Optimal with respect to accuracy methods for evaluating hypersingular integrals

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M), Ω¯ur,γ(Ω,M), Ω=[−1,1]l, l=1,2,…,M=Const, and γ is a real positive number. The functions that belong to classes Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M) have bounded derivatives up to the rth order in domain Ω and derivatives up to the sth order (s=r+⌈γ⌉) in domain Ω∖Γ, Γ=∂Ω. Moduli of derivatives of the vth order (r<v≤s) are power functions of d(x,Γ)−1(1+|lnd(x,Γ)|), where d(x,Γ) is a distance between point x and Γ. The interest in these classes of functions is due to the fact that solutions of singular and hypersingular integral equations are their members. Moreover various physical fields, in particular gravitational and electromagnetic fields belong to these classes as well. We give definitions of optimal with respect to accuracy methods for solving hypersingular integrals. We constructed optimal with respect to order of accuracy quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M).

Realization of fractional band pass filter on reconfigurable analog device

This paper presents a realization of fractional-order Band pass-filter (FOBF) based on the concepts of fractional order inductors and fractional order capacitors. The FOBF is designed and implemented using both simulation and hardware approaches. The proposed filter order is considered up to second order or less with any real positive number. One of the cases is considered when α ≤ 1 and β ≥ 1. In the second case, the filter is designed when β ≤ 1 and α ≥ 1. In order to calculate the optimal filter parameters, the modified Particle Swarm Optimization (mPSO) algorithm has been utilized for coefficient tuning. Also, a generalized approach to design any second order FOBF is discussed in this work. The realization and performance assessment have been carried out in simulation environment as well as in lab experiment with field programmable analog array (FPAA) development board. The experimental results indicate the value of efforts to realize the fractional filter.

Dynamics of Current, Charge and Mass

Electricity plays a special role in our lives and life. The dynamics of electrons allow light to flow through a vacuum. The equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term, the displacement current (of a vacuum). The displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as the entire source of the curl of the magnetic field, as Maxwell did.We show that the Bohm formulation of quantum mechanics allows the easy definition of the total current, and its conservation, without the dificulties implicit in the orthodox quantum theory. The orthodox theory neglects the reality of magnitudes, like the currents, during times that they are not being explicitly measured.We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. We point out that displacement current is handled correctly in electrical engineering by ‘stray capacitances’, although it is rarely discussed explicitly. Matter does not behave as physicists of the 1800’s thought it did. They could only measure on a time scale of seconds and tried to explain dielectric properties and polarization with a single dielectric constant, a real positive number independent of everything. Matter and thus charge moves in enormously complicated ways that cannot be described by a single dielectric constant,when studied on time scales important today for electronic technology and molecular biology. When classical theories could not explain complex charge movements, constants in equations were allowed to vary in solutions of those equations, in a way not justified by mathematics, with predictable consequences. Life occurs in ionic solutions where charge is moved by forces not mentioned or described in the Maxwell equations, like convection and diffusion. These movements and forces produce crucial currents that cannot be described as classical conduction or classical polarization. Derivations of conservation of current involve oversimplified treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave in that simple way-not even approximately-classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved inside atoms. We show that current is conserved exactly in any material no matter how complex are the properties of dielectric, polarization, or conduction currents. Electricity has a special role because conservation of current is a universal law.Most models of chemical reactions do not conserve current and need to be changed to do so. On the macroscopic scale of life, conservation of current necessarily links far spread boundaries to each other, correlating inputs and outputs, and thereby creating devices.We suspect that correlations created by displacement current link all scales and allow atoms to control the machines and organisms of life. Conservation of current has a special role in our lives and life, as well as in physics. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.We surely need successful models as we try to control macroscopic functions by atomic interventions, in technology, life, and medicine. Maxwell’s displacement current lets us see stars. We hope it will help us see how atoms control life.

