Null Vector Research Articles

A study on the spectral properties of covariance type matrices with isotropic log-concave column vectors.

The limiting spectral distribution of matrix [Formula: see text] is considered in this paper. Existing results always focus on the condition of modifying Tn, but for Xn, it is usually assumed to be a matrix composed of n × N independent identically distributed elements. Here we specify the joint distribution of column vectors of Xn. In particular, entries on the same column of Xn are correlated, in contrast with more common independence assumptions. Assuming that the columns of Xn are random vectors following the isotropic log-concave distribution, and under some additional regularity conditions, we prove that the empirical spectral distribution [Formula: see text] of matrix Bn converges to a deterministic probability distribution F almost surely. Moreover, the Stieltjes transformation m = m(z) of F satisfies a deterministic form of equation, and for any [Formula: see text], it is the unique solution of the equation.

Power set of some quasinilpotent weighted shifts on ℓp

For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined kx=limsupz→0ln⁡‖(z−T)−1x‖ln⁡‖(z−T)−1‖ for each nonzero vector x∈X, and call Λ(T)={kx:x≠0} the power set of T. They proved that the power set have a close link with T's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓp for 1≤p<∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓp(N), then Λ(T)={1} when ke0=1, where {en}n=0∞ be the canonical basis for ℓp(N); (2) There is a class of backward unilateral weighted shifts on ℓp(N) whose power set is [0,1]; (3) There exists a bilateral weighted shift on ℓp(Z) with power set [12,1].

General criteria for a stronger notion of lineability

A subset A A of a vector space X X is called α \alpha -lineable whenever A A contains, except for the null vector, a subspace of dimension α \alpha . If X X has a topology, then A A is α \alpha -spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of ( α , β ) (\alpha ,\beta ) -lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of ( α , β ) (\alpha ,\beta ) -lineability/spaceability and, as applications, we extend recent results of different authors.

Some generic hypersurfaces in a Euclidean space

&lt;abstract&gt;&lt;p&gt;In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field $ {\omega } $ on a compact and connected hypersurface $ N $ of the Euclidean space $ R^{m+1} $ with a mean curvature $ \alpha $ constant along the integral curves of $ {\omega } $ and a shape operator $ T $ satisfying $ T({\omega) = \alpha \omega } $ implies that $ \alpha $ is a constant and $ N $ is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field $ \mathbf{v} $ on a compact and connected hypersurface $ N $ of a Euclidean space $ R^{m+1} $ gives a nonzero function $ \sigma = g\left(T \mathbf{v}, \mathbf{v}\right) $ with shape operator $ T $, and the integral of the function $ m\alpha \sigma Ric\left(\mathbf{v}, \mathbf{v}\right) $ has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface $ N $ with support $ \rho $ and basic vector field $ \mathbf{u} $, the integral of the Ricci curvature $ Ric\left(\mathbf{u}, \mathbf{u}\right) $ has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.&lt;/p&gt;&lt;/abstract&gt;

Three-Dimension Space Vector Modulation for Three-Phase Series-End Winding Voltage-Source Inverters

Series-end winding voltage-source inverters (SEW-VSIs) can effectively improve the dc-link voltage utilization and provide the capabilities to handle the zero-sequence loop. However, the existing modulation schemes for SEW-VSIs cannot fully utilize the output capability of SEW-VSIs while guaranteeing simple implementation. To this end, a three-dimensional space vector modulation (3D-SVM) scheme is proposed for driving three-phase SEW-VSIs in this paper. The modulation is conducted in the 3D-space made up of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α-β</i> plane and the zero-sequence-axis, and the reference voltage vector in the 3D-space is synthesized by three adjacent basic voltage vectors and the null voltage vector, which can fully utilize the output capability of three-phase SEW-VSI. Then, a convenient implementation method based on a look-up table is developed to find three basic voltage vectors adjacent to the reference voltage vector. Moreover, in the 3D-space, the feasible region and the linear modulation region of the proposed 3D-SVM are investigated. Accordingly, intuitive overmodulation between different subspaces and the switching sequence generation are developed. A series of simulation and experimental results are presented to validate the effectiveness of the proposed 3D-SVM for three-phase SEW-VSI drives.

THE BEHAVIOUR OF SEMIGROUP OF CONTRACTIONS ON HILBERT SPACE

Abstract. Let be a strongly continuous, one-parameter semigroup of contractions on a complex Hilbert space 𝐻. The generator of is the linear operator with domain defined by Let be a closed densely defined operator on a Hilbert space with domain . For we define to be the set of all for which there exists a neighborhood of with analytic on having values in such that for all . This set is open and contains the resolvent set of . By definition, the local spectrum of at denoted by is the complement of , so it is a closed subset of The set is called local unitary spectrum of at We will say that 𝑻 is a multiple of the identity if there is 𝜆 ∈ ℝ such that 𝑇(𝑡) = 𝑒𝑥𝑝(𝑖𝜆𝑡) for all 𝑡 ≥ 0. Theorem. Let be a strongly continuous, one-parameter semi-group of contractions on a Hilbert space 𝐻. Assume that there exists a vector such that and is at most countable. If is not a multiple of the identity, then there exists a nonzero vector such that

Planar Pseudo-geodesics and Totally Umbilic Submanifolds

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion ι:M↪Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota :M \\hookrightarrow Q$$\\end{document} is totally umbilic if and only if the extrinsic shape of every geodesic of M is planar. This extends a well-known result about surfaces in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{3}$$\\end{document}.

