Mixed problems for the string vibration equation with nonlocal conditions of the general form at the right endpoint and with an inhomogeneous condition at the left endpoint

2017 ◽  
Vol 53 (4) ◽  
pp. 509-515
Author(s):  
I. S. Mokrousov
Keyword(s):  
Nonlocal Conditions ◽  
Left Endpoint ◽  
Vibration Equation ◽  
String Vibration ◽  
Mixed Problems ◽  
The Right ◽  
Inhomogeneous Condition

scholarly journals Boundary value problems with nonlocal conditions for hyperbolic systems of equations with two independent variables

2020 ◽  
Vol 53 (2) ◽  
pp. 159-180
Author(s):  
V. M. Kyrylych ◽  
O. Z. Slyusarchuk
Keyword(s):  
Boundary Value Problems ◽  
A Priori Estimates ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Problems ◽  
Priori Estimates ◽  
Smooth Coefficients

Nonlocal boundary value problems for arbitrary order hyperbolic systems with one spatial variable are considered. A priori estimates for general nonlocal mixed problems for systems with smooth and piecewise smooth coefficients are obtained. The correct solvability of such problems is proved.Examples of additional conditions necessity are provided.

scholarly journals On the Use of Entropy as a Measure of Dependence of Two Events

2021 ◽  
Author(s):  
Valentin Iliev
Keyword(s):  
Probability Space ◽  
Entropy Function ◽  
The Other ◽  
Screening Tests ◽  
Global Maximum ◽  
Left Endpoint ◽  
Positive Dependence ◽  
The Right ◽  
Measure Of Dependence ◽  
The Impact

We define degree of dependence of two events A and B in a probability space by using Boltzmann-Shannon entropy function of an appropriate probability distribution produced by these events and depending on one parameter (the probability of intersection of A and B) varying within a closed interval I. The entropy function attains its global maximum when the events A and B are independent. The important particular case of discrete uniform probability space motivates this definition in the following way. The entropy function has a minimum at the left endpoint of I exactly when one of the events and the complement of the other are connected with the relation of inclusion (maximal negative dependence). It has a minimum at the right endpoint of I exactly when one of these events is included in the other (maximal positive dependence). Moreover, the deviation of the entropy from its maximum is equal to average information that carries one of the binary trials defined by A and B with respect to the other. As a consequence, the degree of dependence of A and B can be expressed in terms of information theory and is invariant with respect to the choice of unit of information. Using this formalism, we describe completely the screening tests and their reliability, measure efficacy of a vaccination, the impact of some events from the financial markets to other events, etc.

GLOBAL CONTROLLABILITY OF A NONLINEAR KORTEWEG–DE VRIES EQUATION

2009 ◽  
Vol 11 (03) ◽  
pp. 495-521 ◽  
Author(s):  
MARIANNE CHAPOULY
Keyword(s):  
Boundary Control ◽  
Vries Equation ◽  
Boundary Value ◽  
Exact Controllability ◽  
Left Endpoint ◽  
Bounded Interval ◽  
First Derivative ◽  
De Vries ◽  
Positive Time ◽  
The Right

We are interested in both the global exact controllability to the trajectories and in the global exact controllability of a nonlinear Korteweg–de Vries equation in a bounded interval. The local exact controllability to the trajectories by means of one boundary control, namely the boundary value at the left endpoint, has already been proved independently by Rosier, and Glass and Guerrero. We first introduce here two more controls: the boundary value at the right endpoint and the right member of the equation, assumed to be x-independent. Then, we prove that, thanks to these three controls, one has the global exact controllability to the trajectories, for any positive time T. Finally, we introduce a fourth control on the first derivative at the right endpoint, and we get the global exact controllability, for any positive time T.

scholarly journals Bridge Dynamic Cable-Tension Estimation with Interferometric Radar and APES-Based Time-Frequency Analysis

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 501
Author(s):  
Jian Wang ◽  
Xiang Wang ◽  
Chongyi Fan ◽  
Yueli Li ◽  
Xiaotao Huang
Keyword(s):  
Frequency Analysis ◽  
Field Experiments ◽  
Displacement Vector ◽  
Frequency Resolution ◽  
Cable Tension ◽  
Time Frequency Analysis ◽  
Vibration Equation ◽  
Time Frequency ◽  
String Vibration ◽  
Analysis Technique

Dynamic cable-tension is an important bridge-health indicator. However, it is difficult to be measured precisely and efficiently. A remote bridge dynamic cable-tension measurement method is proposed. It uses an interferometric radar sensor, a time-frequency analysis technique, and a tension estimation approach based on a string-vibration-equation. One radar can measure the displacements of multiple cables aligned on one side of a bridge, at the same time. By solving the string vibration equation, each cable-tension is calculated from its fundamental frequency, which is obtained by time-frequency analyzing a short section of the cable’s whole displacement vector in an overlapped-piecewise manner. An adaptive amplitude and phase estimation (APES) algorithm is used to solve the frequency resolution deterioration problem due to the short duration. Simulations and field experiments with a K band interferometric radar validate that the proposed method is superior to traditional cable-tension measurements in terms of precision, robustness, and efficiency. The proposed method is of great application value in measuring and monitoring large cable-stayed bridges and cable-suspended bridges.

scholarly journals Matching conditions for values of characteristic oblique derivative at the end of a string, initial data and right-hand side of the wave equation

2020 ◽  
pp. 30-37
Author(s):  
Ekaterina V. Ustilko ◽  
Fiodar E. Lomovtsev
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Initial Conditions ◽  
Time Dependent ◽  
Matching Conditions ◽  
Vibration Equation ◽  
String Vibration ◽  
Boundary Mode ◽  
Critical Characteristic ◽  
Order Smoothness

Sufficient matching conditions the time-dependent characteristic first derivatives in the boundary mode with the initial conditions and the more general vibration equation of a semi-bounded string are derived in the sets of solutions of all higher order smoothness orders. They generalize the previously found sufficient matching conditions in the case of a similar mixed problem for the simplest string vibration equation. The characteristic of non-stationary first oblique derivatives in the boundary mode means that at each moment of time they are directed along the critical characteristic.

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