Mathematical Model for Evaluating Roundness Errors by Maximum Inscribed Circle Method

2011 ◽  
Vol 314-316 ◽  
pp. 393-396 ◽  
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Objective Function ◽  
Optimization Model ◽  
Circle Method ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Two Dimensional ◽  
Minimal Point ◽  
Deviation Measurement ◽  
Theoretical Results

An unconstrained optimization model is established for assessing roundness errors by the maximum inscribed circle method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the two-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function in order to get the wanted roundness errors by the maximum inscribed circle assessment. One example is given to verify the theoretical results presented.

Mathematical Model for Evaluating Roundness Errors by Minimum Circumscribed Circle Method

2011 ◽  
Vol 328-330 ◽  
pp. 380-383 ◽  
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Objective Function ◽  
Optimization Model ◽  
Circle Method ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Two Dimensional ◽  
Minimal Point ◽  
Deviation Measurement ◽  
Theoretical Results

An unconstrained optimization model is established for assessing roundness errors by the minimum circumscribed circle method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the two-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function in order to get the wanted roundness errors by the minimun circumscribed circle assessment. One example is given to verify the theoretical results presented.

Some Properties of Mathematical Model for Cylindricity Errors

2010 ◽  
Vol 37-38 ◽  
pp. 1214-1218 ◽  
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Euclidean Space ◽  
Convex Function ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Deviation Measurement ◽  
Theoretical Results

An unconstrained optimization model, applicable to radial deviation measurement, is established for assessing cylindricity errors by the minimum circumscribed cylinder evaluation. The properties of the objective function in the optimization model are thoroughly investigated. On the basis of the modern theory of convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the four-dimensional Euclidean space R4. Therefore, the minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Thus, any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function to get the wanted values of cylindricity errors by the minimum circumscribed cylinder assessment. An example is given to verify the theoretical results presented.

Mathematical Model for Evaluating Cylindricity Errors by Maximum Inscribed Cylinder

2011 ◽  
Vol 474-476 ◽  
pp. 1418-1422
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Euclidean Space ◽  
Convex Function ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Deviation Measurement ◽  
Theoretical Results

An unconstrained optimization model applicable to radial deviation measurement is established for assessing cylindricity errors by the maximum inscribed cylinder evaluation. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory of convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on a subset of the four-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function to get the wanted values of cylindricity errors by the maximum inscribed cylinder assessment. An example is given to verify the theoretical results presented.

Mathematical Model for Evaluating Cylindricity Errors by Minimum Zone Method

2011 ◽  
Vol 284-286 ◽  
pp. 434-438 ◽  
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Zone Method ◽  
Deviation Measurement ◽  
Minimum Zone ◽  
Theoretical Results

An unconstrained optimization model is established for assessing cylindricity errors by the minimum zone method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on a subset of the four-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Thus, any existing optimization algorithm, so long as it is convergent, can be applied to solve the objective function in order to get the wanted cylindricity errors by the minimun zone assessment. An example is given to verify the theoretical results presented.

Research on the Properties of Coaxality Error Objective Function

2009 ◽  
Vol 16-19 ◽  
pp. 1164-1168 ◽  
Author(s):  
Ping Liu ◽  
San Yang Liu
Keyword(s):  
Euclidean Space ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Zone Method ◽  
Deviation Measurement ◽  
Minimum Zone ◽  
Theoretical Results

The unconstrained optimization model applying to radial deviation measurement is established for assessing coaxality errors by the positioned minimum zone method. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory of convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the four-dimensional Euclidean space R4. Therefore, the global minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Thus, any existing optimization algorithm, as long as it is convergent, can be used to solve the objective function to get the wanted values of coaxality errors by the positioned minimum zone assessment. An example is given to verify the theoretical results presented.

scholarly journals A Mathematical Model of the Transition from the Normal Hematopoiesis to the Chronic and Accelerated Acute Stages in Myeloid Leukemia

