A CLOSED-FORM SOLUTION OF THE EVOLUTION OPERATOR IN TWO-LEVEL SYSTEMS

1994 ◽  
Vol 08 (08n09) ◽  
pp. 505-508 ◽  
Author(s):  
XIAN-GENG ZHAO
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Evolution Operator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Two Level Systems ◽  
Special Case

It is demonstrated by using the technique of Lie algebra SU(2) that the problem of two-level systems described by arbitrary time-dependent Hamiltonians can be solved exactly. A closed-form solution of the evolution operator is presented, from which the results for any special case can be deduced.

Spontaneous potential log response expressed as convolution

Geophysics ◽  
1982 ◽  
Vol 47 (9) ◽  
pp. 1335-1337
Author(s):  
E. A. Nosal
Keyword(s):  
Closed Form ◽  
Signal Analysis ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Individual Contribution ◽  
Spontaneous Potential ◽  
The Earth ◽  
The Individual ◽  
Special Case

A special case of spontaneous potential (SP) logging, which has a closed‐form solution, will be expressed as a convolutional operation. Such a formal demonstration serves two purposes. First, it separates the individual contribution of the tool from that of the earth. Second, it places this logging device within the mathematical context of signal analysis. The special case for which a closed‐form solution is known is that where all resistivities are equal. Fourier analysis applied to this solution leads to a product of two functions, of which one is identified as the contribution of the earth and the other of the tool.

scholarly journals Closed-Form Solution of a Special Case of a Vehicle Longitudinal Motion Model

Vehicles ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Blagojevic ◽  
Djudurovic ◽  
Bajic
Keyword(s):  
Mathematical Model ◽  
Closed Form ◽  
Closed Form Solution ◽  
Large Data ◽  
Form Solution ◽  
Longitudinal Motion ◽  
Data Set ◽  
Engineering Research ◽  
Complex Mathematical Model ◽  
Special Case

One of the most common approaches in modern engineering research, including vehicle dynamics, is to formulate an accurate, but typically complex, mathematical model of a system or phenomenon and then use a software package to solve it. Typically, the solution is obtained in the form of a large data set, which may be difficult to analyse and interpret. This paper represents a purely theoretical analysis of a special case of vehicle longitudinal motion. Starting from a simplified mathematical model, a set of transcendental equations was derived that represents the exact solution of the model (i.e., in a closed form). The equations are analysed and interpreted in terms of what is their physical meaning. Although the equations derived here have only limited application in studying real world problems, due to the simplicity of the mathematical model, they offer a deeper insight into the nature of vehicle longitudinal motion.

scholarly journals Warping Effects in Strongly Perturbed Metrics

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 665-678
Author(s):  
Marco Frasca ◽  
Riccardo Maria Liberati ◽  
Massimiliano Rossi
Keyword(s):  
General Relativity ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Symmetric Case ◽  
Einstein Equations ◽  
Time Dependent ◽  
Warp Factor ◽  
Leading Order ◽  
Numerical Examples

A technique devised some years ago permits us to develop a theory regarding a regime of strong perturbations. This translates into a gradient expansion that, at the leading order, can recover the Belinsky-Kalathnikov-Lifshitz solution for general relativity. We solve exactly the leading order Einstein equations in a spherical symmetric case, assuming a Schwarzschild metric under the effect of a time-dependent perturbation, and we show that the 4-velocity in such a case is multiplied by an exponential warp factor when the perturbation is properly applied. This factor is always greater than one. We will give a closed form solution of this factor for a simple case. Some numerical examples are also given.

A Gas-Surface Interaction Model for Spatial and Time-Dependent Friction Coefficient in Reciprocating Contacts: Applications to Near-Frictionless Carbon

2005 ◽  
Vol 127 (1) ◽  
pp. 82-88 ◽  
Author(s):  
P. L. Dickrell ◽  
W. G. Sawyer ◽  
J. A. Heimberg ◽  
I. L. Singer ◽  
K. J. Wahl ◽  
...  
Keyword(s):  
Friction Coefficient ◽  
Closed Form ◽  
Closed Form Solution ◽  
Interaction Model ◽  
Nitrogen Atmosphere ◽  
Form Solution ◽  
Time Dependent ◽  
Series Expression ◽  
Dependent Model ◽  
Pin On Disk

A closed-form time- and position-dependent model for coverage, based on the adsorption of environmental contaminants and their removal through the pin contact, is developed for reciprocating contacts. The model employs an adsorption fraction and removal ratio to formulate a series expression for the entering coverage at any cycle and location on the wear track. A closed-form solution to the series expression is presented and compared to other coverage models developed for steady-state coverage for pin-on-disk contacts, reciprocating contacts, or the time-dependent center-point model for reciprocating contacts. The friction coefficient is based on the average coverage under the pin contact. The model is compared to position- and time-dependent data collected on near-frictionless carbon self-mated contacts on a reciprocating tribometer in a nitrogen atmosphere. There are many similarities between the model curves and the data, both in magnitude and trends. No new curve fitting was performed in this paper, with all needed parameters coming from previous models of average friction coefficient behavior.

scholarly journals Stochastic time-dependent enzyme kinetics: closed-form solution and transient bimodality

2020 ◽  
Author(s):  
James Holehouse ◽  
Augustinas Sukys ◽  
Ramon Grima
Keyword(s):  
Master Equation ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Delta Function ◽  
Product Formation ◽  
Form Solution ◽  
Chemical Master Equation ◽  
Time Dependent ◽  
Enzyme Molecules

AbstractWe derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations while our approach includes them. We confirm by means of stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

Calculation of the Equilibrium Configuration of Shopping Facility Sizes

1980 ◽  
Vol 12 (9) ◽  
pp. 983-1000 ◽  
Author(s):  
P A Phiri
Keyword(s):  
Closed Form ◽  
Nonlinear Equations ◽  
Gradient Method ◽  
Equilibrium Configuration ◽  
Closed Form Solution ◽  
Theoretical Framework ◽  
Form Solution ◽  
Method Of Solution ◽  
Special Case ◽  
General Method

This paper presents a range of methods for computing the equilibrium configuration of shopping facility sizes. First, the presently available range of quasi-balancing factor methods is considered and built into a theoretical framework in which further algorithms may be defined. Then the use of the gradient method is considered, which is a general method of solution of nonlinear equations. Last, it is shown that a special case exists in which a closed form solution may be obtained.

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