Efficient elementary function generation with multipliers

Abstract The first part of the article presented analytical precision and optimum synthesis methods for linkages for the generation of specified torque histories and applied to the planar 4R four-bar mechanism. This article presents analytical mechanical advantage method (MAM) and integration of power equilibrium method (IPEM) for the synthesis of the planar slider-crank mechanism for the generation of specified input-output force and torque histories. Design equations for one, two, three, and four precision position synthesis are given. They are used to formulate the optimum synthesis technique, which requires no iteration to reach a solution mechanism. Slider-crank mechanisms synthesized also replace pinion-rack drives with noncircular and circular pinions to generate non-uniform and uniform velocity ratios, respectively. The MAM can be applied for discretely defined and continuous force-torque relationships, while IPEM is used with continuous relationships which reduces the force-torque generation into an elementary function generation problem. Application examples are included.

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Protein expression, purification, and elementary function of Dmrt1 in half-smooth tongue sole (Cynoglossus semilae-vis)

Journal of Fishery Sciences of China ◽

10.3724/sp.j.1118.2013.01132 ◽

2013 ◽

Vol 20 (6) ◽

pp. 1132-1138 ◽

Cited By ~ 3

Author(s):

Qiaomu HU ◽

Kailin WANG ◽

Songlin CHEN

Keyword(s):

Protein Expression ◽

Elementary Function ◽

Tongue Sole ◽

Half Smooth Tongue Sole

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Optimal Random Packing of Spheres and Extremal Effective Conductivity

Symmetry ◽

10.3390/sym13061063 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1063

Author(s):

Vladimir Mityushev ◽

Zhanat Zhunussova

Keyword(s):

Effective Conductivity ◽

Dimensional Space ◽

Elementary Function ◽

Voronoi Tessellation ◽

Random Packing ◽

Periodicity Cell ◽

Algebraic Equations ◽

Optimal Packing ◽

Linear Algebraic Equations ◽

Initial Location

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

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W-representation of Rainbow tensor model

Journal of High Energy Physics ◽

10.1007/jhep05(2021)228 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Bei Kang ◽

Lu-Yao Wang ◽

Ke Wu ◽

Jie Yang ◽

Wei-Zhong Zhao

Keyword(s):

Partition Function ◽

Matrix Model ◽

Crucial Role ◽

Elementary Function ◽

Tensor Model ◽

Witt Algebra ◽

Virasoro Constraints ◽

Compact Expression ◽

Non Gaussian ◽

Dual Expression

Abstract We analyze the rainbow tensor model and present the Virasoro constraints, where the constraint operators obey the Witt algebra and null 3-algebra. We generalize the method of W-representation in matrix model to the rainbow tensor model, where the operators preserving and increasing the grading play a crucial role. It is shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operator increasing the grading. We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model. Furthermore, we discuss the case of the non-Gaussian red tensor model and present a dual expression for partition function through differentiation.

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