Controlled differential equations with a parameter and with multivalued impulses

2020 ◽  
pp. 441-447
Author(s):  
Olga V. Filippova
Keyword(s):  
A Priori ◽  
Admissible Pair ◽  
Impulsive System ◽  
Compact Set ◽  
Phase Trajectories ◽  
Phase Constraints ◽  
Admissible Pairs ◽  
The Cauchy Problem ◽  
Parameter Values ◽  
The Given

We study the Cauchy problem for a controlled differential system with a parameter which is an element of some metric space Ξ, containing phase constraints on the control. It is assumed that at the given time instants t_k,k = 1,2,…,p, the solution x is continuous from the left and suffers a discontinuity, the value of which is x(t_k+0)-x(t_k ), belongs to some non-empty compact set of the space R^n. The notions of an admissible pair of this controlled impulsive system are introduced. The questions of continuity of admissible pairs are considered. Definitions of a priori boundedness and a priori collective boundedness on a given set S×K (where S⊂R^n is a set of initial values, K⊂Ξ is a set of parameter values) of the set of phase trajectories are considered. It is proved that if at some point (x_0,ξ)∈R^n×Ξ the set of phase trajectories is a priori bounded, then it will be a priori bounded in some neighborhood of this point.

A network thermodynamic model of salt and water flow across the kidney proximal tubule

1978 ◽  
Vol 235 (6) ◽  
pp. F638-F648 ◽  
Author(s):  
S. R. Thomas ◽  
D. C. Mikulecky
Keyword(s):  
Proximal Tubule ◽  
Water Flow ◽  
Thermodynamic Model ◽  
Circuit Simulation ◽  
Basolateral Membrane ◽  
Kidney Proximal Tubule ◽  
Network Analyses ◽  
Pump Rate ◽  
Parameter Values ◽  
The Given

This network thermodynamic model of kidney proximal tubule epithelium treats coupled salt and water flow across each component membrane of the epithelium. We investigate the effects of various relative internal parameter values on the concentration of transepithelial flow, the concentrations in the cell and interspace, and the distribution of flows between cellular and paracellular routes. Best fit is obtaine if the apical and basolateral membrane reflection coefficients (or) are equal. The measured transepithelial filtration coefficient, Lp, is a function not only of the component Lps but also of the internal concentrations, or's, and permeabilities. For the given system topology (i.e., connectedness), parameters of component membranes must be within a narrow range to be consistent with experimental results. The dependence of the concentration of transported fluid on the balance between the solute pump rate and the transepithelial volume flow driving force is shown. This has implications for the effects of peritubular or lumen oncotic pressure on salt and water flow. With Appendix B of this paper and a user's guide for a circuit-simulation package (e.g., SPICE or PCAP) the reader can perform similar network analyses of transport models himself.

scholarly journals State estimation for bilinear impulsive control systems under uncertainties

2018 ◽  
pp. 35-41 ◽  
Author(s):  
Oxana G. Matviychuk
Keyword(s):  
Differential Equations ◽  
State Estimation ◽  
Control Systems ◽  
Uncertain Systems ◽  
Impulsive Control ◽  
Compact Set ◽  
The Matrix ◽  
Initial States ◽  
Impulsive Control Systems ◽  
The Given

The state estimation problem for uncertain impulsive control systems with a special structure is considered. The initial states are taken to be unknown but bounded with given bounds. We assume here that the coefficients of the matrix included in the differential equations are not exactly known, but belong to the given compact set in the corresponding space. We present here algorithms that allow to find the external ellipsoidal estimates of reachable sets for such bilinear impulsive uncertain systems.

Explaining monotonic ranking functions

2020 ◽  
Vol 14 (4) ◽  
pp. 640-652
Author(s):  
Abraham Gale ◽  
Amélie Marian
Keyword(s):  
Decision Making ◽  
Selection Process ◽  
A Priori ◽  
Data Distribution ◽  
Synthetic Data ◽  
Decision Processes ◽  
Decision Makers ◽  
Ranking Functions ◽  
High School Admissions ◽  
Parameter Values

Ranking functions are commonly used to assist in decision-making in a wide variety of applications. As the general public realizes the significant societal impacts of the widespread use of algorithms in decision-making, there has been a push towards explainability and transparency in decision processes and results, as well as demands to justify the fairness of the processes. In this paper, we focus on providing metrics towards explainability and transparency of ranking functions, with a focus towards making the ranking process understandable, a priori , so that decision-makers can make informed choices when designing their ranking selection process. We propose transparent participation metrics to clarify the ranking process, by assessing the contribution of each parameter used in the ranking function in the creation of the final ranked outcome, using information about the ranking functions themselves, as well as observations of the underlying distributions of the parameter values involved in the ranking. To evaluate the outcome of the ranking process, we propose diversity and disparity metrics to measure how similar the selected objects are to each other, and to the underlying data distribution. We evaluate the behavior of our metrics on synthetic data, as well as on data and ranking functions on two real-world scenarios: high school admissions and decathlon scoring.

