Fundamental Form Research Articles

Similarity invariants of essentially normal Cowen-Douglas operators and Chern polynomials

In this paper, we systematically study a class of essentially normal operators by using the geometry method from the Cowen–Douglas theory and prove a Brown–Douglas–Fillmore theorem in the Cowen–Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.

On the Reconstruction of Geodesic Subspaces of ℝN

We consider the topological and geometric reconstruction of a geodesic subspace of [Formula: see text] both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for the reconstruction problem. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and [Formula: see text]–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of [Formula: see text], we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space.

An analytical framework of the kinematic geometry for general point-contact gears from contact path

Based on the idea that the relative curvature relationships along total contact path will better reflect the contact characteristics of gears, this study attempts to develop an analytical framework for the kinematic geometry of point-contact gears from the contact paths. First, a general method for obtaining the contact path of a gear from a pinion is proposed. Second, a normal curvature and geodesic torsion calculation method in the contact path direction is developed. Then, based on the linearly dependent condition of the tangent vectors of the two contact paths, two relative normal curvature analysis methods are proposed by applying the fundamental form theory. Finally, a sample case is presented to show the application and complete the verification of the kinematic geometry methods. The sample case agrees with the analytical model.

Self-shrinkers with bounded |HA|

In this paper, we study integral and pointwise bounds on the second fundamental form of properly immersed self-shrinkers with bounded |HA|. As applications, we prove gap and compactness results for self-shrinkers.

Large Amplitude Axisymmetric Free Vibration of a Closed Two-Dimensional Spherical Pressure Vessel with a Constrained Volume

The nonlinear axisymmetric vibrational modes of a closed two-dimensional spherical pressure vessel with a constrained volume are presented using the fundamental forms of surfaces. The strain energy of the spherical pressure vessel is derived as a quadratic function of the initial Eulerian, added, and total Lagrangian strains. Two-dimensional bilinear elements described in spherical polar coordinates with the symmetric boundary condition along both sides of the meridian line are used for investigating the nonlinear axisymmetric mode shapes of a spherical pressure vessel. Numerical results were obtained by a nonlinear finite element approach and verified in the case of an empty spherical shell for both lower and upper branches. This study showed that the spherical pressure vessel gave a higher nonlinear axisymmetric natural frequency than the empty spherical shell. The elastic modulus has a large effect on the nonlinear axisymmetric frequencies of both the lower and upper branches for the spherical pressure vessel. Changing the thickness and initial internal pressure significantly affects only the lower branch of the spherical pressure vessel at high mode numbers. The results also indicate that the ratios of the nonlinear natural frequencies of the lower to upper branches decrease when the vibrational modes increase.

New Characterizations of the Whitney Spheres and the Contact Whitney Spheres

In this paper, based on the classical Yano’s formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction \(J\mathbf {H}\) and the norm of the covariant differentiation of the second fundamental form h, where J is the almost complex structure and \(\mathbf {H}\) is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure \((\varphi ,\xi ,\eta ,g)\), we also establish an optimal integral inequality involving the Ricci curvature in the direction \(\varphi \mathbf {H}\) and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant.

Best practice guidelines for reporting spoken discourse in aphasia and neurogenic communication disorders

ABSTRACT Background Spoken discourse is a fundamental form of communication, yet often disrupted in persons with aphasia and other neurogenic communication disorders. It follows that discourse analysis is important for understanding language, its impairments, and language recovery in these populations. Being a highly interdisciplinary field, discourse analysis would benefit from a set of guidelines (best practices) that help streamline reporting and discussion of discourse specific to populations with neurogenic communication disorders. This would enhance the overall quality of ongoing research by improving replicability, allowing meta-analysis, and supporting implementation in clinical practice. Aims To establish clear reporting guidelines for studies that use spoken discourse analysis in aphasia. Methods & Procedures An e-Delphi survey methodology was used. We identified aphasia and neurogenic communication disorder experts using Boolean search criteria and author consensus, then invited these individuals to participate in an online, three-round survey. The three rounds took place between November 2020 and June 2021. In each round, participants were asked to rank necessity of items on a scale of 1 (“not at all necessary”) to 5 (“extremely necessary”). Participants were instructed to rate based on whether the item was necessary to inform the creation of a minimum reporting guideline. Only participants who responded in the prior round were invited to participate in subsequent rounds. Items that met predetermined consensus criteria for ratings greater than “necessary” were brought forward to each subsequent round. New items were added to each round based on qualitative information collected from participants. Outcomes & Results Items that were rated by ≥ 70% of the participants as “highly necessary” or “extremely necessary” in Round 3 were those deemed as “necessary” to be reported in research, whereas items rated < 70% in Round 3 were deemed as “recommended” to be reported in research. Following Round 3, 16 items were deemed “necessary” to report in research. A further five items were “recommended”. Conclusions Consensus by leading researchers in the field established that 20 reporting items were “necessary” or “recommended” as the minimum information reported across all spoken discourse studies. These items included information about sampling, transcription, and analysis. Adherence to rigorously developed reporting guidelines is important for improving the overall quality and replicability of research in the field. We have archived the current recommendations on the Open Science Framework, which is where updates to the recommendations will also be located (https://osf.io/y48n9/).

