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Mathematics plays an important role in the linguistic structure of artificial in- telligence (AI). We describe the linguistic process as a unique structure present both in human cognition and in cognitive computing. The close relationship with both AI and human cognition is due to this unique structure, which paves the way for AI to interfere with the behavior of those who interact with it. We highlight the role of mathematicians in designing algorithms—the core of the AI linguistic process—and in defining steps and instructions for AI. Because al- gorithms, through AI, interfere with the thought of those who interact with AI by providing anticipated solutions that prevent users from making free choices, we investigate how we can apply ethical principles to guide interactions between users and intelligent systems in order to address this issue. We contend that by integrating ethical principles in the mathematical modeling of algorithms, we can avoid manipulation, inequality, and black boxes in the protection of individual rights. As such, ethical considerations are highly important for those working with algorithms, which highlights the humanistic side of mathematics.

The Natural Language for Artificial Intelligence presents the biological and logical structure typical of human language in its dynamic mediating process between reality and the human mind. The book explains linguistic functioning in the dynamic process of human cognition when forming meaning. After that, an approach to artificial intelligence (AI) is outlined, which works with a more restricted concept of natural language that leads to flaws and ambiguities. Subsequently, the characteristics of natural language and patterns of how it behaves in different branches of science are revealed to indicate ways to improve the development of AI in specific fields of science. A brief description of the universal structure of language is also presented as an algorithmic model to be followed in the development of AI. Since AI aims to imitate the process of the human mind, the book shows how the cross-fertilization between natural language and AI should be done using the logical-axiomatic structure of natural language adjusted to the logical-mathematical processes of the machine. Presents a comprehensive approach to natural language and its inherent and complex dynamics Develops language content as the next frontier, identifying the universal structure of language as a common structure that appears in both AI and cognitive computing Explains the standard structure present in cognition and AI, making them interchangeable Offers examples of the application of the universal language model in image analysis and conventional language

The Natural Language for Artificial Intelligence presents the biological and logical structure typical of human language in its dynamic mediating process between reality and the human mind. The book explains linguistic functioning in the dynamic process of human cognition when forming meaning. After that, an approach to artificial intelligence (AI) is outlined, which works with a more restricted concept of natural language that leads to flaws and ambiguities. Subsequently, the characteristics of natural language and patterns of how it behaves in different branches of science are revealed to indicate ways to improve the development of AI in specific fields of science. A brief description of the universal structure of language is also presented as an algorithmic model to be followed in the development of AI. Since AI aims to imitate the process of the human mind, the book shows how the cross-fertilization between natural language and AI should be done using the logical-axiomatic structure of natural language adjusted to the logical-mathematical processes of the machine. Presents a comprehensive approach to natural language and its inherent and complex dynamics Develops language content as the next frontier, identifying the universal structure of language as a common structure that appears in both AI and cognitive computing Explains the standard structure present in cognition and AI, making them interchangeable Offers examples of the application of the universal language model in image analysis and conventional language

The Natural Language for Artificial Intelligence presents the biological and logical structure typical of human language in its dynamic mediating process between reality and the human mind. The book explains linguistic functioning in the dynamic process of human cognition when forming meaning. After that, an approach to artificial intelligence (AI) is outlined, which works with a more restricted concept of natural language that leads to flaws and ambiguities. Subsequently, the characteristics of natural language and patterns of how it behaves in different branches of science are revealed to indicate ways to improve the development of AI in specific fields of science. A brief description of the universal structure of language is also presented as an algorithmic model to be followed in the development of AI. Since AI aims to imitate the process of the human mind, the book shows how the cross-fertilization between natural language and AI should be done using the logical-axiomatic structure of natural language adjusted to the logical-mathematical processes of the machine. Presents a comprehensive approach to natural language and its inherent and complex dynamics Develops language content as the next frontier, identifying the universal structure of language as a common structure that appears in both AI and cognitive computing Explains the standard structure present in cognition and AI, making them interchangeable Offers examples of the application of the universal language model in image analysis and conventional language

The Natural Language for Artificial Intelligence presents the biological and logical structure typical of human language in its dynamic mediating process between reality and the human mind. The book explains linguistic functioning in the dynamic process of human cognition when forming meaning. After that, an approach to artificial intelligence (AI) is outlined, which works with a more restricted concept of natural language that leads to flaws and ambiguities. Subsequently, the characteristics of natural language and patterns of how it behaves in different branches of science are revealed to indicate ways to improve the development of AI in specific fields of science. A brief description of the universal structure of language is also presented as an algorithmic model to be followed in the development of AI. Since AI aims to imitate the process of the human mind, the book shows how the cross-fertilization between natural language and AI should be done using the logical-axiomatic structure of natural language adjusted to the logical-mathematical processes of the machine. Presents a comprehensive approach to natural language and its inherent and complex dynamics Develops language content as the next frontier, identifying the universal structure of language as a common structure that appears in both AI and cognitive computing Explains the standard structure present in cognition and AI, making them interchangeable Offers examples of the application of the universal language model in image analysis and conventional language

