Number of solutions of cubic Thue inequalities with positive discriminant

N. Saradha; Divyum Sharma

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Displaying similar documents to “Number of solutions of cubic Thue inequalities with positive discriminant”

Gauss Sums of the Cubic Character over $G F (2^{m})$ : an Elementary Derivation

Davide Schipani, Michele Elia (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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By an elementary approach, we derive the value of the Gauss sum of a cubic character over a finite field $_{2^{s}}$ without using Davenport-Hasse’s theorem (namely, if s is odd the Gauss sum is -1, and if s is even its value is $- {(- 2)}^{s / 2}$ ).

The reduced ideals of a special order in a pure cubic number field

Abdelmalek Azizi, Jamal Benamara, Moulay Chrif Ismaili, Mohammed Talbi (2020)

Archivum Mathematicum

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Let $K = ℚ (θ)$ be a pure cubic field, with $θ^{3} = D$ , where $D$ is a cube-free integer. We will determine the reduced ideals of the order $𝒪 = ℤ [θ]$ of $K$ which coincides with the maximal order of $K$ in the case where $D$ is square-free and $\neg \equiv \pm 1 (mod 9)$ .

Binomial sums via Bailey's cubic transformation

Wenchang Chu (2023)

Czechoslovak Mathematical Journal

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By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the $_{3} F_{2} (x)$ -series, we examine a class of $_{3} F_{2} (4)$ -series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.

Contribution to construction of global cubic $C^{1}$ or $C^{2}$ -spline on equidistant knotset

Kobza, Jiří

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On $☆$ - associated comonotone functions

Ondrej Hutník, Jozef Pócs (2018)

Kybernetika

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We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to $+$ -associatedness of functions (as proved by Boczek and Kaluszka), but also to their $☆$ -associatedness with $☆$ being an arbitrary...

Rational solutions of certain Diophantine equations involving norms

Maciej Ulas (2014)

Acta Arithmetica

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We present some results concerning the unirationality of the algebraic variety $_{f}$ given by the equation $N_{K / k} (X ₁ + α X ₂ + α ² X ₃) = f (t)$ , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and $_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_{f}$ is k-unirational. A similar result is proved for a broad family of quintic polynomials...

On a cubic Hecke algebra associated with the quantum group $U_{q} (2)$

Janusz Wysoczański (2010)

Banach Center Publications

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We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_{q} (2)$ , which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_{j} : = I_{j} \otimes α \otimes I_{n - 2 - j}$ on ${(ℂ ³)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra $ℋ_{q, n} (2)$ associated with the quantum group $U_{q} (2)$ . The purpose of this note is to present the construction.

$ℱ$ -hypercyclic and disjoint $ℱ$ -hypercyclic properties of binary relations over topological spaces

Marko Kostić (2020)

Mathematica Bohemica

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We examine various types of $ℱ$ -hypercyclic ( $ℱ$ -topologically transitive) and disjoint $ℱ$ -hypercyclic (disjoint $ℱ$ -topologically transitive) properties of binary relations over topological spaces. We pay special attention to finite structures like simple graphs, digraphs and tournaments, providing a great number of illustrative examples.

The cubic nonlinear Dirac equation

Federico Cacciafesta (2012)

Journées Équations aux dérivées partielles

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We present some results obtained in collaboration with prof. Piero D’Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in $H^{1}$ with slight additional assumptions. The new crucial tool is given by the proof of some refined endpoint Strichartz estimates.

From binary cube triangulations to acute binary simplices

Brandts, Jan, van den Hooff, Jelle, Kuiper, Carlo, Steenkamp, Rik

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Cottle’s proof that the minimal number of $0 / 1$ -simplices needed to triangulate the unit $4$ -cube equals $16$ uses a modest amount of computer generated results. In this paper we remove the need for computer aid, using some lemmas that may be useful also in a broader context. One of the $0 / 1$ -simplices involved, the so-called antipodal simplex, has acute dihedral angles. We continue with the study of such acute binary simplices and point out their possible relation to the Hadamard determinant problem. ...

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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Let $ε$ be an algebraic unit of the degree $n \geq 3$ . Assume that the extension $ℚ (ε) / ℚ$ is Galois. We would like to determine when the order $ℤ [ε]$ of $ℚ (ε)$ is $Gal (ℚ (ε) / ℚ)$ -invariant, i.e. when the $n$ complex conjugates $ε_{1}, \dots, ε_{n}$ of $ε$ are in $ℤ [ε]$ , which amounts to asking that $ℤ [ε_{1}, \dots, ε_{n}] = ℤ [ε]$ , i.e., that these two orders of $ℚ (ε)$ have the same discriminant. This problem has been solved only for $n = 3$ by using an explicit formula for the discriminant of the order $ℤ [ε_{1}, ε_{2}, ε_{3}]$ . However, there is no known similar formula for $n > 3$ . In the present paper, we put forward and...

On relative pure cyclic fields with power integral bases

Mohammed Sahmoudi, Mohammed Elhassani Charkani (2023)

Mathematica Bohemica

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Let $L = K (α)$ be an extension of a number field $K$ , where $α$ satisfies the monic irreducible polynomial $P (X) = X^{p} - β$ of prime degree belonging to $𝔬_{K} [X]$ ( $𝔬_{K}$ is the ring of integers of $K$ ). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field...

Lower bounds for Schrödinger operators in H¹(ℝ)

Ronan Pouliquen (1999)

Studia Mathematica

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We prove trace inequalities of type $| | u^{'} {| |}_{L^{2}}^{2} + \sum_{j \in ℤ} k_{j} | u (a_{j}) |^{2} {\geq λ | | u | |}_{L^{2}}^{2}$ where $u \in H^{1} (ℝ)$ , under suitable hypotheses on the sequences ${a_{j}}_{j \in ℤ}$ and ${k_{j}}_{j \in ℤ}$ , with the first sequence increasing and the second bounded.