This article is devoted to sequences ${(u_n)}_n$ of subharmonic functions in ${ℝ}^N$, with finite order, whose means $J_{u_n}(r)$ (over spheres centered at the origin, with radius r) satisfy such a condition as: $\forall r>0$, $\exists A_r>0$ such that $J_{u_n}(r)\le A_r$, $\forall n\in 𝐍$. The paper investigates under which conditions one may extract a pointwise or uniformly convergent subsequence.

Let ${𝕊}^1$ and $𝔻$ be the unit circle and the unit disc in the plane and let us denote by $𝒜({𝕊}^1)$ the algebra of the complex-valued continuous functions on ${𝕊}^1$ which are traces of functions in the Sobolev class $W^{1,2}(𝔻)$. On $𝒜({𝕊}^1)$ we define the following norm $$\parallel u\parallel ={\parallel u\parallel }_{L^{\mathrm{\infty }}({𝕊}^1)}+{\left({\iint }_{𝔻}{|\nabla \tilde{u}|}^2\right)}^{1/2}$$ where is the harmonic extension of $u$ to $𝔻$. We prove that every isomorphism of the functional algebra $𝒜({𝕊}^1)$ is a quasitsymmetric change of variables on ${𝕊}^1$.

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus \{0\}$, $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every...

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is where $\lambda >0$, $k>0$; $f(x)\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, $f(x)\ge 0$ (and so $u\ge 0$). If , we prove the existence of a solution for both the "+" and the "-" signs, while if $k\ge 1$, we prove the existence of a solution for the "+" sign only.

Inspired by a result from Leibov, we find that the supremum defining the $BLO$ norm in $[0,1]$ is actually attained by a specific sub-interval of $[0,1]$ for $f\in VLO([0,1])$

In questa Nota, dati uno spazio metrico perfetto $X$ ed un suo sottoinsieme $K$ chiuso e raro, si dimostra l'esistenza di una funzione continua $f:X\to [0,1]$ tale che $int(f^{-1}(t))=\mathrm{\varnothing }$ per ogni $t\in [0,1]$, $f(x)=0$ per ogni $x\in K$ e $f(y)=1$ per qualche $y\in X\setminus K$. In particolare, ciò permette di dare risposta simultaneamente a due questioni poste in [2]. Si mettono in evidenza, poi, ulteriori conseguenze di tale risultato.

The following results are proved: (i) if $\alpha$, $\beta \in {ℝ}^{+}$ and $A$ is a sectorial operator, then the set $\{t^{\alpha }{A}^{\beta }(t+A);t>0\}$ is bounded; (ii) the same set of operators is R-bounded if $A$ is R-sectorial.

Siano $d$, $g$, $t$ interi con $0\le t\le g$; se esiste in ${\mathbb{P}}_3(ℂ)$ una curva connessa, non singolare di grado $d$ e genere $g$, allora esiste in ${\mathbb{P}}_3(ℂ)$ una curva irriducibile di grado $d$, genere aritmetico $g$ e $t$ nodi.

Let $f(\cdot ,t)$ be the probability density function representing the solution of Kac's Boltzmann-like equation at time $t$, with initial data $f_0$, and let $g_{\sigma }$ be the Gaussian density with zero mean and variance ${\sigma }^2$, ${\sigma }^2$ being the value of the second moment of $f_0$. Henry McKean Jr. put forward the conjecture that the total variation distance between $f(\cdot ,t)$ and $g_{\sigma }$ goes to zero, as $t\to +\mathrm{\infty }$, with an exponential rate equal to $-1/4$. This lecture aims at explaining the main efforts made to a view to validating this conjecture,...

Si considera l’equazione astratta $B{A}_{1}u+A_{0}u=h$, dove $A_i$ $(i=0,1)$ e $B$ sono convenienti operatori lineari chiusi fra spazi di Banach, $A_i$ non è necessariamente invertibile, e $A_0$, $A_1$ non commutano con $B$. Si studiano esistenza ed unicità delle soluzioni. Si indicano alcune applicazioni a certe equazioni differenziali degeneri o singolari.

