Let $f(n)$ be an additive number-theoretical function. A sufficient condition is given for complete additivity of $f(n)$: this condition, involving the behaviour of $\delta (n)=f(n+1)-f(n)$, requires $\delta (u)\to 0$ as $u\to +\mathrm{\infty }$ throughout an increasing sequence of positive integers $u$, having “upper density” I (i.e. ($lim\; sup(\sum I:u\le x)/x=I$) and not necessarily “density” I.

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 $\mathrm{\Psi }$.

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 $H_0^1\times (H_0^1\cap L^p)$.

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...

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.

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.

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

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.

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.

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.

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.

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...

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 $\mathrm{\Psi }$.

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.

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

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.

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.

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

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.

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...

In the first part of the paper we complete the classification of the arithmetical Cohen-Macaulay vector bundles of rank 2 on a smooth prime Fano threefold. In the second part, we study some moduli spaces of these vector bundles, using the decomposition of the derived category of $X$ provided by Kuznetsov, when the genus of $X$ is 7 or 9. This allows to prove that such moduli spaces are birational to Brill-Noether varieties of vector bundles on a smooth projective curve $\mathrm{\Gamma }$. When the second...

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 $X$.

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.

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 $H^1(M,𝒮)$.

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.

L'operatore $$S={\partial }_x+(1+xy){\partial }_y+x+y$$ (in due variabili) è individuato da ogni sua soluzione non nulla, i.e.: se $M$ è un $A_2=ℂ[x,y]⟨{\partial }_x,{\partial }_y⟩$modulo $(sx)$ unitario; se $m\in M$ soddisfa l'equazione $Sm=0$ ($m\ne 0$), allora ogni $T\in A_2$ tale che $Tm=0$ è del tipo $T=RS$. In modo equivalente, si può dire così: $A_{2}S$ è un $A_2$-ideale $(sx)$ massimale. In questo articolo studio le matrici differenziali $M\in A_{(p,q)}$ ($p$ righe e $q$ colonne, con elementi nell'algebra di Weyl $A_n$), tali che $A_{(p,p)}M$ sia massimale in $A_{(p,p)}$.

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 $p(P^n\mathrm{—}Y)=n\mathrm{—}s\mathrm{—}1$.

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 $C$.

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 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.

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.

Sia $\mathrm{\Phi }$ un gruppo di automorfismi del gruppo $(N,+)$ tale che per ogni $\varphi \in \mathrm{\Phi }\setminus \{1\}$ la mappa $T_{\varphi }:N\to N$ $x\mapsto -x+\varphi (x)$ sia biiettiva. In questo lavoro si prova che se $N$ è infinito ed è unione di un numero finito di $\mathrm{\Phi }$-orbite, allora $N$ è abeliano.

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...

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.

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.

In ipotesi molto generali si dimostrano teoremi di completezza nel senso di Picone per l'equazione (1). Come corollario si ottengono teoremi del tipo Runge.

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 $p(P^n\mathrm{—}Y)=n\mathrm{—}s\mathrm{—}1$.

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.

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.

It is proved that Lopatinskii's condition is necessary and sufficient for problem (2.5) to be an index problem. A method is given for the determination of the index.

Necessary and sufficient conditions are given for the existence of smooth solutions of the differential equations (1) with the boundary conditions (2). Coefficients of (1) and (2) are only supposed Hölder-continuous.

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 $A$ 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...

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...