arXiv subjects

Tejinder Neelon

Publications and source records attributed to Tejinder Neelon.

2 recordsLinked to original sources

arXiv2011

Nonlinear Convergence Sets of Divergent Power Series

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh, is introduced. Given a family y=ϕ_{s}(t,x)=sb_{1}(x)t+b_{2}(x)t^{2}+... of analytic curves in C\timesC^{n} passing through the origin, Conv_ϕ(f) of a formal power series f(y,t,x)\inC[[y,t,x]] is defined to be the set of all s\inC for which the power series f(ϕ_{s}(t,x),t,x) converges as a series in (t,x). We prove that for a subset E\subsetC there exists a divergent formal power series f(y,t,x)\inC[[y,t,x]] such that E=Conv_ϕ(f) if and only if E is a F_{σ} set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case ϕ_{s}(t,x)=st.

math.CV↗

arXiv2011

On Boman's Theorem On Partial Regularity Of Mappings

Let Λ\subsetR^{n}\timesR^{m} and k be a positive integer. Let f:R^{n}\rightarrowR^{m} be a locally bounded map such that for each (ξ,η)\inΛ, the derivatives D_{ξ}^{j}f(x):=|((d^{j})/(dt^{j}))f(x+tξ)|_{t=0}, j=1,2,...k, exist and are continuous. In order to conclude that any such map f is necessarily of class C^{k} it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ(ξ,η) which is linear in η=(η_{1},η_{2},...,η_{m}) and homogeneous of degree k in ξ=(ξ_{1},ξ_{2},...,ξ_{n}). This generalizes a result of J. Boman for the case k=1. The statement and the proof of a theorem of Boman for the case k=\infty is also extended to include the Carleman classes C{M_{k}} and the Beurling classes C(M_{k}).

math.CV↗