Let us consider a Young's function $\mathrm{\Phi }:{ℝ}^{+}\to {ℝ}^{+}$ satisfying the ${\mathrm{\Delta }}_2$ condition together with its complementary function $\mathrm{\Psi }$, and let us consider the Dirichlet problem for a second order elliptic operator in divergence form: $B$ the unit ball of ${ℝ}^n$. In this paper we give a necessary and sufficient condition for the $L^{\varphi }$-solvability of the problem, where $L^{\varphi }$ is the Orlicz Space generated by the function $\mathrm{\Phi }$. This means solvability for $f\in L^{\mathrm{\Phi }}$ in the sense of [5], [8], where the case $\mathrm{\Phi }(t)=t^p$ is treated. ...

We study the irreducibility of the Dirichlet polynomial $P_G(s)$ when $G$ is the alternating group on $p+1$ elements with $p$ prime and we prove that $P_G(s)$ is irreducible for infinitely many choiches of $p$.

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\mathrm{\Omega }$ made of two materials. The first one is located in a set $F_{ϵ}$ of cylindrical parallel fibers periodically distributed with a period of size $ϵ$, and the second one is located in the "matrix" $M_{ϵ}=\mathrm{\Omega }\setminus F_{ϵ}$. The ratio between the conductivity coefficients of the two materials is of order $1/{ϵ}^2$. After writing the homogenized problem, we give a corrector result and prove...

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 $D$ 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 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...

The aim of this work is to prove a chain rule and an $L^1$-lower semicontinuity theorems for integral functional defined on $BV(\mathrm{\Omega })$. Moreover we apply this result in order to obtain new relaxation and $\mathrm{\Gamma }$-convergence result without any coerciveness and any continuity assumption of the integrand $f(x,s,p)$ with respect to the variable $s$.

We attempt to generalize conductor degree's results, known in ${\mathbb{P}}^2$, to the case of 0-dimensional schemes of ${\mathbb{P}}^r$. In the first part of this paper, we consider the problem of characterizing the sequences generators's degrees of the conductor which are compatible with a fixed postulation (or Hilbert function) for a set of points in ${\mathbb{P}}^r$ and we determine the conductor degree of every point in a $r$-partial intersection. In addition, we define the separating degree of a point for a 0-dimensional subscheme...

We introduce a new class of spaces, called Hausdorff modulo $ℐ$ or $T_2$ mod $ℐ$ spaces with respect to an ideal $ℐ$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $ℐ$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $ℐ$-compact subsets in Hausdorff modulo $ℐ$ spaces and generalize some results of Hamlett and Jankovic....

Using the theory of adjoints and pluricanonical adjoints, we construct three nonsingular threefolds, as desingularizations of degree six hypersurfaces in ${\mathbb{P}}^4$, having the irregularities $q_1=q_2=0$ and the following periodical sequences of plurigenera respectively $$(p_g,P_2,P_3,\mathrm{\dots },P_m,\mathrm{\dots })=(0,0,1,0,0,1,\mathrm{\dots }),(0,0,0,1,0,0,0,1,\mathrm{\dots }),(0,0,0,0,1,0,0,0,0,1,\mathrm{\dots }).$$In the Appendix, starting from the second above-mentioned example, we construct a threefold of general type with $q{q}_1=q_2=0,p_g=1$, $P_2=2$ whose m-canonical transformation is birational if and only if $m\ge 11$.

We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as $$-\mathrm{div}a(x,\nabla u)=b(x,\nabla u)+\mu$$ where $\mu$ is a measure with bounded total variation. We fix structural conditions on functions $a$, $b$ which ensure existence of solutions. Moreover, if $\mu$ is an $L^1$ function, we prove a uniqueness result under more restrictive hypotheses on the operator.

Let $X\subset {\mathbb{P}}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N\ge 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X)=g(X)-1$ are the del Pezzo surface in ${\mathbb{P}}^4$, in ${\mathbb{P}}^5$ and a conic bundle of degree 5 in ${\mathbb{P}}^4$. We prove that for $N\ge 5$ if $PG(X)=g(X)-1+\lambda$, $\lambda$ a non negative integer, then $g(X)\le \lambda +1+\alpha$ where...

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 ${𝕊}^1$.

We prove that the $\mathrm{\Gamma }$-limit in $L_{\mu }^{\mathrm{\infty }}$ of a sequence of supremal functionals of the form $F_k(u)={\mu -\mathrm{ess}\sup }_{\mathrm{\Omega }}{f}_k(x,u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\mathrm{\Gamma }$-lim $F(\cdot ,B)$ of a sequence of functionals $F_k(u,B)={\mu -\mathrm{ess}\sup }_B{f}_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u,B)={\mu -\mathrm{ess}\sup }_{B}f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope. ...