Let $F\subset C^{\ast }(X)$ be a vector sublattice over $ℝ$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_z$ containing $F$ such that $\omega (Z(F))=\omega (Z(F_z))=e_{F}X$. In particular this implies that $\omega (Z(F))\ge e_{F}X$. Conditions in order to be $\omega (Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an...

We consider the Grassmannian $𝔾r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\mathbb{P}}^1,{𝒪}_{{\mathbb{P}}_1}(n))$. We define ${𝔛}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on ${\mathbb{P}}^1$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute the dimension of ${𝔛}_{k,r,d}$. In the second part we make a link between ${𝔛}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...

Dato un cono $V$ aperto non vuoto, convesso, regolare e affinemente omogeneo in uno spazio vettoriale reale $W$ di dimensione finita si prova che per ogni $v$ appartenente a $V$ esiste un diffeomorfismo $E_v:W\to V$ che soddisfa le condizioni seguenti E1) $E_v(0)=v$; E2) $det(d{E}_v(y))={\mathrm{\Phi }}_V{(E_v(y))}^{-1}$ per ogni $y$ appartenente a $W$ ove ${\mathrm{\Phi }}_V:V\to {𝐑}^{+}$ è la funzione caratteristica di $V$.

Si considerano due spazi $S_{\mu }$ e $S_{\nu }^{\ast }$, Riemanniani e a metrica eventualmente indefinita, riferiti a sistemi di co-ordinate $\mathrm{\varnothing }$ e ${\mathrm{\varnothing }}_{\nu }^{\ast }$; e inoltre un doppio tensore $T_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ associato ai punti ${\mathrm{\varnothing }}^{-1}(x)\in S_{\mu }$ e ${\mathrm{\varnothing }}^{\ast -1}(y)\in S^{\ast }$. Si pensa $T_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ dato da una funzione ${\tilde{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ di $m$ altri tali doppi tensori e di variabili puntuali $x(\in {\mathrm{\Re }}^{\mu })$, $t\in \mathrm{\Re }$ e $y(\in {\mathrm{\Re }}^{\nu })$; poi si considera la funzione composta $${\widehat{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y)={\tilde{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}[\underset{1,\mathrm{\dots },m}{\underbrace{{\breve{H}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y),\mathrm{\dots },{\breve{H}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y)}},x,t,y].$$ Nella Parte I si scrivono due regole per eseguire la derivazione totale di questa, connessa con una mappa $\widehat{ℰ}$ $(={\widehat{ℰ}}_t)$ fra $S_{\nu }^{\ast }$ e $S_{\mu }$; una è a termini generalmente non covarianti...

Ogni logica $L$ genera canonicamente la $L$-equivalenza ${\equiv }_L$e la $L$-immersione $\underset{𝐿}{\to }$ proprio come la logica del primo ordine genera l’equivalenza elementare $\equiv$ e l’immersione elementare $\lesssim$. Astraendo da $L$, è interessante studiare in sè relazioni d’equivalenza e di immersione generali tra strutture. Mostriamo che esiste una corrispondenza biunivoca tra relazioni d’equivalenza con la proprietà di Robinson e relazioni d’immersione con la proprietà di Amalgamazione Forte ($A{P}^{+}$). Caratterizziamo algebricamente...

Siano: $S$ una superficie algebrica proiettiva complessa non singolare, $K$ un divisore canonico ed $H$ un divisore molto ampio su $S$. Questo lavoro ha per oggetto lo studio dell'indice di autointersezione ${(K+H)}^2$. Si dimostra, innanzitutto, la disuguaglianza $${(K+H)}^2\ge 0$$, nell'ipotesi che la superficie $S^{\prime }$ ottenuta immergendo $S$ mediante il sistema lineare completo $|H|$ non sia uno scroll. Questa disuguaglianza è connessa con alcuni risultati di Sommese e Van de Ven sulla generazione del fascio ${𝒪}_s(K+H)$. La dimostrazione...

