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A new approach to the design of uniplanar self-shield gradient coils for fully open magnetic resonance imaging system is proposed to build a specified gradient magnetic field in a target region. In this new design methodology, the surface current density of the coils is represented by a two-dimensional Fourier series expansion. A cost function is constructed in terms of the summation of squared error and penalty functions. Minimizing the cost function, we obtain the continuous current density distribution for the uniplanar gradient coils. Applying the stream function technique to the resulting continuous current distribution, discrete current patterns can be obtained. Employing the Biot-Savart integral equation to the discrete current loops, the gradient magnetic field has been re-evaluated in order to validate the method proposed. Using this approach, uniplanar self-shield x-gradient coils which are capable of creating a gradient field of 40 mT/m have been designed. The results indicate that with this method a magnetic field with high level of linearity and a satisfied screening effect are obtained. [ABSTRACT FROM AUTHOR]

Applying the Fourier path integral formalism to the isothermal-isobaric ensemble, the solid-liquid transition for 13-atom pure Lennard-Jones clusters was characterized. The masses of the clusters were taken as the masses of hydrogen, deuterium and tritium, hence isotopic effects of quantum clusters were considered. The parallel tempering Monte Carlo algorithm was used to solve all multidimensional integral in the FPI method. The volume of the system was defined with respect to the centroids of the quantum particles and a variable constraining potential was used to restrict undesirable thermodynamic events. The maximum value of the constant pressure heat capacity at a given temperature was used to identify the melting temperature. Pressure versus temperature phase diagrams were constructed for these systems with and without the inclusion of quantum effects. A significant difference in the melting temperature was encountered for the different isotopes due to quantum contribution. [ABSTRACT FROM AUTHOR]

Real linear approximation theory is developed further by regarding n as a module over the ring of circlets. By introducing a concept of orthogonality together with the respective Gram-Schmidt orthogonalization process, improved approximations upon the standard complex Hilbert space techniques follow. Related hierarchical bases are devised leading to a new family of rapidly constructible family of unitary matrices. With circlets, so-called oplets are introduced for approximation to improve the singular value decomposition of real matrices. Complex matrix approximation is also considered through finding the nearest real matrix in small rank perturbations. [ABSTRACT FROM AUTHOR]

The method of stationary phase is used to develop an asymptotic expansion of a class of one-dimensional Fourier integrals whose phase function is a general second-degree polynomial. Because of its form, the phase function contains one stationary point that may occur anywhere on the real line. The asymptotic expansion is obtained using the theory of distributions and Fourier transform analysis. In addition to the expansion, a bound on the remainder term is derived. Finally, it is shown that the resulting expansion is, indeed, asymptotic. [ABSTRACT FROM AUTHOR]

Fourier series and related harmonic methods have been demonstrably effective for identifying and characterizing the seasonal behaviour, or phenology, of a variety of terrestrial vegetation communities using Normalized Difference Vegetation Index (NDVI) time series from Earth-orbiting satellites. The ultimate temporal resolution of such applications has been limited, however, by the common practice of truncating, or low pass filtering, harmonic series to relatively low order terms, in order to suppress spurious oscillations in the model results. The temporal resolution of these techniques can be significantly improved if, along with a weighted minimization of the sum of the squared data residuals tracking the upper envelope of observed data, we also enforce an expectation of minimum model roughness to dampen spurious oscillations in predicted values. The resulting annual models have resolutions consistent with the application of special transcendental forms, such as asymmetric Gaussian and logistic (sigmoidal) functions, recently reported in the literature. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator - Δ + q in domains of the form R = {x ∈ n: 0 ≤ xi ≤ ai, i = 1,..., n} with either Dirichlet or Neumann boundary conditions on the faces of R, and study the constraints on q imposed by fixing the spectrum of - Δ + q with these boundary conditions. We work in the space of potentials, q, which become real-analytic on n when they are extended evenly across the coordinate planes and then periodically. Our results have the corollary that there are no continuous isospectral deformations for these operators within that class of potentials. This work is based on new formulas for the trace of the wave group in this setting. In addition to the inverse spectral results these formulas lead to asymptotic expansions for the traces of the wave and heat kernels on rectangular domains. [ABSTRACT FROM AUTHOR]

