Canonical Vector Research Articles

Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

New Characterizations of Hyperspheres and Spherical Hypercylinders in Euclidean Space

Let x be an isometric immersion of a Riemannian n‐manifold M into a Euclidean (n + 1)‐space which does not pass through the origin of . Then, the tangential part of the position vector field x of x is called the canonical vector field, and the normal part gives rise to a scalar function called the support function. Using the canonical vector field, support function, and mean curvature, we establish three new characterizations of hyperspheres. In addition, we prove that if the energy function of M satisfies the static perfect fluid equation, then M has at most two distinct principal curvatures. As an application, we prove that a complete noncompact hypersurface M is a spherical hypercylinder if the energy function of M satisfies the static perfect fluid equation, and it has exactly two distinct nonsimple principal curvatures.

Cup-length of oriented Grassmann manifolds via Gröbner bases

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the Z2-cup-length of the Grassmann manifold G˜n,3 of oriented 3-planes in Rn. Along the way, we calculate the heights of the Stiefel–Whitney classes of the canonical vector bundle over G˜n,3.

Circle-seq based method for eccDNA synthesis and its application as a canonical promoter independent vector for robust microRNA overexpression

Extrachromosomal circular DNA (eccDNA) has recently gained increasing attention due to its significant role in cancer and other pathophysiologic states. The majority of circular DNAs detected by Circle-seq are small-size eccDNAs with enigmatic functions. One major technical hurdle is to synthesize eccDNA for functional identification. Here, we describe CAES (Circle-seq based Artificial EccDNA Synthesis), a promising and reliable method for artificial eccDNA synthesis. Eight eccDNAs carrying different microRNA genes (eccMIR) found in gastric cancer tissues, ranging from 329 bp to 2189 bp in size, were created utilizing the CAES method. Exonuclease V and single restriction-endonuclease digestion identified the circular structure of synthetic eccDNAs. The DNA circularization efficiency afforded by CAES ranged from 15.6% to 31.1%, which was negatively correlated with the eccDNA length. In addition, we demonstrated that CAES-synthesized eccMIRs can express both miRNA-3p and − 5p molecules efficiently independent of a canonical promoter in human cell lines. Further assays proved that these eccMIRs were functional as they were able to repress the luciferase gene containing a miRNA-target sequence in the 3′UTR as well as the endogenous mRNA targets. Finally, kinetics study revealed that eccDNA exhibited a decay rate similar to the standard plasmids and linear DNA in cultured cells. Together, this study offers a rapid and convenient method for Circle-seq users to synthesize artificial eccDNAs. It also demonstrates the promising potential of eccMIR as a bacterial DNA-free vector for safe and robust miRNA overexpression in both basic research and therapeutic applications.

Finite orbits for large groups of automorphisms of projective surfaces

We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $\mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $\mathbf {k} = \mathbf {C}$. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.

Operational rules for a new family of d-orthogonal polynomials of Laguerre type

The aim of this research is to present a new generalization of d-orthogonal polynomials of Laguerre type by utilizing a suitable generating function from Sheffer class and employing operational rules associated with the lowering and raising operators that satisfy d-orthogonality. We derive several properties of these polynomials and establish the recurrence relation. Moreover, we provide explicit, connection and inversion formulas, the -order differential equation, and the canonical d-dimensional functional vector.

Global Dynamics of Planar Piecewise Linear Inelastic systems having straight lines as switching manifolds

In this paper we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider.

On complete Kählerian manifolds endowed with closed conformal vector fields

Let {overline{M}}^{2n}, n>1, be a complete, noncompact Kählerian manifold, endowed with a nontrivial closed conformal vector field xi having at least one singular point. Under a reasonable set of conditions, we show that xi has just one singular point p and that {overline{M}}{setminus }{p} is isometric to a one dimensional cone over a simply connected Sasakian manifold N diffeomorphic to {mathbb {S}}^{2n-1}.As a straightforward consequence, we conclude that if the addition of a single point to the Kählerian cone of a (2n-1)-dimensional Sasakian manifold N has the structure of a complete, noncompact, 2n-dimensional Kählerian manifold whose metric extends that of the cone, and such that the canonical vector field of the cone extends to it having a singularity at the extra point, then N is isometric to mathbb S^{2n-1}, endowed with an appropriate Sasakian structure.

