Physical and Mathematical Foundation

2015 ◽  
pp. 39-63
Author(s):  
Matthias Albert Augustin
Keyword(s):  
Mathematical Foundation

Is the Gaussian Distribution Normal?

1965 ◽  
Vol 18 (1) ◽  
pp. 65-71 ◽  
Author(s):  
E. W. Anderson
Keyword(s):  
Normal Distribution ◽  
Gaussian Distribution ◽  
Mathematical Foundation ◽  
Probability Paper ◽  
Straight Line

It has been said that the mathematician accepts the gaussian distribution because he believes it to be supported in practice, and the practical man accepts it because he believes it has a sound mathematical foundation. Presumably it is for such reasons that the gaussian distribution is known as the ‘normal’ distribution and ‘probability paper’ has scales specially distorted so that a gaussian distribution will appear as a straight line.

A numerical approach for structural system identification by observability techniques

2016 ◽  
Author(s):  
Jose Antonio Lozano Galant ◽  
Maria Nogal ◽  
Jun Lei ◽  
Dong Xu ◽  
José Turmo
Keyword(s):  
System Identification ◽  
Numerical Approach ◽  
Mathematical Foundation ◽  
Structural System ◽  
Numerical Errors ◽  
Definition Of ◽  
Structural System Identification ◽  
Symbolic Approach

Observability techniques enable the structural system identification of static structures from a symbolic approach. The main advantage of this method is its deep mathematical foundation that enables the definition of parametric equations for the estimates. Nevertheless, this symbolic approach is not enough for the application of this method on actual structures. To fill this gap, this article presents the introduction into the symbolic structural system identification by observability techniques of a new numerical approach. This application includes the development of an algorithm that reduces the unavoidable numerical errors produced by the lack of precision of computers. The comparison of the observability technique with other existing methods presented in the literature shows that the number of required measurements is significantly lower. Furthermore, contrary to other analysed methods, no information from the undamaged structure is required.

Finding Dead-Point Positions of Planar Pin-Connected Linkages Through Graph Theoretical Duality Principle

2005 ◽  
Vol 128 (3) ◽  
pp. 599-609 ◽  
Author(s):  
O. Shai ◽  
I. Polansky
Keyword(s):  
Graph Theory ◽  
Discrete Mathematics ◽  
Duality Principle ◽  
Mathematical Foundation ◽  
The Dead

The paper brings another view on detecting the dead-point positions of an arbitrary planar pin-connected linkage by employing the duality principle of graph theory. It is first shown how the dead-point positions are derived through the interplay between the linkage and its dual determinate truss—the relation developed in the previous works by means of graph theory. At the next stage, the process is shown to be performed solely upon the linkage by employing a new variable, the dual of potential, termed face force. Since the mathematical foundation of the presented method is discrete mathematics, the paper points to possible computerization of the method.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Quantum Process ◽  
Hidden Variable ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it could be the stochastic archetype of quantum probability.

scholarly journals Mathematical Model of Rotary Machined Helical Surfaces

2021 ◽  
Vol 346 ◽  
pp. 01038
Author(s):  
Mikhail Popov
Keyword(s):  
Mathematical Model ◽  
Temperature Decrease ◽  
Mathematical Foundation ◽  
Rotary Tool ◽  
High Durability ◽  
High Temperature Alloys ◽  
Cutting Zone ◽  
Rotary Cutting ◽  
Mathematical Calculation ◽  
Laminated Materials

The rotary cutting method of materials has a number of advantages over the existing traditional cutting methods, e.g. temperature decrease in the cutting zone, also noncumulative blade wear. Due to its high durability, the rotary tool allows processing hardened and difficult-to-machine materials, high-temperature alloys, as well as composite and laminated materials. However, this machining method is usually not applied for machining various shaped surfaces, which is mainly due to the lack of mathematical calculation of the resulting profiles, and the absence of a wide variety of methods for rotary tools installation. The article discusses the mathematical foundation of the resulting profile when processing helical surfaces when processing the flanks of rotary tools.

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