On Line Screw Systems and Their Application to Flexure Synthesis

2011 ◽  
Vol 3 (1) ◽  
Author(s):  
Hai-Jun Su ◽  
Hafez Tari
Keyword(s):  
Case Studies ◽  
Computational Algorithm ◽  
Screw Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Constrained Motion ◽  
Motion Patterns ◽  
Linearly Independent ◽  
On Line ◽  
Necessary And Sufficient

Motivated by the problem of synthesizing a pattern of flexures that provide a desired constrained motion, this paper presents a new screw theory that deals with “line screws” and “line screw systems.” A line screw is a screw with a zero pitch. The set of all line screws within a screw system is called a line variety. A general screw system of rank m is a line screw system if the rank of its line variety equals m. This paper answers two questions: (1) how to calculate the rank of a line variety for a given screw system and (2) how to algorithmically find a set of linearly independent lines from a given screw system. It has been previously found that a wire or beam flexure is considered a line screw, or more specifically a pure force wrench. By following the reciprocity and definitions of line screws, we have derived the necessary and sufficient conditions of line screw systems. When applied to flexure synthesis, we show that not all motion patterns can be realized with wire flexures connected in parallel. A computational algorithm based on this line screw theory is developed to find a set of admissible line screws or force wrenches for a given motion space. Two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.

On Line Screw Systems and Their Application to Flexure Synthesis

2010 ◽  
Author(s):  
Hai-Jun Su ◽  
Hafez Tari
Keyword(s):  
Case Studies ◽  
Computational Algorithm ◽  
Necessary Conditions ◽  
Screw Theory ◽  
Motion Pattern ◽  
Linearly Independent ◽  
On Line ◽  
Sufficient And Necessary Conditions ◽  
The Given

This paper presents a new screw theory that deals with “lines” and “line systems”. A line is a screw with a zero pitch. A rank m screw system is called a line screw system if it contains m linearly independent lines. The set of lines within a screw system is called the line variety. A general m-system is a line screw system if the rank of its line variety equals m. This paper will answer two questions: (1) how to calculate the rank of the line variety for a given m-system and (2) how to algorithmically find a set of linearly independent lines from the given screw system. The motivation of this work comes from flexure synthesis whose goal is to find a pattern of flexures to achieve a specified motion. It has been previously found that a wire flexure is considered a line screw, or more specifically a pure force wrench. The synthesis of flexures is to remove undesired motion by applying constraints with flexures. By following reciprocity and definitions of line screws, we have derived the sufficient and necessary conditions of line screw systems. When applied to flexure synthesis, we show that not all motion pattern can be realized with wire flexures connected in parallel. A computational algorithm based on the line screw theory is developed to find all admissible line screws or force wrenches for a given motion space. At last two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.

Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston

2019 ◽  
Vol 63 (1) ◽  
pp. 31-45
Author(s):  
Tapas Chatterjee ◽  
Sonika Dhillon
Keyword(s):  
Linear Independence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sine Function ◽  
Independent Subset ◽  
Linearly Independent ◽  
Cyclotomic Numbers ◽  
Necessary And Sufficient ◽  
Trigonometric Identities ◽  
Rational Arguments

AbstractIn 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

Essential norms of weighted differentiation composition operators between Zygmund type spaces and Bloch type spaces

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2877-2889 ◽  
Author(s):  
Amir Sanatpour ◽  
Mostafa Hassanlou
Keyword(s):  
Composition Operators ◽  
Sufficient Conditions ◽  
Essential Norm ◽  
Necessary And Sufficient Conditions ◽  
Norm Estimates ◽  
Type Spaces ◽  
Necessary And Sufficient

We study boundedness of weighted differentiation composition operators Dk?,u between Zygmund type spaces Z? and Bloch type spaces ?. We also give essential norm estimates of such operators in different cases of k ? N and 0 < ?,? < ?. Applying our essential norm estimates, we get necessary and sufficient conditions for the compactness of these operators.

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