Si supponga che l'anello $𝐑$ ammetta una decomposizione come prodotto subdiretto $𝐑=\underset{\widetilde{\alpha \in 𝚲}}{\times }{𝐑}_{\alpha }$ di anelli ${𝐑}_{\alpha }\ne \mathrm{}$, tali che per ${𝐒}_{\alpha }=𝐑\cap {𝐑}_{\alpha }$ si abbia ${Ann}_{{𝐑}_{\alpha }}{𝐒}_{\alpha }=\mathrm{}$ ($\forall \alpha \in A$), e sia $𝐒={\oplus }_{\alpha \in 𝐀}{𝐒}_{\alpha }$. Si scelga un $𝐑$-modulo (destro) $𝐌$ che sia libero da torsione rispetto ad $𝐒$, cioè ${Ann}_{𝐌}𝐒=\mathrm{}$; allora $𝐌$ può essere rappresentato come prodotto subdiretto irridondante $𝐌\cong \underset{\widetilde{\alpha \in 𝚲}}{\times }{𝐌}_{\alpha }$ degli ${𝐑}_{\alpha }$-moduli ${𝐌}_{\alpha }$ liberi da torsione rispetto ad ${𝐒}_{\alpha }$. Si fa uno studio di un subprodotto generale di una classe $𝐂$ di $𝐑$-moduli ${𝐌}^{(𝐢)}$ $(i\in I)$, dove $𝐂$ è determinato per mezzo di epimorfismi e relazioni. ...

We consider an SPDE in a Hilbert space $H$ of the form $dX(t)=(AX(t)+b(X(t)))dt+\sigma (X(t))dW(t)$, $X(0)=x\in H$ and the corresponding transition semigroup $P_{t}f(x)=𝔼[f(X(t,x))]$. We define the infinitesimal generator $\overline{L}$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\phi =\frac{1}{2}\mathrm{Tr}[\sigma (x){\sigma }^{*}(x)D^2\phi ]+⟨b(x),D\phi ⟩$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\overline{L}$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular...

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.

The object of this note is to study certain 2-dimensional $\lambda$-adic representations of $Gal(\overline{Q}/Q)$; fixed $p_1,\mathrm{\dots },p_n$ distinct primes, we will consider representations $\rho :G\to G{L}_2(A)$, given by the matrix which are unramified outside $p_1,\mathrm{\dots },p_n,\mathrm{\infty }$ and the residue characteristic of $\lambda$, which are a product of $m$ representations over finite extensions of the ring of Witt vectors of the residue field and which are reducible modulo $\lambda$. In analogy with the theory of the modular representations, we will introduce the analogue of Mazur's Hecke...

Let ${\Phi }_1,...,{\Phi }_{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U{\subset }^{N+m}$, let $𝕄$ be a $N$-dimensional $C^k$ submanifold of $U$ and define $$𝕋:=\{z\in 𝕄:{\Phi }_1\left(z\right),...,{\Phi }_{k+1}\left(z\right)\in T_z𝕄\}$$ where $T_z𝕄$ is the tangent space to $𝕄$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $𝕄$) of $𝕋$: If $z_0\in 𝕄$ is a $(N+k)$-density point (relative to $𝕄$) of $𝕋$ then all the iterated Lie brackets of order less or equal to $k$ $${\Phi }_{i_1}\left(z_0\right),[{\Phi }_{i_1},{\Phi }_{i_2}]\left(z_0\right),[[{\Phi }_{i_1},{\Phi }_{i_2}],{\Phi }_{i_3}]\left(z_0\right),...(h,i_h\le k+1)$$ belong to $T_{z_0}𝕄$. Such a property has been proved in [9] for $k=1$ and its proof in the...

