If the minimum problem (${𝒫}_{\mathrm{\infty }}$) is the limit, in a variational sense, of a sequence of minimum problems with obstacles of the type $$\underset{u\ge {\varphi }_h}{\min }{\int }_{\mathrm{\Omega }}\left[f_h(x,Du)+a(x,u)\right]𝑑x,$$ then (${𝒫}_{\mathrm{\infty }}$) can be written in the form $${𝒫}_{\infty }\underset{u}{min}\left\{{\int }_{\Omega }\left[f_{\infty }(x,Du)+a(x,u)\right]dx+{\int }_{\overline{\Omega }}{g}_{\infty }(x,\overline{u}\left(x\right))d{\mu }_{\infty }\left(x\right)\right\}$$ without any additional constraint.

Sia $f=f(x,z)$ quasiconvessa in $z$, quasiperiodica in $x$ nel senso di Besicovitch e soddisfi le disuguaglianze: $${|z|}^p\le f(x,z)\le \mathrm{\Lambda }(1+{|z|}^p).$$ Allora $f$ può essere omogeneizzata: esiste una funzione $\mathrm{\Psi }$ che dipende solo da $z$ tale che i funzionali $${\int }_{\mathrm{\Omega }}f(\frac{x}{ϵ},Du(x))dx\mathit{\hspace{1em}\hspace{1em}}u\in H^{1,p}(\mathrm{\Omega };{ℝ}^m)$$ convergono, per $ϵ$ tendente a $0$ (nel senso della $\mathrm{\Gamma }$-convergenza) a $${\int }_{\mathrm{\Omega }}\mathrm{\Psi }(Du(x))dx.$$ Inoltre si può dare una formula asintotica per $\mathrm{\Psi }$.

Following H. Sato - Y. Okazaky we will prove that: if $X$ is a topological vector space, locally convex and reflexive, and $\mu$ is a gaussian measure on $𝐂(X,X^{\prime })$, then $L^2(\mu )$ is separable.

An existence theorem for 1-atomic standard models of $L_{\omega \omega }(Q_1)$ (more weak than usual “atomic models”) and applications of $L_{\omega \omega }(Q_1)$ to $L_{\omega \omega }$ are the results of this note.

Given an open subset $\mathrm{\Omega }$ of ${ℝ}^n$ and a Borel function $f:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to [0,+\mathrm{\infty }[$, conditions on $f$ are given which assure the lower semicontinuity of the functional ${\int }_{\mathrm{\Omega }}f(x,u,Du)dx$ with respect to different topologies.

Every unit $u$ in the ring ${𝐙}_n$ of the residual classes mod $n$ induces canonically an automorphism $\pi$ of the algebra ${𝐑}_n(q)=GF(q)[z]/(z^n-1)$. Let $𝒞\subset {𝐑}_n(q)$ be a cyclic code, i.e. an ideal. If the numbers $n$ and $q$ are relatively prime then there exists a well-known characterization of the code $\pi (𝒞)$. We extend this characterization to the general case.

Si considera il problema al contorno $-\mathrm{\Delta }u=f(x,u)+ϵ\psi (x,u)$ in $\mathrm{\Omega }$, $u|\partial \mathrm{\Omega }=0$, dove $\mathrm{\Omega }\in {ℝ}^n$ è un aperto limitato e connesso ed $ϵ$ è un parametro reale. Si prova che, se $f(x,s)+ϵ\psi (x,s)$ è «superlineare» ed $ϵ$ è abbastanza piccolo, il problema precedente ha almeno tre soluzioni distinte.

Bubbles are formed in a fluid by inflating a liquid film with a gas in which the pressure $\widehat{\pi }(\nu )$ is a strictly decreasing function of the specific volume, unbounded as $\nu \to 0^{+}$. We show that, if $\widehat{\pi }(\nu )$ grows as fast or faster than ${\nu }^{-2/3}$ as $\nu \to 0^{+}$, then there is at least one stable equilibrium configuration of any such bubble, no matter how much gas has been used to inflate it. On the other hand, if $\widehat{\pi }(\nu )$ grows as slowly or slower than ${\nu }^{-1/3}$ as $\nu \to 0^{+}$, then any such bubble has no equilibrium configuration, when the amount...

