scholarly journals Counting Tensor Rank Decompositions

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 302
Author(s):  
Dennis Obster ◽  
Naoki Sasakura
Keyword(s):  
Degrees Of Freedom ◽  
Symmetric Tensor ◽  
Geometric Interpretation ◽  
Tensor Rank ◽  
Tensor Model ◽  
Symmetric Tensors ◽  
Rank Decomposition ◽  
Similar Quantity ◽  
The Stability ◽  
Unit Norm

Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

scholarly journals Seeking for the Maximum Symmetric Rank

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 247 ◽  
Author(s):  
Alessandro De Paris
Keyword(s):  
State Of The Art ◽  
Symmetric Tensor ◽  
Upper Bounds ◽  
The State ◽  
Tensor Rank ◽  
Symmetric Tensors ◽  
Lower And Upper Bounds ◽  
Symmetric Tensor Rank ◽  
Set Up ◽  
Divided Powers

We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds.

scholarly journals Symmetric Tensors and Symmetric Tensor Rank

2008 ◽  
Vol 30 (3) ◽  
pp. 1254-1279 ◽  
Author(s):  
Pierre Comon ◽  
Gene Golub ◽  
Lek-Heng Lim ◽  
Bernard Mourrain
Keyword(s):  
Symmetric Tensor ◽  
Tensor Rank ◽  
Symmetric Tensors ◽  
Symmetric Tensor Rank

Fast Bilinear Algorithms for Symmetric Tensor Contractions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Symmetric Matrix ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Symmetric Tensor ◽  
Partial Sums ◽  
Symmetric Tensors ◽  
Outer Product ◽  
Bilinear Complexity ◽  
Special Cases ◽  
Tensor Contractions

AbstractIn matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

scholarly journals Velocity and acceleration level inverse kinematic calculation alternatives for redundant manipulators

Meccanica ◽  
2021 ◽  
Author(s):  
Dóra Patkó ◽  
Ambrus Zelei
Keyword(s):  
Degrees Of Freedom ◽  
Direct Method ◽  
Tracking Error ◽  
Inverse Kinematic ◽  
Linear Feedback ◽  
Joint Position ◽  
Stability Properties ◽  
Acceleration Level ◽  
Error Elimination ◽  
The Stability

AbstractFor both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That is when the desired task possesses lower dimensions than the degrees-of-freedom of a redundant manipulator. The IK calculation can be implemented numerically in several alternative ways not only in case of the redundant but also in the non-redundant case. We study the stability properties and the feasibility of a tracking error feedback and a direct tracking error elimination approach of the numerical implementation of IK calculation both on velocity and acceleration levels. The feedback approach expresses the joint position increment stepwise based on the local velocity or acceleration of the desired TCP trajectory and linear feedback terms. In the direct error elimination concept, the increment of the joint position is directly given by the approximate error between the desired and the realized TCP position, by assuming constant TCP velocity or acceleration. We investigate the possibility of the implementation of the direct method on acceleration level. The investigated IK methods are unified in a framework that utilizes the idea of the auxiliary input. Our closed form results and numerical case study examples show the stability properties, benefits and disadvantages of the assessed IK implementations.

Stability of Ring-Type MEMS Gyroscopes Subjected to Stochastic Angular Speed Fluctuation

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Samuel F. Asokanthan ◽  
Soroush Arghavan ◽  
Mohamed Bognash
Keyword(s):  
Angular Velocity ◽  
Degrees Of Freedom ◽  
Microelectromechanical Systems ◽  
Damping Ratio ◽  
Operating Conditions ◽  
System Stability ◽  
System Response ◽  
Ring Type ◽  
Mems Gyroscopes ◽  
The Stability

Effect of stochastic fluctuations in angular velocity on the stability of two degrees-of-freedom ring-type microelectromechanical systems (MEMS) gyroscopes is investigated. The governing stochastic differential equations (SDEs) are discretized using the higher-order Milstein scheme in order to numerically predict the system response assuming the fluctuations to be white noise. Simulations via Euler scheme as well as a measure of largest Lyapunov exponents (LLEs) are employed for validation purposes due to lack of similar analytical or experimental data. The response of the gyroscope under different noise fluctuation magnitudes has been computed to ascertain the stability behavior of the system. External noise that affect the gyroscope dynamic behavior typically results from environment factors and the nature of the system operation can be exerted on the system at any frequency range depending on the source. Hence, a parametric study is performed to assess the noise intensity stability threshold for a number of damping ratio values. The stability investigation predicts the form of threshold fluctuation intensity dependence on damping ratio. Under typical gyroscope operating conditions, nominal input angular velocity magnitude and mass mismatch appear to have minimal influence on system stability.

scholarly journals Galloping Stability and Wind Tunnel Test of Iced Quad Bundled Conductors considering Wake Effect

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Xiaohui Liu ◽  
Ming Zou ◽  
Chuan Wu ◽  
Mengqi Cai ◽  
Guangyun Min ◽  
...  
Keyword(s):  
Wind Tunnel ◽  
Transmission Lines ◽  
Degrees Of Freedom ◽  
Wind Tunnel Test ◽  
Aerodynamic Coefficients ◽  
Wake Effect ◽  
Aerodynamic Coefficient ◽  
Set Up ◽  
The Stability ◽  
Three Degrees Of Freedom

A new quad bundle conductor galloping model considering wake effect is proposed to solve the problem of different aerodynamic coefficients of each subconductor of iced quad bundle conductor. Based on the quasistatic theory, a new 3-DOF (three degrees of freedom) galloping model of iced quad bundle conductors is established, which can accurately reflect the energy transfer and galloping of quad bundle conductor in three directions. After a series of formula derivations, the conductor stability judgment formula is obtained. In the wind tunnel test, according to the actual engineering situation, different variables are set up to accurately simulate the galloping of iced quad bundle conductor under the wind, and the aerodynamic coefficient is obtained. Finally, according to the stability judgment formula of this paper, calculate the critical wind speed of conductor galloping through programming. The dates of wind tunnel test and calculation in this paper can be used in the antigalloping design of transmission lines.

Numerical and Experimental Approaches to Investigate the Stability of a Motorcycle Vehicle

2006 ◽  
Author(s):  
Federico Cheli ◽  
Marco Bocciolone ◽  
Marco Pezzola ◽  
Elisabetta Leo
Keyword(s):  
Degrees Of Freedom ◽  
Structural Parameters ◽  
Experimental Tests ◽  
Roll Angle ◽  
Longitudinal Motion ◽  
Reference Trajectory ◽  
Steering Angle ◽  
Steady State Condition ◽  
The Stability ◽  
Linearized Model

The study of motorcycle’s stability is an important task for the passenger’s safety. The range of frequencies involved for the handling stability is lower than 10 Hz. A numerical model was developed to access the stability of a motorcycle vehicle in this frequency range. The stability is analysed using a linearized model around the straight steady state condition. In this condition, the vehicle’s vertical and longitudinal motion are decoupled, hence the model has only four degrees of freedom (steering angle, yaw angle, roll angle and lateral translation), while longitudinal motion is imposed. The stability was studied increasing the longitudinal speed. The input of the model can be either a driver input manoeuvre (roll angle) or a transversal component of road input able to excite the vibration modes. The driver is introduced in the model as a steering torque that allows the vehicle to follow a reference trajectory. To validate the model, experimental tests were done. To excite the vehicle modes, the driver input was not taken into account considering both the danger for the driver and the repeatability of the manoeuvre. Two different vehicle configurations were tested: vehicle 1 is a motorcycle [7] and vehicle 2 is a scooter. Through the use of the validated model, a sensitivity analysis was done changing structural (for example normal trail, steering angle, mass) and non structural parameters (for example longitudinal speed).

scholarly journals Design of a 2DoF Ankle Exoskeleton with a Polycentric Structure and a Bi-Directional Tendon-Driven Actuator Controlled Using a PID Neural Network

Actuators ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 9
Author(s):  
Taehoon Lee ◽  
Inwoo Kim ◽  
Yoon Su Baek
Keyword(s):  
Neural Network ◽  
Degrees Of Freedom ◽  
Plantar Flexion ◽  
Length Change ◽  
Talocrural Joint ◽  
Lower Limb Exoskeleton ◽  
Exoskeleton Robot ◽  
The Difference ◽  
Ankle Exoskeleton ◽  
The Stability

Lower limb exoskeleton robots help with walking movements through mechanical force, by identifying the wearer’s walking intention. When the exoskeleton robot is lightweight and comfortable to wear, the stability of walking increases, and energy can be used efficiently. However, because it is difficult to implement the complex anatomical movements of the human body, most are designed simply. Due to this, misalignment between the human and robot movement causes the wearer to feel uncomfortable, and the stability of walking is reduced. In this paper, we developed a two degrees of freedom (2DoF) ankle exoskeleton robot with a subtalar joint and a talocrural joint, applying a four-bar linkage to realize the anatomical movement of a simple 1DoF structure mainly used for ankles. However, bidirectional tendon-driven actuators (BTDAs) do not consider the difference in a length change of both cables due to dorsiflexion (DF) and plantar flexion (PF) during walking, causing misalignment. To solve this problem, a BTDA was developed by considering the length change of both cables. Cable-driven actuators and exoskeleton robot systems create uncertainty. Accordingly, adaptive control was performed with a proportional-integral-differential neural network (PIDNN) controller to minimize system uncertainty.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

STUDIES ON HOMOGENEOUS FIRST ORDER GAS REACTIONS: II. THE DECOMPOSITION OF THE ISOMERIC ESTERS BUTYLIDENE DIACETATE AND ETHYLIDENE DIPROPIONATE

1932 ◽  
Vol 6 (4) ◽  
pp. 417-427 ◽  
Author(s):  
C. C. Coffin
Keyword(s):  
Activation Energy ◽  
Degrees Of Freedom ◽  
Maximum Energy ◽  
Velocity Measurements ◽  
First Order ◽  
Gas Reactions ◽  
Points Of View ◽  
The Stability ◽  
Future Work ◽  
Internal Degrees Of Freedom

The gaseous decompositions of the esters butylidene diacetate and ethylidene dipropionate have been studied from points of view previously outlined in papers on the decomposition of ethylidene diacetate (2, 3). The decomposition velocities have been measured at initial pressures of from 5 to 56 cm. of mercury and at temperatures between 211 and 265 °C. The reactions are homogeneous and of the first order. They agree with the Arrhenius equation and give 100% yields (within experimental error) of an aldehyde and an anhydride. The preparation of the compounds and improvements in the technique of the velocity measurements are described.While the specific velocities of the three reactions at any temperature are somewhat different, their activation energies are the same. It is suggested that in the case of such simple reactions, which are strictly localized within the molecular structure, the activation energy can be identified as the maximum energy that the reactive bonds may possess and still exist; i.e., it may be taken as a measure of the stability of the bonds which are broken in the reaction. The suggestion is also made that for a series of reactions which have the same activation energy, the specific velocities can be taken as a relative measure of the number of internal degrees of freedom that contribute to the energy of activation. On the basis of these assumptions it becomes possible to use reaction-velocity measurements for the investigation of intramolecular energy exchange. The theoretical significance of the data is further discussed and the scope of future work in this connection is indicated.The monomolecular velocity constants (sec−1) of the decomposition of ethylidene diacetate, ethylidene dipropionate and butylidene diacetate are given respectively by the equations [Formula: see text], [Formula: see text], and [Formula: see text].

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