Null Vector Research Articles

Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect

We investigate quasi-free Hadamard states defined via characteristic initial data on null cones centred at the axis of symmetry in spherically symmetric space-times. We characterize the necessary singular behaviour of null boundary two-point functions such that one can define non-linear observables at this null boundary and give formulas for the calculation of these observables. These results extend earlier characterizations of null boundary states defining Hadamard states in the bulk of the null cone. As an application of our derived formulas, we consider their implications for the semi-classical Einstein equations and calculate the vacuum polarization associated with Hawking radiation near a collapsing body.

Flat bands and quasi-bound states in the continuum in a photonic Moiré lattice

This paper proposes flat bands and quasi-bound states in the continuum (Q-BIC) in a moiré lattice. In addition to the inter-layer coupling and small rotation angle, we show that the inter-node coupling in a single-layer moiré lattice can be controlled to obtain moiré flat bands. Unlike any flat bands reported in lattices such as kagome, rhombus, Lieb, and stub, we show that the moiré lattice exhibits both singular and non-singular flat bands simultaneously. We investigate the flat band’s eigenmodes to confirm their flatness and assess their singularity. The robustness of the flat bands to variation of background permittivity ε b is studied. Moreover, by further engineering the inter-node distance, the moiré flat band states change to a quasi-bound state at a non-zero wave vector. The Q-BIC exhibit an ultra-high quality factor (Q-factor) of 2.7142 × 10 6 , ultra-low mode volume of 0.214318 µ m 3 , and very narrow linewidth centred at 350.54 THz. We demonstrate the capability of the Q-BIC for lasing and nanocavity applications. Our results show that a single-layer photonic moiré lattice provides a suitable platform to explore fundamental phenomena related to flat band systems.

On concurrent vector fields on Riemannian manifolds

<abstract><p>It is shown that the presence of a non-zero concurrent vector field on a Riemannian manifold poses an obstruction to its topology as well as certain aspects of its geometry. It is shown that on a compact Riemannian manifold, there does not exist a non-zero concurrent vector field. Also, it is shown that a Riemannian manifold of non-zero constant scalar curvature does not admit a non-zero concurrent vector field. It is also shown that a non-zero concurrent vector field annihilates de-Rham Laplace operator. Finally, we find a characterization of a Euclidean space using a non-zero concurrent vector field on a complete and connected Riemannian manifold.</p></abstract>

Two families of circulant nut graphs

A circulant nut graph is a non-trivial simple graph whose adjacency matrix is a circulant matrix of nullity one such that its non-zero null space vectors have no zero elements. The study of circulant nut graphs was originally initiated by Basic et al. [Art Discrete Appl. Math. 5(2) (2021) #P2.01], where a conjecture was made regarding the existence of all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. Later on, it was proved by Damnjanovic and Stevanovic [Linear Algebra Appl. 633 (2022) 127-151] that for each odd t ? 3 such that t ?/10 1 and t ?/18 15, the 4t-regular circulant graph of order n with the generator set {1, 2, 3,..., 2t+1}\{t}) must necessarily be a nut graph for each even n ? 4t + 4. In this paper, we extend these results by constructing two families of circulant nut graphs. The first family comprises the 4t-regular circulant graphs of order n which correspond to the generator sets {1, 2,..., t?1} ? {n/4, n/4 + 1} ? {n/2?(t?1),..., n/2 ? 2, n/2 ? 1}, for each odd t ? N and n ? 4t + 4 divisible by four. The second family consists of the 4t-regular circulant graphs of order n which correspond to the generator sets {1, 2,..., t?1} ? {n+2/4, n+6/4} ? {n/2 ?(t?1),..., n/2?2, n/2?1}, for each t ? N and n ? 4t + 6 such that n ?4 2. We prove that all of the graphs which belong to these families are indeed nut graphs, thereby fully resolving the 4t-regular circulant nut graph order-degree existence problem whenever t is odd and partially solving this problem for even values of t as well.

