Search Results (66)

Ruled surfaces in Minkowski 3-space play a crucial role in differential geometry and have significant applications in physics and engineering. This study explores the fundamental properties of ruled surfaces via orthogonal modified frame in Minkowski space ${\mathrm{E}}_1^3$, focusing on their minimality, developability, and curvature characteristics. We examine the necessary and sufficient conditions for a ruled surface to be minimal, considering the mean curvature and its implications. Furthermore, we analyze the developability of such surfaces, determining the conditions under which they can be locally unfolded onto a plane without distortion. The Gaussian and mean curvatures of ruled surfaces in Minkowski space are computed and discussed, providing insights into their geometric behavior. Special attention is given to spacelike, timelike, and lightlike rulings, highlighting their unique characteristics. This research contributes to the broader understanding of the geometric properties of ruled surfaces within the framework of Minkowski geometry. Full article

In this paper, delay Doppler (DD) domain is utilized for enabling an efficient handover-triggering mechanism in highly dynamic unmanned aerial vehicles (UAVs) or drones to ground networks. In the proposed scheme, the estimated DD channel gains using DD multi-carrier modulation (DDMC), e.g., orthogonal time frequency space (OTFS) modulation, are utilized for triggering the handover decisions. This is motivated by the fact that the estimated DD channel gain is time-invariant throughout the whole OTFS symbol despite the entity speed. This results in more stable handover decisions over that based on the time-varying received-signal strength (RSS) or frequency time (FT) channel gains using orthogonal frequency division multiplexing (OFDM) modulation employed in fifth-generation–new radio (5G-NR) and its predecessors. To mathematically bind the performance of the proposed scheme, we studied its performance under channel estimation errors of the most dominant DD channel estimators, i.e., least square (LS) and minimum mean square error (MMSE), and we prove that they have marginal effects on its performance. Numerical analyses demonstrated the superiority of the proposed DD-based handover-triggering scheme over candidate benchmarks in terms of the handover overhead, the achievable throughput, and ping-pong ratio under different simulation conditions. Full article

Recently, several papers identified technical issues related to equivalent time-domain and frequency-domain “characterization of the n–block or transmission” feedback capacity formula and its asymptotic limit, the feedback capacity, of additive Gaussian noise (AGN) channels, first introduce by Cover and Pombra in 1989 (IEEE Transactions on Information Theory). The main objective of this paper is to derive new results on the Cover and Pombra characterization of the n–block feedback capacity formula, and to clarify the main points of confusion regarding the time-domain results that appeared in the literature. The first part of this paper derives new equivalent time-domain sequential characterizations of feedback capacity of AGN channels driven by non-stationary and non-ergodic Gaussian noise. It is shown that the optimal channel input processes of the new equivalent sequential characterizations are expressed as functionals of a sufficient statistic and a Gaussian orthogonal innovations process. Further, the Cover and Pombra n–block capacity formula is expressed as a functional of two generalized matrix difference Riccati equations (DREs) of the filtering theory of Gaussian systems, contrary to results that appeared in the literature and involve only one DRE. It is clarified that prior literature deals with a simpler problem that presupposes the state of the noise is known to the encoder and the decoder. In the second part of this paper, the existence of the asymptotic limit of the n–block feedback capacity formula is shown to be equivalent to the convergence properties of solutions of the two generalized DREs. Further, necessary and or sufficient conditions are identified for the existence of asymptotic limits, for stable and unstable Gaussian noise, when the optimal input distributions are asymptotically time-invariant but not necessarily stationary. This paper contains an in-depth analysis, with various examples, and identifies the technical conditions on the feedback code and state space noise realization, so that the time-domain capacity formulas that appeared in the literature, for AGN channels with stationary noises, are indeed correct. Full article