The following result is proved: to give two consecutive polynomials $P_n(x)$, $P_{n+1}(x)$ of an orthonormal system is equivalent to assign the first $2n+3$ moments of the Lebesgue-Stieltjes measure associated with the system.

Given a complete and separable metric space $X$, we study the weak convergence of sequences of measures defined on the space $𝒮(X)$ of all real-valued lower semicontinuous functions on $X$ as well as on the space $ℱ(X)$ of all closed subsets of $X$.

We write a cohomological resolution of the sheaf $𝒮$of solutions of the differential operator $\partial /\partial x_n$ on a manifold $M$ and study the cohomology groups $H^0(M,𝒮)$ and $H^1(M,𝒮)$.

Let $G$ be a group and $n$ an integer $\ge 2$. We say that $G$ has the $n$-permutation property $(G\in P_n)$ if, for any elements $x_1,x_2,\mathrm{\cdots },x_n$ in $G$, there exists some permutation $\sigma$ of $\{1,2,\mathrm{\cdots },n\}$, $\sigma \ne id.$ such that $x_1,x_2,\mathrm{\cdots },x_n=x_{\sigma (1)},x_{\sigma (2)},\mathrm{\cdots },x_{\sigma (n)}$. We prouve that every group $G\in P_n$ is an FC-nilpotent group of class $\le n-1$, and that a finitely generated group has the $n$-permutation property (for some $n$) if, and only if, it is abelian by finite. We prouve also that a group $G\in P_3$ if, and only if, its derived subgroup has order at most 2.

This paper shows how some techniques used for the meromorphic functions of one variable can be used for the explicit construction of a solution to the Mittag-Leffler problem for Dolbeault classes of tipe $(0,n-1)$ with singularities in a discrete set of ${𝐂}^{𝐧}$ and $T^n$ (a $n$-dimensional complex torus). A generalisation is given for the Weierstrass $\zeta$ and the Legendre relations.

I give a characterization of the pseudoconvex Hartogs domains $A$ in $C^2$ that satisfy the equation $H^2(A,C)=0$, where $H^2(A,C)$ is the second cohomology group of $A$ with coefficients in the constant sheaf $C$.

Let $Y$ be an 5 dimensional closed subscheme of $P^n\mathrm{—}Y$ and $p(P^n\mathrm{—}Y)$ the largest integer $p$ such that $H^i(P^n\mathrm{—}Y,L)$ is finite dimensional for all and for all locally free sheaves $L$ on $P^n\mathrm{—}Y$. If we introduce the same integer $p(P^n\mathrm{—}{Y}^a)$ in the complex case, i.e. when $L$ runs through the set of all locally free analytic sheaves on $P^n\mathrm{—}{Y}^a$, we show that $p(P^n\mathrm{—}{Y}^a)=n\mathrm{—}s\mathrm{—}1$ if $p(P^n\mathrm{—}Y)=n\mathrm{—}s\mathrm{—}1$.

Let $A$ be an open subset of ${ℝ}^n$, $W_m(A)$ the linear space of $m$-vector valued functions defined on $A$, $G\equiv \{\gamma \}$ a group of orthogonal matrices mapping $A$ onto itself and $T\equiv \{T_{\gamma }\}$ a linear representation of order $m$ of $G$. A suitable group $𝒯(G,T)$ of linear operators of $W_m(A)$ is introduced which leads to a general definition of $T$-invariant linear operator with respect to $G$. When $G$ is a finite group, projection operators are explicitly obtained which define a "maximal" decomposition of the function space into a direct sum...

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

A periodic BVP for a semilinear elliptic-parabolic equation in an unbounded domain $\mathrm{\Omega }$ contained in a half-space of ${ℝ}^n$ is considered, with Dirichlet boundary conditions on the finite part of $\partial \mathrm{\Omega }$. A theorem of uniqueness of periodic solutions is proved by showing that a suitable function of the "energy" $E(x)$ is subharmonic in $\mathrm{\Omega }$ and satisfies a Phragmèn-Lindelöf growth condition at infinity.

