On power series solutions for the Euler equation, and the Behr-Necas-Wu initial datum

We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Necas and Wu in ESAIM: M2AN 35 (2001) 229-238; here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed the power series in time for the solution, determining the first 35 terms by computer algebra. Their calculations suggested for the series a finite convergence radius \tau_3 in the H^3 Sobolev space, with 0.32 < \tau_3 < 0.35; they regarded this as an indication that the solution of the Euler equation blows up. We have repeated the calculations of Behr, Necas and Wu, using again computer algebra; the order has been increased from 35 to 52, using the symmetries of the initial datum to speed up computations. As for \tau_3, our results agree with the original computations of Behr, Necas and Wu (yielding in fact to conjecture that 0.32 < \tau_3 < 0.33). Moreover, our analysis supports the following conclusions: (a) The finiteness of \tau_3 is not at all an indication of a possible blow-up. (b) There is a strong indication that the solution of the Euler equation does not blow up at a time close to \tau_3. In fact, the solution is likely to exist, at least, up to a time \theta_3 > 0.47. (c) Pade' analysis gives a rather weak indication that the solution might blow up at a later time.Comment: 34 pages, 8 figure

Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra

There are many complicated and fussy mathematical analysis processes in geodesy, such as the power series expansions of the ellipsoid’s eccentricity, high order derivation of complex and implicit functions, operation of trigonometric function, expansions of special functions and integral transformation. Taking some typical mathematical analysis processes in geodesy as research objects, the computer algebra analysis are systematically carried out to bread, deep and detailed extent with the help of computer algebra analysis method and the powerful ability of mathematical analysis of computer algebra system. The forward and inverse expansions of the meridian arc in geometric geodesy, the nonsingular expressions of singular integration in physical geodesy and the series expansions of direct transformations between three anomalies in satellite geodesy are established, which have more concise form, stricter theory basis and higher accuracy compared to traditional ones. The breakthrough and innovation of some mathematical analysis problems in the special field of geodesy are realized, which will further enrich and perfect the theoretical system of geodesy

Symbolic finite element analysis using computer algebra: Heat transfer in rectangular duct flow

AbstractThe problem of heat transfer in fully developed laminar flow in a rectangular duct is solved using a symbolic finite element method. The Nusselt number is obtained as a power series of the aspect ratio of the duct. The solution procedure here differs from the conventional finite element method, in that the aspect ratio remains in symbolic form throughout the computation. Part of the computation is done using the computer algebra system Mathematica. However, the most computational intensive part which involves a Gauss elimination in symbolic form is implemented using an ordinary computer program without resorting to a computer algebra system. The agreement between the results from the present work and those from exact numerical procedures is reasonable

Sampling Algebra Structures on Minimal Free Resolutions

Ideals in the ring of power series in three variables can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which algebra structures actually occur; this realizability question was formally raised by Avramov in 2012. We discuss the outcomes of an experiment performed to shed light on Avramov's question: Using the computer algebra system Macaulay2, we classify a billion randomly generated ideals and build a database with examples of ideals of all classes realized in the experiment. Based on the outcomes, we discuss the status of recent conjectures that relate to the realizability question.Comment: 20 p

An algebraic criterion for the onset of chaos in nonlinear dynamic systems

The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one