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

scholarly journals X. On Hamilton's numbers

1887 ◽  
Vol 178 ◽  
pp. 285-312
Keyword(s):  
General Equation ◽  
Minimum Degree ◽  
A Priori ◽  
Senior Author ◽  
System Of Equations ◽  
Absolute Minimum ◽  
Doctor Of Philosophy ◽  
The University ◽  
The Given ◽  
Trinomial Equation

In the year 1786 Erland Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirnhausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University. The process for effecting this reduction seems to have been overlooked or forgotten, and was subsequently re-discovered many years later by Mr. Jerrard. In a report contained in the ‘Proceedings of the British Association’ for 1836, Sir William Hamilton showed that Mr. Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result, known à priori to be impossible, by an extremely valuable discussion of a question raised by Mr. Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees. In the year 1886 the senior author of this memoir showed in a paper in Kronecker'e (better known as Crelle’s ) ‘Journal that the trinomial equation of the 5th degree, upon which by Bring’s method the general equation of that degree can be made to depend, has necessarily imagmaiy coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out method of obtaining the absolute minimum degree M of an equation from which an given number of specified terms can be taken away subject to the condition of no having to solve any equation of a degree higher than M. The numbers furnished be Hamilton’s method, it is to be observed, are not minima unless a more stringer condition than this is substituted, viz., that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible i. e ., of the lowest possible weight and not merely of the lowest order; in the memo: in ‘Crelle,’ above referred to, he has explained in what sense the words weight an order are here employed. He has given the name of Hamilton’s Numbers to these relative minima (minima, i. e ., in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.

scholarly journals Modelling temperature variations in polar snow using DAISY

1997 ◽  
Vol 43 (143) ◽  
pp. 180-191 ◽  
Author(s):  
Ε. M. Morris ◽  
H. -P. Bader ◽  
P. Weilenmann
Keyword(s):  
Field Experiments ◽  
A Priori ◽  
Model Parameters ◽  
Data Set ◽  
International Geophysical Year ◽  
Snow Model ◽  
Using Data ◽  
Polar Snow ◽  
Parameter Values ◽  
The Antarctic

AbstractA physics-based snow model has been calibrated using data collected at Halley Bay, Antarctica, during the International Geophysical Year. Variations in snow temperature and density are well-simulated using values for the model parameters within the range reported from other polar field experiments. The effect of uncertainty in the parameter values on the accuracy of the predictions is no greater than the effect of instrumental error in the input data. Thus, this model can be used with parameters determined a priori rather than by optimization. The model has been validated using an independent data set from Halley Bay and then used to estimate 10 m temperatures on the Antarctic Peninsula plateau over the last half-century.

scholarly journals A New Approach to Newton-Type Polynomial Interpolation with Parameters

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Le Zou ◽  
Liangtu Song ◽  
Xiaofeng Wang ◽  
Thomas Weise ◽  
Yanping Chen ◽  
...  
Keyword(s):  
Polynomial Interpolation ◽  
Information Matrix ◽  
Divided Differences ◽  
Mathematical Representation ◽  
Interpolation Algorithm ◽  
New Approach ◽  
Multiple Parameters ◽  
Parameter Values ◽  
The Given ◽  
Interpolation Functions

Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.

scholarly journals Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 48
Author(s):  
Hongwu Zhang ◽  
Xiaoju Zhang
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Type Equation ◽  
Conditional Stability ◽  
Elliptic Type ◽  
Type Method ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
The Stability ◽  
Regularized Solution

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

scholarly journals An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hao Cheng ◽  
Ping Zhu ◽  
Jie Gao
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Regularization Method ◽  
A Priori ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Regularization Parameters ◽  
The Cauchy Problem

A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. Thea priorianda posteriorirules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method.

An Approach for Generating an Average Bone Template With Semantic Parameters

2017 ◽  
Vol 11 (3) ◽  
Author(s):  
Kunjin He ◽  
Rongli Zhang ◽  
Zhengming Chen ◽  
Junfeng Jiang ◽  
Zhang Yuming
Keyword(s):  
Average Equation ◽  
Bone Model ◽  
Characteristic Points ◽  
Novel Method ◽  
Parameter Values ◽  
The Given ◽  
Average Bone ◽  
Equal Contribution ◽  
Semantic Parameters

To facilitate the design of the serialized implants and to satisfy the requirements of the population, a novel method is put forward for constructing an average bone model (ABM) with semantic parameters as a template. First, the ABM is created from the existing bone models, among which each bone has an equal contribution to the ABM. Second, combined with medical semantics, some characteristic points and semantic parameters are defined on the ABM, and then, parameter values for each bone can be automatically obtained through its registration and deformation to the ABM. Finally, an average bone template (ABT) is constructed by configuring the semantic parameters and by building the constraints between parameters. Taking 100 femur models as samples, we construct the ABT, and the template can be easily extended to generate a new average template through the given average equation.

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