Sierpiński’s partition and its applications

RADON-NIKODYM THEORY FOR OPERATOR VALUED MEASURES AND A THEOREM OF FOURES & SEGAL

I. V. M. Klimkin, "On extension of a vector measure," Izv. Vyssh. Uchebn. Zaved., Mat., ~, 46-53 (1971). 2. V. M. Klimkin, "On certain properties of continuous outer measures," Uch. Zap. Leningr. Gos. Pedagog. Inst., 469, 187-204 (1970). 3. N. S. Gusel'nikov, "On the convergence of a sequence of continuous triangular measures," Nauch. Dokl., Mat., 26, 78-82, Hercen Lectures, Izd. Mosk. 0blast. Pedagog. Inst. im N. K. Krypskoi, Moscow (1973). 4. N. S. Gusel'nikov, "On the extension of quasi-Lipschitz set functions," Mat. Zametki, 17, No. i, 21-31 (1975). 5. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Wiley, New York (1958). 6. J. L. Kelley, General Topology, Van Nostrand, New York (2955). 7. L. Ya. Savel'ev, "On the extension of a mappin E of a Boolean ring into a topological Abelian group," Dokl. Akad. Nauk SSSR, 149, No. 6, 1268-1269 (1963). 8. L. Ya. Savel'ev, "Extension of measures by continuity," Sib. Mat. Zh., ~, No. 3, 639650 (1964). 9. L. Ya. Savel'ev, "Certain conditions for extendability of vector measures," Sib. Mat. Zh., 9, No. 4, 937-950 (1968). 10. M. Kh. Khafizov, "On the extension of a vector-valued measure," in: Current Problems of Mathematical Logic and Set Theory [in Russian], Izd. Mosk. Gos. Pedagog. Inst., Moscow (1975), pp. 291-301. ii. P. R. Halmos, Measure Theory, Van Nostrand, New York (1950).

5. Measure and Integration

Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds. Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae. Methods. Basic results of mathematical analyses, measure theory and differential geometry are used. Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area. Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

Outlier mining task is to discover some unusual objects, and however, most existing methods and their mining results lack pertinence. To address the pertinence of outlier results, we propose a novel outlier detection approach, namely, FOD, which aims at finding anomalies in full dimensions that lack pertinence and understandability. Our key idea is to use fuzzy constraint technology to prune irrelevant objects for outlier detection, during which the nearness measure theory in fuzzy mathematics is used for detecting similarities between objects and constraint information. FOD finds outlier by searching sparse subspace, where genetic algorithms can be extended and incorporated into FOD such that an optimum solution of an anomaly is discovered. While constructing a sparse subspace, we present the sparse threshold concept to describe the sparse levels of data objects in a subspace, where data objects are regarded as anomalies. Then, we demonstrate the effectiveness and scalability of our method on synthetic and UCI datasets. The experiment evaluations reveal that our fuzzy constraint-based outlier detection is superior to two existing full dimensional algorithms. Moreover, FOD algorithm also improves the accuracy of outlier detection.

In this paper, we examine a sequence of uncountable iterated function systems (U.I.F.S.), where each term in the sequence is constructed from an uncountable collection of contraction mappings along with a linear and continuous operator. Each U.I.F.S. within the sequence is associated with an attractor, which represents a set towards which the system evolves over time, a Markov-type operator that governs the probabilistic behavior of the system, and a fractal measure that describes the geometric and measure-theoretic properties of the attractor. Our study is centered on analyzing the convergence properties of these systems. Specifically, we investigate how the attractors and fractal measures of successive U.I.F.S. in the sequence approach their respective limits. By understanding the convergence behavior, we aim to provide insights into the stability and long-term behavior of such complex systems. This study contributes to the broader field of dynamical systems and fractal geometry by offering new perspectives on how uncountable iterated function systems evolve and stabilize. In this paper, we undertake a comprehensive examination of a sequence of uncountable iterated function systems (U.I.F.S.), each constructed from an uncountable collection of contraction mappings in conjunction with a linear and continuous operator. These systems are integral to our study as they encapsulate complex dynamical behaviors through their association with attractors, which represent sets towards which the system evolves over time. Each U.I.F.S. within the sequence is governed by a Markov-type operator that dictates its probabilistic behavior and is described by a fractal measure that captures the geometric and measure-theoretic properties of the attractor. The core contributions of our work are presented in the form of several theorems. These theorems tackle key problems and provide novel insights into the study of measures and their properties in Hilbert spaces. The results contribute to advancing the understanding of convergence behaviors, the interaction of Dirac measures, and the applications of Monge–Kantorovich norms. These theorems hold significant potential applications across various domains of functional analysis and measure theory. By establishing new results and proving critical properties, our work extends existing frameworks and opens new avenues for future research. This paper contributes to the broader field of mathematical analysis by offering new perspectives on how uncountable iterated function systems evolve and stabilize. Our findings provide a foundational understanding that can be applied to a wide range of mathematical and real-world problems. By highlighting the interplay between measure theory and functional analysis, our work paves the way for further exploration and discovery in these areas, thereby enriching the theoretical landscape and practical applications of these mathematical concepts.

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.

The notions of “mathematicalness” and “mathematicability” of nature appear in the context of attempts at explaining the effectiveness of mathematics in the description of the world. Mathematicalness of nature means that structures of the world are mathematical. But is this true? Is nature mathematical? In the paper some reasons for mathematicalness of nature are considered. In the paper some reasons for mathematicalness of nature are considered. In the paper some reasons for mathematicalness of nature are considered. Mathematical analysis is widely used in physics. Its application requires continuity of time and space. There are also different kinds of infinity in the mathematical theories used in physics. This raises the issue: whether the material world is continuous or we “impose” on nature certain properties in order to use convenient mathematical tools. Is mathematics a useful tool, or does it reflect nature? So, is nature mathematical or only mathematicable? The article shows that mathematicalness of nature is only a metaphysical hypothesis.

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.

Hilbert spaces and operator theory : W. Mlak, Kluwer, Dordrecht, 1991. 290 pp., Dfl.235, US$149, UK£81, ISBN 0-7923-1042-X.

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.

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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.