Distribution of Joule heating and the corresponding electromagnetic field in the vicinity of the crack in a thermo- and electro-conductive thin-walled shell carrying a non-stationary electrical current is investigated. Specifically, a circular cylindrical shell with one finitely long crack that is perpendicular to the current direction is considered. The crack is free of mechanical loads. The underlying boundary-value problem is addressed via the use of Fourier series expansion and the solution reduces to that of a singular integral equation with Hilbert-type singular kernel. The influence of the presence of crack, and the effect of non-stationarity of the current on the distribution of current and Joule heating in the shell are investigated and pertinent conclusions are drawn. The results are instrumental toward crack detection and active arrest of crack propagation in structures made of electro-conductive materials or aged structure components in aerospace industry. [ABSTRACT FROM AUTHOR]

We prove that universal approximation (uniform approximation on compact subsets with connected complement) implies almost everywhere approximation in the sense of Menchoff with respect to any given σ-finite Borel measure on [image omitted] (d≥2). [ABSTRACT FROM AUTHOR]

Fourier series containing associated Legendre functions are given for functions related to the Legendre elliptic integrals and for incomplete Epstein-Hubbell integrals. A number of definite and indefinite integrals related to these series are derived. Additional series of Legendre functions and integrals are derived from a binomial expansion of the reciprocal of the delta amplitude. A power series expansion for the elliptic integral of the third kind in terms of the parameter α is obtained. Series of associated Legendre functions are given for general monomials. [ABSTRACT FROM AUTHOR]

A new technique for predicting the harmonic and intermodulation performances of class-AB stages is presented. The proposed technique is based on modelling the transfer characteristic of the class-AB stage using a Fourier-series approximation. The number of terms of the Fourier-series is decided based on the required accuracy and the coefficients of the series are obtainable using simple hand calculations. The results obtained are in excellent agreement with simulation results and confirm that the intermodulation performance is dominant compared to the harmonic performance. [ABSTRACT FROM AUTHOR]

Based on the credibility distributions which are analytic expressions of fuzzy variables, Fourier spectrum of credibility distribution for fuzzy variable, a useful mathematical analytic tool dealing with fuzzy variables, is introduced. Then several elementary properties of Fourier spectrums of credibility distributions for fuzzy variables are studied. Furthermore some relations between credibility distributions and Fourier spectrums such as inversion formula and uniqueness theorem are established. [ABSTRACT FROM AUTHOR]

The Stevenson approach to low-frequency time-harmonic wave scattering, that expanded the electric and magnetic fields in power series of k, essentially the inverse wavelength, is scrutinized. Stevenson's power series approach perforce implies a variable frequency ω, i.e., a variable wave-number k, an assumption challenged here. Presently the three major linear wave physics models: acoustics, electromagnetics, and elastodynamics, are put on an equal footing by introducing the self-consistent system concept. Accordingly any low-frequency series expansion starts with the pertinent Helmholtz equation. Far-field surface-integrals are derived for each case. To verify our approach, an example of low-frequency electromagnetic scattering by a long cylinder is elaborated, the results are compared to, and agree with the exact Hankel-Fourier series solution. [ABSTRACT FROM AUTHOR]

The constitutive laws are obtained for the linear micropolar thermopiezoelectric/ piezomagnetic continuum with one relaxation time. The Maxwell–Cattaneo equation is expressed, as a constitutive law, in a form similar to the classical Fourier law. A new model is introduced, whereas the constitutive equations, not only the heat equation, are modified by the relaxation time. For this model a uniqueness theorem is proved. The constitutive laws, the equations of motion and the boundary conditions are obtained from a Hamilton's type variational principle and the Hamilton's principle is derived. [ABSTRACT FROM AUTHOR]

This note considers the four classes of orthogonal polynomials – Chebyshev, Hermite, Laguerre, Legendre – and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same as that for Fourier series expansions. Each class of polynomials has features which are interesting numerically. Finally a plausibility argument is included showing that this phenomenon for the Gibbs constants should not have been unexpected. These findings suggest further investigations suitable for undergraduate research projects or small group investigations. [ABSTRACT FROM AUTHOR]

