Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^p$-distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

Conformally flat metric $\overline{g}$ is said to be Ricci superosculating with $g$ at the point $x_0$ if $g_{ij}\left(x_0\right)={\overline{g}}_{ij}\left(x_0\right)$, ${\Gamma }_{ij}^k\left(x_0\right)={\overline{\Gamma }}_{ij}^k\left(x_0\right)$, $R_{ij}^k\left(x_0\right)={\overline{R}}_{ij}^k\left(x_0\right)$, where $R_{ij}$ is the Ricci tensor. In this paper the following theorem is proved: ((

Si costruiscono famiglie di curve iperellittiche col $p$—rango della varietà jacobiana uguale a zero. La costruzione sfrutta le proprietà elementari dell’operatore di Cartier e delle estensioni $p$-cicliche dei corpi con la caratteristica $p$ maggiore di zero.

We study some geometric configurations related to projections of an irreducible algebraic curve embedded in $ℂ{\mathbb{P}}^3$ onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve $X$ embedded in $ℂ{\mathbb{P}}^3$, of degree $d$ and genus $g$, can be recovered using its projections from points onto embedded projective planes. The embeddings are unknown. The only input is the defining...

In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space $E_1^4$.

We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation $${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$ where $h$ is a $C^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^4$, and ${\rho }_k$ is a constant such that ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$: $$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$ where $c_0\>0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as...

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve $C$ and the stability of the sheaf of logarithmic vector fields along $C$, the freeness of the divisor $C$ and the Torelli properties of $C$ (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in ${\mathbb{P}}^3$. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$, we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in ${\mathbb{P}}^N$ and $E$ to be the image of the double curve of a ${\mathbb{P}}^3$-model of $X$. In the classical Segre theory, a...

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable...