scholarly journals Coordinate-free exponentials of general multivector in Cl(p,q) algebras for p+q=3

2021 ◽  
Author(s):  
Arturas Acus ◽  
Adolfas Dargys
Keyword(s):  
Signal Processing ◽  
Differential Equations ◽  
Automatic Control ◽  
Closed Form ◽  
Exponential Function ◽  
Free Form ◽  
Hyperbolic Function ◽  
Main Difficulty ◽  
Geometric Algebras ◽  
And Robotics

Closed form expressions in real Clifford geometric algebras Cl(0,3), Cl(3,0), Cl(1,2), and Cl(2,1) are presented in a coordinate-free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components a and a in a form (a-a), i≠ j≠ k . After disentanglement, the obtained formulas simplify to the well-known Moivre-type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.

scholarly journals Exponentials of general multivector in 3D Clifford algebras

2022 ◽  
Vol 27 (1) ◽  
pp. 179-197
Author(s):  
Adolfas Dargys ◽  
Artūras Acus
Keyword(s):  
Image Processing ◽  
Differential Equations ◽  
Automatic Control ◽  
Closed Form ◽  
Series Expansion ◽  
Clifford Algebras ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Geometric Algebras ◽  
And Robotics

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

scholarly journals A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ji Juan-Juan ◽  
Guo Ye-Cai ◽  
Zhang Lan-Fang ◽  
Zhang Chao-Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Space Time ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Table Lookup ◽  
Exact Analytical Solutions ◽  
Function Solution

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

scholarly journals Demonstration of a portable intracortical brain-computer interface

2019 ◽  
Author(s):  
Jeffrey M. Weiss ◽  
Robert A. Gaunt ◽  
Robert Franklin ◽  
Michael Boninger ◽  
Jennifer L. Collinger
Keyword(s):  
Signal Processing ◽  
Brain Computer Interface ◽  
Computer Interface ◽  
Free Form ◽  
Small Signal ◽  
Standard Laboratory ◽  
Tablet Pc ◽  
Computer Interfaces ◽  
Investigational Device Exemption ◽  
Human Participant

AbstractWhile recent advances in intracortical brain-computer interfaces (iBCI) have demonstrated the ability to restore motor and communication functions, such demonstrations have generally been confined to controlled experimental settings and have required bulky laboratory hardware. Here, we developed and evaluated a self-contained portable iBCI that enabled the user to interact with various computer programs. The iBCI, which weighs 1.5 kg, consists of digital headstages, a small signal processing hub, and a tablet PC. A human participant tested the portable iBCI in laboratory and home settings under an FDA Investigational Device Exemption (NCT01894802). The participant successfully completed 96% of trials in a 2D cursor center-out task with the portable iBCI, a rate indistinguishable from that achieved with the standard laboratory iBCI. The participant also completed a variety of free-form tasks, including drawing, gaming, and typing.

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

Exponential function for calculating saturable enzyme kinetics.

1988 ◽  
Vol 34 (12) ◽  
pp. 2486-2489 ◽  
Author(s):  
F Keller ◽  
C Emde ◽  
A Schwarz
Keyword(s):  
Steady State ◽  
Enzyme Kinetics ◽  
Exponential Function ◽  
Association Constant ◽  
Time Dependent ◽  
Mathematical Representation ◽  
Hyperbolic Function ◽  
Maximal Velocity ◽  
Michaelis Constant ◽  
General Meaning

Abstract Enzyme kinetics are usually described by the Michaelis-Menten equation, where the time-dependent decrease of substrate (-dS/dt) is a hyperbolic function of maximal velocity (Vmax), Michaelis constant (Km), and amount of substrate (S). Because the Michaelis-Menten function in its most general meaning requires an assumption of steady-state, it is less curvilinear than true enzyme kinetics. A saturation-type exponential function is more curvilinear than the hyperbolic function and more closely approximates enzyme kinetics: -dS/dt = Vmax [1 - exp(-S/Km)]. The mathematical representation of enzyme kinetics can be further improved by introducing a deceleration term (Vdec), to make the assumption of a steady state unnecessary. For the action of chymotrypsin on N-acetyltyrosylethylester, the Michaelis-Menten equation yields the following: Vmax = 3.74 mumol/min and Km = 833 mumol. According to decelerated enzyme kinetics, the values Vmax = 4.80 mumol/min, Vdec = 0.0118 mumol/min, and the association constant (Ka) = 0.00111/mumol are more nearly accurate for this reaction (where 1/Ka = 901 mumol approximately Km).

A note on the summation of divergent power series

1969 ◽  
Vol 66 (1) ◽  
pp. 109-114 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Severe Restriction ◽  
Mathematical Problems ◽  
Divergent Power Series ◽  
Euler Transformation ◽  
Direct Use

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

Closed-Form Approximations to the Solution of V-Belt Force and Slip Equations

1985 ◽  
Vol 107 (2) ◽  
pp. 292-300 ◽  
Author(s):  
J. P. Dolan ◽  
W. S. Worley
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Design Practice ◽  
Tension Distribution ◽  
Analytical Approximations ◽  
Current Design

A method for generating accurate numerical solutions of the exact differential equations describing tension distribution and radial penetration of a flexible V-belt on driveN and driveR sheaves is presented and results are compared with approximate solutions reported in the literature. Analytical approximations for these solutions of higher accuracy than any previously published have been found and are presented. They suggest important modifications of current design practice for belt tensioning and life appraisal.

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