MATHEMATICALNESS OR MATHEMATICABILITY OF NATURE?

In this series of articles, the meaning of ‘φ-science’ or φ-ness has been discussed from the angle of its material aspect, especially the role of measurement as a mediator between mathematical theories and the material world the theories are to describe. The previous part discussed quantities, or properties of the material world treated as numerisable and used as variables in laws of the sciences. This article extends this topic by focussing on units as an indispensable component of measurement that is considered appropriate for exact sciences—extensive or additive measurement. The definitions and the inherently technical realisations of units will be discussed to determine their φ-ness and their role in what makes a scientific discipline φ. The focus is thereby on the base, or fundamental, units that other units of the system of units are derived from. Their relation to what early Vihalemm considered the essence and measure in phaenomena of the exact sciences will be analysed. Technicality-experimentality and abstraction of φ-ness and measurement will be conclusively posited.

In very general terms, the physical scientist is concerned to identify and define certain regularities in the material world, and to embody them in laws or theories which both furnish an explanation of them and enable him to predict and to some extent to control particular kinds of future event. It is noticeable at once that the elements of prediction and control are largely absent from Greek speculation, and this is due partly to certain of its inherent characteristics (to which I return in a moment), and partly to its lack of three further particular features which are an essential part of the natural sciences today. These are briefly that they are experimental, quantitative, and laboratory-based. This generalization is, of course, at once too wide and too narrow. Not all scientific theories rely on experiment in the narrow sense or on laboratory operations; the observations of the astronomer and the activities of the field-biologist can, up to a point, dispense with both. But it remains true that the experimental test and the laboratory and its instruments are essential elements in modern science, and neither the astronomer nor the biologist can in the last resort dispense with them. Similarly, quantitative measurement plays an indispensable role in the physical sciences and is the basis on which they have been built up, even though the theoretical physicist may spend little time measuring or experimenting and be mainly concerned with the mathematical analysis which is so vital an element in modern physical theory.

The paper is devoted to the presentation of Chwistek’s philosophical ideas concerning logic and mathematics. The main feature of his philosophy was nominalism, which found full expression in his philosophy of mathematics. He claimed that the object of the deductive sciences, hence in particular of mathematics, is the expression being constructed in them according to accepted rules of construction. He treated geometry, arithmetic, mathematical analysis and other mathematical theories as experimental disciplines, and obtained in this way a nominalistic interpretation of them. The fate of Chwistek’s philosophical conceptions was similar to the fate of his logical conceptions. The system of rational meta-mathematics was not developed by him in detail. He worked on his own ideas without any collaboration with other logicians, mathematicians or philosophers. His investigations were not in the mainstream of the development of logic and philosophy of mathematics.

Typically in electrical engineering a network modelling approach for the simulation of devices and their surrounding circuitry is taken, where each device is considered by a voltage-to-current relation. For some applications, however, this simplification does not yield the required accuracy. In these cases, refined modelling can be performed, where a spatially distributed partial differential equation modelling the required physical quantity is coupled to the classic network equations. The resulting coupled system of equations often exhibits a multiscale, multirate and even multiphysical behaviour that is tackled with involved algorithms so as to efficiently simulate it. Its structural analysis is therefore important, to numerically treat the system appropriately and to ensure that the algorithms converge properly. This thesis deals with the mathematical analysis of these type of systems as well as their simulation. The systems of equations obtained from circuits with semidiscrete refined models are typically differential algebraic equations. Their numerical and analytical difficulties is studied in the context of their differential algebraic index. For that, three generalised circuit element definitions are given, that allow the classification of the refined models. Hereby, the index of the entire coupled system can be specified by means of topological properties of the circuit. Several approximations to Maxwell’s equations are classified with the generalised element definitions to obtain the index properties of the field-circuit coupled systems. For the simulation two algorithms are studied. First the co-simulation waveform relaxation method is analysed for field-circuit coupled systems arising from magnetoquasistatic fields with eddy current effects on superconducting coils. The convergence of the algorithm is sped up by means of optimised Schwarz methodologies. Here, the information exchange between both subsystems is improved by a linear combination of the coupling conditions. To further speed up simulation time, the parallel-in-time method Parareal is analysed. The algorithm is investigated in the context of differential algebraic equations by studying its applicability to nonlinear higher index systems arising e.g. from circuit simulation. Finally, two approaches are proposed for the combination of Parareal and waveform relaxation. One of them is specifically designed for field-circuit coupled systems and yields a micro-macro-like Parareal algorithm. However, the idea behind it can be applied to other type of coupled systems. Numerical tests of field-circuit coupled systems are made to underline the results obtained from the mathematical theory as well as test the efficiency of the proposed algorithms.

