Torsion Functions Research Articles

The Pedal Curves Generated by Alternative Frame Vectors and Their Smarandache Curves

In this paper, pedal-like curves are defined resulting from the orthogonal projection of a fixed point on the alternative frame vectors of a given regular curve. For each pedal curve, the Frenet vectors, the curvature and the torsion functions are found to provide the common relations among the main curve and its pedal curves. Then, Smarandache curves are defined by using the alternative frame vectors of each pedal curve as position vectors. The relations of the Frenet apparatus are also established for the pedal curves and their corresponding Smarandache curves. Finally, the expressions of the alternative frame apparatus of each Smarandache curves are given in terms of the alternative frame elements of the pedal curves. Thus, a set of new symmetric curves are introduced that contribute to the vast curve family.

Curves of stationary acceleration according to alternative frame

This paper investigates curves of stationary acceleration by using alternative frame which includes the principal normal vector, the derivative of principal normal vector and the Darboux vector. Mentioned curves are studied by the way of rigid body motions, that is to say a point in the moving body follows the curve and the alternative frame in the moving body stays aligned with the members of frame. It is determined that in which condition this special motion becomes to stationary acceleration motion. The matrix representations of a constant vector related to velocity vector of the motion which is used to characterize stationary acceleration is obtained by means of alternative frame curvatures. Some examinations are developed with some solutions of differential equations. The main result is attained as: general helix curves with linear curvature and torsion functions are curves of stationary acceleration which are curves in the rigid body motions group SE(3) correlated with robotics. The paths designed as stationary acceleration curves can lead the way to control the end-effectors of robots. Finally, some explanatory examples are imputed.

Ray-Singer Torsion and the Rumin Laplacian on Lens Space

We express explicitly the analytic torsion functions associated with the Rumin complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. Moreover, we have a formula between this torsion and the Ray-Singer torsion.

Families torsion and Morse functions for covering spaces

In this paper we prove the Cheeger–Müller theorem for \(L^2\)-analytic torsion form under the assumption that there exists a fiberwise Morse function and the Novikov–Shubin invariants are positive.

Additive manufactured thermal metamaterial devices for manipulating heat flow: geometric configuration design and SLM-assisted fabrication

Thermal metamaterials, benefitting from ingenious design and construction of artificial structures, commonly exhibit amazing heat conduction characteristics beyond naturally occurring materials in conventional scenarios. Based on the transformation thermodynamics theory and the effective medium theory, two types of metamaterial structure devices aiming to manipulate heat flow were artificially designed and fabricated with the aid of selective laser melting additive manufacturing technology in this work. The well-designed thermal metamaterial structures respectively achieved the thermal self-focus and thermal torsion functions. The relation of the configuration geometric parameters and the heat flux changes of the thermal metamaterial devices was carefully dissected. Following the geometric optimization design, the heat transfer directions achieved to be precisely manipulated as the design wishes. The changes in both heat flux and temperature distribution on the thermal self-focus and thermal torsion devices were traced through the numerical simulation and experimental measurement methods. The thermal simulation results were highly consistent with the experimental findings of the design-intended manipulation effects on the heat flow. The research work verified the applicability of the thermal metamaterial structures fabricated with the assist of selective laser melting in the thermal engineering fields.

Curvature- and Torsion-Based Consensus for the Markovian Switching Multiagent System with Communication of Noise Disturbance and Time Delay

This paper investigates the consensus problem for the distributed multiagent system (MAS), where the trajectory of each agent is displayed by curvature and torsion, and the communication behaviors among agents are influenced by time delay and corrupted by noises. According to the Frenet–Serret formulas, a class of consensus protocols is designed for all agents, and a closed-loop system is obtained. Based on the Lyapunov method, several consensus criteria are derived, where the consensus criteria are characterized by curvature functions and torsion functions. Finally, one example shows the reliability of the proposed methods.

