Null Vector Research Articles

Sensitivity analysis to geometrical imperfections in shell buckling via a mixed generalized path-following method

An efficient continuation method is proposed for a direct sensitivity analysis of the critical point to geometrical imperfections, e.g. expressed as combination of given shapes, for thin-walled structures prone to buckling. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc-length equation is added to follow a path of critical points in the imperfection space. Remarkable novelties are achieved. Firstly, the Jacobian of the extended system is entirely computed analytically by means of a solid-shell model and a strain-based modeling of the geometrical deviation. Moreover, a mixed scheme with independent element-wise stress variables is devised for a more efficient and robust iterative solution compared to the standard null vector method. The mixed algorithm speeds up the sensitivity analysis allowing larger imperfection steps and much fewer factorizations of the condensed stiffness matrix, only one per imperfection in its modified version with constant matrix over the step. Finally, the critical point derivatives with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient-based critical path for a quick search of the worst-case imperfection.

Late time tails and nonlinear memories in asymptotically de Sitter spacetimes

We study the propagation of a massless scalar wave in de Sitter spacetime perturbed by an arbitrary central mass. By focusing on the late time limit, this probes the portion of the scalar signal traveling inside the null cone. Unlike in asymptotically flat spacetimes, the amplitude of the scalar field detected by an observer at timelike infinity does not decay back to zero but develops a spacetime constant shift---both at zeroth and at first order in the central mass $M$. This indicates that massless scalar field propagation in asymptotically de Sitter spacetimes exhibits both linear and nonlinear tail-induced memories. On the other hand, for sufficiently late retarded times, the monopole portion of the scalar signal measured at null infinity is found to be amplified relative to its timelike infinity counterpart, by its nonlinear interactions with the gravitation field at first order in $M$.

Dimension Reduced Channel Feedback for Reconfigurable Intelligent Surface Aided Wireless Communications

Reconfigurable intelligent surface (RIS) has recently received extensive research interest due to its capability to intelligently change the wireless propagation environment. For RIS-aided wireless communications in frequency division duplex (FDD) mode, channel feedback at the user equipment (UE) is essential for the base station (BS) to acquire the downlink channel state information (CSI). In this paper, a dimension reduced channel feedback scheme is proposed to reduce the channel feedback overhead by exploiting the single-structured sparsity of BS-RIS-UE cascaded channel. Since different UEs share the same sparse BS-RIS channel but have their respective RIS-UE channels, there are only limited non-zero column vectors in the BS-RIS-UE cascaded channel matrix, and different UEs share the same indexes of the non-zero columns. Thus, the downlink CSI can be decomposed into <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">user-independent</i> channel information (i.e., the indexes of non-zero columns) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">user-specific</i> channel information (i.e., the non-zero column vectors), where the former for all UEs can be fed back by only one UE, while the latter can be fed back with a fairly low overhead by different UEs, respectively. Simulation results show that, compared with the conventional method, the proposed scheme can reduce channel feedback overhead by more than 80% for RIS-aided wireless communications.

The Rigging Technique for Null Hypersurfaces

Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null hypersurfaces, with emphasis in null cones, totally geodesic, totally umbilic, and compact null hypersurfaces. On the other hand, we show the interplay with the ambient space, including its influence in causality theory.

Asymptotic vectors of entire curves

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors.&#x0D; A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t&lt;\infty $, $\left|z\left(t\right)\right|&lt;\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|&lt;\infty \right\}$.&#x0D; We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.

A Composite Initialization Method for Phase Retrieval

Phase retrieval is a classical inverse problem with respect to recovering a signal from a system of phaseless constraints. Many recently proposed methods for phase retrieval such as PhaseMax and gradient-descent algorithms enjoy benign theoretical guarantees on the condition that an elaborate estimate of true solution is provided. Current initialization methods do not perform well when number of measurements are low, which deteriorates the success rate of current phase retrieval methods. We propose a new initialization method that can obtain an estimate of the original signal with uniformly higher accuracy which combines the advantages of the null vector method and maximal correlation method. The constructed spectral matrix for the proposed initialization method has a simple and symmetrical form. A lower error bound is proved theoretically as well as verified numerically.

Electromagnetic and gravitational radiation in all dimensions: A classical field theory treatment

How long does a light bulb shine in odd dimensional flat spacetimes, according to a distant observer? This question is non-trivial because electromagnetic and gravitational waves, despite being comprised of massless particles, can develop tails: they travel inside the light cone. To this end, I attempt to close a gap in the literature by first deriving, strictly within classical field theory, the real-time electromagnetic dipole and gravitational quadrupole energy and angular momentum radiation formulas in all relevant dimensions. The even dimensional case, where massless signals travel strictly on the null cone, depends on the time derivatives of the dipoles and quadrupoles solely at retarded time; whereas the odd dimensional ones involve an integral over their retarded histories. Despite the propagation of light inside the null cone, however, I argue that a monochromatic light bulb of some intrinsic duration in odd dimensions remains approximately the same apparent duration to a distant detector, though the tail effect does produce a phase shift and adds to the signal several transitory non-oscillatory inverse square roots in time.