One Can Hear the Area of a Torus by Hearing the Eigenvalues of the Polyharmonic Operators

This paper considers the asymptotic properties for the spectrum of a positive integer power l of the Laplace-Beltrami operator acting on an n-dimensional torus T. If N(λ) is the number of eigenvalues counted with multiplicity, smaller than a real positive number, we establish a Weyl-type asymptotic formula for the spectral problem of the polyharmonic operators on T, that is, as λ → +∞N (λ) ~ ωwhere ω

Simple Models of Coastal-Trapped Waves Based on the Shape of the Bottom Topography

Solutions of barotropic coastal-trapped waves in the shallow-water context are discussed for different shapes of the bottom topography. In particular, an infinite family of topographic waves over continental shelves characterized by a shape parameter is considered. The fluid depth is proportional to xs, where x is the offshore coordinate and s is a real, positive number. The model assumes the rigid-lid approximation and a semi-infinite domain 0 ≤ x ≤ ∞. The wave structure and the dispersion relation depend explicitly on the shape parameter s. Essentially, waves over steeper shelves possess higher frequencies and phase speeds. In addition, the wave frequency is independent of the alongshore wavenumber k, implying a zero group velocity component along the coast. The advantages and limitations of this formulation, as well as some comparisons with other models, are discussed in light of numerical simulations for waves over arbitrary topography within a finite domain. The numerical calculations show that the frequency of the waves present a nondispersive regime at small wavenumbers (observed by several authors), followed by a constant value predicted by the analytical solutions for larger k. It is concluded that these frequencies can be considered as an upper limit reached by barotropic coastal-trapped waves over the infinite family of xs-bottom profiles, regardless of the horizontal and vertical scales of the system. The modification of the dispersion curves in a stratified ocean is briefly discussed.

Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications

A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so-obtained can be expressed explicitly in terms of multivariate Mittag-Leffler functions. In the case where the multiorders are multiples of a common real positive number, the solutions can be reduced to linear combinations of Mittag-Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatory under certain conditions. This technique is illustrated in detail by two concrete examples, namely, the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions and simulations of the corresponding solutions are given.

HELICAL MAGNETIC FIELDS FROM INFLATION

We analyze the generation of seed magnetic fields during de Sitter inflation considering a noninvariant conformal term in the electromagnetic Lagrangian of the form [Formula: see text], where I(ϕ) is a pseudoscalar function of a nontrivial background field ϕ. In particular, we consider a toy model that could be realized owing to the coupling between the photon and either a (tachyonic) massive pseudoscalar field or a massless pseudoscalar field nonminimally coupled to gravity, where I follows a simple power law behavior I(k,η) = g/(-kη)β during inflation, while it is negligibly small subsequently. Here, g is a positive dimensionless constant, k the wave number, η the conformal time, and β a real positive number. We find that only when β = 1 and 0.1 ≲ g ≲ 2 can astrophysically interesting fields be produced as excitation of the vacuum, and that they are maximally helical.

Fractional Integrative Operator and Its FPGA Implementation

In this paper the fractional order integrative operator s−m, where m is a real positive number, is approximated via a mathematical formula and then an hardware implementation of fractional integral operator is proposed using Field Programmable Gate Array (FPGA). Digital hardware implementation of fractional-order integral operator requires careful consideration of issue of system performance, hardware cost, and hardware speed. FPGA-based implementation are up to one hundred times faster than implementations based on micro-processors; this extra speed can be exploited to allow higher performance in terms of digital approximations of fractional-order systems.

Bernstein's Inequality in the Bivariate Case

SummaryIf (Xl, X2,…, Xn), is a set of n independent random variables, such that EXi=0, Var and if t is a real positive number and , then Bernstein [2] has given an upper bound for Pr when the X's are bounded. The best English language discussion of Bernstein's work is probably by Bennett [1].

On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems

Free vibrations in normal modes are examined for a system consisting of two unequal (or equal) masses, interconnected by a nonlinear coupling spring, and each mass connected by nonlinear unequal (or equal) anchor springs to fixed points. All spring forces are odd functions, and proportional to the k’th power, of the spring deflections, where k is a real, positive number. The frequency-amplitude relations for the in and out-of-phase modes are derived without approximation, the stability of these modes is analyzed, and several numerical examples are worked out. A surprising feature of these systems is that they may have a greater number of normal modes than they have degrees of freedom.