scholarly journals Some Associated Curves of Normal Indicatrix of a Regular Curve

2020 ◽  
Vol 12 (1) ◽  
pp. 92
Author(s):  
Nidal ECHABBI ◽  
Amina OUAZZANI CHAHDI
Keyword(s):  
Vector Field ◽  
Regular Curve ◽  
New Approach ◽  
Integral Curves

In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.

scholarly journals Some Associated Curves of Normal Indicatrix of a Regular Curve

2020 ◽  
Vol 12 (1) ◽  
pp. 84
Author(s):  
Nidal ECHABBI ◽  
Amina OUAZZANI CHAHDI
Keyword(s):  
Vector Field ◽  
Regular Curve ◽  
New Approach ◽  
Integral Curves

In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

Convergence of stochastic algorithms: from the Kushner–Clark theorem to the Lyapounov functional method

1996 ◽  
Vol 28 (04) ◽  
pp. 1072-1094
Author(s):  
Jean-Claude Fort ◽  
Gilles Pagès
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Functional Method ◽  
Stochastic Algorithms ◽  
Smoothness Assumption ◽  
New Approach ◽  
The Mean ◽  
Path Dependent ◽  
Mean Vector ◽  
Theoretical Results

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.

scholarly journals A New Approach on Helices in Pseudo-Riemannian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Evren Zıplar ◽  
Yusuf Yaylı ◽  
İsmail Gök
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Frenet Frame ◽  
New Approach ◽  
Nonzero Constant ◽  
Slant Helix

A proper curveαin then-dimensional pseudo-Riemannian manifold(M,g)is called aVn-slant helix if the functiong(Vn,X)is a nonzero constant alongα, whereXis a  parallel vector field alongαandVnisnth Frenet frame. In this work, we study such curves and give important characterizations about them.

scholarly journals A new approach to anomalous axial vector field theory – II

2021 ◽  
pp. 2150155
Author(s):  
A. K. Kapoor
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vector Field ◽  
Axial Vector ◽  
Quantum Field ◽  
Stochastic Quantization ◽  
New Approach ◽  
Four Dimensions

This work is continuation of a stochastic quantization program reported earlier. In this paper, we propose a consistent scheme of doing computations directly in four dimensions using conventional quantum field theory methods.

The geometry of the trajectories

2020 ◽  
Vol 17 (04) ◽  
pp. 2050051
Author(s):  
Mohammadreza Molaei
Keyword(s):  
Black Hole ◽  
Vector Field ◽  
Vector Fields ◽  
Second Step ◽  
Curvature Scalar ◽  
Schwarzschild Black Hole ◽  
Integral Curves ◽  
Trajectory Manifold

In this paper, we use of the geometry of a class of the nature flows to define trajectory manifolds. Trajectory connections as a generalization of the Levi-Civita connections are considered. A method for determining the geometry of the flows created by the integral curves of a vector field is presented. The method contains two steps, the first step is finding the connection by the trajectories of a vector field, and the second step is finding a trajectory metric corresponding to the deduced connection. We show that doing the first step is possible, but for some of the vector fields, the second step may not be possible. In the case of existence of a trajectory manifold a new kind of curvature which we called it “trajectory curvature scalar” appears. We calculate trajectory connections for some vector fields and by an example we show that the trajectory curvature scalar for a trajectory manifold may not be equal to the curvature scalar of it. We find trajectory connection for a vector field close to the Schwarzschild black hole.

Stochastic 4D Flow Vector-Field Signatures: A New Approach for Comprehensive 4D Flow MRI Quantification

2021 ◽  
pp. 215-224
Author(s):  
Mohammed S. M. Elbaz ◽  
Chris Malaisrie ◽  
Patrick McCarthy ◽  
Michael Markl
Keyword(s):  
Vector Field ◽  
4D Flow Mri ◽  
4D Flow ◽  
New Approach ◽  
Flow Vector ◽  
Flow Mri

scholarly journals A NEW APPROACH TO LINEAR SHELL THEORY

2005 ◽  
Vol 15 (08) ◽  
pp. 1181-1202 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE
Keyword(s):  
Vector Field ◽  
Minimization Problem ◽  
Shell Theory ◽  
Displacement Vector ◽  
Constrained Minimization ◽  
New Approach ◽  
Quadratic Minimization ◽  
Displacement Vector Field ◽  
Constrained Minimization Problem ◽  
Linear Shell Theory

We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

scholarly journals Complete lift of vector fields and sprays to T∞M

2015 ◽  
Vol 12 (10) ◽  
pp. 1550113 ◽  
Author(s):  
Ali Suri ◽  
Somaye Rastegarzadeh
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Dimensional Manifold ◽  
Closed Curves ◽  
Infinite Dimensional ◽  
Integral Curves ◽  
Complete Lift ◽  
Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

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