The orthogonality among sub-carriers and the minimum frequency span which satisfies the Nyquist criterion in frequency domain are foundational principles of conventional OFDM (Orthogonal Frequency Division Multiplexing) system. Although reducing the minimum frequency span in OFDM can improve the spectral efficiency, it is not practical previously because of large detection complexity which is derived from the non-orthogonality and considerable inter carrier interference among sub-carriers. Gram-Schmidt orthogonalization procedure proposed in the paper can transform the non-orthogonal sub-carriers into orthogonal waveforms and the bit error rate is not deteriorated in additive white Gaussian noise channel. Meanwhile, the total signal bandwidth becomes smaller than that of conventional OFDM, i.e., the spectral efficiency can be increased. In the paper, Gram-Schmidt orthogonalization procedure is explained by dirty paper coding in information theory and functional analysis in mathematics. Simulation results also verify the validity and correctness of this signal processing method - Gram-Schmidt orthogonalization procedure.

Passive dynamic walking refers to the dynamical behavior of mechanical devices that are able to naturally walk down a shallow slope in a stable manner, without using actuation or sensing of any kind. Such devices can attain motions that are remarkably human-like by purely exploiting their natural dynamics. This suggests that passive dynamic walking machines can be used to model and study human locomotion; however, there are two major limitations: they can be difficult to design, and they cannot walk on level ground or uphill without some kind of actuation. This thesis presents a mechanism design optimization framework that allows the designer to find the best design parameters based on the chosen performance metric(s). The optimization is formulated as a convex problem, where its solutions are globally optimal and can be obtained efficiently. To enable locomotion on level ground and uphill, this thesis studies a robot based on a passive walker: the rimless wheel with an actuated torso. We design and validate two control policies for the robot through the use of scalable methodology based on tools from mathematical analysis, optimization theory, linear algebra, differential equations, and control theory.

On the Lupaş <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-transform

The article introduces the methodology used for analyzing networked linguistic data, regarded as a basis of global online discursive fields. The authors scrutinized the discursive fields, which emerged during 2020 US Presidential Election.The research methodology, which combines natural science methods (mathematical analysis, graph theory, network analysis and relational analysis) and modern methods of linguistic research (complex linguo-discursive analysis, and methods of network linguistics), makes it possible to analyze discursive fields as social graphs, identify narratives and discourses that form the basis of the modern global communication space, and their potential for manipulating. The empirical base of the study is comprised of bulks of networked data that include the messages published by ordinary users and D. Trump's team on Twitter platform in March – October 2020. The application of the authors' technique has resulted in discursive fields visualization, abnormal discursive activity areas identification, the interaction of discourses within the discursive field description, the mode of messages and their recurrence level determination. It is shown that the analysis of Internet communication using the developed methodology contribute to understanding the essence of socio-political and socio-economic processes and deepening the predictive analytics of their development, and can also be used for discursive management in order to strengthen constructive and neutralize destructive social practices in online space.

We investigate interrelationships among different notions from mathematical analysis, effective topology, and classical computability theory. Our main object of study is the class of computable functions defined over an interval with the boundary being a left-c.e. real number. We investigate necessary and sufficient conditions under which such functions can be computably extended. It turns out that this depends on the behavior of the function near the boundary as well as on the class of left-c.e. real numbers to which the boundary belongs, that is, how it can be constructed. Of particular interest a class of functions is investigated: sawtooth functions constructed from computable enumerations of c.e. sets.

Additive manufacturing as the technology of the future brings new challenges. One of them is the economic efficiency of production. This paper focuses on the mathematical analysis and structural design of extrusion screw used for additive manufacturing. The primary objective is to analyze the screw tool geometry and determine a procedure for its design, specifically the theory involved with the pressing tool and force relations which are necessary for the verification of the proposed tool geometry and its strength analysis. Procedures for determining frictional performance of the screw press are used in designing the drive of the screw extruder of 3D printer. Familiarity with the above mentioned procedures forms the basis for research into new tool - screw that will improve the service life and competitiveness of the technology.