Analytic torsions associated with the Rumin complex on contact spheres

We explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

The cosmological principle in theories with torsion: the case of Einstein-Cartan-Dirac-Maxwell gravity

We address the implementation of the cosmological principle, that is, the assumption of homogeneity and isotropy in the spatial distribution of matter in the Universe, within the context of Einstein-Cartan theory including minimal couplings of both Dirac and Maxwell fields to torsion. This theory gives rise to new physical effects in environments of high spin densities while leaving the vacuum dynamics unaffected. Our approach is to impose the cosmological principle from the onset to the geometrical degrees of freedom (metric and torsion functions), which constrains the torsion components and the corresponding correction terms in the Friedmann-like equations and in the resulting fermionic and bosonic (non-linear) dynamics. We derive the corresponding cosmological dynamics for the geometrical and matter degrees of freedom and discuss the validity of this approach.

Deformation of beams in the grade consistent theory of microstretch elastic solids

A microstretch continuum is a material with microstructure in which the microelements can stretch and contract independently of their translations and rotations. This paper is concerned with the grade consistent theory of microstretch elastic solids, where the second-order displacement is added to the classical set of independent constitutive variables. We study the equilibrium of a homogeneous and isotropic elastic beam loaded by tractions distributed over its plane ends. First, the problem of bending and extension is investigated. It is shown that the solution of the problem can be expressed in terms of solutions of three plane strain problems. Then, we study the problem of torsion in the framework of the grade consistent theory of microstretch elastic solids. This problem is solved with the help of three torsion functions. The results are used to investigate the torsion of a right circular cylinder.

Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3

In the present work, we report a detailed description of the symmetry properties of the eight-atomic molecule ethane, with the aim of facilitating the variational calculations of rotation-vibration spectra of ethane and related molecules. Ethane consists of two methyl groups CH 3 where the internal rotation (torsion) of one CH 3 group relative to the other is of large amplitude and involves tunnelling between multiple minima of the potential energy function. The molecular symmetry group of ethane is the 36-element group G 36 , but the construction of symmetrised basis functions is most conveniently done in terms of the 72-element extended molecular symmetry group G 36 (EM). This group can subsequently be used in the construction of block-diagonal matrix representations of the ro-vibrational Hamiltonian for ethane. The derived transformation matrices associated with G 36 (EM) have been implemented in the variational nuclear motion program TROVE (Theoretical ROVibrational Energies). TROVE variational calculations are used as a practical example of a G 36 (EM) symmetry adaptation for large systems with a non-rigid, torsional degree of freedom. We present the derivation of irreducible transformation matrices for all 36 (72) operations of G 36 (M) (G 36 (EM)) and also describe algorithms for a numerical construction of these matrices based on a set of four (five) generators. The methodology presented is illustrated on the construction of the symmetry-adapted representations both of the potential energy function of ethane and of the rotation, torsion and vibration basis set functions.

Non-local torsion functions and embeddings

ABSTRACTGiven , we discuss the embedding of in . In particular, for we deduce its compactness on all open sets on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in in a suitable weak sense, for every open set . The proofs make use of a non-local Hardy-type inequality in , involving the fractional torsion function as a weight.

Prolongation of Tanaka structures: an alternative approach

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated with second-order differential equations, and structures defined by gradings of Lie algebras (in the framework of parabolic geometries). Tanaka’s prolongation procedure associates with a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description of local invariants, investigation of the automorphism group, and equivalence problem. In this paper, we develop an alternative constructive approach for Tanaka’s prolongation procedure, based on the theory of quasi-gradations of filtered vector spaces, G-structures, and their torsion functions.

