Rational Exponents Research Articles

Generalized power series on a non-Archimedean field

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.

Optimal Gevrey classes for the existence of solution operators for linear partial differential operators in three variables

For a constant coefficient linear partial differential operator acting on all infinitely differentiable functions or ω-ultradifferentiable functions of Beurling type on euclidean 3-space, the existence of a continuous linear solution operator is investigated. It is shown that there is an optimal weight ω in the sense that a solution operator exists for a weight σ if and only if ω = O ( σ ) , provided that such an operator exists for at least one weight. Furthermore, the optimal class is either a Gevrey class of rational exponent or the class of all infinitely differentiable functions.

Fractals for functions with rational exponent

This paper investigates the fractals generated by the iteration of functions of the form f( z)= z α + c, where α is a rational non-integer number larger than 2, and z and c are complex numbers. We will examine both the dynamic and parameter spaces and discuss the consequences of the discontinuity of the function f( z). The images of these fractals allow for remarkable insight into the difficulty encountered in trying to extend results from polynomial cases to more generalized functions.

Fermat's Last Theorem for Rational Exponents

where the first 15/6 is really (e2,i)5/6 e5 ri/3, the second is (el0ri )5/6 eri/3, and the third is (e0)5/6 = 1. Thus the equation becomes the more believable e5i/3 + eri/3 -= 1. As equation (1) makes most of us uncomfortable (and certainly leads to confusion), it behooves us to rewrite the equation an/rm + bn/m= cn/m in the form (al/m)n + (bl/m)n =(cl/m)n and then ask: For what mth roots of positive integers a, b, and c and for what n > 2 with gcd(m, n) = I do we have an + b = cn? Our main theorem is:

Fermat's Last Theorem for Rational Exponents

Fermat's Last Theorem for Rational Exponents

Effective spin quantum numbers in iron, cobalt and nickel

The known macroscopic data for the temperature dependence of the normalized spontaneous magnetization, m s( T), of iron, cobalt and nickel are reconsidered and compared with our normalized zero-field hyperfine field data, h hf( T), obtained by Mössbauer spectroscopy or nuclear magnetic resonance. It is observed that for all three metals h hf( T) is not proportional to m s( T) and systematically smaller than m s( T). This is attributed to smaller magnetic interactions in the domain walls compared to the volume of the domains. The temperature dependence of m s( T) and h hf( T) can be described in the whole range 0< T< T C by a few temperature power functions with analytical changes (crossovers) at the intersections of these functions. Below the critical power law, m s∼( T C− T) β , further power functions of the absolute temperature according to m s( T)= A− cT ε , follow. From the observed rational exponents ε it can be concluded whether the effective spin is integer or half-integer. Using additionally the known data for the saturation magnetic moments and the effective paramagnetic Bohr magneton numbers it is suggested that for nickel the effective spin is S=1/2 for T→0 but S=1 for higher temperatures including the paramagnetic phase. For iron and cobalt the corresponding values are S=3/2 and 2 .

Units of measurement

It is shown that the symbols for the fundamental units of mechanics, namely length, time and mass, are capable of a meaningful interpretation as positive real parameters. Then a suitable parameter domain allows one to take the derived units into account. The formal manipulations usually carried out with symbols of physical quantities, involving real coefficients and units of different kinds affected by rational exponents, are consistent with that interpretation. The note is expected to serve as a well-motivated introduction to a deeper study of dimensional analysis.

Special factors, periodicity, and an application to Sturmian words

Let w be a finite word and n the least non-negative integer such that w has no right special factor of length \(n\) and its right factor of length n is unrepeated. We prove that if all the factors of another word v up to the length n + 1 are also factors of w, thenv itself is a factor ofw. A similar result for ultimately periodic infinite words is established. As a consequence, some ‘uniqueness conditions’ for ultimately periodic words are obtained as well as an upper bound for the rational exponents of the factors of uniformly recurrent non-periodic infinite words. A general formula is derived for the ‘critical exponent’ of a power-free Sturmian word. In particular, we effectively compute the ‘critical exponent’ of any Sturmian sequence whose slope has a periodic development in a continued fraction.