The sequence of balancing numbers $\left(B_n\right)$ is defined by the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ for $n\ge 2$ with initial conditions $B_0=0$ and $B_1=1.$ $B_n$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_n$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6)$. Namely, $B_2=6={\left(11\right)}_5$ and $B_3=35={\left(55\right)}_6.$

Let $X$, $X^{\prime }$, $Y$ be smooth projective complex curves with $Y$ curve of genus $\ge 1$. Let $d$ be an integer $\ge 3$, let $\underset{¯}{e}=(e_1,\mathrm{\dots },e_r)$ be a partition of $d$ and let $|e|={\sum }_{i=1}^r(e_i-1)$. Let $X\stackrel{𝜋}{\to }{X}^{\prime }\stackrel{𝑓}{\to }Y$ be a sequence of coverings where $\pi$ is a degree 2 branched covering and $f$ is a degree $d$ covering, with monodromy group $S_d$, branched in $n_2+1$ points, one of which is special point $c$ whose local monodromy has cycle type given by $\underset{¯}{e}$. Moreover the branch locus of the covering $\pi$ is contained in $f^{-1}(c)$. In this paper we prove the irreducibility...

Se $X$ ed $Y$ sono spazi topologici, un'applicazione $f:X\to Y$ dicesi semicontinua [3] se, per ogni insieme aperto $V$ di $Y$, $f^{-1}(V)$ risulta un insieme semiaperto di $X$. Qui si studiano tali applicazioni e si dànno tre loro caratterizzazioni.

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

We fix $\mathrm{\ell }$ a prime and let $M$ be an integer such that $\mathrm{\ell }|̸M$; let $f\in S_2({\mathrm{\Gamma }}_1(M{\mathrm{\ell }}^2))$ be a newform supercuspidal of fixed type at $\mathrm{\ell }$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$, of discriminant dividing the level of $f$, there is a local quaternionic Hecke algebra $T$ associated to $f$. The algebra $T$ acts on a module $M_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global...

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

Si dimostra che il funzionale ${\int }_{\mathrm{\Omega }}f(u,Du)𝑑x$ è semicontinuo inferiormente su $W_{loc}^{1,1}(\mathrm{\Omega })$, rispetto alla topologia indotta da $L_{loc}^1(\mathrm{\Omega })$, qualora l’integrando $f(s,p)$ sia una funzione non-negativa, misurabile in $s$, convessa in $p$, limitata nell’intorno dei punti del tipo $(s,0)$, e tale che la funzione $s\mapsto f(s,0)$ sia semicontinua inferiormente su $𝐑$.

In this paper we determine all squarefree completely lexsegment ideals which have a linear resolution. Let $M_d$ denote the set of all squarefree monomials of degree $d$ in a polynomial ring $k[x_1,\mathrm{\dots },x_n]$ in $n$ variables over a field $k$. We order the monomials lexicographically such that $x_1>x_2>\mathrm{\dots }>x_n$, thus a lexsegment (of degree $d$) is a subset of $M_d$ of the form $L(u,v)=\{w\in M_d:u\ge w\ge v\}$ for some $u,v\in M_d$ con $u\ge v$. An ideal generated by a lexsegment is called a lexsegment ideal. We describe the procedure to determine when such an ideal has a linear...

Let $G$ be a group and $m$ be an integer greater than or equal to $2$. $G$ is said to be $m$-permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$, then $G$ is $(1+[z/2])$-permutable. More bounds are given on the least $m$ such that $G$ is $m$-permutable.

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain...

Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti $u\in 𝐂([0,T],E)$ dell'equazione di evoluzione non autonoma $u^{\prime }(t)-A(t)u(t)=f(t)$ con il dato iniziale $u(0)=x$, in uno spazio di Banach $E$. Gli operatori $A(t)$ sono generatori infinitesimali di semi-gruppi analitici ed hanno dominio indipendente da $t$ e non necessariamente denso in $E$. Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità hölderiana della soluzione e della sua derivata.