Analysis of the fetal heart rate during pregnancy is essential for monitoring the proper development of the fetus. Current fetal heart monitoring techniques lack the accuracy in fetal heart rate monitoring and features acquisition, resulting in diagnostic medical issues. The challenge lies in the extraction of the fetal ECG from the mother's ECG during pregnancy. This approach has the advantage of being a reliable and non-invasive technique. For this aim, we propose in this paper a wavelet/multi-wavelet method allowing to extract perfectly the feta ECG parameters from the abdominal mother ECG. The method is essentially due to the exploitation of Clifford wavelets as recent variants in the field. We prove that these wavelets are more efficient and performing against classical ones. The experimental results are therefore due to two basic classes of wavelets and multi-wavelets. A first-class is the classical Haar Schauder, and a second one is due to Clifford valued wavelets and multi-wavelets. These results showed that wavelets/multiwavelets are already good bases for the FECG processing, provided that Clifford ones are the best., Comment: 21 pages, 8 figures, 1 table

Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these wavelets for solving nonlinear SBVPs. These wavelets can deal with singularities easily and efficiently. To deal with the nonlinearity, we use both Newton's quasilinearization and the Newton-Raphson method. To show the importance and accuracy of the proposed methods, we solve the Lane-Emden type of problems and compare the computed solutions with the known solutions. As the resolution is increased the computed solutions converge to exact solutions or known solutions. We observe that the proposed technique performs well on a class of Lane-Emden type BVPs. As the paper deals with singularity, non-linearity significantly and different wavelets are used to compare the results.

The fractional derivative of the Dirichlet eta function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function on the critical strip. Its convergence is studied. In particular, its half-plane of convergence gives the possibility to better understand the fractional derivative of the Riemann zeta function and its critical strip. As an application, two signal processing networks, corresponding to the fractional derivative of the eta function and to its Fourier transform, respectively, are shortly described.

A method, based on a multiscale (wavelet) decomposition of the solution is proposed for the analysis of the Poisson problem. The solution is approximated by a finite series expansion of harmonic wavelets and is based on the computation of the connection coefficients. It is shown, how a sourceless Poisson's problem, solved with the Daubechies wavelets, can also be solved in presence of a localized source in the harmonic wavelet basis.

In the following are considered the harmonic wavelet solutions of the fundamental equations of acoustic waves in isotropic media. It is defined an adaptive analysis of waves which are parametrized by the scale (or level) factor of wavelets. The finite harmonic solution of acoustic waves in isotropic media [14], is rediscovered as the wavelet solution at the lowest level of approximation. There follows that the harmonic wavelet analysis gives a more general framework for the investigation of acoustic waves.

In the following is given a method [2 10] for representing partial differential (evolution) operators using Haar wavelet bases [8, 10, 12]- Since the Haar wavelets are not smooth, they are regularized using auxiliary polynomial splines and defining suitable discrete operators [8, 3] acting on the spaces of the piecewise functions $V_n \subset L^2(lR)$. Thus the projection of the partial differential operators is done into the Spline-Haar Space, subspace of $V_n$, obtaining discrete operators. Althought these discrete operators depend both on the order of the splines and on the space resolution level [12], they are univocally defined (with respect to the projection) in the Spline-Haar Space. A comparison of the approximate wavelet solution with a classical problem of heat propagation is also given.

In this paperwe will approach the analysis of time series by the discrete Haar wavelet trasnform and by the energy distribution. it is shown that the wavelet coefficients are strictly related to the scheme of finite differences, thus giving information on the first order properties of the data. In particular, this method is tested on financial data, such as stock pricings, by characterizing the trends and the abrupt changes, by means of the Shannon's information function (entropy), which is defined on the local energy. AMS Classification: Primary 35A35, 42C40, 41A15, 47A58. Secondary 65T60. 65D07. Jel Classification: C63, E43. Key words and phrases: Wavelets, haar function, time series, entropy.

Nell'ambito della decomposizione 3+1 dello spazio-tempo viene considerato il problema di Cauchy per le equazioni gravitazionali, nel caso del vuoto.

