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On the lower semicontinuity of certain integral functionals

Ennio De Giorgi, Giuseppe Buttazzo, Gianni Dal Maso (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si dimostra che il funzionale Ω f ( u , D u ) d x è semicontinuo inferiormente su W l o c 1 , 1 ( Ω ) , rispetto alla topologia indotta da L l o c 1 ( Ω ) , qualora l’integrando f ( s , p ) sia una funzione non-negativa, misurabile in s , convessa in p , limitata nell’intorno dei punti del tipo ( s , 0 ) , e tale che la funzione s f ( s , 0 ) sia semicontinua inferiormente su 𝐑 .

On the mappings 𝒵 A and A in intermediate rings of C ( X )

Mehdi Parsinia (2018)

Commentationes Mathematicae Universitatis Carolinae

In this article, we investigate new topological descriptions for two well-known mappings 𝒵 A and A defined on intermediate rings A ( X ) of C ( X ) . Using this, coincidence of each two classes of z -ideals, 𝒵 A -ideals and A -ideals of A ( X ) is studied. Moreover, we answer five questions concerning the mapping A raised in [J. Sack, S. Watson, C and C * among intermediate rings, Topology Proc. 43 (2014), 69–82].

On the non-existence of norms for some algebras of functions

Bertram Yood (1994)

Studia Mathematica

Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for Ω = n where ℝ is the reals.

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