The wrong kind of gravity

The KPZ formula [V.G. Knizhnik, A.M. Polyakov, and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819] shows that coupling central charge c≤1 spin models to 2D quantum gravity dresses the conformal weights to get new critical exponents, where the relation between the original and dressed weights depends only on c. At the discrete level the coupling to 2D gravity is effected by putting the spin models on annealed ensembles of Φ 3 planar random graphs or their dual triangulations, where the connectivity fluctuates on the same time-scale as the spins. Since the sole determining factor in the dressing is the central charge, one could contemplate putting a spin model on a quenched ensemble of 2D gravity graphs with the “wrong” c value. We might then expect to see the critical exponents appropriate to the c value used in generating the graphs. In such cases the KPZ formula could be interpreted as giving a continuous line of critical exponents which depend on this central charge. We note that rational exponents other than the KPZ values can be generated using this procedure for the Ising, tricritical Ising and 3-state Potts models.

Global Optimization of Nonconvex Polynomial Programming Problems HavingRational Exponents

This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.

Distance measurements, splitting of electromagnetic waves caused by the dispersion and GPS retrieval of the model atmosphere

An atmosphere above a half space is considered with the dispersion represented by a relation between the electric field and the induction which contains derivatives of rational order and is similar to the empirical formula of Cole and Cole (1941), commonly used in experimental physics, and to the formula used by Jacquelin (1991) in studying the dispersion of energy in electric networks. The consequent index of refractionn contains a rational power of the imaginary frequencyif and is a polymorphic function off; this function, for each frequency, gives a set of different velocity fields whose number depends on the rational exponent ofif. Each electromagnetic wave leaving the source, with givenf and direction, is split in a number of waves with different velocities; ifn is a function of position, the paths of the waves are different and reach a given elevation at different points and times. Ifn is independent of the position, the paths of the waves coincide although the waves have different velocities. The length of a path and the travel time of electromagnetic waves in the atmosphere of a flat Earth model are computed. It is found that the difference between the arc length of the ray and the chord is nil to the second order of the refractivity. It is also seen that a change of water content in layers of the atmosphere, leaving the average velocity to a given elevation unchanged, may change the length of the ray paths to that elevation. It is found that the separation of the rays with the same frequency and direction at the source, causes small uncertainties in electromagnetic distance measurements which increase with the frequency. In the Liebe (1985) atmospheric model we considered frequencies on the range 1 GHz to 2 GHz and found that the arrival of the phases of the rays, with the same frequency in this range, with a zenithal angle smaller than 2π/5 and at a distance of about 104 km, are spread in less than 0.01 ns or 0.3 cm; which does not influence the accuracy presently achieved in distance measurements with electromagnetic waves. The dissipation of energy of the rays in the atmospheric model used, for zenithal angles smaller than 2π/5, is negligible for any length of the path. Formulae are given for the retrieval of a spherical model of the atmosphere of the Earth from a set of differences of the times of arrival, at two observing stations, of the waves emitted from satellites of known orbits.

Classification of oscillators in the Heisenberg-matrix representation

For an arbitrary finite and not too singular (presumably, phenomenological) superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential Schrodinger bound-state problem into its matrix equivalent with the minimalized number L of non-zero diagonals. The construction-with proofs-is performed via the standard power-series method in its non-Hermitean matrix (sometimes called Hill-determinant) version. All the simplest and (in this sense) 'algebraizable' interactions are then displayed up to L=5. An example of application in perturbation theory is included.

Distance measurements and splitting of electromagnetic waves in water caused by dispersion.