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Let Sm(c) denote a sphere with a positive constant curvature c and Mn(n≥3) be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space Sm(c)×R with a nonzero parallel mean curvature vector (PMC), where R is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of Mn, which allow us to generalize some classical curvature pinching results in spheres.

Eigenvectors of the De-Rham Operator

We aim to examine the influence of the existence of a nonzero eigenvector ζ of the de-Rham operator Γ on a k-dimensional Riemannian manifold (Nk,g). If the vector ζ annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field ζ on (Nk,g), there are two operators Tζ and Ψζ associated with ζ, called the basic operator and the associated operator of ζ, respectively. We show that the existence of an eigenvector ζ of Γ on a compact manifold (Nk,g), such that the integral of Ric(ζ,ζ) admits a certain lower bound, forces (Nk,g) to be isometric to a k-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field ζ on a connected and complete Riemannian space (Nk,g), having divζ≠0 and annihilating the associated operator Ψζ, forces (Nk,g) to be isometric to the k-dimensional Euclidean space, provided that the squared length of the covariant derivative of ζ possesses a certain lower bound.

Rigidity of three-dimensional steady internal waves with constant vorticity governed by the [formula omitted]-plane approximation

This paper investigates the structural implications of constant vorticity for steady three-dimensional internal water waves governed by the equatorial f-plane approximation. Under the assumption that the first components of the non-zero constant vorticity vectors in the upper and lower layers are zeros, it is proved that every three-dimensional traveling internal wave with bounded velocity is either two-dimensional or a trivially three-dimensional shear flow with the pressures in both layers are identically equal.

Comprehensive analysis of multiple machine learning techniques for rock slope failure prediction

In this study, twelve machine learning (ML) techniques are used to accurately estimate the safety factor of rock slopes (SFRS). The dataset used for developing these models consists of 344 rock slopes from various open-pit mines around Iran, evenly distributed between the training (80%) and testing (20%) datasets. The models are evaluated for accuracy using Janbu's limit equilibrium method (LEM) and commercial tool GeoStudio methods. Statistical assessment metrics show that the random forest model is the most accurate in estimating the SFRS (MSE = 0.0182, R2 = 0.8319) and shows high agreement with the results from the LEM method. The results from the long-short-term memory (LSTM) model are the least accurate (MSE = 0.037, R2 = 0.6618) of all the models tested. However, only the null space support vector regression (NuSVR) model performs accurately compared to the practice mode by altering the value of one parameter while maintaining the other parameters constant. It is suggested that this model would be the best one to use to calculate the SFRS. A graphical user interface for the proposed models is developed to further assist in the calculation of the SFRS for engineering difficulties. In this study, we attempt to bridge the gap between modern slope stability evaluation techniques and more conventional analysis methods.

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

Algebraic Geometry and Elliptic Integrals Approach for Calculation of the Propagation Time of a Signal Between GPS Satellites with Account of General Relativity Theory

The approach for calculating the propagation time of a signal between GPS satellites will be summarized, based on the proposed new theoretical approach in several previous publications, as well as the perspectives for future development of the theory. Topics include: 1. Basic notions of inter-satellite communications. 2. Shapiro delay formulae in General Relativity Theory - basic formalism and the necessity to extend the formalism by taking into account the satellite motion on a plane elliptic or space-distributed elliptic orbit. 3. Basic facts about the disturbed motion in celestial mechanics and the necessity to incorporate it in the theory of inter-satellite communications, accounting for General Relativity Effects. 4. Propagation time of a signal, emitted by a satellite on a plane and also space-distributed elliptical orbit in terms of zero-order elliptic integrals and respectively of higher order integrals. Proof of the real-valuedness of the propagation time for all cases as one of the criteria for the correctness of the theoretical approach. 5. New analytical algorithms for calculation of zero-order elliptic integrals in the Legendre form. Relation to two representations in the Weierstrass form. 6. The new formalism of intersecting four-dimensional null cones and the resulting physical notions of the (intersecting)) space-time interval (with the property of being positive, negative or equal to zero) and the (intersecting) geodesic distance (being only positive, because is related to the distance, travelled by light or radio signals). Proof of these properties in the general case and in some partial cases. New numerical estimate E lim > 45.002510943228 [deg], above which the space-time interval is positive and thus inter-satellite communications between satellites on one plane elliptical orbit are possible. The angular distance of 45[ deg] is typical for the disposition of 8 satellites on one orbit in the Russian satellite constellation GLON ASS, so it might be claimed that such a configuration is favourable from the point of view of inter-satellite communications (with account of GRT effects).

