Torsion Elements Research Articles

Realization of graded monomial ideal rings modulo torsion

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.

Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

We use recent advances in the local evaluation of Brauer elements to study the role played by odd torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over arbitrary number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, under which the odd torsion does not obstruct weak approximation. We also note a systematic way to produce K 3 K3 surfaces over Q 2 \mathbb {Q}_2 with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.

Torsion in 1-Cusped Picard Modular Groups

We present a systematic effective method to construct coarse fundamental domains for the action of the Picard modular groups $PU(2,1,\mathcal {O}_{d})$ where $\mathcal {O}_{d}$ has class number one, i.e., d = 1,2,3,7,11,19,43,67,163. The computations can be performed quickly up to the value d = 19. As an application of this method, we classify conjugacy classes of torsion elements, deduce short presentations for the groups, and construct neat subgroups of small index.

Generalized torsion for hyperbolic 3‐manifold groups with arbitrary large rank

AbstractLet be a group and a non‐trivial element in . If some non‐empty finite product of conjugates of equals to the trivial element, then is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic 3‐manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer there are infinitely many closed hyperbolic 3‐manifolds which enjoy the property: (i) the Heegaard genus of is , (ii) the rank of is , and (ii) has a generalized torsion element. Furthermore, we may choose as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.

Extending periodic maps on surfaces over the 4-sphere

Let [Formula: see text] be the closed orientable surface of genus [Formula: see text]. We address the problem to extend torsion elements of the mapping class group [Formula: see text] over the 4-sphere [Formula: see text]. Let [Formula: see text] be a torsion element of maximum order in [Formula: see text]. Results including: (1) For each [Formula: see text], [Formula: see text] is periodically extendable over [Formula: see text] for some non-smooth embedding [Formula: see text], and not periodically extendable over [Formula: see text] for any smooth embedding [Formula: see text]. (2) For each [Formula: see text], [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if and only if [Formula: see text]. (3) Each torsion element of order [Formula: see text] in [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if either (i) [Formula: see text] and [Formula: see text] is even; or (ii) [Formula: see text] and [Formula: see text]; or (iii) [Formula: see text]. Moreover, the conditions on [Formula: see text] in (i) and (ii) cannot be removed.

Identification of Torsional Modes in Complex Molecules Using Redundant Internal Coordinates: The Multistructural Method with Torsional Anharmonicity with a Coupled Torsional Potential and Delocalized Torsions.

Identification of internal-rotation modes in the normal-mode analysis of complex molecules is important for accurately describing the thermodynamic properties and kinetics of complex molecules when it is necessary to treat the anharmonicity of torsions and the multiconformer anharmonicity caused by the internal rotations. However, identifying and distinguishing torsional modes are very challenging because they are coupled to one another. In this work, we present a new strategy to automatically identify torsional vibrations and separate them from the other vibrational modes. By combining a redundant-internal-coordinate auto-generation procedure with torsional projection techniques, we automate the procedure of identifying and separating the coupled torsions, and we show that we can obtain robust and consistent results with various reasonable definitions of redundant-internal-coordinate sets. This model has been implemented in a new development version of the MSTor program to reduce the user input needed for multistructural and torsional anharmonicity (MS-T) calculations. The new method is called multistructural and torsional anharmonicity with a coupled torsional potential and delocalized torsions ([MS-T(CD)]. As example applications, we consider MS-T(CD) calculations on three molecules (2-hexyl radical, n-propylbenzene, and 5-hydroperoxy-6-oxohexanoylperoxy radical) that have multiple rotors and that provide challenges to choosing good sets of nonredundant-internal coordinates, and we compare the performance of the new strategy to five other torsion identification methods. The new strategy is demonstrated to be efficient in separating the torsional and nontorsional elements in the Hessian matrix, as well as in providing reasonable projected nontorsional frequencies to be used for calculations of partition function and thermochemistry.

DLP in semigroups: Algorithms and lower bounds

Abstract The discrete logarithm problem (DLP) in semigroups has attracted some interests and serves as the foundation of many cryptographic schemes. In this work, we study algorithms and lower bounds for DLP in semigroups. First, we propose a variant of the deterministic algorithm for solving the cycle length of torsion elements and show the lower bound of computing the DLP in a semigroup. Then, we propose an algorithm for solving the multiple discrete logarithm (MDL) problem in the semigroup and give the lower bound for solving the MDL problem by considering the MDL problem in the generic semigroup model. Besides, we solve the multidimensional DLP and product DLP in the semigroup.