In this paper the stability of a new class of exact symmetrical solutions in the Newtonian gravitational (n + 1) -body problem is studied. This class of solution follows from a suitable geometric distribution of the (n+1) -bodies, and initial conditions, so that the solution is represented geometrically by an oscillating regular polygon with n sides rotating non-uniformly about its center. The body having a mass m0 is at the center of the polygon, while n bodies having the same mass m are at the vertices of the polygon and move about the central body in identical elliptic orbits. It is proved that for n = 2 and for regular polygons 3 <= n <= 6 each corresponding solution is unstable for any value of the central mass m0 . For n => 7 the solution is linearly stable if both

The paper can be referred to that direction in the wavelet theory, which was called by Kaiser "the physical wavelets". He developed the analysis of first two kinds of physical wavelets - electromagnetic (optic) and acoustic wavelets. Newland developed the technique of application of harmonic wavelets especially for studying the harmonic vibrations. Recently Cattani and Rushchitsky proposed the 4th kind of physical wavelets - elastic wavelets. This proposal was based on three main elements: 1. Kaiser's idea of constructing the physical wavelets on the base of specially chosen (admissible) solutions of wave equations. 2. Developed by one of authors theory of solitary waves (with profiles in the form of Chebyshov-Hermite functions) propagated in elastic dispersive media. 3. The theory and practice of using the wavelet "Mexican Hat" system, the mother and farther wavelets (and their Fourier transforms) of which are analytically represented as the Chebyshov-Hermite functions of different indexes. An application of elastic wavelets to studying the evolution of solitary waves of different shape during their propagation through composite materials is shown on many examples.

In this paper a wavelet technique to study time series by discrete wavelet coefficients, is proposed. Since time series are mainly represented by histograms we will use the Haar waveletes and the Haar wavelet interpolation [1,2]. Moreover we propose an improvement (from computational point of view) of the discrete Haar wavelet transform, called reduced Haar wavelet transform [1,2], allowing a local windowing of the transform. Due to their localization property, the wavelet coefficients of the histogram, together with the wavelet coefficients of the generalized derivative of the histogram, allow to study, with high level of accuracy, the rate of change as well as other characteristics of the time series. Numerical experiments with two kind of (financial) time series show the effectiveness of this technique. AMS Classification: 35A35, 47A58, 65N13, 65F10, 76W05. JEL Classification: C63, E43. Key words and phrases: Wavelets, Haar function, interpolation, time series.

The paper can be referred to that direction in the wavelet theory, which was called by Kaiser "the physical wavelets". He developed the analysis of first two kinds of physical wavelets - electromagnetic (optic) and acoustic wavelets. Newland developed the technique of application of harmonic wavelets especially for studying the harmonic vibrations. Recently Cattani and Rushchitsky proposed the 4th kind of physical wavelets - elastic wavelets. This proposal was based on three main elements: 1. Kaiser's idea of constructing the physical wavelets on the base of specially chosen (admissible) solutions of wave equations. 2. Developed by one of authors theory of solitary waves (with profiles in the form of Chebyshov-Hermite functions) propagated in elastic dispersive media. 3. The theory and practice of using the wavelet "Mexican Hat" system, the mother and farther wavelets (and their Fourier transforms) of which are analytically represented as the Chebyshov-Hermite functions of different indexes. An application of elastic wavelets to studying the evolution of solitary waves of different shape during their propagation through composite materials is shown on many examples.

In the following are considered the harmonic wavelet solutions of the fundamental equations of acoustic waves in isotropic media. It is defined an adaptive analysis of waves which are parametrized by the scale (or level) factor of wavelets. The finite harmonic solution of acoustic waves in isotropic media [14], is rediscovered as the wavelet solution at the lowest level of approximation. There follows that the harmonic wavelet analysis gives a more general framework for the investigation of acoustic waves.

A method, based on a multiscale (wavelet) decomposition of the solution is proposed for the analysis of the Poisson problem. The solution is approximated by a finite series expansion of harmonic wavelets and is based on the computation of the connection coefficients. It is shown, how a sourceless Poisson's problem, solved with the Daubechies wavelets, can also be solved in presence of a localized source in the harmonic wavelet basis.

In this paperwe will approach the analysis of time series by the discrete Haar wavelet trasnform and by the energy distribution. it is shown that the wavelet coefficients are strictly related to the scheme of finite differences, thus giving information on the first order properties of the data. In particular, this method is tested on financial data, such as stock pricings, by characterizing the trends and the abrupt changes, by means of the Shannon's information function (entropy), which is defined on the local energy. AMS Classification: Primary 35A35, 42C40, 41A15, 47A58. Secondary 65T60. 65D07. Jel Classification: C63, E43. Key words and phrases: Wavelets, haar function, time series, entropy.

In this paper a wavelet technique to study time series by discrete wavelet coefficients, is proposed. Since time series are mainly represented by histograms we will use the Haar waveletes and the Haar wavelet interpolation [1,2]. Moreover we propose an improvement (from computational point of view) of the discrete Haar wavelet transform, called reduced Haar wavelet transform [1,2], allowing a local windowing of the transform. Due to their localization property, the wavelet coefficients of the histogram, together with the wavelet coefficients of the generalized derivative of the histogram, allow to study, with high level of accuracy, the rate of change as well as other characteristics of the time series. Numerical experiments with two kind of (financial) time series show the effectiveness of this technique. AMS Classification: 35A35, 47A58, 65N13, 65F10, 76W05. JEL Classification: C63, E43. Key words and phrases: Wavelets, Haar function, interpolation, time series.