The growth of simulations of particle systems has been aided by advances in computer speed and algorithms. The adoption of algorithms to solve N-body simulation problems has been less rapid due to the fact that such scaling was only competitive for relatively large N. Our work seeks to find algorithmic modifications and practical implementations for intermediate values of N in typical use for molecular simulations. This article reviews fast multipole techniques for calculation of electrostatic interactions in molecular systems. The basic mathematics behind fast summations applied to long ranged forces is presented along with advanced techniques for accelerating the solution, including our most recent developments. The computational efficiency of the new methods facilitates both simulations of large systems as well as longer and therefore more realistic simulations of smaller systems. [ABSTRACT FROM AUTHOR]

A general method is proposed to analyze aperiodic or periodic Longitudinally Inhomogeneous Waveguides (LIWs). In this method, the electric permittivity function is expanded in the Fourier series. Then, the eigenvalues of periodic LIWs and so the S parameters of aperiodic LIWs are obtained. The accuracy of the method is studied using a comprehensive example. [ABSTRACT FROM AUTHOR]

The equation , given in the domain G={z: r 0 , is considered. We formally construct a full solution and prove an analogy of the Liouville theorem when R=∞ . [ABSTRACT FROM AUTHOR]

If we divide the interval [0,1] into N sub-intervals, then hybrid Fourier and block-pulse functions on each sub-interval can approximate any function. This ability helps us to have more accurate approximations of piecewise continuous functions. Hence we obtain more accurate solutions to problems in the calculus of variations. In this article, we use a combination of Fourier and block-pulse functions on the interval [0, 1] to solve a variational problem in the solution of algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Simple Fourier series in terms of the amplitude φ are derived for the common Legendre elliptic integrals. The coefficients in the various series contain Legendre functions of the second kind and half-integral degree, also known as toroidal functions. For the Legendre elliptic integrals of the first and second kinds, F (φ, k ) and E (φ, k ), repectively, the expansions given are simple sine series in the amplitude φ and an additional aperiodic term proportional to φ. These series are valid for φ∈ℝ. The complete elliptic integral of the third kind Π(α 2 , k ) can be expressed in terms of Heuman’s lambda function Λ 0 (β, k ) and the Jacobi zeta function Z (β, k ), which in turn can be expressed in terms of the integrals of the first and second kinds. This enables simple sine series in terms of β to be derived for Λ 0 (β, k ) and Z (β, k ). The series for Λ 0 (β, k ) has an aperiodic term in β but the series for Z (β, k ) does not. The various series are obtained by first expanding the delta amplitude and Δ -1 (φ, k ) as cosine series in the amplitude φ and integrating term by term with repect to φ. The recurrence relation for the Legendre functions is frequently used to simplify or rearrange the various series. [ABSTRACT FROM AUTHOR]

This paper presents a computational approach to the imaging of a perfectly conducting cylinder buried in a slab medium. Both cubic-spline method and trigonometric series for shape description are used and compared. We measure the scattered field in the first and third layers. Based on the boundary condition and the recorded scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is employed to find the global extreme solution of the object function. It is found that the shape described by cubic-spline can be reconstructed. In such a case, Fourier series expansion will fail. Even when the initial guess is far away from the exact one, the cubic-spline expansion and genetic algorithm can avoid the local extreme and converge to a global extreme solution. Numerical results are given to show that the shape description by using cubic-spline method is much better than that by the Fourier series. In addition, the effect of Gaussian noise on the reconstruction is investigated. [ABSTRACT FROM AUTHOR]

The analysis of axisymmetric mechanical and thermal stresses for a long hollow cylinder made of functionally graded material, as functions of radial and longitudinal directions, is developed. The material properties are graded along the radial direction according to power functions of radial direction. The temperature and Navier equations are solved analytically, using the generalized Bessel function and complex form of Fourier integral. A direct method of solution of Navier equations is presented, rather than potential functions method. [ABSTRACT FROM AUTHOR]

