a) that the HCS is locally flat, and (b) that its magnetic topology is described by a modified Harris field.Aims.A revision of HYTARO by including the spacecraft’s trajectory and developing a more precise procedure able to reduce the number of the required fit parameters is presented in this paper.Methods.This magnetic topology is proposed for the HCS’s local field at 1 AU, and it was applied as a fit function to a collection of 40 HCS crossings detected by the WIND spacecraft during the solar minimum. The estimated HCS local parameters (i.e., orientation and width) are consistent with previous works, which used the minimum variance analysis or the coplanarity variance analysis, and this is inferred from HYTARO results. One of the points of discussion about HYTARO was determining if it was necessary to apply two or three rotations to characterize the field’s relative orientation with respect to the current sheet (CS) plane. As principal component analysis (PCA) is a method to decompose a set of data into no correlated components by identifying eigenvalues and eigenvectors, it is feasible to analyze the possibility of reducing the dimensionality of the problem.Results.HYTARO gives a qualitative description of the observed magnetic field during an HCS crossing. The number of free parameters from the original HYTARO method has been reduced including the relative spacecraft trajectory through the CS and adding a background field related to each Cartesian component. Further, PCA has been implemented to justify applying two rotations with the HYTARO model. The dimensionality is another way to explore the number of rotations needed to relate the field in the local system with the field in GSE coordinates.

Trees with the reciprocal eigenvalue property

It was shown in 2006 that among the nonsingular trees T (whose adjacency matrix A(T) is nonsingular), the corona trees (trees that are obtained by taking any tree T and then adding a new pendant vertex at each vertex of T) are the only ones which satisfy the reciprocal eigenvalue property (λ is an eigenvalue of A(T) if and only if is an eigenvalue of A(T), where their multiplicities are allowed to be different). A general question remained open. Can there be a tree which has at least one zero eigenvalue and whose nonzero eigenvalues satisfy the reciprocal eigenvalue property? In this note, we show that there are no such trees with at least two vertices. The proof is a beautiful application of the product of graphs.

On the inverse of unicyclic 3-coloured digraphs

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight and with a unique perfect matching, denoted by , is considered in this article. We characterize the 3-coloured digraphs in whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight satisfying the strong reciprocal eigenvalue property.

An Algorithm for the Factorization of Split Quaternion Polynomials

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.

Classes of nonbipartite graphs with reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A(G) is nonsingular and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity, for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

The classical and quantum formulations for open systems related to dissipative dynamics are constructed on a complex hyperbolic ring, following universal symmetry principles, and considering the double thermal fields approach for modeling the system of interest, and the environment. The hyperbolic rotations are revealed as an underlying internal symmetry for the dissipative dynamics, and a chemical potential is identified as conjugate variable to the charge operator, and thus a grand partition function is constructed. As opposed to the standard scheme, there are not patologies associated with the existence of many unitarity inequivalent representations on the hyperbolic ring, since the whole of the dissipative quantum dynamics is realized by choosing only one representation of the field commutation relations. Entanglement entropy operators for the subsystem of interest and the environment, are constructed as a tool for study the entanglement generated from the dissipation. The holographic perspectives of our results are discussed.

GH CORDIC-Based Architecture for Computing $N$ th Root of Single-Precision Floating-Point Number

This article presents hardware implementation for computing arbitrary roots of a single-precision floating-point number. The proposed architecture is based on Generalized Hyperbolic COordinate Rotation Digital Computer (GH CORDIC) algorithm. Benefiting from the wide range of floating-point numbers, our design is able to compute the ${N}$ th root ( ${N} {\ge 2}$ ) of a single-precision floating-point number. After implementation, a series of tests have been carried out, including accuracy, power consumption, performance comparison, and so on. Simulation results indicate that our proposed method is capable of calculating the ${N}$ th root of a positive single-precision floating-point number with a relative error of 10−7 approximately and promises an error-flatten performance. Synthesized results from a design compiler under TSMC-40-nm CMOS technology show that our design can achieve the highest frequency of 2.38 GHz with the area consumption of ${{{140894.44 ~\mu \text {m}}}}^{2}$ and power consumption of 86.9573 mW.