In the following is given a method [2 10] for representing partial differential (evolution) operators using Haar wavelet bases [8, 10, 12]- Since the Haar wavelets are not smooth, they are regularized using auxiliary polynomial splines and defining suitable discrete operators [8, 3] acting on the spaces of the piecewise functions $V_n \subset L^2(lR)$. Thus the projection of the partial differential operators is done into the Spline-Haar Space, subspace of $V_n$, obtaining discrete operators. Althought these discrete operators depend both on the order of the splines and on the space resolution level [12], they are univocally defined (with respect to the projection) in the Spline-Haar Space. A comparison of the approximate wavelet solution with a classical problem of heat propagation is also given.

In this paper the stability of a new class of exact symmetrical solutions in the Newtonian gravitational (n + 1) -body problem is studied. This class of solution follows from a suitable geometric distribution of the (n+1) -bodies, and initial conditions, so that the solution is represented geometrically by an oscillating regular polygon with n sides rotating non-uniformly about its center. The body having a mass m0 is at the center of the polygon, while n bodies having the same mass m are at the vertices of the polygon and move about the central body in identical elliptic orbits. It is proved that for n = 2 and for regular polygons 3 <= n <= 6 each corresponding solution is unstable for any value of the central mass m0 . For n => 7 the solution is linearly stable if both

Nell'ambito della decomposizione 3+1 dello spazio-tempo viene considerato il problema di Cauchy per le equazioni gravitazionali, nel caso del vuoto.

The paper can be referred to that direction in the wavelet theory, which was called by Kaiser "the physical wavelets". He developed the analysis of first two kinds of physical wavelets - electromagnetic (optic) and acoustic wavelets. Newland developed the technique of application of harmonic wavelets especially for studying the harmonic vibrations. Recently Cattani and Rushchitsky proposed the 4th kind of physical wavelets - elastic wavelets. This proposal was based on three main elements: 1. Kaiser's idea of constructing the physical wavelets on the base of specially chosen (admissible) solutions of wave equations. 2. Developed by one of authors theory of solitary waves (with profiles in the form of Chebyshov-Hermite functions) propagated in elastic dispersive media. 3. The theory and practice of using the wavelet "Mexican Hat" system, the mother and farther wavelets (and their Fourier transforms) of which are analytically represented as the Chebyshov-Hermite functions of different indexes. An application of elastic wavelets to studying the evolution of solitary waves of different shape during their propagation through composite materials is shown on many examples.

The paper can be referred to that direction in the wavelet theory, which was called by Kaiser "the physical wavelets". He developed the analysis of first two kinds of physical wavelets - electromagnetic (optic) and acoustic wavelets. Newland developed the technique of application of harmonic wavelets especially for studying the harmonic vibrations. Recently Cattani and Rushchitsky proposed the 4th kind of physical wavelets - elastic wavelets. This proposal was based on three main elements: 1. Kaiser's idea of constructing the physical wavelets on the base of specially chosen (admissible) solutions of wave equations. 2. Developed by one of authors theory of solitary waves (with profiles in the form of Chebyshov-Hermite functions) propagated in elastic dispersive media. 3. The theory and practice of using the wavelet "Mexican Hat" system, the mother and farther wavelets (and their Fourier transforms) of which are analytically represented as the Chebyshov-Hermite functions of different indexes. An application of elastic wavelets to studying the evolution of solitary waves of different shape during their propagation through composite materials is shown on many examples.

The paper can be referred to that direction in the wavelet theory, which was called by Kaiser "the physical wavelets". He developed the analysis of first two kinds of physical wavelets - electromagnetic (optic) and acoustic wavelets. Newland developed the technique of application of harmonic wavelets especially for studying the harmonic vibrations. Recently Cattani and Rushchitsky proposed the 4th kind of physical wavelets - elastic wavelets. This proposal was based on three main elements: 1. Kaiser's idea of constructing the physical wavelets on the base of specially chosen (admissible) solutions of wave equations. 2. Developed by one of authors theory of solitary waves (with profiles in the form of Chebyshov-Hermite functions) propagated in elastic dispersive media. 3. The theory and practice of using the wavelet "Mexican Hat" system, the mother and farther wavelets (and their Fourier transforms) of which are analytically represented as the Chebyshov-Hermite functions of different indexes. An application of elastic wavelets to studying the evolution of solitary waves of different shape during their propagation through composite materials is shown on many examples.