A transparent, exhaustive, and self-contained method for the calculation of the demagnetization tensor of the uniformly magnetized ellipsoid is presented. The method is an alternative to the established Maxwell derivation and is based on a Fourier-space approach to the micromagnetics of magnetized bodies. The key to the success of the procedure lies in the convenient treatment of shape effects through the Fourier representation. The scaled form of the demagnetization factors which depends on two dimensionless aspect ratios is argued to be their natural integral representation. Amongst other advantages, it allows for the immediate implementation of symmetry arguments such that only one of the principal factors needs to be computed. The oblate and prolate ellipsoids of revolution are examined from the same general point of view. The demagnetization factors for these distinct types of spheroid are seen to be related by analytic continuation of well-known Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper presents a trend analysis of monthly riverflows recorded at the Apalachicola River at Chattahoochee, FL during the 1930–2001 water years. The environmental concern is that increasing urbanization upstream (Atlanta and Columbus, GA) increases downstream riverflows. In statistical terms, our objective is to test whether there are months in which riverflows are significantly increasing. For this purpose, a parsimonious periodic autoregressive moving-average time series model is fitted to the error structure of the series. The covariances of this time series model are then used to assess the significance of monthly and yearly linear trend estimators. In the end, we do not find strong evidence of increasing riverflows. [ABSTRACT FROM AUTHOR]

In a previous paper (part I), it has been shown that a random wavefield from a randomly rough half-plane for a TM plane wave incidence is written in terms of a Wiener-Hermite expansion with three types of Fourier integrals. This paper studies a concrete representation of the random wavefield by an approximate evaluation of such Fourier integrals, and statistical properties of scattering and diffraction. For a Gaussian roughness spectrum, intensities of the coherent wavefield and the first-order incoherent wavefield are calculated and shown in figures. It is then found that the coherent scattering intensity decreases in the illumination side, but is almost invariant in the shadow side. The incoherent scattering intensity spreads widely in the illumination side, and have ripples at near the grazing angle. Moreover, a major peak at near the antispecular direction, and associated ripples appear in the shadow side. The incoherent scattering intensity increases rapidly at near the random half-plane. These new phenomena for the incoherent scattering are caused by couplings between TM guided waves supported by a slightly random surface and edge diffracted waves excited by a plane wave incidence or by free guided waves on a flat plane without any roughness. [ABSTRACT FROM AUTHOR]

This paper deals with a scattering and diffraction problem by a one-dimensionally random rough half-plane with perfect conductivity when a TM plane wave is incident. Since a divergence difficulty appears in a perturbation analysis for such a problem, a new formulation by a combination of the stochastic functional approach and the Wiener-Hopf technique is presented. By the Dα-Fourier transformation, an inhomogeneous scattered wavefield is transformed into an s-homogeneous random field, where s is a complex wavenumber parameter. Such an s-homogeneous random field is represented in terms of a Wiener-Hermite expansion with unknown coefficients called Wiener kernels. Hierarchical equations for the Wiener kernels are derived by the boundary condition. First, a Wiener-Hopf equation is derived from the hierarchical equations by approximate evaluations of scattering processes due to the surface roughness. Next, the Wiener-Hopf equation is solved by the Wiener-Hopf technique, and then the Wiener kernels are determined. By the inverse Dα-Fourier transformation, the inhomogeneous random wavefield is made up of three types of Fourier integrals representing effects of multiple scattering due to surface roughness, single edge diffraction by the edge and interactions between them. As a result, any statistical properties of the random wavefield may be calculated reasonably. [ABSTRACT FROM AUTHOR]

An asymmetry estimate for extremal functions in Pólya–Szegö type inequalities for Dirichlet integrals is established in presence of critical points and nonstrictly convex integrands. [ABSTRACT FROM AUTHOR]

The uniqueness theorem for non-harmonic Fourier series is proved under a much weaker condition that for a number γ > 0 and a constant h > 0. [ABSTRACT FROM AUTHOR]

