Theory Of Differentiable Manifolds Research Articles

Quotient and transversal mappings for topological quasigroups

This article is devoted to studying the structure of topological left (or right) quasigroups, which play a great role in noncommutative geometry. Quotient and transversal mappings are important in the theory of differentiable manifolds and topological manifolds. Their transversal and quotient mappings are investigated. Necessary and sufficient conditions for their continuity are scrutinized. Examples are given. Homogeneous spaces are investigated related to topological quasigroups and their subquasigroups. For this purpose, the products of special types of topological left (or right) quasigroups, which are called smashed, are investigated. They are used to describe an extensive family of topological nondiscrete left (or right) quasigroups for which transversal mappings are continuous.

Research on Network Behavior Risk Measurement Method Based on Traffic Analysis

At present, the network security problem is facing a serious threat, and network security events continue to occur. It has become an important link to prevent network attacks and ensure network security. According to the network security protection measures and security technical requirements, it has become an urgent need to establish appropriate security measurement methods and strengthen the monitoring and analysis of network security status. This study proposes a network behavior risk measurement method based on traffic analysis to accurately and objectively evaluate the security state of the network. Traffic is the most basic behavior of the network and the basis of security risk measurement. Firstly, we regard the traffic data as network behavior to build scenarios. Through differential manifold modeling, the traffic data and topology of the network system are semantically described to form a matrix. Then, after manifold dimensionality reduction, the objective risk assessment value can be obtained by manifold mapping and Riemann metric. In this study, the differential manifold theory is applied to network behavior risk measurement, and the innovation of differential manifold in the field of network behavior risk measurement is given. After giving the network behavior risk measurement theory, we first verify the effectiveness of the proposed method through the simulation experiments. Secondly, the public CIC-IDS-2017 data set is used for analysis and calculation to prove the accuracy of the proposed method.

Numerical strategies for variational updates in large strain inelasticity with incompressibility constraint

AbstractIn finite deformation inelasticity, one often has to deal with the incompressibility constraint. In the past, this was dealt with using, for example, an exponential mapping approach, which yields exact volume preservation in plastic deformations. In this article however, the special‐linear update approach by Hurtado et al. The special‐linear update: An application of differential manifold theory to the update of isochoric plasticity flow rules, Int J Numer Methods Eng. 2014;97(4):298‐312, which utilizes a projection method to fulfill the incompressibility constraint is used. The model is applied to isotropic plasticity by a novel approximation of the logarithm and treats kinematic hardening without losing the symmetry of the internal variables. The model results are compared to models utilizing an exponential mapping approach in numerical experiments.

Dynamics of Traveling Waves for the Perturbed Generalized KdV Equation

This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.

Aircraft safety analysis based on differential manifold theory and bifurcation method

Loss of control (LOC) is considered one of the leading causes of fatal aircraft accidents worldwide. Reducing LOC is critical to improve flight safety. Although it is still vaguely defined, LOC is generally associated with a flight state that is outside the safety envelope, with nonlinear influences of aircraft dynamics and incorrect handling by the flight crew. We have studied how nonlinear factors and pilot operations contribute to LOC. In this study, the stall point and bifurcation point are confirmed using the bifurcation analysis, and the results show that the aircraft will stall when excessive elevator movement is commanded. Moreover, even though there may be an equilibrium state in one of the elevator deflections, the flight state may still be outside the flight safety envelope. When the flight state is near the edge of the flight safety envelope, the strategy to regulate the elevator deflection is super-sensitive, and a slight change in the elevator deflection may contribute to a flight state outside the safety envelope. To solve this issue, the differential manifold theory is introduced to determine the safety envelope. Examples are provided using NASA’s generic transport model.

The special‐linear update: An application of differential manifold theory to the update of isochoric plasticity flow rules

SUMMARYThe evolution of plastic deformations in metals, governed by incompressible flow rules, has been traditionally solved using the exponential mapping. However, the accurate calculation of the exponential mapping and its tangents may result in computationally demanding schemes in some cases, while common low‐order approximations may lead to poor behavior of the constitutive update because of violation of the incompressibility condition. Here, we introduce the special‐linear (SL) update for isochoric plasticity, a flow‐rule integration scheme based on differential manifolds concepts. The proposed update exactly enforces the plastic incompressibility condition while being first‐order accurate and consistent with the flow rule, thus bearing all the desirable properties of the now standard exponential mapping update. In contrast to the exponential‐mapping update, we demonstrate that the SL update can drastically reduce the computing time, reaching one order of magnitude speed‐ups in the calculation of the update tangents. We demonstrate the applicability of the update by way of simulation of single‐crystal plasticity uniaxial loading tests. We anticipate that the SL update will open the way to efficient constitutive updates for the solution of complex multiscale material models, thus making it a very promising tool for large‐scale simulations. Copyright © 2013 John Wiley & Sons, Ltd.

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.