A half space is considered with the dispersion represented by a relation, between the electric field and the induction, which contains derivatives of fractional order and has a Fourier Transform similar to the empirical formula of Cole and Cole (1941), commonly used in experimental physics, and to the formula used by Jacquelin (1991) in studying the dispersion of energy in electric networks. The consequent index of refractionn contains a rational power of the imaginary frequencyif and is a polymorphic function off; this function, for each frequency, gives a set of different velocity fields whose number depends on the rational exponent ofif. Each electromagnetic wave leaving the surface of the half space, directed into the half space and with an assignedf, is split in a number of waves with different velocities; ifn is a function of the position the paths of the waves are different and reach a given depth at different points and times. In the case when the medium is water we assume for the dielectric parameter the expression most commonly used in the literature and containing a rational power ofif and the parameter values of Hasted (1973). It is found that the amplitude of each set of waves with the samef and direction at the source, due to the phase difference of the single components of the set and independently of the dissipation of energy, fluctuates with successive maxima whose amplitude decreases with increasing distance from the source; also the rate of decrease of the maxima is augmented by the dissipation. It is finally estimated that the measurement of a length of 1 km, using a frequency of 10 MHz, may be effected by an uncertainty of 38 ns or 1.22 m.

Spiked harmonic oscillator and Hill determinants

For an arbitrary superposition of the weakly singular power-law anharmonicities with rational exponents, we propose and prove a new Hill-determinant construction of bound states. In contrast to its predessors, our prescription is “minimal” - each spike in the potential adds just one nonzero diagonal to the quasi-Hamiltonian.

Algebraic approximants: a new method of series analysis

The authors propose a new method of series analysis in which the available series coefficients are fitted to an algebraic equation. Such a proposal restricts solutions to have algebraic singularities with rational critical exponents. The advantage of the method over the method of differential approximants is that in those cases where the two methods are exact, the algebraic approximant method usually (though not invariably) requires far fewer series coefficients for its exact specification, and hence discovery. The method is therefore potentially valuable in the study of two-dimensional critical systems in particular.

5 - REMOVED: Rational Exponents and Roots

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Calculus and Analytic Geometry