A new layer by layer/discrete layer model is presented for the linearly elastic and thermoelastic responses of multilayered composite plates. The transverse variation of the displacement field is defined in terms of a one-dimensional finite element representation, using quadratic Lagrangian interpolations. The laminate thickness is subdivided into a series of one-dimensional finite elements (i.e., thickness subdivisions) whose nodes correspond to planes of constant transverse normal coordinates in the undeformed configuration of the laminate. Each of the displacements and loads is expanded in a double Fourier series in the cartesian surface coordinates. Interface and boundary conditions are exactly verified and exploited to reduce the number of independent generalized displacements, before solving the boundary value problem from the standard variational principle for displacements. This leads to a new model, whose size 3N in 3D, is less than the classic layer by layer approach requiring 3(2N + 1), also in 3D. In addition, this model allows satisfying exactly all the interface and boundary conditions. Numerical evaluations and comparisons with exact 3D elasticity solutions confirm the efficacy of the proposed approach. Finally, the proposed modelling has been extended to the thermoelastic response of laminates and the results are compared with the exact 3D solutions. Temperature distributions required for the thermoelastic analysis are determined from the constrained discrete layer approach previously applied to heat conduction. [ABSTRACT FROM AUTHOR]

The application of Fourier Series to cartography and GIS, in particular to shape analysis, line simplification and enhancement, and the measurement of cartographic scale, is explored. Vector representations of the Australian coastline are manipulated using Fourier techniques and the output appraised in terms of its ability to encapsulate the shape of the original coastline, or as a simplified or enhanced version of the original. Using Fourier techniques on vector representations of parts of the Australian coastline at six different cartographic scales, a relationship between cartographic scale and line complexity is found to exist, and an empirical expression of that relationship developed. The intent of developing an empirical relationship is to bring greater consistency to the process of assigning cartographic scale to data, and thereby to the process of map production, data simplification and GIS analysis. Fourier techniques are found to be a poor encapsulator of cartographic shape, a good method of line simplification and enhancement, and a promising method of measuring cartographic scale. As background, some revision of the history and mathematics of Fourier Series, and previous research into the application of Fourier Series to cartography, is given. [ABSTRACT FROM AUTHOR]

We give conditions on the functions σ and u on R″ such that if u is given by the convolution of σ and u, then u is harmonic on R″. [ABSTRACT FROM AUTHOR]

The article focuses on the connections that links Fourier series, catalan numbers and random walks on trees. Catalan numbers have many combinatorial interpretations They satisfy the recursion relations. A tree is a connected graph with no loops such that each vertex is an endpoint of only finite edges. Two distinct vertices of the tree are called neighbors if there is edge connecting them. A path in the tree is a finite or infinite sequences of vertices. It is frequently convenient to single out a specific vertex of a tree and declare it to be a root of the tree. There is a natural distance function defined on the set of vertices of a tree. Transient random walks have the property that with probability one an infinite path will visit any particular vertex only finitely many times. There is a simple test for transience of a random walk on an arbitrary tree with a nearest-neighbor probability. The generalized Catalan numbers have numerous combinatorial interpretations. The generalized Catalan numbers can be derived by means of a Pascal-like triangle.

The article presents a two-parameter trigonometric series. It derives a function such that the given trigonometric series is its Fourier series.

This article gives the main representation theorems for harmonic functions in the spaces bωp(B) on the unit ball B in R″. These spaces depend on a parameter function ω and are arbitrarily large. We receive the integral representation for the functions of bωp(B) over the unit ball. The article also gives a representation connected with the natural isometry between bω²(B) and the ordinary space L² on the unit sphere, which is explicitly given in the form of an integral operator along with its inversion. [ABSTRACT FROM AUTHOR]

In this article, we discuss the problem of reconstructing a bandlimited signal f from the samples of the integral of the signal. We first apply the trigonometric polynomials to approximate the integral of the signal and then reconstruct instantaneous samples from the signal integral. [ABSTRACT FROM AUTHOR]