Review and preliminary design of 3-D printed ABS polymer torsion element with parametric internal structures

Various components utilized in the automobile dashboard controls are manufactured from plastic by a conventional molding process. With the recent trend and continuously changing designs of the components due to frequently upgraded models, the conventional process of manufacture of these components proves costly owing to the high investment in molds. The technology which has taken attention for these demanding conditions is the Fused Deposition or 3-D printing technology in development and manufacture of plastic components for ready use direct assembly which bypasses the process of mold manufacture and then subsequent plastic molding which is many times involves a considerable amount of production time and high production cost. The strength of the components produced by the 3-D printing method is seen to be greatly influenced by the infill density, print quality, and layer thickness.The depositing parameters have been studied in various studies but they failed to characterize the geometrical arrangement permutations through the addition of internal support structures to enhance the torsion strength of the components. The paper discusses a review of research work over the years proposes the first stage design and analysis of the plain component using Ansys workbench and also the manufacturing of these plain elements with various permutations of infill density, layer thickness, and print quality. The components thus produced will be tested and compared for optimal deposition parameters. By considering the gathered data, an optimization model that realizes the material usage, build time, strength characteristics, and their related variables will be presented and used to assist the components with optimal deposition parameters which will be further considered for design with internal support structures with the incorporation of varied parametric internal structure matrix.

Concept and electromechanical-coupling modeling of a torsional vibration excitation method

The sandwiched type torsional piezoelectric actuators have widely been employed in the fields of ultrasonic cutting, ultrasonic drilling, ultrasonic welding, ultrasonic wire drawing, ultrasonic spraying, and microdroplet generation, due to their unique vibration form. However, the circular piezoelectric ceramic plate polarized in the circumferential direction holds a complicated manufacturing process, a long processing period, and a high price. It can only be used for the excitation of the torsional vibration of circular rods, and cannot be used for square cross-section beams, severely restricting its applications. In order to solve the above problems, a novel excitation method is proposed for generating torsional vibration in the square cross-section beam in this study, which uses the d15 vibration mode of rectangular piezoelectric ceramic plates to produce the torsional vibration, and a novel transfer matrix model of square torsional piezoelectric element is created. Then a case study is conducted in this paper to confirm the proposed excitation method and the developed model, in which two novel sandwiched type torsional piezoelectric actuators are designed. Moreover, electromechanical coupling models of these two torsional piezoelectric actuators are developed using the transfer matrix method, in order to reveal their dynamics behaviors. Finally, the finite element method and the experimental investigations are adopted to analyze and test the frequency response characteristics and vibration shapes of the proposed piezoelectric actuators. Both simulation analysis and experimental tests verify the effectiveness of the proposed excitation method and the correctness of the developed electromechanical coupling dynamic models. The proposed excitation method provides a new idea for the design of sandwiched type torsional piezoelectric actuators, and the developed theoretical model lays a theoretical foundation for the further research of torsional piezoelectric actuators.

The Biomechanical Profile of an Osseo-Integrated Rectangular Block Implant: A Pilot In Vivo Strain Analysis

Aim: To load-test the osseo-integrated rectangular block implant (RBI), measure the generated cortical peri-implant strains, and relate these findings to known human physiological parameters. Materials and methods: Two RBIs were placed into the posterior mandibular saddle in a mature greyhound dog and allowed to osseo-integrate. The half mandible (implants in situ) was mounted in a servohydraulic system. Four triple-stacked rosette gauges were placed cortically (mesial, distal, buccal, and lingual). A modified ISO-14801 protocol was used (1000 N, 300, 2 Hz, 1 h) and the generated principal strains (ep, eq) and their angular orientations (F), were calculated. Results: (1) Bucco-lingual “horizontal” dimension: dominant “horizontal” compressive stresses were on the lingual aspect and “horizontal” tensile stresses on the buccal aspect. The buccal cortex was elastically tensile-stretched, while the lingual cortex was elastically compressed. (2) Bucco-lingual “vertical” dimension: dominant vertical torsional stresses were oriented buccally and apically, with an overall buccally inclined torsional effect. This was also evidenced on the lingual aspect, where there remained high torsional rotation elements (high F and e2). (3) Mesio-distal “horizontal” dimension: dominant torsional stresses oriented as a distal-lingual “counter-clockwise” rotation. Conclusions: The applied off-axial loads generated a heterogeneous pattern of bucco-lingual and mesio-distal cortical strains, both vertically and horizontally. The short dimensioned osseo-integrated RBI design appeared to biomechanically withstand the applied loads and to maintain the strains generated to levels that were within physiological limits. More studies and statistical analyses are needed to confirm these findings.

Unrestricted virtual braids and crystallographic braid groups

We show that the crystallographic braid group \(B_n/[P_n,P_n]\) embeds naturally in the group of unrestricted virtual braids \(UVB_n\), we give new proofs of known results about the torsion elements of \(B_n/[P_n,P_n]\), and we characterise the torsion elements of \(UVB_n\).

Stable Torsion Length

Abstract The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the 1st exact computations of stable torsion length for nontrivial examples.

On torsion elements and their annihilators

Let R be a commutative ring with identity, and let M be an R-module. In this paper, we focus on the ideals of R that are annihilators of torsion elements of M. In analogy with definitions and results on zero-divisor and annihilator graphs of rings, we define the annihilator graph of a module. We investigate the structure, the diameter, and the girth of this graph and the closely related torsion graph of a module introduced by Ghalandarzadeh and Malakooti Rad. Given the right definitions, the properties of the modules are reflected in the graph theoretic properties of the graphs. We thus modify and extend results on zero divisors of rings to the much more general setting of modules and their torsion elements. In addition, we significantly strengthen the known results on the torsion graphs of modules. Some of the results will be refined further for the cases when M is a multiplication module or a reduced module.