We consider the problem of finding a couple of solutions $(u^{+},u^{-})$ satisfying the following conditions (4) and (5) for a couple of two uniformly elliptic partial differential operators $A^{+}$ and $A^{-}$ of order $2m$ in a non regular situation.

Given an open subset $\mathrm{\Omega }$ of ${ℝ}^n$ and a Borel function $f:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to [0,+\mathrm{\infty }[$, conditions on $f$ are given which assure the lower semicontinuity of the functional ${\int }_{\mathrm{\Omega }}f(x,u,Du)𝑑x$ with respect to different topologies.

An existence theorem for 1-atomic standard models of $L_{\omega \omega }(Q_1)$ (more weak than usual “atomic models”) and applications of $L_{\omega \omega }(Q_1)$ to $L_{\omega \omega }$ are the results of this note.

Following H. Sato - Y. Okazaky we will prove that: if $X$ is a topological vector space, locally convex and reflexive, and $\mu$ is a gaussian measure on $𝐂(X,X^{\prime })$, then $L^2(\mu )$ is separable.

Siano $C\in {𝐏}^3$ una curva ridotta ed irriducibile, ed $f_1,f_2,\mathrm{\dots },f_m$ un sistema minimale di generatori dell'ideale omogeneo $I(C)$. Nel § 2 determiniamo una condizione necessaria e sufficiente perché due superfici $F_i$, $F_j$, aventi equazioni $f_i=0$, $f_j=0$ $(i,j=1,2,\mathrm{\dots },m)$, si sechino trasversalmente lungo $C$. Nel § 3 applichiamo questo risultato alle curve aritmeticamente di Cohen-Macaulay.

In this paper we prove a theorem which extends a result due to H. Heineken. We prove that if $n(G)\simeq n(H)$ ($G$ hypercentral not locally cyclic p-group with property (P) in no. 1, $H^{\prime }$ hypercentral group) then $H$ is a hypercentral p-group. More generally: if $n(G)\simeq n(H)$ (G hypercentral torsion group, $H$ soluble group) then $H$ is a locally finite group.

Given a complete and separable metric space $X$, we study the weak convergence of sequences of measures defined on the space $𝒮(X)$ of all real-valued lower semicontinuous functions on $X$ as well as on the space $ℱ(X)$ of all closed subsets of $X$.

Sia $p$ un numero primo, $n$ un intero $\ge 1$, e $G$ un $p$-gruppo finito non abeliano e non hamiltoniano. Si dice che $G$ appartiene ad $S(p^n)$ se i sottogruppi non normali di $G$ hanno tutti ordine $p^n$. In un Nota precedente [3] sono stati determinati tutti i gruppi appartenenti a $S(p^n)$ ($p$ dispari, $n\ge 1$), tutti quelli appartenenti ad $S(2)$ e tutti i gruppi di esponente $4$ appartenenti ad $S(4)$. Nella presente Nota si determinano tutti i gruppi appartenenti ad $S(2^n)$ ($n\ge 2$) e di esponente $>4$, e in tal modo è completata la classificazione...

In this paper we prove that the simple groups $PSL(2,p^f)$ ($p$ a prime, $p^f\ge 4$) are strongly lattice determined.

The paper gives a characterization of the traces, that are indicated in the title, in the case of a compact manifold M.

Bubbles are formed in a fluid by inflating a liquid film with a gas in which the pressure $\widehat{\pi }(\nu )$ is a strictly decreasing function of the specific volume, unbounded as $\nu \to 0^{+}$. We show that, if $\widehat{\pi }(\nu )$ grows as fast or faster than ${\nu }^{-2/3}$ as $\nu \to 0^{+}$, then there is at least one stable equilibrium configuration of any such bubble, no matter how much gas has been used to inflate it. On the other hand, if $\widehat{\pi }(\nu )$ grows as slowly or slower than ${\nu }^{-1/3}$ as $\nu \to 0^{+}$, then any such bubble has no equilibrium configuration, when the amount...