With the term microinstabilities, we indicate a class of plasma drift waves such as the ion temperature gradient (ITG) mode, the trapped electron mode (TEM), and the electromagnetic Alfvénic ion temperature gradient (AITG) mode. They are usually driven by spatial gradients in the plasma and they are held responsible for anomalous transport and enhanced heat flux in tokamaks. Their understanding is therefore crucial for the development of magnetically confined plasma fusion devices. Our approach is based on the gyrokinetic model (Hahm 1988) for particle dynamics. The gyrokinetic nonlinear Vlasov equation, in the presence of an electromagnetic perturbation (Brizard 1989, 1995), is derived using Lie perturbation theory (Cary and Littlejohn 1983) to average out the fast particle motion around the gyrocenter. We have linearized the gyrokinetic Vlasov equation and added the quasi‐neutrality equation and Ampère's law, providing closure to the system. A temporal spectral formulation is used, ensuring the absence of numerical instabilities due to high frequency modes. The equations are simplified with a large aspect ratio approximation for an axisymmetric toroidal configuration. The fields are expanded in a Fourier series in the three spatial directions (radial, poloidal, and toroidal) reducing the problem in a matrix form. The code EM‐GLOGYSTO (Brunner et al. 1998; Falchetto et al. 2003; Joint Varenna‐Lausanne International Workshop 2002) has been developed to look at its solutions. Recently, we have included in the model an electrostatic equilibrium potential in order to provide E × B sheared flow. The code has been optimized for running in a parallel environment using MPI routines, and it allows us to study frequencies, growth rates, and mode structures of electromagnetic microinstabilities. Results on an AITG mode are shown to illustrate these features. [ABSTRACT FROM AUTHOR]

This article deals with some special cases which are extension of the strong summability of Fourier series with constant factor. We obtain a new equivalent form of inequalities [ABSTRACT FROM AUTHOR]

Presents a recursive formula for computing the sums of harmonic series of even order. Fourier series expansion of some periodic functions; Polynomials; Binomial coefficient.

The purpose of this article is to analyze the harmonic spectrum of the three-phase PWM inverter drives used in a power system. The analytical methods are based on a 3-D modulation model and double Fourier series (DFS). In this article, with Matlab, we analyzed the harmonic model created by DFS, and used time-domain simulation tools, Pspice and Simulink, to construct the model and simulation of the three-phase DC to AC PWM inverter found in the system. Results show that the harmonics generated by the simulator tools and that generated by DFS method are almost identical, and that the harmonic model constructed by means of the 3-D modulation model proposed is highly accurate. [ABSTRACT FROM AUTHOR]

Presents musical examples and computer tools that could be used in an introductory partial differential equation and Fourier Series class, where students work with the heat and wave equations. Analysis of the wave forms and harmonics of recorded sounds; Relation of a wave form composed of a summation of Fourier Series basis functions to the frequency domain; Wave equation with Dirichlet boundary conditions; Modes of a vibrating string.

In theory, there are many methods for the representation of signals. In practice, however, Fourier analysis involving the resolution of signals into sinusoidal components is used widely. There are several methods for Fourier analysis available for representation of signals. If the signal is periodic, then the Fourier series is used to represent the signal in terms of a set of harmonically related sinewaves. In this article new formulae for evaluating the trigonometric Fourier series coefficients when the signal under consideration is polynomial are developed by changing the integration to a derivation form. The solution presented here is an extension of the formulae proposed by Al-Smadi and Wilkes to the trigonometric Fourier series. The proposed technique is a powerful tool that can be used for solving practical science and engineering problems without excessive tedium. [ABSTRACT FROM AUTHOR]

In this paper there is proved a generalization of a theorem of Stein and Weiss concerning metric properties of the conjugate characteristic functions of given sets on the interval [0, 2π]. As an application for the Hilbert transform of a bounded function, the optimal correlation where the Favard's constant : is also established. [ABSTRACT FROM AUTHOR]

In this article, two dynamic models of the Torus machine are discussed. The first model uses the Fourier series expansion for the back emf and is implemented with PSpice. The second model uses the neural network estimation of the back emf and is implemented with Matlab/Simulink. The models may be used to study the steady-state as well as the transient performances of the machine operating as a motor or as a generator. Experimental results are compared with Matlab/Simulink and PSpice predictions for a number of operating conditions, including the complete starting transient. Excellent correlation between the experimental and simulated results is obtained which validates the Torus machine modeling approach. [ABSTRACT FROM AUTHOR]

Our aim in this paper is almost periodic solutions (ap-solutions) originating from an equilibrium state when the parameters of the difference equation are varied. We consider the bifurcation of ap-solutions for an ap-difference equation with parameters of x n +1 = f ( n , x n ; μ ), using the Green's function for regular ap-operators and Σ operators. [ABSTRACT FROM AUTHOR]