Underwater acoustic channel (UWAC) is characterized by significant multipath effects, strong time-varying properties and complex noise environments, which make achieving high-rate and reliable underwater communication a formidable task. To address the above adverse challenges, this study first presents a novel, robust and efficient polar code construction (NREPCC) method using the base-adversarial polarization weight (BPW) algorithm tailored for typical ocean channel models, including invariable sound velocity gradient (ISVG) channels, negative sound velocity gradient (NSVG) channels, and positive sound velocity gradient (PSVG) channels. Subsequently, a robust and reliable polar-coded UWAC system model based on the orthogonal frequency division multiplexing (OFDM) technique is designed using the $t$-distribution noise model in conjunction with real sea noise data fitting. Then, an enhanced time synchronization and packet detection algorithm based on $t$-distribution is proposed for the performance optimization of the polar-coded UWAC OFDM system. Finally, extensive numerical simulation results confirm the excellent performance of the proposed NREPCC method and polar-coded UWAC OFDM system under a variety of channel conditions. Specifically, the NREPCC method outperforms low-density parity-check (LDPC) codes by approximately 0.5~1 dB in PSVG and ISVG channels while maintaining lower encoding and decoding complexity. Moreover, the robustness of the NREPCC method under $t$-distribution noise with varying degrees of freedom is rigorously validated, which renders vital technical support for the design of high-precision and high-robustness UWAC systems. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article

Time-varying underwater acoustic (UWA) channels are the key challenge of underwater acoustic communication (UAC). Although UAC exhibits time-variance characteristics significantly in time domains, its delay-Doppler (DD) domain representation tends to be time-invariant. Orthogonal time–frequency space (OTFS) modulation has recently been proposed and has acquired widespread interest due to its excellent performance over time-varying channels. In the UWA OTFS system, the novel DD domain channel estimation algorithm that employs a multidirectional adaptive moving average scheme is proposed. Specifically, the proposed scheme is cascaded by a channel estimator and moving average filter. The channel estimator can be employed to estimate the time-invariant channel of the DD domain multidirectionally, improving proportionate normalized least mean squares (IPNLMS). Meanwhile, the moving average filter is used to reduce the output noise of the IPNLMS. The performance of the proposed method is verified by simulation experiments and real-world lake experiments. The results demonstrate that the proposed channel estimation method can outperform those of benchmark algorithms. Full article

The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional cases. Additionally, we investigate the stochastic case by considering a normal distribution, analyzing the asymptotic normality of random matrices and its implications on the convergence properties of the proposed measure. Full article

The field of multiplicative analysis has recently garnered significant attention, particularly in the context of solving multiplicative differential equations (MDEs). The symmetry concept in MDEs facilitates the determination of invariant solutions and the reduction of these equations by leveraging their intrinsic symmetrical properties. This study is motivated by the need for efficient methods to address MDEs, which are critical in various applications. Our novel contribution involves leveraging the fundamental properties of orthogonal polynomials, specifically Laguerre polynomials, to derive new solutions for MDEs. We introduce the definitions of Laguerre multiplicative differential equations and multiplicative Laguerre polynomials. By applying the power series method, we construct these multiplicative Laguerre polynomials and rigorously prove their basic properties. The effectiveness of our proposed solution is validated through illustrative examples, demonstrating its practical applicability and potential for advancing the field of multiplicative analysis. Full article

Conventional global navigation satellite system receivers typically employ a two-step positioning procedure (2SP) by first independently estimating the synchronization parameters and then using these parameters to solve a system of superdeterministic equations derived from multilateration to accomplish positioning. Direct position estimation (DPE) has emerged as a promising alternative that utilizes a single-step procedure to obtain the maximum likelihood estimate of a position. This approach has been shown to effectively mitigate biases incurred by the second estimation step in 2SP. However, for code-division multiple-access systems, the pseudo-orthogonality of the spreading codes causes the estimation problem not to be mapped to a perfectly orthogonal space. Additionally, the cross-correlation interference between satellites renders the maximum likelihood invariant theory untenable in the first estimation step of the 2SP. This study presents the derivation of the Cramér–Rao bound constraint for both the 2SP and DPE, evaluating the performance degradation of the 2SP compared to that of the DPE with the consideration of cross-correlation. Furthermore, a more stringent result is proven, indicating that the 2SP is not as asymptotically efficient as the DPE in all scenarios. The derived bounds are validated using realistic scenarios, and the root-mean-square error performance of the respective maximum likelihood estimators is compared. Full article

