A differential equation related to the $l^{p}$-norms

Jacek Bojarski; Tomasz Małolepszy; Janusz Matkowski

Displaying similar documents to “A differential equation related to the $l^{p}$ -norms”

Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision

Jeremy Yirmeyahu Kaminski, Michael Fryers, Mina Teicher (2005)

Journal of the European Mathematical Society

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We study some geometric configurations related to projections of an irreducible algebraic curve embedded in $ℂ ℙ^{3}$ 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 $ℂ ℙ^{3}$ , 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...

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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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 $ℙ^{3}$ . 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 $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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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.

The Mordell-Weil bases for the elliptic curve $y^{2} = x^{3} - m^{2} x + m^{2}$

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

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Let $D_{m}$ be an elliptic curve over $ℚ$ of the form $y^{2} = x^{3} - m^{2} x + m^{2}$ , where $m$ is an integer. In this paper we prove that the two points $P_{- 1} = (- m, m)$ and $P_{0} = (0, m)$ on $D_{m}$ can be extended to a basis for $D_{m} (ℚ)$ under certain conditions described explicitly.

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Existence problem of the osculating planes of a curve in $R_{n}$

S. Topa (1964)

Annales Polonici Mathematici

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Joint distribution for the Selmer ranks of the congruent number curves

Ilija S. Vrećica (2020)

Czechoslovak Mathematical Journal

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We determine the distribution over square-free integers $n$ of the pair $({dim}_{𝔽_{2}} {Sel}^{Φ} (E_{n} / ℚ), {dim}_{𝔽_{2}} {Sel}^{\hat{Φ}} (E_{n}^{'} / ℚ))$ , where $E_{n}$ is a curve in the congruent number curve family, $E_{n}^{'} : y^{2} = x^{3} + 4 n^{2} x$ is the image of isogeny $Φ : E_{n} \to E_{n}^{'}$ , $Φ (x, y) = (y^{2} / x^{2}, y (n^{2} - x^{2}) / x^{2})$ , and $\hat{Φ}$ is the isogeny dual to $Φ$ .

Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque (2024)

Archivum Mathematicum

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We consider the Lebesgue-Ramanujan-Nagell type equation $x^{2} + 5^{a} 13^{b} 17^{c} = 2^{m} y^{n}$ , where $a, b, c, m \geq 0, n \geq 3$ and $x, y \geq 1$ are unknown integers with $gcd (x, y) = 1$ . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$ -integral points on a class of elliptic curves.

On the complexity of braids

Ivan Dynnikov, Bert Wiest (2007)

Journal of the European Mathematical Society

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We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $_{i j}$ , which are Garside-like half-twists involving strings $i$ through $j$ , and by counting powered generators $Δ_{i j}^{k}$ as $log (| k | + 1)$ instead of simply $| k |$ . The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism....

A curve of genus $q$ with a Half-Canonical embedding in $\mathbf{P}^{3}$

Sevin Recillas (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si costruiscono curve di genere $g=4n—3$ , $n\geq 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^{0}(L)=4$ . Per $n+3$ si dimostra che gli $L$ sono molto ampi.

A curve of genus $q$ with a Half-Canonical embedding in $\mathbf{P}^{3}$

Sevin Recillas (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Si costruiscono curve di genere $g=4n—3$ , $n\geq 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^{0}(L)=4$ . Per $n+3$ si dimostra che gli $L$ sono molto ampi.

The end curve theorem for normal complex surface singularities

Walter D. Neumann, Jonathan Wahl (2010)

Journal of the European Mathematical Society

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We prove the “End Curve Theorem,” which states that a normal surface singularity $(X, o)$ with rational homology sphere link $Σ$ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function $(X, o) \to (ℂ, 0)$ whose zero set intersects $Σ$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” $(X, o)$ is described by giving an explicit set of...

Shells of monotone curves

Josef Mikeš, Karl Strambach (2015)

Czechoslovak Mathematical Journal

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We determine in $ℝ^{n}$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $Ω$ of affinities containing the translation group is a geodesic with respect to $\nabla$ . Special attention is paid to the case that $Ω$ contains many dilatations or that $C$ is a curve in $ℝ^{3}$ . If $C$ is a curve in $ℝ^{3}$ and $Ω$ is the translation group then we calculate not only the components of the curvature and the Weyl...

Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca, Edoardo Sernesi (2014)

Journal de l’École polytechnique — Mathématiques

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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.

Integral points on the elliptic curve $y^{2} = x^{3} - 4 p^{2} x$

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

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Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E : y^{2} = x^{3} - 4 p^{2} x$ . Further, let $N (p)$ denote the number of pairs of integral points $(x, \pm y)$ on $E$ with $y > 0$ . We prove that if $p \geq 17$ , then $N (p) \leq 4$ or $1$ depending on whether $p \equiv 1 (mod 8)$ or $p \equiv - 1 (mod 8)$ .

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $C$ be a smooth curve of genus $g$ . For each positive integer $r$ the birational $r$ -gonality $s_{r} (C)$ of $C$ is the minimal integer $t$ such that there is $L \in {Pic}^{t} (C)$ with $h^{0} (C, L) = r + 1$ . Fix an integer $r \geq 3$ . In this paper we prove the existence of an integer $g_{r}$ such that for every integer $g \geq g_{r}$ there is a smooth curve $C$ of genus $g$ with $s_{r + 1} (C) / (r + 1) > s_{r} (C) / r$ , i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.

Displaying similar documents to “A differential equation related to the l p -norms”

Displaying similar documents to “A differential equation related to the $l^{p}$ -norms”