Mechanical properties of pristine and nanoporous graphene

We present molecular dynamics simulations of monolayer graphene under uniaxial tensile loading. The Morse, bending angle, torsion and Lennard-Jones potential functions are adopted within the mdFOAM library in the OpenFOAM software, to describe the molecular interactions in graphene. A well-validated graphene model using these set of potentials is not yet available. In this work, we investigate the accuracy of the mechanical properties of graphene when derived using these simpler potentials, compared to the more commonly used complex potentials such as the Tersoff-Brenner and AIREBO potentials. The computational speed up of our approach, which scales O(1.5N), where N is the number of carbon atoms, enabled us to vary a larger number of system parameters, including graphene sheet orientation, size, temperature and concentration of nanopores. The resultant effect on the elastic modulus, fracture stress and fracture strain is investigated. Our simulations show that graphene is anisotropic, and its mechanical properties are dependant on the sheet size. An increase in system temperature results in a significant reduction in the fracture stress and strain. Simulations of nanoporous graphene were created by distributing vacancy defects, both randomly and uniformly, across the lattice. We find that the fracture stress decreases substantially with increasing defect density. The elastic modulus was found to be constant up to around 5% vacancy defects, and decreases for higher defect densities.

Torsion Functions and the Cheeger Problem: A Fractional Approach

Abstract Let Ω be a Lipschitz bounded domain of ℝ N ${\mathbb{R}^{N}}$ , N ≥ 2 ${N\geq 2}$ . The fractional Cheeger constant h s ⁢ ( Ω ) ${h_{s}(\Omega)}$ , 0 < s < 1 ${0<s<1}$ , is defined by h s ⁢ ( Ω ) = inf E ⊂ Ω ⁡ P s ⁢ ( E ) | E | , where P s ⁢ ( E ) = ∫ ℝ N ∫ ℝ N | χ E ⁢ ( x ) - χ E ⁢ ( y ) | | x - y | N + s ⁢ d x ⁢ d y , $h_{s}(\Omega)=\inf_{E\subset{\Omega}}\frac{P_{s}(E)}{|E|},\quad\text{where}% \quad P_{s}(E)=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|\chi_{E}(x)-% \chi_{E}(y)|}{|x-y|^{N+s}}\,\mathrm{d}x\,\mathrm{d}y,$ with χ E ${\chi_{E}}$ denoting the characteristic function of the smooth subdomain E. The main purpose of this paper is to show that lim p → 1 + | ϕ p s | L ∞ ⁢ ( Ω ) 1 - p = h s ( Ω ) = lim p → 1 + | ϕ p s | L 1 ⁢ ( Ω ) 1 - p , $\lim_{p\rightarrow 1^{+}}\lvert\phi_{p}^{s}|_{L^{\infty}(\Omega)}^{1-p}=h_{s}(% \Omega)=\lim_{p\rightarrow 1^{+}}\lvert\phi_{p}^{s}|_{L^{1}(\Omega)}^{1-p},$ where ϕ p s ${\phi_{p}^{s}}$ is the fractional ( s , p ) ${(s,p)}$ -torsion function of Ω, that is, the solution of the Dirichlet problem for the fractional p-Laplacian: - ( Δ ) p s ⁢ u = 1 ${-(\Delta)_{p}^{s}\,u=1}$ in Ω, u = 0 ${u=0}$ in ℝ N ∖ Ω ${\mathbb{R}^{N}\setminus\Omega}$ . For this, we derive suitable bounds for the first eigenvalue λ 1 , p s ⁢ ( Ω ) ${\lambda_{1,p}^{s}(\Omega)}$ of the fractional p-Laplacian operator in terms of ϕ p s ${\phi_{p}^{s}}$ . We also show that ϕ p s ${\phi_{p}^{s}}$ minimizes the ( s , p ) ${(s,p)}$ -Gagliardo seminorm in ℝ N ${\mathbb{R}^{N}}$ , among the functions normalized by the L 1 ${L^{1}}$ -norm.

Torsion functions on moduli spaces in view of the cluster algebra

We introduce non-acyclic $PGL_n(\mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Porti's $PGL_2(\mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(\mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichm\"{u}ler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.

Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars

The Saint-Venant torsion of linearly elastic anisotropic cylindrical bars with solid and hollow cross-section is treated. The shear flexibility moduli of the non-homogeneous bar are given functions of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsion problem of non-homogeneous anisotropic bar is expressed in terms of the torsion and Prandtl's stress functions of the corresponding homogeneous anisotropic bar having the same cross-section as the non-homogeneous bar.