A multi-locus linear mixed model methodology for detecting small-effect QTLs for quantitative traits in MAGIC, NAM, and ROAM populations

Although multi-parent populations (MPPs) integrate the advantages of linkage and association mapping populations in the genetic dissection of complex traits and especially combine genetic analysis with crop breeding, it is difficult to detect small-effect quantitative trait loci (QTL) for complex traits in multiparent advanced generation intercross (MAGIC), nested association mapping (NAM), and random-open-parent association mapping (ROAM) populations. To address this issue, here we proposed a multi-locus linear mixed model method, namely mppQTL, to detect QTLs, especially small-effect QTLs, in these MPPs. The new method includes two steps. The first is genome-wide scanning based on a single-locus linear mixed model; the P-values are obtained from likelihood-ratio test, the peaks of negative logarithm P-value curve are selected by group-lasso, and all the selected peaks are regarded as potential QTLs. In the second step, all the potential QTLs are placed on a multi-locus linear mixed model, all the effects are estimated using expectation-maximization empirical Bayes algorithm, and all the non-zero effect vectors are further evaluated via likelihood-ratio test for significant QTLs. In Monte Carlo simulation studies, the new method has higher power in QTL detection, lower false positive rate, lower mean absolute deviation for QTL position estimate, and lower mean squared error for the estimate of QTL size (r2) than existing methods because the new method increases the power of detecting small-effect QTLs. In real dataset analysis, the new method (19) identified five more known genes than the existing three methods (14). This study provides an effective method for detecting small-effect QTLs in any MPPs.

Exposure to acoustic waves on viscoelastic cylinder

The problem of the impact of acoustic waves in a homogeneous viscoelastic cylinder is considered. The investigation aims to investigate the diffraction of acoustic harmonic waves in a viscoelastic cylinder. The body is assumed to be in an infinite acoustic space filled with an ideal fluid. Numerical calculations of the angular and frequency characteristics of the scattered field for viscoelastic cylinders under the action of harmonic acoustic waves are carried out. In the case of steady waves, the Helmholtz equation describes the propagation of small disturbances in an acoustic medium. And in a viscoelastic homogeneous isotropic cylinder, scalar and vector Helmholtz equations with complex coefficients, the solution of which is described by Bessel and Neumann functions with complex arguments. A technique and algorithm have been developed for solving the problem of diffraction of acoustic harmonic waves in a viscoelastic cylinder. It has been established that the stress and displacement of a point of a viscoelastic cylinder take on a maximum value in the region of long waves. It was also found that considering the material’s viscous properties reduces the stress components to 10%.

Geometric and o-minimal Littlewood–Offord problems

The classical Erdős–Littlewood–Offord theorem says that for nonzero vectors a1,…,an∈Rd, any x∈Rd, and uniformly random (ξ1,…,ξn)∈{−1,1}n, we have Pr(a1ξ1+⋯+anξn=x)=O(n−1/2). In this paper, we show that Pr(a1ξ1+⋯+anξn∈S)≤n−1/2+o(1) whenever S is definable with respect to an o-minimal structure (e.g., this holds when S is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.

Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

In this paper, we define the curve at a constant distance from the edge of regression on a curve r(s) with arc length parameter s in Galilean 3-space. Here, d is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve r(s) in 3-dimensional Galilean space. We build the Frenet frame of the constructed curve with respect to two types of the vector d and we indicate the properties related to the curvatures of the curve . Also, for the curve , we give the conditions to be a circular helix. Furthermore, we discuss ruled surfaces of type A generated via the curve and the vector D which is defined as tangent of the curve in 3-dimensional Galilean space. The constructed ruled surfaces also appear in two ways. The first is constructed with the curve and the non-isotropic vector D. The second is formed by the curve and the non-isotropic vector D. We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable. Finally, we provide examples and visuals to back up our research.

Null vectors, Schur complements, and Parter vertices

Null vectors, Schur complements, and Parter vertices

The averaging problem on the past null cone in inhomogeneous dust cosmologies

Cosmological models typically neglect the complicated nature of the spacetime manifold at small scales in order to hypothesize idealized general relativistic solutions for describing the average dynamics of the Universe. Although these solutions are remarkably successful in accounting for data, they introduce a number of puzzles in cosmology, and their foundational assumptions are therefore important to test. In this paper, we go beyond the usual assumptions in cosmology and propose a formalism for averaging the local general relativistic spacetime on an observer's past null cone: we formulate average properties of light fronts as they propagate from a cosmological emitter to an observer. The energy-momentum tensor is composed of an irrotational dust source and a cosmological constant -- the same components as in the $\Lambda$CDM model for late cosmic times -- but the metric solution is not \emph{a priori}constrained to be locally homogeneous or isotropic. This generally makes the large-scale dynamics depart from that of a simple Friedmann--Lema\^\i tre--Robertson--Walker solution through \emph{backreaction} effects. Our formalism quantifies such departures through a fully covariant system of area-averaged equations on the light fronts propagating towards an observer, which can be directly applied to analytical and numerical investigations of cosmic observables. For this purpose, we formulate light front averages of observable quantities, including the effective angular diameter distance and the cosmological redshift drift and we also discuss the backreaction effects for these observables.