If the second order problem $\ddot{u}+\dot{u}+Au=f$ has $L^p$ maximal regularity for some $p\in (1,\mathrm{\infty })$, then it has $L^p$ maximal regularity for every $p\in (1,\mathrm{\infty })$.

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.

The range of the bigenus $P_2$ is one of the unsolved problems concerning smooth complex projective regular threefolds of general type with $p_g=0$: The examples in the literature have $P_2\le 5$. In the present paper we present a non-singular threefold with $p_g=q_1=q_2=0$; $P_2=6$; the bicanonical map is stably birational.

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.

The Tango bundle $T$ is defined as the pull-back of the Cayley bundle over a smooth quadric $Q_5$ in ${\mathbb{P}}_6$ via a map $f$ existing only in characteristic 2 and factorizing the Frobenius $\phi$. The cohomology of $T$ is computed in terms of $S\otimes C$, ${\phi }^{*}(C)$, ${\text{Sym}}^2(C)$ and $C$, which we handle with Borel-Bott-Weil theorem.

Per ogni soluzione della (1) nel dominio limitato $\mathrm{\Omega }$,, appartenente a $H_0^2(\mathrm{\Omega })$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^0$ del contorno; si consente a $\partial \mathrm{\Omega }$ di essere singolare in $x^0$.

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.

We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove inverse approximation theorems for functions $f$ such that $w_{2m}f\in L^p(R)$, where $1\le p\le \mathrm{\infty }$ and $w_{2m}(x)={(1+x^{2m})}^{-1}$, $m\in N_0$.

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...

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,...

In this note we show a relative version of ${\gamma }_k$-set introduced and studied in [12]. We give several characterizations of this property; in particular one of the characterizations is Ramsey theoretical. Also we give a result involving a property of the corresponding mapping between function spaces.

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 $\mathrm{\Delta }$ be thick affine building of type ${\tilde{G}}_2$ the Laplace operators of $\mathrm{\Delta }$; associated with a pair $(y_1,y_2)$; is the Poisson transform of a suitable finitely additive measure on the maximal boundary $\mathrm{\Omega }$ of $\mathrm{\Delta }$; by using only the combinatorial structure of $\mathrm{\Delta }$.

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...

Let $\gamma ,\tau :[a,b]\to R^{k+1}$ be a couple of Lipschitz maps such that ${\gamma }^{\prime }=\pm |{\gamma }^{\prime }|\tau$ almost everywhere in $[a,b]$. Then $\gamma ([a,b])$ is a $C^2$-rectifiable set, namely it may be covered by countably many curves of class $C^2$ embedded in $R^{k+1}$. As a conseguence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a $C^2$-rectifiable set.

Let $I=[a,b]\subset ℝ$, let , let $u$ and $v$ be positive functions with $u\in L_{p^{\prime }}(I)$ e $v\in L_q(I)$ and let $T:L_p(I)\to L_q(I)$be the Hardy-type operator given by $$\left(Tf\right)\left(x\right)=v\left(x\right){\int }_a^{x}f\left(t\right)u\left(t\right)dt,x\in I.$$ We show that the asymptotic behavior of the eigenvalues$\lambda$of the non-linear integral system $$g\left(x\right)=\left(TF\right)\left(x\right){\left(f\left(x\right)\right)}_{\left(p\right)}=\lambda \left(T^{*}{g}_{\left(p\right)}\right))\left(x\right)$$ (where, for example,$t_{(p)}={|t|}^{p-1}\mathrm{sgn}(t)$is given by $$\begin{array}{cc}& \underset{n\to \infty }{lim}n{\widehat{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^r{)}^{1/r}dt\right)}^{1/r},\text{for}1<p<q<\infty \hfill \\ & \underset{n\to \infty }{lim}n{\check{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^{r}dt\right)}^{1/r}\text{for}1<q<p<\infty \hfill \end{array}$$ Here$r=\frac{1}{p^{\prime }}+\frac{1}{p}$, $c_{p,q}$ is an explicit constant depending only on $p$ and $q$, $\widehat{\lambda }(T)=\max (s{p}_n(T,p,q))$, ${\check{\lambda }}_n(T)=\min (s{p}_n(T,p,q))$ where $s{p}_n(T,p,q)$ stands for the set of all eigenvalues $\lambda$ corresponding to eigenfunctions $g$ with $n$ zeros.

Richness of the class of $ℛ$-maps [3] is investigated. Several consequences for the theory of quasi-$\mathscr{H}$-closed spaces are indicated.

We use notations: $[x,y]=[x_{,1}y]$ and $[x_{,k+1}y]$ e $[[x_{,k}y],y]$. We consider groups generated by $x$, $y$ satisfying relations $x=[x_{,n}y],y=[y_{,n}x]$ or $[x,y]=[x_{,n}y]$, $[y,x]=[y_{,n}x]$. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if $p$ is a prime then the group $S{l}_2(p)$ is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation $⟨x,y\mid [x,y]=[x_{,2}y],[y,x]=[y_{,2}x],x^n,y^m⟩$ are finite.

Si stabiliscono due condizioni sufficienti per un germe di ipersuperficie reale di classe $c^{\mathrm{\infty }}$ in ${𝐂}^n$ affinchè esistano coordinate olomorfe rispetto alle quali l'ipersuperficie risulti essere il luogo di zeri di una funzione di variabili e $k$ sia minimale rispetto a questa proprietà. In altre parole si vuole che l'ipersuperficie, a meno di una trasformazione bi-olomorfa, sia l’unione di sottovarietà lineari complesse, parallele di dimensione $n\mathrm{—}k$.

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...

The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism $u$ a Jacobian $J(C)$ has connected kernel, the principal polarization on $J(C)$ induces a multiple of the principal polarization on the image of $u$. We reformulate and complete this theorem proving "constructively" the following: Theorem. Let $Z\subset J(C)$ be an abelian subvariety and $Y$ its complementary variety. $Z$ is a Prym-Tyurin variety with respect to $J(C)$ if and only if the following sequence...

Si considera l’equazione non lineare nell'incognita $u(t,x)$ ((1,1) del testo) soddisfatta in un cilindro $G=(o,T)\times \mathrm{\Omega }$ ($\mathrm{\Omega }$ dominio limitato di ${𝐑}^{𝐧}$) con condizioni al contorno tipo Dirichlet o Neumann sulla superficie laterale di $G$ e con relazioni omogenee fra $u$ e $u_t$ sulle basi. Si stabiliscono per la (1) e nel caso di risonanza alcuni teoremi di perturbazione.

We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot )}$ whenever the exponent function $p(\cdot )$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot )$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_a$, , maps $L^{p(\cdot )}$ into $L^{q(\cdot )}$, where $1/p(\cdot )-1/q(\cdot )=a/n$.

In this paper we consider a random variable $h$ arising from an intersection problem between a fixed convex body $K_0$ and a system of random independent and uniformly distributed ovaloids in $E_3$.

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.

Sfruttando i risultati di [1], si prova che le derivate spaziali $D^{\alpha }u$ di ordine $|\alpha |$ con delle soluzioni in $Q$ di un sistema parabolico quasilineare di ordine $2m$ con andamenti strettamente controllati, sono parzialmente hölderiane in $Q$ con esponente di hölderianità decrescente al crescere di $|\alpha |$.

Per ogni soluzione della (1) nel dominio limitato $\mathrm{\Omega }$,, appartenente a $H_0^2(\mathrm{\Omega })$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^0$ del contorno; si consente a $\partial \mathrm{\Omega }$ di essere singolare in $x^0$.

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...

Se $A$ è un operatore in uno spazio di Hilbert e $V$ è un sotto insieme di questo spazio, in molti problemi si è indotti a modificare $A$ sul «bordo» di $V$ in modo da ottenere un operatore $A$ tale che le soluzioni dell'equazione differenziale associata $$0\in U^{\prime }+\tilde{A}(U)$$ non escano da $V$. Se $V$ non è convesso, l'operatore $A$ non rientra nei casi classici esaminati, ad esempio, in [1]. In questo lavoro introduciamo alcune classi di operatori che contengono, in qualçhe caso significativo, quelli del genere sopra...

Si dimostra il seguente risultato. Sia $X$ una superficie proiettivamente rigata, non iperpiana, di ${𝐏}^{\mathrm{𝟒}}$; allora $X$ è la rigata cubica oppure è una rigata quintica ellittica. Si descrive inoltre una nuova generazione proiettiva delle rigate quintiche ellittiche di ${𝐏}^{\mathrm{𝟒}}$.

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...

Sia $F$ un endomorfismo olomorfo della palla unitaria aperta $B_n$ di ${𝐂}^{𝐧}$. In questa nota proviamo che certe ipotesi minimali, relative al comportamento di $F$ su un orociclo e vicino ad un punto del bordo, assicurano che $F$ è un automorfismo olomorfo di $B_n$.

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 $V$.