Squeezing for Bloch four-hyperboloid via the non-compact Hopf map

We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between the squeeze operation and hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac and the Schwinger types, are introduced. We clarify the underlying hyperbolic geometry and representations of the squeezed states along the line of the first non-compact Hopf map. Following the geometric hierarchy of the non-compact Hopf maps, we extend the analysis to —the isometry of a split-signature four-hyperboloid. We explicitly construct the squeeze operators in the Dirac and Schwinger types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type squeezed one-photon state is equal to an entangled superposition state of two squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the squeezed coherent states are also investigated. In particular, we show that the squeezed vacuum naturally realizes a generalized squeezing in a 4D manner.

Low Complexity Generic VLSI Architecture Design Methodology for $N^{th}$ Root and $N^{th}$ Power Computations

In this paper, we propose a low complexity architecture design methodology for fixed point root and power computations. The state of the art approaches perform the root and power computations based on the natural logarithm-exponential relation using Hyperbolic COordinate Rotation DIgital Computer (CORDIC). In this paper, any root and power computations have been performed using binary logarithm-binary inverse logarithm relation. The designs are modeled using VHDL for fixed point numbers and synthesized under the $TSMC 40$ -nm CMOS technology @ 1 GHz frequency. The synthesis results shows that the proposed ${ { N}}^{\text {th}}$ root computation saves 19.38% on chip area and 15.86% power consumption when compared with the state of the art architecture for root computation without compromising the computational accuracy. Similarly, the proposed ${ { N}}^{\text {th}}$ power computation saves 38% on chip area, 35.67% power consumption when compared with the state of the art power computation with out loss in accuracy. The proposed root and power computation designs save 8 clock cycle latency when compared with the state of the art implementations.

Configurable Rotation Matrix of Hyperbolic CORDIC for Any Logarithm and Its Inverse computation

In this paper, we propose a configurable rotation matrix for Hyperbolic COordinate Rotation DIgital Computer (CORDIC) to compute any logarithm and inverse logarithm. The rotation matrix of the conventional Hyperbolic CORDIC computes only natural logarithm and natural inverse logarithm whereas the proposed rotation matrix configures the Hyperbolic CORDIC to compute any logarithm and inverse logarithm. Subsequently, an architecture for configurable Hyperbolic CORDIC has been designed based on the proposed rotation matrix. An extensive vector matching is performed to validate the proposed architecture. Finally, an ASIC implementation for configurable Hyperbolic CORDIC has been performed using TSMC 45-nm CMOS technology @ 1 GHz frequency. The accuracy of the proposed configurable Hyperbolic CORDIC is same as conventional Hyperbolic CORDIC. The proposed approach saves 19.22% on chip area and 17.52% power consumption for decimal logarithm computation and 18.55% on chip area and 16.12% power consumption for inverse logarithm computation when compared with the state of the art approaches.

A New Variable Forgetting Factor and Variable Regularized Square Root Extended Instrumental Variable PAST Algorithm With Applications

This article proposes a square-root (SR) extended instrumental variable (EIV) projection approximation subspace tracking (PAST) algorithm with variable forgetting factor (VFF) and variable regularization (VR). A new local polynomial modeling (LPM) based VFF is proposed by minimizing the mean squares deviation of the EIV linear model and the IV-PAST algorithm. A new variable $\ell 2$ regularization term is also derived to reduce the variance of the estimator resulting from possibly ill conditioned covariance matrix at low input signal level. An SR version of the algorithm is developed to improve the numerical stability of the algorithm and avoid the problem of loss of positive definiteness of the inverse covariance matrix. The proposed LOFF-VR-SREIV-PAST algorithm can be implemented by both the conventional EIV-PAST algorithm and numerically more stable hyperbolic rotations. Furthermore, the convergence of the proposed VFF-EIV-PAST algorithm using the ordinary differential equation method is analyzed. Its application to the estimation and tracking of direction of arrival under spatial color sensor noise in both stationary and nonstationary scenarios are presented. Simulations demonstrate that the proposed algorithm yields improved performance over the conventional PAST and EIV-PAST algorithms, especially at medium to low signal-to-noise ratio, which is more frequently encountered in practical situations.