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Latest Published Articles

\(L^p-\) boundedness for integral transforms associated with singular partial differential operators

OMA-Vol. 2 (2018), Issue 2, pp. 53–77 | Open Access Full-Text PDF
Lakhdar T. Rachdi, Samia Sghaier
Abstract:We define fractional transforms \(\mathscr{R}_\mu\) and \(\mathscr{H}_\mu\), \(\mu>0\) on the space \(\mathbb{R}\times\mathbb{R}^n\). First, we study these transforms on regular function spaces and we establish that these operators are topological isomorphisms and we give the inverse operators as integro differential operators. Next, we study the \(L^p\)-boundedness of these operators. Namely, we give necessary and sufficient condition on the parameter \(\mu\) for which the transforms \(\mathscr{R}_\mu\) and \(\mathscr{H}_\mu\) are bounded on the weighted spaces \(L^p([0,+\infty[\times\mathbb{R}^n,r^{2a}dr\otimes dx)\) and we give their norms.
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Remarks on Fractional Locally Harmonious Coloring

OMS-Vol. 2 (2018), Issue 1, pp. 301–306 Open Access Full-Text PDF
Wei Gao
Abstract:Locally harmonious coloring is a relax version of standard harmonious coloring which only needs that the color pairs for adjacent edges are different. In this remark, we introduce the concept of fractional locally harmonious coloring, and present some basic facts for this coloring.
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Necessary and sufficient condition for a surface to be a sphere

OMA-Vol. 2 (2018), Issue 2, pp. 51–52 | Open Access Full-Text PDF
Alexander G. Ramm
Abstract:Let \(S\) be a \(C^{1}\)-smooth closed connected surface in \(\mathbb{R}^3\), the boundary of the domain \(D\), \(N=N_s\) be the unit outer normal to \(S\) at the point \(s\), \(P\) be the normal section of \(D\). A normal section is the intersection of \(D\) and the plane containing \(N\). It is proved that if all the normal sections for a fixed \(N\) are discs, then \(S\) is a sphere. The converse statement is trivial.
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BOOK-foundations-of-mathematical-analysis-and-semigroups-theory
BOOK - NULL CONTROLLABILITY OF DEGENERATE AND NON-DEGENERATE SINGULAR PARABOLIC