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.

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.

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.

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...

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.

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...

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...

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.

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

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 $\mathrm{\Psi }$.

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.

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

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

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.

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.

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 $X$.

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.

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.

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.

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 $\mathrm{\Psi }$.

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 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 $X$.

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

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...

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.

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 $G$ 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.

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

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.

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 $A$ 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...

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 $C$.

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 $A$ 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...

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.

In this note we show that every smooth “general enough” hypersurface of bidegree $(p,2)$ with $p>1$ in ${𝐏}^2\times {𝐏}^2$ is an ordinary non rational conic-bundle. Moreover we construct an example, for $p=3$, which is unirational.

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.

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...

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.

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.

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.

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.