Generalized Killing Ricci tensor for real hypersurfaces in the complex quadric

AbstractFirst we introduce a new notion of generalized Killing Ricci tensor which is equivalent to the notion of cyclic parallel Ricci tensor for real hypersurfaces in the complex quadric . Next, in terms of ‐principal or ‐isotropic unit normal vector fields we give a complete classification of real hypersurfaces in with cyclic parallel Ricci tensor.

Direct Torque Control for Dual Three-Phase Permanent Magnet Motor With Improved Torque and Flux

The conventional duty ratio modulation based direct torque control (DR-DTC) for the dual three-phase permanent magnet (DTP-PM) motor reduces the torque ripple using duty ratio modulation in torque control. However, the flux was controlled by a hysteresis comparator, resulting in deficient steady-state performance. This paper proposes a novel DR-DTC strategy to improve torque and flux simultaneously. Firstly, a switching table with the multiple vector selection is newly established. Two adjacent virtual voltage vectors and a null vector are selected in each control period. Then, the duty ratios of selected adjacent virtual voltage vectors are optimized with the dual objective modulation considering torque performance and flux performance. In addition, the combination of adjacent virtual voltage vectors for the DTP-PM motor generates non-standard symmetrical switching sequences. Two switching sequences generated methods are further investigated to address the issue. So, the switching sequences of the proposed strategy can also be generated easily. Finally, the conventional strategy and the proposed strategy are compared through experiments, and the experimental results verify the effectiveness of the proposed strategy.

ON RELATIVISTIC HARMONIC OSCILLATOR

A relativistic quantum harmonic oscillator in 3+1 dimensions is derived from a quaternionic non-relativistic quantum harmonic oscillator. This quaternionic equation also yields the Klein-Gordon wave equation with a covariant (space-time dependent) mass. This mass is quantized and is given by $m_{*n}^2=m_\omega^2\left(n_r^2-1-\beta\,\left(n+1\right)\right)\,,$ where $m_\omega=\frac{\hbar\omega}{c^2}\,,$ $\beta=\frac{2mc^2}{\hbar\,\omega}\, $, $n$, is the oscillator index, and $n_r$ is the refractive index in which the oscillator travels. The harmonic oscillator in 3+1 dimensions is found to have a total energy of $E_{*n}=(n+1)\,\hbar\,\omega$, where $\omega$ is the oscillator frequency. A Lorentz invariant solution for the oscillator is also obtained. The time coordinate is found to contribute a term $-\frac{1}{2}\,\hbar\,\omega$ to the total energy. The squared interval of a massive oscillator (wave) depends on the medium in which it travels. Massless oscillators have null light cone. The interval of a quantum oscillator is found to be determined by the equation, $c^2t^2-r^2=\lambda^2_c(1-n_r^2)$, where $\lambda_c$ is the Compton wavelength. The space-time inside a medium appears to be curved for a massive wave (field) propagating in it.

Model Predictive Torque Control for a Dual Three-phase PMSM Using Modified Dual Virtual Vector Modulation Method

Single voltage vectors applied in the conventional model predictive torque control (MPTC) for multiphase motors do not only suffer from serious torque and stator flux ripples but also cause the large harmonic current. To address the aforementioned challenges, an MPTC using a modified dual virtual vector modulation method is proposed to improve the operational performance of a dual three-phase permanent magnet synchronous motor. Virtual voltage vectors are synthesized as the candidate control set to restrain the harmonic current. A transformation method is introduced to consider both the stator flux and torque in the duty cycle modulation. The torque and stator flux ripples are simultaneously reduced by addressing the limitations of nonuniform units. Furthermore, the null voltage vector is then inserted to expand the modulation range and improve the steady-state performance. Moreover, the sawtooth carrier is adopted to address the challenge of the asymmetric switch sequence caused by the modified modulation. Finally, the experimental results are presented to verify the effectiveness and superiority of the proposed MPTC method.