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials. Full article

The time–energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the ${\mathsf{\ell }}_1$ norm of energy coherence $\mathcal{C}$ is invariant with respect to the unitary evolution of the quantum state. Thus, the ${\mathsf{\ell }}_1$ norm of energy coherence $\mathcal{C}$ of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal ${\mathsf{\ell }}_1$ norm of energy coherence, $\mathcal{C}<1$, the Mandelstam–Tamm and Margolus–Levitin relations generate instances of infinite “time uncertainty” that is devoid of physical meaning. Only for quantum states with the maximal ${\mathsf{\ell }}_1$ norm of energy coherence, $\mathcal{C}=1$, the Mandelstam–Tamm and Margolus–Levitin relations avoid infinite “time uncertainty”, but they both reduce to a strict equality that expresses the Einstein–Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance. Full article

We propose a new hierarchy of the vector derivative nonlinear Schrödinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves $\Gamma =\left\{\right(\mu ,\lambda \left)\right\}$ of the simplest multiphase solutions have a quite simple symmetry. They are invariant with respect to holomorphic involution $\tau$. The type of this involution depends on the genus of the spectral curve. Or the involution has the form $\tau :(\mu ,\lambda )\to (\mu ,-\lambda )$, or $\tau :(\mu ,\lambda )\to (-\mu ,-\lambda )$. The presence of symmetry leads to the fact that the dynamics of the solution is determined not by the entire spectral curve $\Gamma$, but by its factor $\Gamma /\tau$, which has a smaller genus. Secondly, it turned out that the dynamics of the two-component vector $\mathbf{p}={(p_1,p_2)}^t$ is determined, first of all, by the dynamics of its length $\left|\mathbf{p}\right|$. Independent equations determine the dependence of the direction of the vector $\mathbf{p}$ from its length. In cases where the direction of the vector $\mathbf{p}$ is fixed, the corresponding spectral curve splits into separate components. In conclusion, we note that, as in the case of the Manakov system, the equation of the spectral curve is invariant with respect to the orthogonal transformation of the vector solutions. I.e., the solution can be found from the spectral curve up to the orthogonal transformation. This fact indicates that the spectral curve does not depend on the individual components of the solution, but on their symmetric functions. Thus, the spectral data of multiphase solutions have two symmetries. These symmetries make it difficult to reconstruct signals from their spectral data. The work contains examples illustrating these statements. Full article

In this work the matrix exponential function is solved analytically for the special orthogonal groups $SO\left(n\right)$ up to $n=9$. The number of occurring k-th matrix powers gets limited to $0\le k\le n-1$ by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of $SO\left(3\right)$, a quadratic equation needs to be solved for $n=4,5$, a cubic equation for $n=6,7$, and a quartic equation for $n=8,9$. As an interesting subgroup of $SO\left(7\right)$, the exceptional Lie group $G_2$ of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, $\xi =1$. The traces of the $SO\left(n\right)$-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities. Full article

In this paper, we introduce a Reduced-Dimension Multiple-Signal Classification (RD-MUSIC) technique via Higher-Order Orthogonal Iteration (HOOI), which facilitates the estimation of the target range and angle for Frequency-Diverse Array Multiple-Input–Multiple-Output (FDA-MIMO) radars in the unfolded coprime array with unfolded coprime frequency offsets (UCA-UCFO) structure. The received signal undergoes tensor decomposition by the HOOI algorithm to get the core and factor matrices, then the 2D spectral function is built. The Lagrange multiplier method is used to obtain a one-dimensional spectral function, reducing complexity for estimating the direction of arrival (DOA). The vector of the transmitter is obtained by the partial derivatives of the Lagrangian function, and its rotational invariance facilitates target range estimation. The method demonstrates improved operation speed and decreased computational complexity with respect to the classic Higher-Order Singular-Value Decomposition (HOSVD) technique, and its effectiveness and superiority are confirmed by numerical simulations. Full article