Generating the mapping class group of a nonorientable surface by three torsions

We prove that the mapping class group $$\mathcal {M}(N_g)$$ of a closed nonorientable surface of genus g different than 4 is generated by three torsion elements. Moreover, for every even integer $$k\ge 12$$ and g of the form $$g=pk+2q(k-1)$$ or $$g=pk+2q(k-1)+1$$ , where p, q are non-negative integers and p is odd, $$\mathcal {M}(N_g)$$ is generated by three conjugate elements of order k. Analogous results are proved for the subgroup of $$\mathcal {M}(N_g)$$ generated by Dehn twists.

Abelian quotients of the Y –filtration on the homology cylinders via the LMO functor

We construct a series of homomorphisms from the $Y$-filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3\mathcal{IC}_{g,1}/Y_4$ of the $Y$-filtration.

Generating the Mapping Class Group by Two Torsion Elements

We prove that the mapping class group of a closed connected orientable surface of genus \(g\ge 6\) is generated by two elements of order g. Moreover, for \(g\ge 7\), we obtain a generating set of two elements, of order g and \(g'\), where \(g'\) is the least divisor of g greater than 2. We also prove that the mapping class group is generated by two elements of order \(g/\gcd (g,k)\) for \(g\ge 3k^2+4k+1\) and any positive integer k.

ФИЗИЧЕСКАЯ СУТЬ СОПРОТИВЛЕНИЯ БЕТОНА И ЖЕЛЕЗОБЕТОНА ОТ ДИСЛОКАЦИЙ ДО ТРЕЩИН

The physical essence of resistance of concrete and reinforced concrete from dislocations, micro-cracks to macro-cracks and its experimental justification is investigated. For the "eight" structure of crystals of different materials (concrete and steel) a general model in the form of a sphere was developed. For it the summation of volume sectors, levels - radii from the matrix of sliding planes (including octahedral and pure shear) is written down. This uses an alternative to the theory of plasticity in the form of energy interpretation on the surface of the sphere and determining the integral of the mean square of the tangential stresses. It is important to obtain dislocations in the microcrack, angular and linear deformations, and displacements in a representative volume of the concrete cube. As the intensity increases, the deformation process proceeds already to the mainline cracks, where the double-concole elements of tension, compression, transverse shear and torsion (its internal parameters) are refined. Significant issues are the dilatation modulus and transverse coefficient, for which functions have been developed at the stages of the stress-strain state of concrete during the evolution of the transition from crack formation to main cracks. Concrete compression and tension diagrams for strain intensity or minimum pure shear use shear stresses. The fundamental difference of the stress diagram in the downward section is the use of the ultimate resistance of the concrete. Stress reduction in a material whose failure has a "tear-off" character is an unnatural phenomenon, and the limiting resistance of concrete at and reduction of prism strength at the i-th step is . The deformation pattern of concrete during the formation of earlier microcracks and then later main cracks is oriented along for compression or across the loading line for tensile force.

Generalized torsion for knots with arbitrarily high genus

AbstractLet G be a group, and let g be a nontrivial element in G. If some nonempty finite product of conjugates of g equals the identity, then g is called a generalized torsion element. We say that a knot K has generalized torsion if $G(K) = \pi _1(S^3 - K)$ admits such an element. For a $(2, 2q+1)$ -torus knot K, we demonstrate that there are infinitely many unknots $c_n$ in $S^3$ such that p-twisting K about $c_n$ yields a twist family $\{ K_{q, n, p}\}_{p \in \mathbb {Z}}$ in which $K_{q, n, p}$ is a hyperbolic knot with generalized torsion whenever $|p|> 3$ . This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the $(-2, 3, 7)$ -pretzel knot, have generalized torsion. Because generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

The cohomology rings of homogeneous spaces

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of G and K is invertible in a given principal ideal domain k. It is known that in this case the cohomology of the homogeneous space G / K with coefficients in k and the torsion product of H ∗ ( B K ) and k over H ∗ ( B G ) are isomorphic as k-modules. We show that this isomorphism is multiplicative and natural in the pair ( G , K ) provided that 2 is invertible in k. The proof uses homotopy Gerstenhaber algebras in an essential way. In particular, we show that the normalized singular cochains on the classifying space of a torus are formal as a homotopy Gerstenhaber algebra.

The body-language of the elephant trunk

The elephant trunk functions as a muscular hydrostat: it can achieve elaborate movements without the support of any bone. Elephants have evolved unique biomechanical strategies to manage the complexity of this organ. Its 3D trajectories are composed of basic elements of torsion, curvature and elongation (motion primitives), emerging from the orchestrated actions of antagonist muscle groups. - submission by Paule Dagenais, Michel C. Milinkovitch