We write a cohomological resolution of the sheaf $𝒮$of solutions of the differential operator $\partial /\partial x_n$ on a manifold $M$ and study the cohomology groups $H^0(M,𝒮)$ and $H^1(M,𝒮)$.

We study a class of functions which differ essentially from those which are the sum of a convex function and a regular one and which have interesting properties related to $\mathrm{\Gamma }$-convergence and to problems with non-convex constraints. In particular some results are given for the associated evolution equations.

Every unit $u$ in the ring ${𝐙}_n$ of the residual classes mod $n$ induces canonically an automorphism $\pi$ of the algebra ${𝐑}_n(q)=GF(q)[z]/(z^n-1)$. Let $𝒞\subset {𝐑}_n(q)$ be a cyclic code, i.e. an ideal. If the numbers $n$ and $q$ are relatively prime then there exists a well-known characterization of the code $\pi (𝒞)$. We extend this characterization to the general case.

Considerata un'equazione iperbolica del secondo ordine in due variabili indipendenti, si studia il problema di Goursat nel caso singolare relativo a due archi assegnati ${\mathrm{\Gamma }}_1$ e ${\mathrm{\Gamma }}_2$ con la stessa tangente nel punto $0$ di intersezione. Vengono fornite condizioni di esistenza e di unicità per la soluzione. Queste condizioni dipendono dall'ordine del contatto che ${\mathrm{\Gamma }}_1$ e ${\mathrm{\Gamma }}_2$ hanno in $0$.

The following result is proved: to give two consecutive polynomials $P_n(x)$, $P_{n+1}(x)$ of an orthonormal system is equivalent to assign the first $2n+3$ moments of the Lebesgue-Stieltjes measure associated with the system.

Let $C$ be an algebraic projective smooth and trigonal curve of genus $g\ge 5$. In this paper we define, in a way equivalent to that followed by A. Maroni in [1], an integer $m$, called the species of $C$, which is a birational invariant having the property that $0\le m\le \frac{g+2}{3}$ and $m\mathrm{—}g\equiv 0$ mod(2). In section 1 we prove that for every $g(\ge 5)$ and every $m$, as before, there are trigonal curves of genus $g$ and species $m$. In section 2 we define the space ${ℳ}_{g,3;m}^1$ of moduli of trigonal curves of genus $g$ and species $m$. We note that ${ℳ}_{g,3;m}^1$ is...

Si considera il problema al contorno $-\mathrm{\Delta }u=f(x,u)+ϵ\psi (x,u)$ in $\mathrm{\Omega }$, $u|\partial \mathrm{\Omega }=0$, dove $\mathrm{\Omega }\in {ℝ}^n$ è un aperto limitato e connesso ed $ϵ$ è un parametro reale. Si prova che, se $f(x,s)+ϵ\psi (x,s)$ è «superlineare» ed $ϵ$ è abbastanza piccolo, il problema precedente ha almeno tre soluzioni distinte.

Let $Y$ be an 5 dimensional closed subscheme of $P^n\mathrm{—}Y$ and $p(P^n\mathrm{—}Y)$ the largest integer $p$ such that $H^i(P^n\mathrm{—}Y,L)$ is finite dimensional for all and for all locally free sheaves $L$ on $P^n\mathrm{—}Y$. If we introduce the same integer $p(P^n\mathrm{—}{Y}^a)$ in the complex case, i.e. when $L$ runs through the set of all locally free analytic sheaves on $P^n\mathrm{—}{Y}^a$, we show that $p(P^n\mathrm{—}{Y}^a)=n\mathrm{—}s\mathrm{—}1$ if $p(P^n\mathrm{—}Y)=n\mathrm{—}s\mathrm{—}1$.

This paper shows how some techniques used for the meromorphic functions of one variable can be used for the explicit construction of a solution to the Mittag-Leffler problem for Dolbeault classes of tipe $(0,n-1)$ with singularities in a discrete set of ${𝐂}^{𝐧}$ and $T^n$ (a $n$-dimensional complex torus). A generalisation is given for the Weierstrass $\zeta$ and the Legendre relations.