Damage Index Analysis of Retaining Wall Structures Based on the Impulse Response Function and Virtual Impulse Response Function

To identify the damage within retaining wall structures, the Hilbert–Huang Transforms of the impulse response function and virtual impulse response function were performed. The Hilbert marginal energy ratio spectrums of the impulse response function and virtual impulse response function were acquired. To reflect damage information effectively, those bands with stronger damage sensitivity were extracted via the threshold value ε0. Then, the Hilbert feature bands, which were more sensitive to damage within retaining walls, were selected by considering the contribution of the residual band to the damage identification. Based on the feature bands, the Hilbert damage feature vector, which reflects the variations of Hilbert marginal energy ratio caused by damage, was created. Based on the damage feature vector, two damage identification indexes (the energy ration standard deviation and Energy Ration Standard Deviation), which were based on the impulse response function and virtual impulse response function, respectively, were proposed to identify damage within retaining walls. To investigate the validity of the damage indexes, vibration tests on a pile plate retaining wall were done. The test results show that the damage feature vector is a zero vector or the value of damage index is zero when the wall is undamaged. The damage feature vector is a nonzero vector or the value of the damage index is more than zero when the wall is damaged. Thus, the damage state of the wall can be detected sensitively via the damage feature vector or damage indexes. Partial damage causes greater fluctuation of trend surface of the damage index. The location of partial damage can be diagnosed validly via the coordinate of peak value in the trend surface. The quantitative relationship formula between the damage index and damage intensity is established. The damage intensity of the wall can be calculated reversely, when the damage index is available. Either the energy ration standard deviation or Energy Ration Standard Deviation can be used to detect the damage state, diagnose the damage location, and identify the damage intensity. In comparison with the energy ration standard deviation, the stability and damage sensitivity of the Energy Ration Standard Deviation is much better.

Extending Nirenberg–Spencer’s question on holomorphic embeddings to families of holomorphic embeddings

Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan–Fubini-type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.

Bone Morphogenetic Protein 7 Gene Delivery Improves Cardiac Structure and Function in a Murine Model of Diabetic Cardiomyopathy.

Diabetes is a major contributor to the increasing burden of heart failure prevalence globally, at least in part due to a disease process termed diabetic cardiomyopathy. Diabetic cardiomyopathy is characterised by cardiac structural changes that are caused by chronic exposure to the diabetic milieu. These structural changes are a major cause of left ventricular (LV) wall stiffness and the development of LV dysfunction. In the current study, we investigated the therapeutic potential of a cardiac-targeted bone morphogenetic protein 7 (BMP7) gene therapy, administered once diastolic dysfunction was present, mimicking the timeframe in which clinical management of the cardiomyopathy would likely be desired. Following 18 weeks of untreated diabetes, mice were administered with a single tail-vein injection of recombinant adeno-associated viral vector (AAV), containing the BMP7 gene, or null vector. Our data demonstrated, after 8 weeks of treatment, that rAAV6-BMP7 treatment exerted beneficial effects on LV functional and structural changes. Importantly, diabetes-induced LV dysfunction was significantly attenuated by a single administration of rAAV6-BMP7. This was associated with a reduction in cardiac fibrosis, cardiomyocyte hypertrophy and cardiomyocyte apoptosis. In conclusion, BMP7 gene therapy limited pathological remodelling in the diabetic heart, conferring an improvement in cardiac function. These findings provide insight for the potential development of treatment strategies urgently needed to delay or reverse LV pathological remodelling in the diabetic heart.

Rotating black holes in general relativity coupled to nonlinear electrodynamics

We find an exact spherically symmetric magnetically charged black hole solution to general relativity (GR) coupled to nonlinear electrodynamics (NED) with an appropriate Lagrangian density. In turn, starting with this spherical black hole as a seed metric, we construct a rotating spacetime, a modification of Kerr black hole, using the revised Newman–Janis algorithm that depends on mass, spin, and a NED parameter g. We find an exact expression for thermodynamic quantities of the black holes like the mass, Hawking temperature, entropy, heat capacity, and free energy expressed in terms of horizon radius, and they show significant deviations from the Kerr case owing to NED. We also calculate analytical expressions for effective Komar mass and angular momentum for the rotating black hole and demonstrate that the Komar conserved quantity corresponding to the null Killing vector at horizon obeys Kχ=2S+T+. The radiating counterpart renders a generalization of Carmeli’s spacetime as well as Vaidya’s spacetime in the appropriate limits.

