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Convex Symmetrization: The equality case in Pòlya-Szegö Inequality
No full textLet ||•|| denote an arbitrary norm on |Rn and let ||•||_0 be its dual norm . Relative to this arbitrary norm, the perimeter of a set E of R^n is defined in a standard way that reduces to the usual classical perimeter in the case of the Euclidean norm. For u an arbitrary function defined on R^n let u^# be the convex rearrangement of u. The main result of this paper is the following: If u is a nonnegative function in the Sobolev space W^{1,p}(Rn), (1<p<∞), with the property that the set of critical points of |u^#| has measure zero and realize the equality in the Pòlya-Szegö inequality then u= u# a.e. in Rn (up to translations). The Euclidean version of this result was established by J. E. Brothers and the W.P. Ziemer and the original proof, even if based on a geometrical clear approach, the rigorous part justification of the argumants is accomplished after overcoming serious technical difficulties by means of results from geometric measure theory. In this paper is given a proof based on arguments from the classical theory of Sobolev spaces. The very worth of this method is that it is more flexible and can be easily adapted to more general problems. Indeed it has been used from other authors to prove quantitative version of this inequality
Enantioselective Synthesis of Polycyclic Ketones by Desymmetrisation of Bis(phenylsulfonyl)alkenes with Chiral Alcoholates. Control of the Absolute Configuration by a Simple Modification of the Chiral Auxiliary
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A priori estimates for elliptic equations with gradient dependent term and zero order term
No full textIn this paper we prove an existence result for “solution obtained as limit of approximations” to a class of Dirichlet boundary value problems whose prototype is {−Δpu=β(1+|∇u|)q+c(x)|u|p−2u+finΩu=0on∂Ω, where Ω is a bounded open subset of RN, N≥2, 1<2, Δpu=div(|∇u|p−2∇u), [Formula presented], β is a positive constant, c with c≥0, c≠0 and f are measurable functions satisfying suitable summability conditions depending on q. We further assume smallness assumptions on β, c and f. Our approach is based on Schauder's fixed point theorem. Similar results can be proved also for 2≤
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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