Regularity for parabolic equations with singular non-zero divergence vector fields

We establish two-sided Gaussian bounds on the heat kernel of divergence-form parabolic equation with singular time-inhomogeneous vector field satisfying some minimal assumptions.

Elementary coupling coefficients for the Wigner supermultiplet symmetry

An algorithm for evaluating Wigner coefficients of U(4) ⊃ SUS(2) ⊗ SUT(2) for the coupling [n14,n24,n34,n44]⊗[1,0,0,0]↓[m14,m24,m34,m44] with arbitrary irreducible representation [n14,n24,n34,n44] is provided based on the algebraic expressions with the expansion coefficients of the U(4)⊃SUS(2)⊗SUT(2) states in terms of those in the U(4) canonical basis and the Clebsch-Gordan coefficients of U(4) in the canonical basis. The state expansion coefficients are evaluated as the components of the null space vectors of the spin-isospin projection matrices. Program summaryProgram Title: Wigner-U4-SU2xSU2CPC Library link to program files:https://doi.org/10.17632/4b7gf2phh2.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 9.0.1.0 and beyondNature of problem: Numerical evaluation of the elementary Wigner coefficients (EWCs) of U(4)⊃SUS(2)⊗SUT(2).Solution method: Mathematica implementation of the EWCs of U(4)⊃SUS(2)⊗SUT(2) [1] using the expansion coefficients of U(4)⊃SUS(2)⊗SUT(2) states in terms of those of the U(4) canonical basis determined as components of the null space vectors of the spin-isospin projection matrices [2].

Sparks of symmetric matrices and their graphs

The spark of a matrix is the smallest number of nonzero coordinates of any nonzero null vector. For real symmetric matrices, the sparsity of null vectors is shown to be associated with the structure of the graph obtained from the off-diagonal pattern of zero and nonzero entries. The smallest possible spark of a matrix corresponding to a graph is defined as the spark of the graph. Connections are established between graph spark and well-known concepts including minimum rank, forts, orthogonal representations, Parter and Fiedler vertices, and vertex connectivity.

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (B) in particular. We demonstrate that any B-differentiable function gives rise to a null shear-free congruence (NSFC) on the B-vector space CM and on its Minkowski subspace M. Making use of the Kerr–Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of B-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in CM. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the “Unique Worldline” and discuss the connections of the theory with the Feynman–Wheeler concept of “One-Electron Universe”.

Toward Optimal Synthesis of Discrete-Time Hopfield Neural Network.

In this research brief, the relationship between eigenvectors (with {+1, -1} components) of a synaptic weight matrix W and the stable/anti-stable states of discrete-time Hopfield associative memory (HAM) is established. Also, the synthesis of W with desired stable/anti-stable states using spectral representation of W in even/odd dimension is discussed when the threshold vector is a non-zero vector. Freedom in choice of eigenvalues is capitalized to improve the noise immunity of the Hopfield neural network (HNN). Also, the problem of optimal synthesis of Hopfield Associative memory is formulated.

Conditional-Extreme-Point Based Fault-Tolerant Predictive Control of Five-Phase PMSM With Low Switching Frequency

The virtual voltage vector (VVV) based fault-tolerant control has been widely concerned, but suffering from the increased switching frequency and the reduced bus voltage utilization at the faulty operation. Also, the harmonic space is uncontrollable. This paper proposes a conditional-extreme-point based fault-tolerant model predictive control for a five-phase permanent magnet synchronous motor. The conditional extreme point is used to predict the optimal reference voltage vector. Firstly, to suppress the harmonics, dual control sets are established without the sacrifice of bus voltage utilization. Then, the unconditional extreme point and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</i> -axis voltage are predicted to select the appropriate control set, and two adjacent vectors are pre-selected for generating the optimal reference voltage vector. Subsequently, the conditional extreme point of the cost function is calculated to obtain the optimal reference voltage vector. Due to the unemployment of the null vector, the switching frequency is greatly reduced. Finally, the experimental results are presented to verify the proposed control scheme.

A Low Common-Mode SVPWM for Two-Level Three-Phase Voltage Source Inverters

In order to reduce the common-mode voltage (CMV) generated by the use of space vector pulse width modulation (SVPWM) in two-level three-phase voltage source inverters, a low common-mode SVPWM method is proposed. In this method, the voltage plane is divided into 12 sectors, and on each sector, two non-zero vectors of the same class and one single zero vector are adopted for synthesis. The action time of the zero vector is placed at both ends of each switching cycle, the currents are sampled at the beginning of each switching cycle, and the action time and sequence of vectors on each sector is provided. Simulation and experimental results show that, in the vector control system of a permanent magnet synchronous motor fed by the inverter, compared with the conventional SVPWM, the proposed method reduces the CMV peak-to-valley value by 33.333%, the CMV jump frequency by three times, and the performance of the line voltage and line current. The electromagnetic torque and rotor speed remain good, which has good application value in high-performance drives.