A system of disjoint representatives of line segments with given k directions

We prove that for all positive integers n and k, there exists an integer N=N(n,k) satisfying the following. If U is a set of k nonzero vectors in the plane and JU is the set of all line segments in direction u for some u∈U, then for every N families F1,…,FN, each consisting of n mutually disjoint segments in JU, there is a set {A1,…,An} of n disjoint segments in ⋃1≤i≤NFi and distinct integers p1,…,pn∈{1,…,N} satisfying that Aj∈Fpj for all j∈{1,…,n}. We generalize this property for underlying lines on fixed k directions to k families of simple curves with certain conditions.

Traversable phantom wormholes via conformal symmetry in f(R,φ,χ) gravity

This analysis explores the solutions for wormhole in [Formula: see text] gravity, where [Formula: see text], and [Formula: see text] represent the kinetic term, scalar potential, and Ricci scalar, respectively. For this study, we use the spherically symmetric spacetime with the anisotropic source of matter. Further, we use the linear equation of state to complete this current investigation in the background of conformal symmetry motion. By plugging non-zero conformal Killing vectors, we discuss the feasible phantom wormhole configurations. Under the linear equation of state, the stability of wormhole solutions with specific values of parameters is also checked by using the Tolman–Oppenheimer–Volkoff equation. Further, the energy conditions are also discussed with conformal motions. Moreover, it is concluded that our inquired solutions are physically viable in [Formula: see text] gravity.

Linear bijective maps preserving fixed values of products of matrices at fixed vectors

For a fixed integer let be the algebra of all n × n matrices over the complex field Let be two fixed nonzero vectors, and fix also In this paper, we characterize linear bijective maps having the property that whenever

Character Polynomials and the Restriction Problem

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

Mutual information superadditivity and unitarity bounds

We derive the property of strong superadditivity of mutual information arising from the Markov property of the vacuum state in a conformal field theory and strong subadditivity of entanglement entropy. We show this inequality encodes unitarity bounds for different types of fields. These unitarity bounds are precisely the ones that saturate for free fields. This has a natural explanation in terms of the possibility of localizing algebras on null surfaces. A particular continuity property of mutual information characterizes free fields from the entropic point of view. We derive a general formula for the leading long distance term of the mutual information for regions of arbitrary shape which involves the modular flow of these regions. We obtain the general form of this leading term for two spheres with arbitrary orientations in spacetime, and for primary fields of any tensor representation. For free fields we further obtain the explicit form of the leading term for arbitrary regions with boundaries on null cones.

Topological theory of inversion-breaking charge-density-wave monolayer 1T-TiSe2

A charge density wave (CDW) of a nonzero ordering vector q couple electronic states at k and k + q statically, giving rise to a reduced Brillouin zone due to the band folding effect. Its structure, referred to an irreducible representation of the little group of q, would change the symmetry of the system and electronic structure accompanying possible change of band inversion, offering a chance of the topological phase transition. Monolayer 1T-TiSe2 is investigated for it shows an unconventional CDW phase having a triple-q structure. Moreover, the coupling between the triple-q component of the CDW will inevitably produces a small CDW. The CDW yields a band inversion in 1T-TiSe2 but different types of CDW can affect the electronic structure and system topology differently. The impact of CDW of different types was studied by utilizing a symmetrization–antisymmetrization technique to extract the and CDW contributions in the density functional theory-based tight-binding model and study their effects. The results are consistent with the parity consideration, improving understanding of topology for a CDW system with and without parity.

Satins, lattices, and extended Euclid's algorithm

Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L(m,a)=langle (1,a),(0,m)rangle where 1le a<m are integers with gcd (m,a)=1. We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a.

Singular tuples of matrices is not a null cone (and the symmetries of algebraic varieties)

Abstract The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING n , m {{\rm SING}_{n,m}} , consisting of all m-tuples of n × n {n\times n} complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SING n , m {{\rm SING}_{n,m}} will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: SING n , m {{\rm SING}_{n,m}} is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of SING n , m {{\rm SING}_{n,m}} . To prove this result, we identify precisely the group of symmetries of SING n , m {{\rm SING}_{n,m}} . We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case m = 1 {m=1} , and suggests a general method for determining the symmetries of algebraic varieties.

Collective Coordinate Model of Kink-Antikink Collisions in ϕ^{4} Theory.

The fractal velocity pattern in symmetric kink-antikink collisions in ϕ^{4} theory is shown to emerge from a dynamical model with two effective moduli: the kink-antikink separation and the internal shape mode amplitude. The shape mode usefully approximates Lorentz contractions of the kink and antikink, and the previously problematic null vector in the shape mode amplitude at zero separation is regularized.