Results on a class of analytic functions with finitely many fixed coefficients related to a generalised multiplier transformation

A new class of univalent functions denoted by Tm(ξ,ϖ,σ,en) is defined and studied in this work. The class of functions Tm(ξ,ϖ,σ,en) generalises certain previously introduced classes of functions in the literature. Geometric properties of the class of functions in the class such as coefficient estimate, radius of starlikenes and radius convexity were studied.

Virtual Sociability

Since the 2020 COVID-19 pandemic, the term virtual has become one of the most used in media and everyday speech. There is an increasing amount of research done on this new reality, and the results are still to be published. However, it is insufficiently known in scientific periodicals that the concept of virtual reality, enabled by information technology, has existed in the sociological literature since the 1990s when Castells introduced it to the theory of network society. Therefore, the paper's primary goal is to consider basic concepts such as virtual reality, real virtuality, virtual communities, virtual sociability, and networked individualism. We will also briefly look at the forgotten classical notion of community as a fundamental form of sociability defined by the sociologist F. Tönnies, which re-enters the focus of scientific interest, without a clear and sociologically known classical definition. Starting from this framework, the paper examines the key question of whether these new virtual communities are indeed communities in the classical sense of the term or whether it is a form of networked individualism.

Dendritic autophagy degrades postsynaptic proteins and is required for long-term synaptic depression in mice

The pruning of dendritic spines during development requires autophagy. This process is facilitated by long-term depression (LTD)-like mechanisms, which has led to speculation that LTD, a fundamental form of synaptic plasticity, also requires autophagy. Here, we show that the induction of LTD via activation of NMDA receptors or metabotropic glutamate receptors initiates autophagy in the postsynaptic dendrites in mice. Dendritic autophagic vesicles (AVs) act in parallel with the endocytic machinery to remove AMPA receptor subunits from the membrane for degradation. During NMDAR-LTD, key postsynaptic proteins are sequestered for autophagic degradation, as revealed by quantitative proteomic profiling of purified AVs. Pharmacological inhibition of AV biogenesis, or conditional ablation of atg5 in pyramidal neurons abolishes LTD and triggers sustained potentiation in the hippocampus. These deficits in synaptic plasticity are recapitulated by knockdown of atg5 specifically in postsynaptic pyramidal neurons in the CA1 area. Conducive to the role of synaptic plasticity in behavioral flexibility, mice with autophagy deficiency in excitatory neurons exhibit altered response in reversal learning. Therefore, local assembly of the autophagic machinery in dendrites ensures the degradation of postsynaptic components and facilitates LTD expression.

Axillary Bud Fate Shapes Plant Architecture in Horticultural Crops

Above-ground plant architecture is dictated to a large extent by the fates and growth rates of aerial plant meristems. Shoot apical meristem gives rise to the fundamental plant form by generating new leaves. However, the fates of axillary meristems located in leaf axils have a great influence on plant architecture and affect the harvest index, yield potential and cultural practices. Improving plant architecture by breeding facilitates denser plantations, better resource use efficiency and even mechanical harvesting. Knowledge of the genetic mechanisms regulating plant architecture is needed for precision breeding, especially for determining feasible breeding targets. Fortunately, research in many crop species has demonstrated that a relatively small number of genes has a large effect on axillary meristem fates. In this review, we select a number of important horticultural and agricultural plant species as examples of how changes in plant architecture affect the cultivation practices of the species. We focus specifically on the determination of the axillary meristem fate and review how plant architecture may change even drastically because of altered axillary meristem fate. We also explain what is known about the genetic and environmental control of plant architecture in these species, and how further changes in plant architectural traits could benefit the horticultural sector.

Bergman kernel and period map for curves

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative—in a suitable sense—of this meromorphic form.

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a constant mean curvature surface with free boundary in C. We provide a pinching condition on the length of the traceless second fundamental form on $$\Sigma $$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if C is a geodesic ball of three-dimensional space forms, then $$\Sigma $$ is either a spherical cap or a Delaunay surface.

A subclass of meromorphic Janowski-type multivalent q-starlike functions involving a q-differential operator

Keeping in view the latest trends toward quantum calculus, due to its various applications in physics and applied mathematics, we introduce a new subclass of meromorphic multivalent functions in Janowski domain with the help of the q-differential operator. Furthermore, we investigate some useful geometric and algebraic properties of these functions. We discuss sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikeness, radius of convexity, inclusion property, and convex combinations via some examples and, for some particular cases of the parameters defined, show the credibility of these results.

Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

On 𝐶² umbilical hypersurfaces

We show by an elementary argument that the second fundamental form of a connected, totally umbilical hypersurface of class C 2 C^2 is a constant multiple of the metric tensor. It follows that the hypersurface is smooth and it is either a piece of a hyperplane or of a sphere.

Geometry of contact skew CR-warped product submanifolds of Sasakian manifolds

In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products. We establish a lower bound relationship between the squared norm of the second fundamental form and the warping function. The equality case of the inequality is investigated and some special cases of derived inequality are given. Furthermore, we provide non-trivial examples of such submanifolds.

A geometric flow on null hypersurfaces of Lorentzian manifolds

Abstract We introduce a geometric flow on a screen integrable null hypersurface in terms of its local second fundamental form. We use it to give an alternative proof to the vorticity free Raychaudhuri’s equation for null hypersurface, as well as establishing conditions for the existence of constant mean curvature (CMC) null hypersurfaces, and leaves of constant scalar curvatures.