Hermitian Metrics on the Resolvent Set and Extremal Arc Length

Author(s):  
Mai Tran ◽  
Rongwei Yang
1997 ◽  
Author(s):  
J. Farley Norman ◽  
Joseph S. Lappin ◽  
Hideko F. Norman

2020 ◽  
pp. 26-32
Author(s):  
M. I. Kalinin ◽  
L. K. Isaev ◽  
F. V. Bulygin

The situation that has developed in the International System of Units (SI) as a result of adopting the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities, is analyzed. It is shown that the basis for such a solution was a misunderstanding of the mathematical formula relating the arc length of a circle with its radius and corresponding central angle, as well as of the expansions of trigonometric functions in series. From the analysis presented in the article, it follows that a plane angle does not depend on any of the SI quantities and should be assigned to the base quantities, and its unit, the radian, should be added to the base SI units. A solid angle, in this case, turns out to be a derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.


2019 ◽  
Vol 13 (1) ◽  
pp. 69-74
Author(s):  
Wang Yazhou ◽  
Xiao Junfeng ◽  
Liu Yongping ◽  
An Jianmin

Background: Various relevant patents and papers which have reported noncircular gears synthesize the advantages of circular gears and cam mechanisms, and are widely used in many types of mechanical instruments. Hobbing is a better method for fabricating noncircular gears. There are 4 linkagemethods to hob noncircular gears. However, which linkage method should be chosen practically has not yet been reported. Objective: The goal of this work is to choose the best linkage method for hobbing noncircular gears. Method: Firstly, the hobbing models of noncircular gears was deduced. Then, based on the model, hobbing linkage methods of noncircular gears were obtained. Thirdly, under different hobbing linkage methods, their aspects (developing regularity of hobbing cutter trace, arc length of program blocks and motion axes of machine tools) were compared. Results: Finally, with the best characteristics of a high density of shaping cutter trace, high uniformity of arc length of program blocks and ease of control, the equal arc-length of gear billet (EALGB) is obtained. It has been proven that EALGB is an excellent linkage method to hob noncircular gears. Conclusion: It has been proven that EALGB is an excellent linkage method to hob noncircular gears.


2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Leonardo Alese

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.


2021 ◽  
Vol 11 (11) ◽  
pp. 4959
Author(s):  
Peng Guo ◽  
Yijie Wu ◽  
Guang Yang ◽  
Zhebin Shen ◽  
Haorong Zhang ◽  
...  

The curvature of the NURBS curve varies along its trajectory, therefore, the commonly used feedrate-planning method, which based on the acceleration/deceleration (Acc/Dec) model, is difficult to be directly applied in CNC machining of a NURBS curve. To address this problem, a feedrate-planning method based on the critical constraint curve of the feedrate (CCC) is proposed. Firstly, the problems of existing feedrate-planning methods and their causes are analyzed. Secondly, by considering both the curvature constraint and the kinematic constraint during the Acc/Dec process, the concept of CCC which represents the relationship between the critical feedrate-constraint value and the arc length is proposed. Then the CCC of a NURBS curve is constructed, and it has a concise expression conforming to the Acc/Dec model. Finally, a feedrate-planning method of a NURBS curve based on CCC and the Acc/Dec model is established. In the simulation, a comparison between the proposed method and the conventional feedrate-planning method is performed, and the results show that, the proposed method can reduce the Acc/Dec time by over 40%, while little computational burden being added. The machining experimental results validate the real-time performance and stability of the proposed method, and also the machining quality is verified. The proposed method offers an effective feedrate-planning strategy for a NURBS curve in CNC machining.


Author(s):  
Yoshinobu Kamishima

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .


2021 ◽  
Vol 112 (1) ◽  
Author(s):  
Christine Rademacher ◽  
Hans-Bert Rademacher

AbstractFor a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Vikram Jakkamsetti ◽  
William Scudder ◽  
Gauri Kathote ◽  
Qian Ma ◽  
Gustavo Angulo ◽  
...  

AbstractTime-to-fall off an accelerating rotating rod (rotarod) is widely utilized to evaluate rodent motor performance. We reasoned that this simple outcome could be refined with additional measures explicit in the task (however inconspicuously) to examine what we call movement sub-structure. Our goal was to characterize normal variation or motor impairment more robustly than by using time-to-fall. We also hypothesized that measures (or features) early in the sub-structure could anticipate the learning expected of a mouse undergoing serial trials. Using normal untreated and baclofen-treated movement-impaired mice, we defined these features and automated their analysis using paw video-tracking in three consecutive trials, including paw location, speed, acceleration, variance and approximate entropy. Spectral arc length yielded speed and acceleration uniformity. We found that, in normal mice, paw movement smoothness inversely correlated with rotarod time-to-fall for the three trials. Greater approximate entropy in vertical movements, and opposite changes in horizontal movements, correlated with greater first-trial time-to-fall. First-trial horizontal approximate entropy in the first few seconds predicted subsequent time-to-fall. This allowed for the separation, after only one rotarod trial, of different-weight, untreated mouse groups, and for the detection of mice otherwise unimpaired after baclofen, which displayed a time-to-fall similar to control. A machine-learning support vector machine classifier corroborated these findings. In conclusion, time-to-fall off a rotarod correlated well with several measures, including some obtained during the first few seconds of a trial, and some responsive to learning over the first two trials, allowing for predictions or preemptive experimental manipulations before learning completion.


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