Exact de Sitter solutions in quadratic gravitation with torsion

Several exact cosmological solutions of quadratic gravitation with two torsion functions are presented. These solutions give an essentially different explanation from the one in most of previous works to the cause of the accelerating cosmological expansion and the origin of the torsion of the spacetime. These solutions can be divided into two classes. The solutions in the first class define the critical points of a dynamic system representing an asymptotically stable de Sitter spacetime. The solutions in the second class have exact analytic expressions which have never been found in the literature. The acceleration equation of the universe in general relativity is only a special case of them. These solutions indicate that even in vacuum the spacetime can be endowed with torsion, which means that the torsion of the spacetime has an intrinsic nature and a geometric origin. In these solutions the acceleration of the cosmological expansion is due to either the scalar torsion or the pseudoscalar torsion function. Neither a cosmological constant nor dark energy is needed. It is the torsion of the spacetime that causes the accelerating expansion of the universe in vacuum. All the effects of the inflation, the acceleration and the phase transformation from deceleration to acceleration can be explained by these solutions. Furthermore, the energy and pressure of the matter without spin can produce the torsion of the spacetime and make the expansion of the universe decelerate as well as accelerate.

Limiting energy density and a regular accelerating universe in Riemann-Cartan spacetime

Isotropic cosmology built in the Riemann-Cartan spacetime by using sufficiently general expression of gravitational Lagrangian is investigated. It is shown that cosmological equations obtained by certain restrictions on indefinite parameters of gravitational Lagrangian lead to limiting energy density at the beginning of cosmological expansion and all cosmological models filled with usual gravitating matter satisfying standard energy conditions are regular with respect to energy density, spacetime metrics with its time derivative and torsion functions. At asymptotics cosmological solutions of spatially flat models coincide with that of standard ΛCDM-model for accelerating Universe.

Solutions of the Cheeger problem via torsion functions

The Cheeger problem for a bounded domain Ω ⊂ R N , N > 1 consists in minimizing the quotients | ∂ E | / | E | among all smooth subdomains E ⊂ Ω and the Cheeger constant h ( Ω ) is the minimum of these quotients. Let ϕ p ∈ C 1 , α ( Ω ¯ ) be the p-torsion function, that is, the solution of torsional creep problem − Δ p ϕ p = 1 in Ω, ϕ p = 0 on ∂ Ω, where Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplacian operator, p > 1 . The paper emphasizes the connection between these problems. We prove that lim p → 1 + ( ‖ ϕ p ‖ L ∞ ( Ω ) ) 1 − p = h ( Ω ) = lim p → 1 + ( ‖ ϕ p ‖ L 1 ( Ω ) ) 1 − p . Moreover, we deduce the relation lim p → 1 + ‖ ϕ p ‖ L 1 ( Ω ) ⩾ C N lim p → 1 + ‖ ϕ p ‖ L ∞ ( Ω ) where C N is a constant depending only of N and h ( Ω ) , explicitely given in the paper. An eigenfunction u ∈ BV ( Ω ) ∩ L ∞ ( Ω ) of the Dirichlet 1-Laplacian is obtained as the strong L 1 limit, as p → 1 + , of a subsequence of the family { ϕ p / ‖ ϕ p ‖ L 1 ( Ω ) } p > 1 . Almost all t-level sets E t of u are Cheeger sets and our estimates of u on the Cheeger set | E 0 | yield | B 1 | h ( B 1 ) N ⩽ | E 0 | h ( Ω ) N , where B 1 is the unit ball in R N . For Ω convex we obtain u = | E 0 | − 1 χ E 0 .

DE SITTER SPACETIME WITH TORSION AS PHYSICAL SPACETIME IN THE VACUUM AND ISOTROPIC COSMOLOGY

Homogeneous isotropic models with two torsion functions built in the framework of the Poincaré gauge theory of gravity (PGTG) based on general expression of gravitational Lagrangian without cosmological constant are analyzed. It is shown that the physical spacetime in the vacuum in the framework of PGTG can have the structure of flat de Sitter spacetime with torsion. Some physical consequences of obtained conclusion are discussed.