On the Discretization of Double-Bracket Flows

This paper extends the method of Magnus series to Lie-algebraic equations originating in double-bracket flows. We show that the solution of the isospectral flow Y'=[[Y,N],Y] , Y(0)=Y ₀ ∈\Sym(n) , can be represented in the form Y(t)=e ^Ω(t) Y ₀ e ^-Ω(t) , where the Taylor expansion of Ω can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolor leaves. This approach is extended to other Lie-algebraic equations that can be appropriately expressed in terms of a finite ``alphabet.'     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Fourier series and elliptic functions

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.     

Convergence for Fourier series solutions of the forced harmonic oscillator II

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when the forcing function is bounded, piecewise continuous, and piecewise smooth. The series solution and its term-by-term derivative converge uniformly over the entire real line. The term-by-term differentiation produces a series for the second derivative that converges pointwise and uniformly over any interval not containing a jump discontinuity of the forcing function.     

Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0

This paper deals with the solutions of the differential equation u?+λ²zu+(α?1)λ²u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α?1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.     

Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (?y/?α, ?²y/?α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which ‘build-in’ the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the ‘secular’ components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.     

The Gibbs' phenomenon from a signal processing point of view

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫₀ ^π (sin x / x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function ? (x) is ‘very nearly’ equal to the convolution integral 1/π ∫ _-∞ ^+∞ ? (x - t)(sin nt / t) dt.     

SYMMETRIC GROUP ACTIONS ON TENSOR PRODUCTS OF HOPF ALGEBROIDS

We describe an action of the symmetric group Σ _n on A ^{? n ? 1}, the n ? 1-fold tensor product of A over K, for (K,A) a Hopf algebroid. This arises in a natural way in stable homotopy theory: when A = E _* E, the ‘co-operations’ in the cohomology theory associated to a suitable ring spectrum E, this action is induced from the natural action on the n-fold smash product E ⁽ⁿ⁾. The case n = 2 is classical: the switch action of Σ₂ on E ∧ E induces the canonical conjugation of E _* E. Therefore we may think of the symmetric group actions as ‘higher order conjugation maps’.     

Conversion of a rhotrix to a ‘coupled matrix’

In this note, a method of converting a rhotrix to a special form of matrix termed a ‘coupled matrix’ is proposed. The special matrix can be used to solve various problems involving n?×?n and (n?–?1)?×?(n?–?1) matrices simultaneously.     

The auto‐correlation equation on the finite interval

Applying the Fourier cosine transformation, the quadratic auto‐correlation equation on the finite interval [0,T] of the positive real half‐axis ?₊ is reduced to a problem for the modulus of the finite complex Fourier transform of the solution. From the solutions of this problem L²‐solutions of the auto‐correlation equation are obtained in closed form. Moreover, as in the case of the equation on ?₊ a Lavrent'ev regularization procedure for the auto‐correlation equation is suggested. Copyright © 2003 John Wiley & Sons, Ltd.     

Inverse Scattering for Schrödinger Operators with Miura Potentials,II. Different Riccati Representatives

This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u′ + u ² for some u ∈ L ²(?). We consider potentials for which there exist ‘left’ and ‘right’ Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev–Marchenko potentials in L ¹(?, (1 + |x|)dx) generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.     

Radially symmetric global classical solutions of non-linear wave equations

The Cauchy problem for semilinear wave equations u_tt ? Δu + h(|x|)u^p = 0 with radially symmetric smooth ‘large’ data has a unique global classical solution in arbitrary space dimensions if h is non-negative and p any odd integer provided the smooth factor h vanishes with sufficiently high order at the origin and is bounded together with its derivatives.     

Solution of a Gellerstedt problem for the Lavrent’ev-Bitsadze equation

We write out the solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation in the form of a series for the case in which the elliptic part of the domain is a half-strip and the boundary data are nonzero only on the characteristics in the hyperbolic parts of the domain. We obtain new results on the basis property, completeness, and minimality of the system of sines with discontinuous phase used in the series representation of the solution of the Gellerstedt problem. We prove the uniform convergence and justify the possibility of term-by-term differentiation of the series.     

Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on ? ⁿ . Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform $\tilde f$ cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ? ⁿ , the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.     

On the numerical integration of an odd periodic function over a half-period

A method is described for the numerical evaluation of integrals of the form ∫ ₀ ^p f(t)dt, wheref is an odd periodic function with period 2p. The method is based on term-by-term integration of the Fourier sine series forf(t).     

Analytic solution of an exterior Neumann problem in a non‐convex domain

A new method, based on the Kelvin transformation and the Fokas integral method, is employed for solving analytically a potential problem in a non‐convex unbounded domain of ?², assuming the Neumann boundary condition. Taking advantage of the property of the Kelvin transformation to preserve harmonicity, we apply it to the present problem. In this way, the exterior potential problem is transformed to an equivalent one in the interior domain which is the Kelvin image of the original exterior one. An integral representation of the solution of the interior problem is obtained by employing the Kelvin inversion in ?² for the Neumann data and the ‘Neumann to Dirichlet’ map for the Dirichlet data. Applying next the ‘reverse’ Kelvin transformation, we finally obtain an integral representation of the solution of the original exterior Neumann problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Algebraic optimization: The Fermat-Weber location problem

The Fermat-Weber location problem is to find a point in ⁿ that minimizes the sum of the (weighted) Euclidean distances fromm given points in ⁿ . In this work we discuss some relevant complexity and algorithmic issues. First, using Tarski's theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem, where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one-dimensional polynomial. Secondly, we exhibit an explicit solution to the strong separation problem associated with the Fermat-Weber model. This separation result shows that an-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.     

The Lagrange inversion theorem in the smooth case

The classical Lagrange inversion theorem is a concrete, explicit form of the implicit function theorem for real analytic functions. An explicit construction shows that the formula is not true for all merely smooth functions. The authors modify the Lagrange formula by replacing the smooth function by its Maclaurin polynomials. The resulting modified Lagrange series is, in analogy to the Maclaurin polynomials, an approximation to the solution function accurate to o(x^N) as x→0.     

Formally Real Involutions on Central Simple Algebras

An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r ^# r where r ? R is nonzero. Suppose that R is a central simple algebra (i.e., R = M _n (D) for some integer n and central division algebra D) and # is an involution on R of the form r ^# = a ^?1 r? a, where ? is some transpose involution on R and a is an invertible matrix such that a? = ±a. In Section 1 we characterize formal reality of # in terms of a and ?| _D . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebra D ? _F K ? M _n (K). Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x ? tr(x ^# x) is not positive semidefinite.