The main object of this paper is to prove the following theorem: Theorem . Let S ⊆ [a,b ] Then there exists f : [a,b ] → R such that f is differentiable on [a,b ], f ′ is bounded on [a,b ] and C = { x ∈ [a,b ], f ′ is continuous at x} = S if and only if S is a G δ set dense in [a,b ]. In section 5, consequences of the above theorem explore how badly discontinuous a derivative can be. Among these it will be shown that there exists a function f : [a,b ] → R such that f ′ is a Dirichlet function, and it will be shown that there exists no function f : [a,b ] → R such that D f′ = [ a , b ] - C f′ is an interval, and an example of a 'worst case' scenario is provided. The arguments stay within the realm of elementary classical analysis and are thus accessible to students who have encountered a first proof course in the subject. [ABSTRACT FROM AUTHOR]

A geometrically nonlinear theory of magnetothermoelasticity of electroconductive anisotropic plates in a magnetic field is developed. In this context, the Kirchhoff hypothesis is adopted for the plate modeling and the geometrical nonlinearities are considered in the von Kármán sense. In addition, the assumptions related to the distribution of electric and magnetic field disturbances through the plate thickness as proposed by Ambartsumyan and his collaborators are adopted. Based on the electromagnetic equations (i.e., the ones by Faraday, Ampère, Ohm, Maxwell, and Lorentz), on the modified Fourier law of heat conduction, and elastokinetic field equations, the three-dimensional coupled problem is reduced to an equivalent two-dimensional one appropriate to the theory of plates. The theory developed herein enables one to investigate the interacting effects among the magnetic, thermal, and elastic fields in orthotropic thin plates. As a special case, the problem of the free vibration of simply supported plate strips immersed in a transversal magnetic field is considered. Effects of the directionality property of the constituent material, magnetic and temperature fields, and electric conductivity, as well as thermal expansion coefficients, on the characteristics of vibrational behavior of the plate strips are investigated. [ABSTRACT FROM AUTHOR]

Of concern is the steady-state thermoelastodynamics of a helix, whose material is governed by a coupled thermoelastic model. Heat transfer follows either a parabolic (Fourier) or a hyperbolic (Maxwell-Cattaneo) rule. As a result, in the one-dimensional setting, there are three coupled differential equations that lead to a fast (primarily axial) and a slow (primarily torsional) wave. Particularly for harmonic motions, we find that the thermal diffusivity and the relaxation time (of the Maxwell-Cattaneo model) have minor effects, whereas the thermoelastic coupling constant is dominant--it speeds up these waves--and overall there is damping. Additionally, we review and note differences from solutions of equations governing plane harmonic waves in a non-centrosymmetric, thermoelastic micropolar continuum in three dimensions. [ABSTRACT FROM AUTHOR]

Considers problems of inverse scattering in a half space with passive boundary. Reduced wave equation; Physical meaning of the Robin boundary conditions; Fourier integrals.

Describes a trigonometric-series model of star junction consisting of H-plane coupled rectangular waveguides. Expression of the field in the form of simple trigonometric-series expansions which are constructed using the domain-product technique and the symmetry properties of the structure; Validity and efficiency of the solution; Application of the approach to the scattering characteristics of the unit.

Non-linear second-order differential equations whose solutions are the elliptic functions sn ( t, k ), cn ( t, k ) and dn ( t, k ) are investigated. Using Mathematica , high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions. [ABSTRACT FROM AUTHOR]

A result is discussed which permits the summing of series whose terms have more complicated sign patterns than simply alternating plus and minus. The Alternating Series Test, commonly taught in beginning calculus courses, is a corollary. This result, which is not difficult to prove, widens the series summable by beginning students and paves the way for understanding more advanced questions such as convergence of Fourier series. An elementary exposition is given of Dirichlet's Test for the convergence of a series and an elementary example suitable for a beginning calculus class and a more advanced example involving a Fourier series which is appropriate for an advanced calculus class are provided. Finally, two examples are discussed for which Dirichlet's Test does not apply and a general procedure is given for deciding the convergence or divergence of these and similar examples. [ABSTRACT FROM AUTHOR]