On the Lupaş q-transform

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of
 topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$
 and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations.

In this paper we give a solution of the discrete and continuous versions of the problem where is the class of absolutely continuous functions on with common majorizing modulus of continuity . We also discuss applications of the results obtained to mathematical economics (the Kantorovich-Monge mass transfer problem), approximation theory and numerical differentiation (Chebyshev -polynomials and splines), the constructive theory of functions (inequalities for -rearrangements), graph theory (graphs of rearrangements), and optimal control theory (the theory of total control and the Fel'dbaum-Bushaw problem).

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

A new generic nonlinear feature extraction map f is presented based on concepts from approximation theory. Let f map an input data vector x/spl isin//spl Rfr//sup n/, where n is high, to an appropriate feature vector y/spl isin//spl Rfr//sup m/, where m is sufficiently low. Also let X={X/sub 1/,...,X/sub N/} denote an available training set in /spl Rfr//sup n/. In this paper f is derived by requiring that the geometric structure (metric space attributes) of the points f(X)={f(X/sub 1/),...,f(X/sub N/)} in the feature space /spl Rfr//sup m/ be as similar as possible to the structure of the points X in the data space /spl Rfr//sup n/. This is accomplished by selecting first an appropriate dimension m for the feature space /spl Rfr//sup m/ according to the size N of the available training set X, subject to bounds on the distortion of the data structure caused by f and on the error in the estimation of the underlying likelihood functions in the feature space. The map f(i) is designed by a multi-dimensional scaling (MDS) approach that minimizes the Sammon's cost function. This approach uses graph-theoretic (minimal spanning tree) and genetic algorithmic concepts to search efficiently the optimal structure-preserving point-to-point mapping of the training samples X/sub 1/,...,X/sub N/ to their images in /spl Rfr//sup m/,Y/sub 1/,...,Y/sub N/. Finally, an optimal interpolating (OI) artificial neural network is used to recover the entire function f:/spl Rfr//spl rarr//spl Rfr//sup m/ by interpolating the values y/sub i/=1,...,N at X/sub i/, i=1,...,N. Preliminary simulation results based on this approach are also given.

Contributors

The approximate calculation of integrals is a common part of research in various fields of mathematics, primarily in the theory of approximations and computational mathematics. In its turn, within the theory of numerical integration, distinctions take place depending on the features of the integrand, the scale of which is “Quadrature (cubature) formulas” and “Theory of oscillations”. This article belongs to the second type of problems and consists in the approximate calculation of the Chebyshev coefficients.

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 article analyses the methods of enhancing the combat capability of small police tactical groups, in particular, with regard to their participation in hostilities during martial law in Ukraine. The main attention is paid to the importance of the optimal location of the commander in a small tactical group, which ensures the efficiency of performance of service and combat tasks, prompt decision-making, mobility and security of the group. The paper uses a scientific approach to improving the combat capability and tactics of small tactical groups. The following methods are used in the article: analysis, analysis of primary sources, comparative analysis, statistical and mathematical analysis, survey, expert survey, optimisation and approximation, graphical method of formations and method of optimisation of the commander’s location in a small tactical group, and method of experimental research. Different types of combat formations are studied, in particular, such as line, column, wedge, rhombus, star, snake and others, which are basic tactical elements. For each type of formation, experiments were conducted to determine the optimal location of the commander. In particular, the speed of command transmission and the risks of information loss depending on the commander’s position and the distance between the soldiers were analysed. It is established that the most optimal location of the commander is in the centre of the formation, as this ensures fast communication and minimises the risk of losses during an attack. The criterion-based approach is proposed, which includes quantitative, spatial and tactical criteria for choosing the commander’s location. The main criteria include the commander’s personal safety, efficiency of command transfer, reduction of command transformations, and minimisation of movements during re-arrangements. The empirical results confirm the importance of a comprehensive consideration of these factors to increase the combat capability of a small tactical group. The practical significance of the study lies in its possible use in the training system of the National Police of Ukraine, which will help to ensure the smooth performance of service and combat tasks and reduce personnel losses. Based on the results obtained, recommendations aimed at optimising the location of the commander in different types of combat formations have been developed.

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.

Destiny is the most complicated subject in our world and cannot be comprehended without taking a global view in both space and time. Therefore this paper takes a multi disciplinary approach using mathematics, physics, engineering, economics, and philosophy. The paper has two major related objectives (A) Provide a theory for the existence of destiny and (B) Explain away the counter logic. For part A we take the following approaches. (a1) The idea of destiny is well recognized by Newtonian physics. In our universe everything happens because of simultaneous interactions of everything. Thus destiny is not an individual concept. It is not my destiny or your destiny, it is our destiny. (a2) From this global concept of our destiny, we derive how the destiny plays out locally for an individual; how humans are really part of nature, work in cooperation with nature, and according to its dictates. In particular we show that our first reaction to every natural event is planned. (a3) Reincarnation theory of soul is an integral part of destiny theory. We prove giving real life examples, that reincarnation is a law of nature, and we are all reincarnated souls. Thus destiny is not only global in space; it is also global in time, covering all past lives. For part B we cover several important opposing views. (b1) The uncertainty principle of quantum mechanics says nothing can be certain, and therefore destiny cannot be true. We point out that the core idea behind Heisenberg's original mathematical proof is based on Fourier Transform theory. Fourier Transform uses infinity, which as we show is not meaningful in nature. Replacing infinity by any finite number changes the uncertainty to certainty, just like the Newtonian physics. (b2) Destiny is not in the mainstream, therefore it must be wrong. We show that it is the opposite. All Mathematical and physical theories have assumptions that are invalid in nature and therefore they cannot describe nature. These theories are not approximate theories, they are outright wrong. As a natural consequence, the real life engineering which uses these incorrect theories is full of kludges and patches. (b3) We must also understand money, money power, and the way it is controlled by the Central Bank. It is the same money that remote controls every activity and the mindset of every individual in our world. If anyone anywhere is allowed to find the truth on any subject, including destiny, then people will eventually discover the truth behind the central bank, which cannot be allowed. The central bank is a very secret private organization, cannot be controlled by any government, as is well known in economics. In the hierarchy of all activities, however, the soul theory and the laws of nature are at the top, above the central bank.

The incompatible numerical manifold method (INMM) is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given.