(NOTE: Every chapter ends with Questions to Guide Your Review, Practice Exercises, and Additional Exercises.) P. Preliminaries. Real Numbers and the Real Line. Coordinates, Lines, and Increments. Functions. Shifting Graphs. Trigonometric Functions. 1. Limits and Continuity. Rates of Change and Limits. Rules for Finding Limits. Target Values and Formal Definitions of Limits. Extensions of the Limit Concept. Continuity. Tangent Lines. 2. Derivatives. The Derivative of a Function. Differentiation Rules. Rates of Change. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation and Rational Exponents. Related Rates of Change. 3. Applications of Derivatives. Extreme Values of Functions. The Mean Value Theorem. The First Derivative Test for Local Extreme Values. Graphing with y e and y . Limits as x a a, Asymptotes, and Dominant Terms. Optimization. Linearization and Differentials. Newton's Method. 4. Integration. Indefinite Integrals. Differential Equations, Initial Value Problems, and Mathematical Modeling. Integration by Substitution--Running the Chain Rule Backward. Estimating with Finite Sums. Riemann Sums and Definite Integrals. Properties, Area, and the Mean Value Theorem. Substitution in Definite Integrals. Numerical Integration. 5. Applications of Integrals. Areas Between Curves. Finding Volumes by Slicing. Volumes of Solids of Revolution--Disks and Washers. Cylindrical Shells. Lengths of Plan Curves. Areas of Surfaces of Revolution. Moments and Centers of Mass. Work. Fluid Pressures and Forces. The Basic Pattern and Other Modeling Applications. 6. Transcendental Functions. Inverse Functions and Their Derivatives. Natural Logarithms. The Exponential Function. ax and logax. Growth and Decay. L'Hopital's Rule. Relative Rates of Growth. Inverse Trigonomic Functions. Derivatives of Inverse Trigonometric Functions Integrals. Hyperbolic Functions. First Order Differential Equations. Euler's Numerical Method Slope Fields. 7. Techniques of Integration. Basic Integration Formulas. Integration by Parts. Partial Fractions. Trigonometric Substitutions. Integral Tables and CAS. Improper Integrals. 8. Infinite Series. Limits of Sequences of Numbers. Theorems for Calculating Limits of Sequences. Infinite Series. The Integral Test for Series of Nonnegative Terms. Comparison Tests for Series of Nonnegative Terms. The Ratio and Root Tests for Series of Nonnegative Terms. Alternating Series, Absolute and Conditional Convergence. Power Series. Taylor and Maclaurin Series. Convergence of Taylor Series Error Estimates. Applications of Power Series. 9. Conic Sections, Parametrized Curves, and Polar Coordinates. Conic Sections and Quadratic Equations. Classifying Conic Sections by Eccentricity. Quadratic Equations and Rotations. Parametrizations of Plan Curves. Calculus with Parametrized Curves. Polar Coordinates. Graphing in Polar Coordinates. Polar Equations for Conic Sections. Integration in Polar Coordinates. 10. Vectors and Analytic Geometry in Space. Vectors in the Plane. Cartesian (Rectangular) Coordinates and Vectors in Space. Dot Products. Cross Products. Lines and Planes in Space. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates. 11. Vector-Valued Functions and Motion in Space. Vector-Valued Functions and Space Curves. Modeling Projectile Motion. Arc Length and the Unit Tangent Vector T. Curvature, Torison, and the TNB Frame. Planetary Motion and Satellites. 12. Multivariable Functions and Partial Derivatives. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials. The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives, Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange Multipliers. Taylor's Formula. 13. Multiple Integrals. Double Integrals. Areas, Moments, and Centers of Mass. Double Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and Moments in Three Dimensions. Triple Integrals in Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals. 14. Integration in Vector Fields. Line Integrals. Vector Fields, Work, Circulation, and Flux. Path Independence, Potential Functions, and Conservative Fields. Green's Theorem in the Plane. Surface Area and Surface Integrals. Parametrized Surfaces. Stokes's Theorem. The Divergence Theorem and a Unified Theory. Appendices. Mathematical Induction. Proofs of Limit Theorems in Section 1.2. Complex Numbers. Simpson's One-Third Rule. Cauchy's Mean Value Theorem and the Stronger Form of L'Hopital's Rule. Limits that Arise Frequently. The Distributive Law for Vector Cross Products. Determinants and Cramer's Rule. Euler's Theorem and the Increment Theorem.

5 - WITHDRAWN: Rational Exponents and Roots

This Book/Chapter has been withdrawn as the rights have been returned to the author

Extended continued fractions and energies of the anharmonic oscillators

We describe the analytic solution to the Schrödinger eigenvalue problem for the class of the central potentials V(r)=∑δ∈Zaδrδ, where a−2&amp;gt;−1/4, amax δ &amp;gt;0, Z is an arbitrary finite set of the integer or rational exponents, −2≤δ1&amp;lt;δ2&amp;lt;⋅⋅⋅&amp;lt;δI, and the couplings aδ satisfy only one auxiliary (formal, ‘‘superconfinement’’) restriction of the type aδI−1 &amp;gt;0. The formalism is based on an analysis of the asymptotic behavior of the explicit regular solution ψ(r) and issues in the formulation of the ‘‘secular’’ equation 1/L1(E)=0 which determines the binding energies. The main result is the rigorous construction of L1(E) as a generalized (‘‘extended’’) and convergent continued fraction. The method cannot be applied to the aδI−1 &amp;lt;0 cases—this disproves the closely related Hill-determinant approach as conjectured recently by Singh et al. for the simplest potentials with Z={−2,2,4,6} and Z={−2,−1,1,2}.

5 - WITHDRAWN: Rational Exponents and Roots

This Book/Chapter has been withdrawn as the rights have been returned to the author