Error Self-Calibration of Phase Current Reconstruction Based on Random Pulsewidth Modulation

Three-phase current is the cornerstone parameter in vector control. Restricted by minimum sampling time of the traditional space vector pulsewidth modulation (PWM), the problem of immeasurable areas (IAs) exists in one dc bus current sensor (ODBCS) three-phase current reconstruction. In addition, affected by actual conditions such as temperature drift, aging, and noise, the problem of offset error (OE) in the three-phase current sampling path is the essential factor that restricts the sampling accuracy. This article is the first to investigate the influence of IAs and OE of ODBCS on the reconstructed current. An error self-calibration (ESC) random PWM reconstruction strategy is proposed, which is self-detection and self-calibration of current OE accomplished by dual sampling of the complementary nonzero vector dynamic current. The proposed ESC of phase current reconstruction based on random PWM does not require the insertion of redundant voltage vectors and major computation and is not limited to a particular PWM technology. The experimental results verify the accuracy and practicability of the proposed method.

On Nonlinearization of 3-parameter Eigenvalue Problems

In this paper, the linear 3-parameter eigenvalue problem (3PEP) in terms of matrix equations is considered. Using the Rayleigh Quotient iteration method, any one of the three parameters can be fixed to transform the problem into a linear 2-parameter eigenvalue problem (2PEP). This admits a family nonlinear eigenvalue problems (NEP) in one parameter. The transformation results from the elimination of the second equation of respective 2PEP, which is re-arranged as a generalized eigenvalue problem (GEP) of the form \(Ey=\lambda Fy\), where \(E\) and \(F\) are matrices \(n\times n\) over \(\mathbb{C}\), \(y\in \mathbb{C}^n\) is a non-zero vector and \(\lambda\) is a scalar. A review of some results of the condition number of NEP is also presented in this paper.

Twelve-Sector Direct Torque Control without Flux Closed-Loop for PMSM

The direct torque control (DTC) of permanent magnet synchronous motor (PMSM), which includes flux and torque double closed-loops, requires constant flux and high calculation accuracy. Besides, uncertain flux estimation will cause torque ripples. To solve these problems, in this paper, twelve-sector direct torque control (DTC) without flux closed-loop for PMSM is proposed. To reduce the requirement on constant flux and high calculation accuracy and to simplify the structure of the system, the flux closed-loop is removed from the control system. To suppress the torque ripple, a twelve-sector space voltage vector control scheme is adopted which increases the switch states of the inverter. The number of nonzero voltage vectors is increased from 6 to 12 without increasing the hardware equipment of the system. The sections of the voltage vectors are redefined and the torque ripples are reduced. The hardware-in-the-loop simulation results show the effectiveness and feasibility of the proposed method.

Polystability in positive characteristic and degree lower bounds for invariant rings

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf’s theory of optimal subgroups and we make some improvements and simplify the theory from a computational perspective. We exhibit our technique in many examples and in particular, give an algorithm to decide if a symmetric polynomial in n -variables has a closed \operatorname{SL}_n -orbit. As an important application, we prove exponential lower bounds on the maximal degree of a system of generators of invariant rings for two actions that are important from the perspective of Geometric Complexity Theory (GCT). The first is the action of \operatorname{SL}(V) on S^3(V)^{\oplus 3} , the space of 3 -tuples of cubic forms, and the second is the action of \operatorname{SL}(V) \times \operatorname{SL}(W) \times \operatorname{SL}(Z) on the tensor space (V \otimes W \otimes Z)^{\oplus 5} . In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone.

Kundt three-dimensional left invariant spacetimes

Kundt spacetimes are of great importance to general relativity. We show that a Kundt spacetime is a Lorentz manifold with a non-singular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kundt structures on Lie groups. Finally, we classify all left invariant Kundt structures on three-dimensional simply connected unimodular Lie groups.

Thermodynamics of Taub-NUT and Plebanski solutions

We observe the parallel between the null Killing vector on the horizon and degenerate Killing vectors at both north and south poles in Kerr-Taub-NUT and general Plebanski solutions. This suggests a correspondence between the pairs of the angular momentum/velocity and the NUT charge/potential. We treat the time as a real line such that the Misner strings are physical. We find that the NUT charge spreads along the Misner strings, analogous to that the mass in the Schwarzschild black hole sits at its spacetime singularity. We develop procedures to calculate all the thermodynamic quantities and we find that the results are consistent with the first law (Wald formalism), the Euclidean action and the Smarr relation. We also apply the Wald formalism, the Euclidean action approach, and the (generalized) Komar integration to the electric and magnetic black holes in a class of EMD theories, and also to boosted black strings and Kaluza-Klein monopoles in five dimensions, to gain better understandings of how to deal with the subtleties associated with Dirac and Misner strings.