共查询到20条相似文献,搜索用时 15 毫秒
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In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family. 相似文献
3.
Mathematical Notes - In this paper, we deal with the equation $$(a^{n}-2)(b^{n}-2)=x^{2}$$ , $$2\leq a3$$ is odd and $$P_{k},Q_{k}$$ are the Pell and Pell Lucas numbers, respectively. We also... 相似文献
4.
Christophe Dupont 《Mathematische Annalen》2011,349(3):509-528
Let f be an endomorphism of
\mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ
1, . . . , λ
k
). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dimHm 3 [(log d)/(l1)] + [(log d)/(l2)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of
the ν-generic inverse branches of f
n
in
\mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in
\mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f
n
. 相似文献
5.
Brieden 《Discrete and Computational Geometry》2002,28(2):201-209
Abstract. Maximizing geometric functionals such as the classical l
p
-norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions
can be computed or at least approximated.
While the maximum of the l
∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l
p
-norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes.
Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R
n
, numbers c
ij
, 1≤ i≤ j≤ n , are given and the goal is to maximize Σ
1≤ i≤ j≤ n
c
ij
x
i
x
j
over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP.
Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed,
in this paper it is proved that analogous inapproximability results hold for computing the l
p
-norm-maximum and various l
p
-radii of centrally symmetric polytopes for 2≤ p < ∞. 相似文献
6.
A. Melakhessou K. Guenda T. A. Gulliver M. Shi P. Solé 《Journal of Applied Mathematics and Computing》2018,57(1-2):375-391
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R. 相似文献
7.
Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic. 相似文献
8.
We construct the category of quotients of
-spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990. 相似文献
9.
The Ramanujan Journal - In this paper, we prove two supercongruences conjectured by Sun via the Wilf–Zeilberger method. One of them is, for any prime $$p>3$$ , $$\begin{aligned}... 相似文献
10.
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics
and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper,
six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over are presented. These polynomials have simple shapes, and they are related to planar functions.
This work was supported by Australian Research Council (Grant No. DP0558773), National Natural Science Foundation of China
(Grant No. 10571180) and the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No. 612405) 相似文献
11.
E. Mukhin V. Schechtman V. Tarasov A. Varchenko 《Proceedings of the Steklov Institute of Mathematics》2007,258(1):155-177
A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum
Gaudin model is presented.
Dedicated to V.I. Arnold on the occasion of his 70th birthday 相似文献
12.
A reducible representation of the Temperley-Lieb algebra is constructed on a tensor product of n-dimensional spaces. As a
centralizer of this action, we obtain a quantum algebra (quasi-triangular Hopf algebra) with the representation ring that
is equivalent to the representation ring of the Lie algebra. Bibliography: 23 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 167–177. 相似文献
13.
设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s
相似文献
14.
Theoretical and Mathematical Physics - The relativistic wave equation considered to mathematically describe the Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the... 相似文献
15.
Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P
3-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P
3-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs. 相似文献
16.
Designs, Codes and Cryptography - We classify all permutation polynomials of the form $$x^3g(x^{q-1})$$ of $${\mathbb F}_{q^2}$$ where $$g(x)=x^3+bx+c$$ and $$b,c \in {\mathbb F}_q^*$$ . Moreover... 相似文献
17.
We prove that the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is exactly the class of \(\mathbb {Z}_2\)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial \(\mathbb {Z}_2\mathbb {Z}_2[u]\) structure. Moreover, we exhibit some examples of \(\mathbb {Z}_2\)-linear codes which are not \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear. Also, we state that the duality of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is the same as the duality of \(\mathbb {Z}_2\)-linear codes. Finally, we prove that the class of \(\mathbb {Z}_2\mathbb {Z}_4\)-linear codes which are also \(\mathbb {Z}_2\)-linear is strictly contained in the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes. 相似文献
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Maria Donten-Bury 《Annals of Combinatorics》2016,20(3):549-568
We study phylogenetic invariants of general group-based models of evolution with group of symmetries \({\mathbb{Z}_3}\). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal \({I'}\) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that \({I = I'}\). 相似文献
20.
Nurettin Irmak 《Periodica Mathematica Hungarica》2016,73(1):130-136
Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).
相似文献
$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$