Geometric properties of nonlinear networks containing capacitor-only cutsets and/or inductor-only loops. Part II: Symmetries

This paper is the last in a two-part sequence which studies nonlinear networks, containing capacitor-only cutsets and/or inductor-only loops, from the geometric coordinate-free point of view of the theory of differentiable manifolds. For such circuits, it is shown that (subject to certain assumptions) there is a naturally defined Lie group action of\(\mathbb{R}^{\delta _0 }\) on the state space ∑ ofN, whereδ0 is the sum of the number of independent capacitor-only cutsets and the number of independent inductor-only loops. Circuit theoretic sufficient conditions on the reactive constitutive relations are derived for the circuit dynamics to be invariant under this Lie group action.

Solutions of singular constrained differential equations: A generalization of circuits containing capacitor-only loops and inductor-only cutsets

Employing the theory of differentiable manifolds we give a geometric coordinate-free description of constrained differential equations (CDE), which are usually thought of as systems of simultaneous differential and algebraic equations. We regard the algebraic constraints as defining a differentiable manifold, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> , and regard solutions of the CDE as curves in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> . Our main contribution in this paper is to characterize geometrically a particular class of singular constrained differential equations for which consistency forces solutions to lie in a proper subset of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> . Constrained differential equations describing electric circuits having capacitor-only loops and/or inductor-only cutsets are shown to be of the above type.

The Klein-Gordon operator in conformal space

We consider the Minkowski space M 4 as a local chart of a compact differentiable pseudo-Riemannian manifold M 4 c , on which the whole conformal group O(2, 4) Z 2 acts continuously. We investigate the conditions under which functions or differential operators on the space M 4 can be uniquely continued to the conformal manifold M 4 c . This is done by using methods well-known in the theory of differentiable manifolds. In particular, we show that the Klein-Gordon operator □+ m 2 can be uniquely continued to the space M 4 c and we discuss the conformal invariance of the Klein-Gordon equation on the manifold M 4 c .

On the General Theory of Differentiable Manifolds with Almost Tangent Structure

Some of the most important G-Structures of the first kind [1] are those defined by linear operators satisfying algebraic relations. If the linear operator J acting on the complexified space of a differentiable manifold V satisfies a relation of the formwhere I is the identity operator, the manifold has an almost complex structure ([2] [3]). The structures defined byare the almost product structures ([3] [4]).

On Transformations in Rn And a Theorem of Sard

Extensions of Theorem B under less restrictive conditions have been given by Rado and Reichelderfer, and in particular it has been shown that Theorem B holds if f is merely differentiable on D ([2], pp. 339, 343). Here, however, we restrict ourselves to the case stated above, which is the case most often used in the theory of differentiable manifolds. The result of Theorem B is most naturally viewed as an extension of the inequality of Theorem A to the case in which J(x) vanishes at points of D, and indeed Theorem B shows that (1.1) continues to hold in this case. It is also clear that we can remove the hypothesis in Theorem A that f is (1-1), for if f is not (1-1) the integral fBI J(x) j dx is equal to the measure of f(E) with multiplycovered volumes being counted multiply. We are therefore led to the following theorem, which contains both Theorems A and B.

Algebras of Finite Differential Order and the Operational Calculus

The notions of topological spaces and of continuous functions are well established in mathematics. The related concepts of differentiable spaces and of differentiable functions have not as yet been introduced in a comparable broad and definitive form, as their study is classically confined mostly to locally euclidean spaces. The present work arose from wishes to extend part of the known results of the theory of differentiable manifolds, due to Whitney, to a larger class of spaces which are not required to be locally euclidean a priori. As a motivation to the question studied here, let us mention that an n-differentiable structure on a topological space X is definable by a topological algebra A of real continuous functions on X, A being the algebra of n-differentiable real functions on X and the topology of A being that of uniform convergence of a function and all its derivatives of order ? n on compact sets. A has to satisfy suitable requirements, among them the following two conditions. If d e X, let M(e) be the closed ideal of all f e A such that f (a) = 0. Then all the powers Ml(e), i > n + 1, are identical and A/Mn+,(;) is a local algebra whose maximal ideal has (n+-1)power equal to 0. Moreover, A is locally convex (? 3) with respect to the collection {Mn+,(f)}, a e X. Such a local convexity assumption expresses the local nature of A (rather its infinitesimal nature). Another property of A is that it has an operational calculus with ordinary differentiable functions, i.e., that f e A implies rp(f) e A for every n-differentiable real function q on the real line R, the mapping ((p, f ) --'p(f ) being continuous; and similarly for functions q of several variables. Since this second property of A can be shown to follow from the first, provided A is complete, we consider the question, when do suitable local convexity assumptions imply existence of an operational calculus? the answer to which is given by the theorem stated in ? 7. This article is devoted to the proof of the assertion that, if C is a category of pure separated algebras whose radicals are nilpotent and n > 0 is an integer, in order that every topological algebra which is