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## Singular integral operators on 53-0153-0153-01-spaces

### Annali di Matematica Pura ed Applicata (1923 -) (1988-12-01) 153: 53-62 , December 01, 1988

### Summary

It is shown that certain singular integral operators with variable kernels leave invariant the*L*^{v,Φ}-spaces studied by Campanato, Stampacchia, Peetre and others. Our results extend Peetre's work on convolution operators.

## Meromorphic approximation theorem in a Stein space

### Annali di Matematica Pura ed Applicata (1923 -) (2005-06-01) 184: 263-274 , June 01, 2005

We prove the meromorphic version of the Weil–Oka approximation theorem in a reduced Stein space *X* and give some characterizations of meromorphically
$\mathcal{O}(X)$
-convex open sets of *X*. As an application we prove that for every meromorphically
$\mathcal{O}(X)$
-convex open set *D* of a reduced Stein space *X* with no isolated points there exists a family
$\mathcal{F}$
of holomorphic functions on *X* such that the normality domain
$D(\mathcal{F})$
of
$\mathcal{F}$
coincides with *D*.

## Double loop algebras and elliptic root systems

### Annali di Matematica Pura ed Applicata (1923 -) (2017-04-01) 196: 743-771 , April 01, 2017

In this note, we describe an elliptic root system and elliptic Weyl group, due to Saito (Publ RIMS Kyoto Univ 21:75–179, 1985), from view point of double loop algebra and its group. A natural action of the double loop group will be introduced on a trivial $$\mathbb {C}^*$$ -bundle over the space of $$\overline{\partial }$$ -connections on a $$C^\infty $$ -trivial principal bundle over an elliptic curve that would be constructed from 2-dimensional central extension of a double loop algebra. The invariant theory of the elliptic Weyl group will be also discussed.

## Semilinear delay evolution equations with measures subjected to nonlocal initial conditions

### Annali di Matematica Pura ed Applicata (1923 -) (2016-10-01) 195: 1639-1658 , October 01, 2016

We prove a global existence result for bounded solutions to a class of abstract semilinear delay evolution equations with measures subjected to nonlocal initial data of the form $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)=\{Au(t)+f(t,u_t)\}\mathrm{d}t+\mathrm{d}g(t),&{}\quad t\in \mathbf{R}_+,\\ u(t)=h(u)(t),&{}\quad t\in [\,-\tau ,0\,], \end{array}\right. \end{aligned}$$ where $$\tau \ge 0$$ , $$A:D(A)\subseteq X\rightarrow X$$ is the infinitesimal generator of a $$C_0$$ -semigroup, $$f:\mathbf{R}_+\times \mathcal {R} ([\,-\tau ,0\,];X)\rightarrow X$$ is continuous, $$g\in BV_{\mathrm{loc}}(\mathbf{R}_+;X)$$ , and $$h:\mathcal {R} _b(\mathbf{R}_+;X)\rightarrow \mathcal {R} ([\,-\tau ,0\,];X)$$ is nonexpansive.

## The minimal number of generators of a Togliatti system

### Annali di Matematica Pura ed Applicata (1923 -) (2016-12-01) 195: 2077-2098 , December 01, 2016

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree *d* in
$$n+1$$
variables, for any
$$d\ge 2$$
and
$$n\ge 2$$
. We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if
$$n=2$$
(resp.
$$n=2,3$$
) all range between the lower and upper bound is covered, while if
$$n\ge 3$$
(resp.
$$n\ge 4$$
) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for
$$n=2$$
the Mumford–Takemoto stability of the syzygy bundle associated with smooth monomial Togliatti systems.