Let $G$ be a group and $n$ an integer $\ge 2$. We say that $G$ has the $n$-permutation property $(G\in P_n)$ if, for any elements $x_1,x_2,\mathrm{\cdots },x_n$ in $G$, there exists some permutation $\sigma$ of $\{1,2,\mathrm{\cdots },n\}$, $\sigma \ne id.$ such that $x_1,x_2,\mathrm{\cdots },x_n=x_{\sigma (1)},x_{\sigma (2)},\mathrm{\cdots },x_{\sigma (n)}$. We prouve that every group $G\in P_n$ is an FC-nilpotent group of class $\le n-1$, and that a finitely generated group has the $n$-permutation property (for some $n$) if, and only if, it is abelian by finite. We prouve also that a group $G\in P_3$ if, and only if, its derived subgroup has order at most 2.

If $\{{\phi }_h\}$ and $\{{\psi }_h\}$ are sequences of arbitrary functions from ${𝐑}^n$ into $\overline{𝐑}$, with ${\phi }_h\le {\psi }_h$, then there exist two subsequences $\{{\phi }_{h_k}\}$ and $\{{\psi }_{h_k}\}$, a function $f(x,u)$ convex in $u$, and two positive Radon measures $\mu$ and $\nu$, with $\mu \in H^{-1}({𝐑}^n)$, such that for every “admissible” open set $A$ and Borei set $B$, with $B\subseteq A$, and for every $g\in L^2(A)$, the sequences $\{m_k\}$ and $\{u_k\}$ of the minima and of the minimum points of the functional $${\int }_A\left[{|Du|}^2+{|u|}^2+gu\right]𝑑x,$$ with constraints of the type $\{{\phi }_{h_k}\}\le u\le \{{\psi }_{h_k}\}$ on $B$, converge respectively to the minimum $m_0$ and to the minimum point $u_0$ of the functional $${\int }_A\left[{|Du|}^2+{|u|}^2+gu\right]𝑑x+{\int }_{B}f(x,u)𝑑\mu +\nu (B),$$ without...

I give a characterization of the pseudoconvex Hartogs domains $A$ in $C^2$ that satisfy the equation $H^2(A,C)=0$, where $H^2(A,C)$ is the second cohomology group of $A$ with coefficients in the constant sheaf $C$.

A periodic BVP for a semilinear elliptic-parabolic equation in an unbounded domain $\mathrm{\Omega }$ contained in a half-space of ${ℝ}^n$ is considered, with Dirichlet boundary conditions on the finite part of $\partial \mathrm{\Omega }$. A theorem of uniqueness of periodic solutions is proved by showing that a suitable function of the "energy" $E(x)$ is subharmonic in $\mathrm{\Omega }$ and satisfies a Phragmèn-Lindelöf growth condition at infinity.

We consider an incompressible elastic solid which admits a configuration of equilibrium having the shape of a rectangular parallelepiped $C^{*}$ when external forces are absent. We look for a thermoelastic transformation mapping that configuration onto a cylindrical wedge $C$. The problem we consider is analogous to the one where both $C^{*}$ and $C$ are cylindrical crowns. This case has been considered by T. Manacorda${}^{(1)}$ Referring $C$ to a system of cylindrical coordinates $r$, $\vartheta$ and $\zeta$, we show that the...

A proof is given of an abstract characterization of operators of the form $$\sum _{i,j=1}^n\frac{\partial }{\partial x_i}\left(a_{ij}\frac{\partial }{\partial x_j}\right)+\sum _{i=1}^n{b}_i\frac{\partial }{\partial x_i}+c\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}(a_{ij}=a_{ji})$$ with $a_{ij}\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, $b_i\in L^r(\mathrm{\Omega })$ $(r>2)$, $c\in L^q(\mathrm{\Omega })$ $(q>1)$, $\mathrm{\Omega }$ open set of ${\mathrm{R}}^n$, by particular continuous forms on $H_0^1\times (H_0^1\cap L^p)$.