<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math> solutions of semialgebraic or definable equations

Periodic Wave Shock solutions of Burgers equations

In this paper we investigate the exact peroidic wave shock solutions of the Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr ε and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock type solution for the cauchy problem of the in...

Definable Representations of Definable C Groups

Let G be a compact affine definable C group and let r be∞ or ω. We prove that the representative definable C functions on G is dense in the space of continuous functions on G. Moreover we compare the category of compact affine definable C groups D with that of compact real algebraic groups A.

Improving the Accuracy of the Solutions of Riccati Equations

Simultaneous Quadratic Equations with Few or No Solutions

In this paper, we completely solve the simultaneous Diophantine equations x − az = 1, y − bz = 1 provided the positive integers a and b satisfy b−a ∈ {1, 2, 4}. Further, we show that these equations possess at most one solutions in positive integers (x, y, z) if b − a is a prime or prime power, under mild conditions. Our approach is (relatively) elementary in nature and relies upon classical re...

On meromorphic solutions of certain type of difference equations

‎We mainly discuss the existence of meromorphic (entire) solutions of‎ ‎certain type of non-linear difference equation of the form‎: ‎$f(z)^m+P(z)f(z+c)^n=Q(z)$‎, ‎which is a supplement of previous‎ ‎results in [K‎. ‎Liu‎, ‎L. Z‎. ‎Yang and X‎. ‎L‎. ‎Liu‎, ‎Existence of entire solutions of nonlinear difference‎ ‎equations‎, ‎Czechoslovak Math. J. 61 (2011)‎, no. 2, ‎565--576‎, and X‎. ‎G‎. ‎Qi‎...