2020 ◽  
Author(s):  
Lorand Gabriel Parajdi ◽  
Radu Precup ◽  
Eduard Alexandru Bonci ◽  
Ciprian Tomuleasa
Keyword(s):  
Mathematical Model ◽  
Stem Cells ◽  
Bone Marrow ◽  
Stability Analysis ◽  
Myeloid Leukemia ◽  
Transition Process ◽  
Two Dimensional ◽  
The Stability ◽  
Theoretical Results

A mathematical model given by a two - dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

scholarly journals A Mathematical Model of the Transition from Normal Hematopoiesis to the Chronic and Accelerated-Acute Stages in Myeloid Leukemia

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 376
Author(s):  
Lorand Gabriel Parajdi ◽  
Radu Precup ◽  
Eduard Alexandru Bonci ◽  
Ciprian Tomuleasa
Keyword(s):  
Mathematical Model ◽  
Stem Cells ◽  
Bone Marrow ◽  
Stability Analysis ◽  
Myeloid Leukemia ◽  
Transition Process ◽  
Two Dimensional ◽  
The Stability ◽  
Theoretical Results

A mathematical model given by a two-dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated-acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated-acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

A Quasi-Two-Dimensional Method for the Rotordynamic Analysis of Centered Labyrinth Liquid Seals

1999 ◽  
Vol 121 (1) ◽  
pp. 144-152 ◽  
Author(s):  
M. Arghir ◽  
J. Freˆne
Keyword(s):  
Mathematical Model ◽  
Coordinate Transformation ◽  
Present Method ◽  
Dynamic Regime ◽  
Test Case ◽  
Two Dimensional ◽  
First Order ◽  
Simplifying Assumptions ◽  
Theoretical Results ◽  
Dynamic Perturbation

The work presents a method for analyzing the dynamic regime of labyrinth liquid seals. By using the traditional simplifying assumptions for the centered seal (sinusoidal, harmonically varying, first order dynamic perturbation), the approach can be addressed as “quasi” two-dimensional. A numerical coordinate transformation capable to treat displacement perturbations is introduced. The first order mathematical model is then deduced following the same steps as in a previously published work (Arghir and Freˆne, 1997b). From this standpoint, the present method can be regarded as an extension of the above mentioned approach which was able to deal only with stator-grooved seals. The method is validated by comparisons with Nordmann and Dietzen’s (1988) theoretical results for a seal with grooves on both stator and rotor and with the experimental results of Staubli’s (1993) test case concerning a general seal.

Two Dimensional Unsteady State Mathematical Model for dispersion of Chlorine in Water in a Channel

2011 ◽  
Vol 3 (8) ◽  
pp. 503-505
Author(s):  
Jaipal Jaipal ◽  
◽  
Rakesh Chandra Bhadula ◽  
V. N Kala V. N Kala
Keyword(s):  
Mathematical Model ◽  
Two Dimensional ◽  
Unsteady State

COMPARISON OF TWO OPTIMIZATION MODELS FOR CONSTRUCTING PATTERNS IN THE METHOD OF LOGICAL ANALYSIS OF DATA

2020 ◽  
pp. 10-13
Author(s):  
Р.И. Кузьмич ◽  
А.А. Ступина ◽  
С.Н. Ежеманская ◽  
А.П. Шугалей
Keyword(s):  
Objective Function ◽  
Optimization Model ◽  
Logical Analysis ◽  
Optimization Models ◽  
Logical Analysis Of Data

Предлагаются две оптимизационные модели для построения информативных закономерностей. Приводится эмпирическое подтверждение целесообразности использования критерия бустинга в качестве целевой функции оптимизационной модели для получения информативных закономерностей. Информативность, закономерность, критерий бустинга, оптимизационная модель Comparison of two optimization models for constructing patterns in the method of logical analysis of data Two optimization models for constructing informative patterns are proposed. An empirical confirmation of the expediency of using the boosting criterion as an objective function of the optimization model for obtaining informative patterns is given.

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