CCA identifies a neurophysiological marker of adaptation capacity that is reliably linked to internal locus of control of cognition in amnestic MCI.

Locus of control (LOC) describes whether an individual thinks that they themselves (internal LOC) or external factors (external LOC) have more influence on their lives. LOC varies by domain, and a person's LOC for their intellectual capacities (LOC-Cognition) may be a marker of resilience in older adults at risk for dementia, with internal LOC-Cognition relating to better outcomes and improved treatment adherence. Vagal control, a key component of parasympathetic autonomic nervous system (ANS) regulation, may reflect a neurophysiological biomarker of internal LOC-Cognition. We used canonical correlation analysis (CCA) to identify a shared neurophysiological marker of ANS regulation from electrocardiogram (during auditory working memory) and functional connectivity (FC) data. A canonical variable from root mean square of successive differences (RMSSD) time series and between-network FC was significantly related to internal LOC-Cognition (β = 0.266, SE = 0.971, CI = [0.190, 4.073], p = 0.031) in 65 participants (mean age = 74.7, 32 female) with amnestic mild cognitive impairment (aMCI). Follow-up data from 55 of these individuals (mean age = 73.6, 22 females) was used to show reliability of this relationship (β = 0.271, SE = 0.971, CI = [0.033, 2.630], p = 0.047), and a second sample (40 participants with aMCI/healthy cognition, mean age = 72.7, 24 females) showed that the canonical vector biomarker generalized to visual working memory (β = 0.36, SE = 0.136, CI = [0.023, 0.574], p = 0.037), but not inhibition task RMSSD data (β = 0.08, SE = 1.486, CI = [- 0.354, 0.657], p = 0.685). This canonical vector may represent a biomarker of autonomic regulation that explains how some older adults maintain internal LOC-Cognition as dementia progresses. Future work should further test the causality of this relationship and the modifiability of this biomarker.

Hard and soft clustering of categorical time series based on two novel distances with an application to biological sequences

Two novel distances between categorical time series are introduced. Both of them measure discrepancies between extracted features describing the underlying serial dependence patterns. One distance is based on well-known association measures, namely Cramer’s v and Cohen’s κ. The other one relies on the so-called binarization of a categorical process, which indicates the presence of each category by means of a canonical vector. Binarization is used to construct a set of innovative association measures which allow to identify different types of serial dependence. The metrics are used to perform crisp and fuzzy clustering of nominal series. The proposed approaches are able to group together series generated from similar stochastic processes, achieve accurate results with series coming from a broad range of models and are computationally efficient. Extensive simulation studies show that both hard and soft clustering algorithms outperform several alternative procedures proposed in the literature. Two applications involving biological sequences from different species highlight the usefulness of the introduced techniques.

The standard cohomology of regular Courant algebroids

For any regular Courant algebroid E over a smooth manifold M with characteristic distribution F and ample Lie algebroid AE, we prove that there exists a canonical homological vector field on the graded manifold AE[1]⊕(TM/F)⁎[2] such that the resulting dg manifold ME, which we call the minimal model of the Courant algebroid E, encodes all cohomological information of E. Indeed, the standard cohomology of E can be identified with the cohomology of the function space on ME, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.

Ventricular segmentation and modeling using topological watershed transformation and harmonic state model

This paper proposes an adapted ventricular segmentation method based on topological watershed transform. Segmentation will allow spatio-temporal modeling of trajectories of the different points belonging to the borders of the ventricle using a harmonic motion model that is able to describe such motion over the entire cardiac cycle. In addition, extraction of the adopted canonical state vector and the corresponding state equations guarantees an optimal efficacy and a gradual transition from order n to order n + 1. To validate the proposed approach, an intern-image base was used. Our results show a promising ability to discern whether subjects are healthy or pathological with an 80% success rate.