The present section is devoted to a number of results and statements which are closely connected with the classical Sierpinski partition of the Euclidean plane R 2. It turns out that these results and statements are rather useful in various situations. In particular, they can be successfully applied to many questions and problems from mathematical analysis, measure theory and general topology.

This paper proposes an efficient technique of shape feature extraction based on the application of mathematical morphology theory. A new shape complexity index for preclassification of machine printed Chinese Character Recognition (CCR) is also proposed. For characters represented in different fonts/sizes or in a low resolution environment, a more stable local feature such as shape structure is preferred for character recognition. Morphological valley extraction filters are applied to extract the protrusive strokes from four sides of an input Chinese character. The number of extracted local strokes reflects the shape complexity of each side. These shape features of characters are encoded as corresponding shape complexity indices. Based on the shape complexity index, data base is able to be classified into 16 groups prior to recognition procedures. The performance of associating with shape feature analysis reclaims several characters from misrecognized character sets and results in an average of 3.3% improvement of recognition rate from an existing recognition system. In addition to enhance the recognition performance, the extracted stroke information can be further analyzed and classified its own stroke type. Therefore, the combination of extracted strokes from each side provides a means for data base clustering based on radical or subword components. It is one of the best solutions for recognizing high complexity characters such as Chinese characters which are divided into more than 200 different categories and consist more than 13,000 characters.

This paper describes the development of an automatic segmentation method and a novel navigation system that detect polyps using advanced image processing techniques. The segmentation method described here outlines the colon wall of the CT data set of abdomen by combining the active contouring model - level set method and the minima detection using mathematical morphology theory. The polyp detection is achieved by analysing the surface curvature and the texture information surrounded on the colon wall. As such, the polyp candidates are examined not only by their shape, size but also by their texture appearance which improves the current existing methods. The major contribution of our method is the detailed information analysis using the active contour model assisted by mathematical morphology operations and texture analysis.

Objectives . Identification of the propagation patterns of surface waves in inhomogeneous and nonlinear crystal structures using mathematical models is an important fundamental problem in condensed matter physics, specifically waveguide optics. Models of waveguide structures used to establish an exact analytical solution are of particular significance. The aim of this work is to carry out a theoretical study of transversely polarized surface electric waves propagating along a photonic crystal with a certain refractive index profile. Methods . The methods of mathematical physics, analysis, differential equations, and theory of special functions, as well as physical models of waveguide optics, were used in this study. Results . A generalized hyperbolic permittivity profile was proposed to describe the spatially inhomogeneous distribution of the optical properties of a photonic crystal. This profile has a wide range of possibilities for varying its shape, allowing it to be used for a wide range of problems not limited to waveguide optics. An exact analytical solution of the wave equation with the selected permittivity profile was found in terms of the Whittaker function. Frequent cases of the generalized profile for which exact analytical solutions were indicated were also considered. These are expressed through the Whittaker and Macdonald functions. The study also describes surface transverse electric waves, where the field is localized near the surface of the photonic crystal and decreases with distance from it. The solution obtained also describes waveguide modes in which the field decreases with distance from the surface of the photonic crystal with oscillations. New features of surface wave localization were established. These were caused by a change in the parameters of the generalized hyperbolic profile modeling the dependence of the permittivity. It was also established that the maximum intensity of the surface wave is located in the photonic crystal. Conclusions . The results of the description of the characteristics of surface waves obtained expand the theoretical concepts of waveguide optics. They can be useful in predicting the optical properties of various photonic crystal structures, as well as in designing various waveguide structures with the required dispersion-optical characteristics.

In this paper, we presented a novel family of heavy-tailed continuous probability distributions tailored to model data exhibiting trimodal features with various kurtosis levels in asymmetric, under-dispersed datasets across various categories of risk. The proposed model enhanced the alpha-skew logistic distribution, providing significant adaptability to represent asymmetry and various tail characteristics. Key statistical properties were established, and additional characterizations were offered. Mathematical and numerical analyses indicated that the model efficiently handled asymmetric data, especially under conditions of under-dispersion. The associated hazard rate function displayed various forms, demonstrating the model's versatility in reliability and survival analysis. Parameter estimation was conducted using the maximum likelihood method, and simulation studies were employed to assess the performance of estimators in small samples. The practical applicability of the model was further demonstrated through analyses of two real datasets. Finally, comparative analyses with alternative models employing a likelihood ratio test substantiated its better performance.