Propagation of Gevrey singularities for solutions of linear differential operators with constant coefficients

Summary In [4]L. Hörmander has given sufficient conditions for propagation of C singularities for solutions of linear differential operators P with constant coefficients in terms of limit operators called «localization of P at infinity». In this paper a result (Theorem 1.2) of the same type concerning the propagation of Gevrey singularities is given.     

Complementi per una formula di rappresentazione per funzioni analitiche in un cono aperto diR n

Riassunto Si prova una formula di rappresentazione per funzioni analitiche in un cono aperto Ω⊂R ⁿ

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Summary An integral representation formula for analytic functions in any open cone Ω⊂R ⁿ is proved.

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Lavoro eseguito nell’ambito dell’attività del Gruppo Nazionale per l’Analisi Funzionale e le sue Applicazioni.     

Sull’integrazione frazionaria anisotropa

Riassunto In questa nota si indica il tipo della singolarità nell'origine della trasformata di Fourier di certe funzioni; mediante, tale risultato si possono rapidamente ritrovare alcune maggiorazioni circa i nuclei di integrazione frazionaria anisotropa già stabiliti daP. I. Lizorkin.

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Summary In this paper the form of the singularity near the origine of the Fourier trasform of certain functions is shown; with the aid of this result some estimates for the kernel of non-isotropic fractional integration previously proved byP. I. Lizorkin are established in a very simple way.

     

Quasi-linear weakly hyperbolic equations with Gevrey-Levi conditions

This paper is devoted to the study of the analytic regularity for real solutions of a non-linear weakly hyperbolic equation of the form: $$\sum\limits_{m - (1 - e)r< |a| \leqslant m} {a_a (y,u^{(\beta )} } )\partial _y^a u = g(y,u^{(\beta )} ){\text{, |}}\beta | \leqslant m - (1 - e)r,$$ wherea _α andg are analytic functions of their arguments,r≥2 denotes the largest multiplicity of the characteristic roots of (*) and ?∈(0,(r?1)/r). Assuming that 026 the characteristic roots are of constant multiplicity and that the generalized Levi’s conditions related to the index ? are satisfied, we prove that, ifu is a solution of (*) 026 and belongs to a Gevrey class of index σ<1/?, then the analyticity of Cauchy data propagates according to the geometry of the influence domains of the equation.     

The Cauchy problem for quasilinear SG‐hyperbolic systems

We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × ?ⁿ and presenting a linear growth for |x | → ∞. We prove well‐posedness in the Schwartz space ?? (?ⁿ ). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On adaptive step-size selections in gradient methods

In recent years several proposals for the step-size selection have largely improved the gradient methods, in the case of both constrained and unconstrained nonlinear optimization. We introduce a new step-size rule with some crucial properties. We design step-size selection strategies where the new rule and a standard Barzilai-Borwein (BB) rule can be adaptively alternated to get meaningful convergence rate improvements in comparison with other BB-like gradient schemes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementazione di tests probabilistici di primalità

Sunto In questo lavoro si considerano due tests probabilistici di primalità per interi disparim di forma qualsiasi. Gli algoritmi sono tali che sem è dichiarato composto allora lo è certamente, mentre sem è dichiarato primo il risultato ha un certo margine di errore, che può essere reso arbitrariamente piccolo. Un programma scritto e compilato in linguaggio FORTRAN, applicabile ad interi fino a 10²⁰⁰⁰ ed adatto anche a personal computer, permette un confronto dei due tests sulla base del tipo dei risultati e del tempo di elaborazione, fornendo diverse opzioni ed una stima del limite superiore per l’eventuale errore in una dichiarazione di probabile primalità.

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Summary In this paper we consider two probabilistic tests for determining the primality of any odd integerm. Both algorithms have the property that ifm is declared to be composite than it is guaranteed to be so. On the other hand, ifm is declared to be prime than it is sure to be prime only within a certain margin of error which can be made arbitrarily small. A program, written and compiled in the computer language FORTRAN enable this tests to be applied to integers up to 10²⁰⁰⁰, allowing a comparison between the two tests with respect to their running time and with an esplicit bound for the maximum possible error whenm is declared prime. The program is sufficiently compact to be implemented on a personal computer such as an IBM PS/2 with an arithmetic co-processor

     

Well-posedness in Sobolev spaces for semi-linear 3-evolution equations

We prove local in time well-posedness of the Cauchy problem in Sobolev spaces for semi-linear 3-evolution equations of the first order. We require real principal part, but complex valued coefficients for the lower order terms. Therefore decay conditions on the imaginary parts are needed, as \(x\rightarrow \infty \) .