Large-deflection and post-buckling behavior of slender beam-columns with non-linear end-restraints

The large-deflection analysis and post-buckling behavior of laterally braced or unbraced slender beam-columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation, including the effects of out-of-plumbness, are developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by Aristizabal-Ochoa [1]. The proposed method can be used in the large-deflection analysis and post-buckling behavior of elastic slender beam-columns with rigid, semi-rigid, and simple flexural connections at both ends including linear and non-linear inelastic connections like those that suffer from flexural degradation (such as flexural cracking and elasto-plastic connections) or flexural stiffening. Only bending strains are considered in the proposed analysis. Results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for laterally braced or unbraced slender beam-columns under bending caused by end loads. The large-deflection analysis and post-buckling behavior of slender beam-columns with both supports partially restrained against rotation and with sway inhibited or uninhibited are complex problems requiring the simultaneous solution of two coupled non-linear equations with elliptical integrals whose unknowns are the limits of the integrals. The validity of the proposed method and equations are verified against solutions available in the technical literature. Three comprehensive examples are included that show the effects of linear and non-linear connections at both ends on the large-deflection analysis and post-buckling behavior of slender beam-columns.     

Dynamics about Neural Array with Simple Lateral Inhibitory Connections

Lateral inhibitory effect is a well-known feature of information processing in neural systems. This paper presents a neural array model with simple lateral inhibitory connections. After detailed examining into the dynamics of this kind of neural array, the author gives the sufficient conditions under which the outputs of the network will tend to a special stable pattern called spatial sparse pattern in which if the output of a neuron is 1, then the outputs of the neurons in its neighborhood are 0. This ability called spatial sparse coding plays an important role in self-coding, self-organization and associative memory for patterns and pattern sequences. The main conclusions about the dynamics of this kind of neural array which is related to spatial sparse coding are introduced.     

Abelian Chern-Simons theory, Stokes’ theorem, and generalized connections

Generalized connections and their calculus have been developed in the context of quantum gravity. Here we apply them to abelian Chern-Simons theory. We derive the expectation values of holonomies in U(1) Chern-Simons theory using Stokes’ theorem, flux operators and generalized connections. A framing of the holonomy loops arises in our construction, and we show how, by choosing natural framings, the resulting expectation values nevertheless define a functional over gauge invariant cylindrical functions.The abelian theory considered in the present article is the test case for our method. It can also be applied to the non-abelian theory. Results will be reported in a companion article.     

Using the K5Connected Cognition Diagram to analyze teachers’ communication and understanding of regions in three-dimensional space

This paper reports on a study that introduces and applies the K₅Connected Cognition Diagram as a lens to explore video data showing teachers’ interactions related to the partitioning of regions by axes in a three-dimensional geometric space. The study considers “semiotic bundles” ( Arzarello, 2006), introduces “semiotic connections,” and discusses the fundamental role each plays in developing individual understanding and communication with peers. While all teachers solved the problem posed, many failed to make or verbalize connections between the types of semiotic resources introduced during their discussions.     

Classification of Kantowski-Sachs and Bianchi Type III Space-Times According to Their Killing Vector Fields in Teleparallel Theory of Gravitation

In this paper we classify Kantowski-Sachs and Bianchi type Ⅲ space-times according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the telepaxallel Killing vector fields are 4 or 6, which are the same in numbers as in general relativity. In case of 4 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 6 Killing vector fields the metric functions become constants and the Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case.     

Topological probability and connection strength induced activity in complex neural networks

Recent experimental evidence suggests that some brain activities can be assigned to small-world networks. In this work, we investigate how the topological probability p and connection strength C affect the activities of discrete neural networks with small-world (SW) connections. Network elements are described by two-dimensional map neurons (2DMNs) with the values of parameters at which no activity occurs. It is found that when the value of p is smaller or larger, there are no active neurons in the network, no matter what the value of connection strength is; for a given appropriate connection strength, there is an intermediate range of topological probability where the activity of 2DMN network is induced and enhanced. On the other hand, for a given intermediate topological probability level, there exists an optimal value of connection strength such that the frequency of activity reaches its maximum. The possible mechanism behind the action of topological probability and connection strength is addressed based on the bifurcation method. Furthermore, the effects of noise and transmission delay on the activity of neural network are also studied.     

Structure of the space of reducible connections for Yang-Mills theories

The geometrical structure of the gauge equivalence classes of reducible connections are investigated. The general procedure to determine the set of orbit types (strata) generated by the action of the gauge group on the space of gauge potentials is given. In the so obtained classification, a stratum, containing generically certain reducible connections, corresponds to a class of isomorphic subbundles given by an orbit of the structure and gauge group. The structure of every stratum is completely clarified. A nonmain stratum can be understood in terms of the main stratum corresponding to a stratification at the level of a subbundle.     

Flat connections,geometric invariants and energy of harmonic functions on compact Riemann surfaces

A geometric invariant is associated to the space of flat connections on a G-bundle over a compact Riemann surface and is related to the energy of harmonic functions.     

On the Unification of Geometric and Random Structures through Torsion Fields: Brownian Motions,Viscous and Magneto-fluid-dynamics

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes equations for viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations.     

Cartan–Weyl Dirac and Laplacian Operators,Brownian Motions: The Quantum Potential and Scalar Curvature,Maxwell’s and Dirac-Hestenes Equations,and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q