## Maximal subgroups of finite soluble groups in general position

### Annali di Matematica Pura ed Applicata (1923 -) (2016-08-01) 195: 1177-1183 , August 01, 2016

For a finite group *G* we investigate the difference between the maximum size
$${{\mathrm{MaxDim}}}(G)$$
of an “independent” family of maximal subgroups of *G* and maximum size *m*(*G*) of an irredundant sequence of generators of *G*. We prove that
$${{\mathrm{MaxDim}}}(G)=m(G)$$
if the derived subgroup of *G* is nilpotent. However,
$${{\mathrm{MaxDim}}}(G)-m(G)$$
can be arbitrarily large: for any odd prime *p*, we construct a finite soluble group with Fitting length two satisfying
$$m(G)=3$$
and
$${{\mathrm{MaxDim}}}(G)=p$$
.

## Un teorema di unicità per una equazione a derivate parziali non lineare del secondo ordine

### Annali di Matematica Pura ed Applicata (1923 -) (1966-12-01) 72: 105-131 , December 01, 1966

### Sunto

Nel presente lavoro si considera l'equazione a derivate parziali del secondo ordine non lineare di tipo iperbolico $$F(x,y,z,p,q,r,s,t) = 0$$ nell'ipotesi che la funzione F(x, y, z, p, q, r, s, t) sia continua con le derivate prime lipschitziane e si dimostra che su ogni superficie integrale z=z(x, y), dove z(x, y) è una funzione continua con le derivate prime e con le derivate seconde lipschitziane, vale (in senso generalizzato) il sistema delle equazioni differenziali delle strisce caratteristiche considerato nelle ipotesi classiche.

Nelle stesse ipotesi si dimostra inoltre un teorema di unicità della soluzione del problema di*Darboux* relativo all'equazione (*I*).

## Sul comportamento asintotico e numerico delle soluzioni dell' equazione diA. J. Lerner $$y\frac{{dy}}{{dx}} + y + \sqrt {|x|} $$ sgnx=0sgnx=0

### Annali di Matematica Pura ed Applicata (1923 -) (1966-12-01) 72: 79-95 , December 01, 1966

### Summary

The asymptotic and numerical behavior of the solutions of the equation $$y\frac{{dy}}{{dx}} + + y + \sqrt {|x|} $$ sgn x=0, by A. I. Lerner, are studied.

## Extensions of degree $$p^\ell $$ p ℓ of a p-adic field

### Annali di Matematica Pura ed Applicata (1923 -) (2017-04-01) 196: 457-477 , April 01, 2017

Given a *p*-adic field *K* and a prime number
$$\ell $$
, we count the total number of the isomorphism classes of
$$p^\ell $$
-extensions of *K* having no intermediate fields. Moreover, for each group that can appear as Galois group of the normal closure of such an extension, we count the number of isomorphism classes that contain extensions whose normal closure has Galois group isomorphic to the given group. Finally, we determine the ramification groups and the discriminant of the composite of all
$$p^\ell $$
-extensions of K with no intermediate fields. The principal tool is a result, proved at the beginning of the paper, which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree
$$p^\ell $$
of *K* having no intermediate extensions and the irreducible *H*-submodules of dimension
$$\ell $$
of
$$F^*{/}{F^*}^p$$
, where *F* is the composite of certain fixed normal extensions of *K* and *H* is its Galois group over *K*.

## Parabolic problems in highly oscillating thin domains

### Annali di Matematica Pura ed Applicata (1923 -) (2015-08-01) 194: 1203-1244 , August 01, 2015

In this work, we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a region of
$${\mathbb {R}}^2$$
that degenerates into a line segment when a positive parameter
$$\epsilon $$
goes to zero (a *thin domain*). Here we also allow that its boundary presents highly oscillatory behavior with different orders and variable profile. We take thin domains possessing the same order
$$\epsilon $$
to the thickness and amplitude of the oscillations, but assuming different order to the period of oscillations on the top and the bottom of the boundary. Combining methods from linear homogenization theory and the theory on nonlinear dynamics of dissipative systems, we obtain the limit problem establishing convergence properties for the nonlinear semigroup, as well as the upper semicontinuity of the attractors and stationary states.