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On the notions of absolute continuity for functions of several variables

Stanislav Hencl (2002)

Fundamenta Mathematicae

Absolutely continuous functions of n variables were recently introduced by J. Malý [5]. We introduce a more general definition, suggested by L. Zajíček. This new absolute continuity also implies continuity, weak differentiability with gradient in Lⁿ, differentiability almost everywhere and the area formula. It is shown that our definition does not depend on the shape of balls in the definition.

On weak Hessian determinants

Luigi D'Onofrio, Flavia Giannetti, Luigi Greco (2005)

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

We consider and study several weak formulations of the Hessian determinant, arising by formal integration by parts. Our main concern are their continuity properties. We also compare them with the Hessian measure.

Polynomials and degrees of maps in real normed algebras

Takis Sakkalis (2020)

Communications in Mathematics

Let 𝒜 be the algebra of quaternions ℍ or octonions 𝕆 . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f ( t ) ∈ 𝒜 [ t ] has a root in 𝒜 . As a consequence, the Jacobian determinant | J ( f ) | is always non-negative in 𝒜 . Moreover, using the idea of the topological degree we show that a regular polynomial g ( t ) over 𝒜 has also a root in 𝒜 . Finally, utilizing multiplication ( * ) in 𝒜 , we prove various results on the topological degree of products...

Spectral synthesis and the Pompeiu problem

L. Brown, B. Schreiber, B. A. Taylor (1973)

Annales de l'institut Fourier

It is shown that every closed rotation and translation invariant subspace V of C ( R n ) or δ ( R n ) , n ≥ 2 , is of spectral synthesis, i.e. V is spanned by the polynomial-exponential functions it contains. It is a classical problem to find those measures μ of compact support on R 2 with the following property: (P) The only function f ∈ C ( R 2 ) satisfying ∫ R 2 f ∘ σ d μ = 0 for all rigid motions σ of R 2 is the zero function. As an application of the above result a characterization of such measures is obtained in terms of their Fourier-Laplace transforms....

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