Restrictions of Fourier transforms to curves

S. W. Drury

Displaying similar documents to “Restrictions of Fourier transforms to curves”

Complex orientation formulas for $M$ -curves of degree $4 d + 1$ with 4 nests

S.Yu. Orevkov (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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On démontre la formule d’orientations complexes pour les $M$ -courbes dans ${ℝ P}^{2}$ de degré $4 d + 1$ ayant $4$ nids. Cette formule généralise celle pour les $M$ -courbes à nid profond. C’est un pas vers la classification des $M$ -courbes de degré $9$ .

Timelike $B_{2}$ -slant helices in Minkowski space $E_{1}^{4}$

Ahmad T. Ali, Rafael López (2010)

Archivum Mathematicum

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We consider a unit speed timelike curve $α$ in Minkowski 4-space $𝐄_{1}^{4}$ and denote the Frenet frame of $α$ by ${𝐓, 𝐍, 𝐁_{1}, 𝐁_{2}}$ . We say that $α$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of $𝐄_{1}^{4}$ . In this work we study those helices where the function $〈 𝐁_{2}, U 〉$ is constant and we give different characterizations of such curves.

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

Jacek Bojarski, Tomasz Małolepszy, Janusz Matkowski (2011)

Annales Polonici Mathematici

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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.

On the arithmetic of the hyperelliptic curve $y^{2} = x^{n} + a$

Kevser Aktaş, Hasan Şenay (2016)

Czechoslovak Mathematical Journal

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We study the arithmetic properties of hyperelliptic curves given by the affine equation $y^{2} = x^{n} + a$ by exploiting the structure of the automorphism groups. We show that these curves satisfy Lang’s conjecture about the covering radius (for some special covering maps).

Optimal curves differing by a 5-isogeny

Dongho Byeon, Taekyung Kim (2014)

Acta Arithmetica

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For i = 0,1, let $E_{i}$ be the $X_{i} (N)$ -optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational...

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.

A group law on smooth real quartics having at least $3$ real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

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Let $C$ be a smooth real quartic curve in $ℙ^{2}$ . Suppose that $C$ has at least $3$ real branches $B_{1}, B_{2}, B_{3}$ . Let $B = B_{1} \times B_{2} \times B_{3}$ and let $O \in B$ . Let $τ_{O}$ be the map from $B$ into the neutral component Jac $(C) {(ℝ)}^{0}$ of the set of real points of the jacobian of $C$ , defined by letting $τ_{O} (P)$ be the divisor class of the divisor $\sum P_{i} - O_{i}$ . Then, $τ_{O}$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac $(C) {(ℝ)}^{0}$ . It generalizes the classical geometric description of the group law on the neutral component of the set of real...

A Remark on a Paper of Crachiola and Makar-Limanov

Robert Dryło (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line $¹_{k}$ , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

Osculating curves in 4-dimensional semi-Euclidean space with index 2

Kazim İlarslan, Nihal Kiliç, Hatice Altin Erdem (2017)

Open Mathematics

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In this paper, we give the necessary and sufficient conditions for non-null curves with non-null normals in 4-dimensional Semi-Euclidian space with indeks 2 to be osculating curves. Also we give some examples of non-null osculating curves in [...] E24 $𝔼_{2}^{4}$ .

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 the Neumann-Poincaré operator

Josef Král, Dagmar Medková (1998)

Czechoslovak Mathematical Journal

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Let $Γ$ be a rectifiable Jordan curve in the finite complex plane $ℂ$ which is regular in the sense of Ahlfors and David. Denote by $L_{C}^{2} (Γ)$ the space of all complex-valued functions on $Γ$ which are square integrable w.r. to the arc-length on $Γ$ . Let $L^{2} (Γ)$ stand for the space of all real-valued functions in $L_{C}^{2} (Γ)$ and put $L_{0}^{2} (Γ) = {h \in L^{2} (Γ) \int_{Γ} h (ζ) | d ζ | = 0} .$ Since the Cauchy singular operator is bounded on $L_{C}^{2} (Γ)$ , the Neumann-Poincaré operator $C_{1}^{Γ}$ sending each $h \in L^{2} (Γ)$ into $C_{1}^{Γ} h (ζ_{0}) : = ℜ {(π i)}^{- 1} P . V . \int_{Γ} \frac{h (ζ)}{ζ - ζ_{0}} d ζ, ζ_{0} \in Γ,$ is bounded on $L^{2} (Γ)$ . We show that the inclusion $C_{1}^{Γ} (L_{0}^{2} (Γ)) \subset L_{0}^{2} (Γ)$ characterizes the circle in the...