See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\mathrm{\Sigma }$, $\chi$ and $M$ satisfy conditions (11.7) when $𝒬$ is a polynomial in $\dot{\gamma }$, conditions (C)-i.e. (11.8) and (11.7) with $𝒬\equiv 0$-are proved to be necessary for treating satisfactorily $\mathrm{\Sigma }$'s hyper-impulsive motions (in which positions can suffer first order discontinuities)....

See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case $\mathrm{\Sigma }$, $\chi$ and $M$ satisfy conditions (11.7) when $𝒬$ is a polynomial in $\dot{\gamma }$, conditions (C)-i.e. (11.8) and (11.7) with $𝒬\equiv 0$-are proved to be necessary for treating satisfactorily $\mathrm{\Sigma }$'s hyper-impulsive motions (in which positions can suffer first order discontinuities)....

In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot )$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot )$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\mathrm{\Sigma }$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy...

Let $\mathrm{\Sigma }$ be a constrained mechanical system locally referred to state coordinates $(q^1,\mathrm{\dots },q^N,{\gamma }^1,\mathrm{\dots },{\gamma }^M)$. Let $({\tilde{\gamma }}^1\mathrm{\dots }{\tilde{\gamma }}^M)(\cdot )$ be an assigned trajectory for the coordinates ${\gamma }^{\alpha }$ and let $u(\cdot )$ be a scalar function of the time, to be thought as a control. In [4] one considers the control system ${\mathrm{\Sigma }}_{\widehat{\gamma }}$, which is parametrized by the coordinates $(q^1,\mathrm{\dots },q^N)$ and is obtained from $\mathrm{\Sigma }$ by adding the time-dependent, holonomic constraints ${\gamma }^{\alpha }={\widehat{\gamma }}^{\alpha }(t):={\tilde{\gamma }}^{\alpha }(u(t))$. More generally, one can consider a vector-valued control $u(\cdot )=(u^1,\mathrm{\dots },u^M)(\cdot )$ which is directly identified with $\widehat{\gamma }(\cdot )=({\widehat{\gamma }}^1,\mathrm{\dots },{\widehat{\gamma }}^M)(\cdot )$. If one denotes the momenta...

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

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].

In [1] I and II various equivalence theorems are proved; e.g. an ODE $(ℰ)\dot{z}=F(t,z,u,\dot{u})(\in {ℝ}^m)$ with a scalar control $u=u(\cdot )$ is linear w.r.t. $\dot{u}$ iff $(\alpha )$ its solution $z(u,\cdot )$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta )$ $z(u,\cdot )$ satisfies certain conditions by which $1^{st}$-order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z=(q,p)$ equation $(ℰ)$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

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...

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 $f\in VLO([0,1])$

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 $f$.

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.

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. ...

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot )$. However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot )$, where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\mathrm{\Sigma }$, referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy...

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.

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.

A metacyclic group $H$ can be presented as $\langle \alpha ,\beta :{\alpha }^n=1$, ${\beta }^m={\alpha }^t$, $\beta \alpha {\beta }^{-1}={\alpha }^r\rangle$ for some $n$, $m$, $t$, $r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma \left(\alpha \right)={\alpha }^{x_1}{\beta }^{y_1}$, $\sigma \left(\beta \right)={\alpha }^{x_2}{\beta }^{y_2}$ for some integers $x_1$, $x_2$, $y_1$, $y_2$. We give sufficient and necessary conditions on $x_1$, $x_2$, $y_1$, $y_2$ for $\sigma$ to be an automorphism.

Let $X$ be a zero-dimensional space and $C_c\left(X\right)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K\left(X\right)$ (resp., $C_c^{\psi }\left(X\right))$ the set of all functions in $C_c\left(X\right)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K\left(X\right)=O_c^{{\beta }_{0}X\setminus X}$ (resp., $C_c^{\psi }\left(X\right)=M_c^{{\beta }_{0}X\setminus {\upsilon }_{0}X}$), where ${\beta }_{0}X$ is the Banaschewski compactification of $X$ and ${\upsilon }_{0}X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c\left(X\right)$ is equal to $C_c^K\left(X\right)$, i.e., $M_c^{{\beta }_{0}X\setminus X}=C_c^K\left(X\right)$. By applying...

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...

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 $V$.