On Type IIA AdS3 solutions and massive GK geometries

We give necessary and sufficient conditions for warped AdS3 (and Mink3) solutions of Type II supergravities to preserve \U0001d4a9 = (2, 0) supersymmetry, in terms of geometric conditions on their internal space M7. Such solutions possess a canonical ten-dimensional Killing vector that can be either time-like or null. In this work we classify the null case in massive Type IIA supergravity which necessitates that M7 decomposes as a circle fibration over a six-dimensional base with orthogonal SU(2)-structure containing a complex four-manifold. We narrow our focus to solutions for which M7 becomes \U0001d54b2 fibred over a foliation of a Kähler manifold over an interval. We find a class of solutions which are the massive Type IIA version of GK geometries and present an extremal problem which computes the central charge of the solution using just topology. Finally, we present geometric conditions for AdS3 solutions to preserve arbitrary extended chiral supersymmetry.

Orders of the canonical vector bundles over configuration spaces of finite graphs

We prove that the order of the canonical vector bundle over the configuration space is $2$ for a general planar graph, and is $4$ for a nonplanar graph.

Joint modulation format identification and OSNR estimation method based on trajectory information analysis

We propose a joint modulation format identification (MFI) and optical signal-to-noise ratio (OSNR) estimation method via analyzing the trajectory information of the received adjacent symbols. A uniform grid in the first quadrant of constellation diagram is designed for converting the trajectory information of the received adjacent symbols to the adjacent matrix of corresponding mapped graph structure. Then, we translate the OSNR-independent MFI scheme to an appropriate canonical vector searching optimization problem via improving on the classical cosine similarly algorithm, where the eigenvector associated with the largest eigenvalue of the adjacent matrix is served as discriminated-feature to identify five commonly-used modulation format types, i.e., polarization division multiplexing (PDM)-quadrature phase shift keying (QPSK), PDM=8 quadrature amplitude modulation (QAM), PDM-16QAM, PDM-32QAM, and PDM-64QAM. After exact modulation format type is obtained, we build the relationship between the second-largest eigenvalue of the adjacent matrix and the OSNR value via polynomial regression method, and the least-squares principle (LSP) is utilized to find the best fit line for obtaining the optimal polynomial coefficient. The proposed method is verified by 28 GBaud PDM-QPSK, PDM-8QAM, PDM-16QAM, PDM-32QAM, and PDM-64QAM optical signal numerical simulations, and the results show that could achieve excellent performance on various parameter analysis. Besides, a proof-of-concept experiment is also conducted. More specifically, the proposed method only need one training process and one matrix decomposition, which would provide a superior convenience for future optical performance monitoring.

On a Vector-Valued Measure of Multivariate Skewness

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector.

Some Characterization of Spiral Surfaces

In this paper we will study special spiral surfaces in the three dimensional Euclidean space and we give some characterization of these surfaces. More specifically, we investigate the Chang-Yau operator acting on the Gauss map of spiral surfaces. We also give some results about canonical vector field of these surfaces, i.e., we study incomperssibility of canonical vector field in two types of spiral surfaces. Moreover, we give some necessary conditions for a spiral surface to be a Weingarten surface. Existence of umbilical point is another problem that we investigate about it for a special case of spiral surfaces of the first type.

A note on the height of the third Stiefel--Whitney class of the canonical vector bundles over some Grassmann manifolds

We compute the height of the third Stiefel--Whitney characteristic class of the canonical bundles over some infinite classes of Grassmann manifolds of five dimensional vector subspaces of real vector spaces.

Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

We study a variational Ginzburg–Landau type model depending on a small parameter \(\varepsilon >0\) for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As \(\varepsilon \rightarrow 0\), these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus \({\mathfrak {g}}\) of S is different than 1. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the \(2{\mathfrak {g}}\)-dimensional space of harmonic 1-forms on S if \({\mathfrak {g}}\geqq 1\). Our second main result determines the interaction energy (called renormalized energy) between vortex points as a \(\Gamma \)-limit (at the second order) as \(\varepsilon \rightarrow 0\). The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of S as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg–Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces \(S\subset {{\mathbb {R}}}^3\), in particular, to a physical model for non-tangent maps to S coming from micromagnetics.

Riemann-Hilbert Boundary Value Problem with Piecewise Constant Transition Function

We prove that, for any Fuchsian system of differential equations on the Riemann sphere, there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsian system. From this, we obtain solution of the Riemann-Hilbert boundary value problem for piecewise constant matrix function in terms of holomorphic sections of vector bundle and calculate the partial indices of the problem.