scholarly journals The generators of $3$-class group of some fields of degree $6$ over $\mathbb{Q}$

2021 ◽  
Vol 39 (3) ◽  
pp. 37-52
Author(s):  
Siham Aouissi ◽  
Moulay Chrif Ismaili ◽  
Mohamed Talbi ◽  
Abdelmalek Azizi
Keyword(s):  
Prime Number ◽  
Class Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Let be k=Q(\sqrt[3]{p},\zeta_3), where p is a prime number such that p \equiv 1 (mod  9), and let C_{k,3} the 3-component of the class group of k. In his work [7], Frank Gerth III proves a conjecture made by Calegari and Emerton which gives a necessary and sufficient conditions for C_{k,3} to be of rank two. The present work display a consideration steps towards determination of generators of C_{k,3}, when C_{k,3} is isomorphic to Z/9Z \times Z/3Z.

Arithmetical Semigroup Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
Bonnie R. Hardy ◽  
Thomas S. Shores
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Forthcoming Paper ◽  
Semigroup Ring ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Rings ◽  
Principal Ideal Ring ◽  
Arithmetical Semigroup ◽  
Necessary And Sufficient

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

scholarly journals Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Khuanchanok Chaichana ◽  
Pradthana Jaipong
Keyword(s):  
Prime Number ◽  
Fuchsian Group ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Minimal Length ◽  
Necessary And Sufficient Conditions ◽  
Fuchsian Groups ◽  
Suborbital Graphs ◽  
Necessary And Sufficient ◽  
Recursive Representation

In this study, we work on the Fuchsian group Hm where m is a prime number acting on mℚ^ transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.

scholarly journals The distribution of prime ideals of a Dedekind domain

1974 ◽  
Vol 11 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Anne P. Grams
Keyword(s):  
Class Group ◽  
Sufficient Conditions ◽  
Torsion Group ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Class Groups ◽  
Maximal Ideals ◽  
The Family ◽  
Necessary And Sufficient

Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.

On the Hausdorff and Trigonometric Moment Problems

1961 ◽  
Vol 13 ◽  
pp. 454-461
Author(s):  
P. G. Rooney
Keyword(s):  
Moment Problem ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Moment Problems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that(1)For various collections K this problem has been solved—see (3, Chapter III)By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that(2)For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

scholarly journals The Determination of Jupiter's Mass from Large Perturbations on Cometary Orbits in Jupiter's Sphere of Action

1972 ◽  
Vol 45 ◽  
pp. 227-232 ◽  
Author(s):  
E. I. Kazimirchak-Polonskaya
Keyword(s):  
Electronic Computer ◽  
Sufficient Conditions ◽  
Great Accuracy ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Necessary and sufficient conditions are formulated for determining the mass of Jupiter from large perturbations induced in cometary orbits in the sphere of action of Jupiter. A procedure for the investigation has been developed and programmed for an electronic computer. Comparison of heliocentric and jovicentric computations shows that the perturbations on P/Wolf could be determined with great accuracy when this comet passed through Jupiter's sphere of action in 1922. The first attempt has been made to determine the mass of Jupiter using this passage and the observations of the comet in 1925. The resulting value for the reciprocal mass is 1047.345.

scholarly journals Existence and determination of the set of Metzler matrices for given stable polynomials

2012 ◽  
Vol 22 (2) ◽  
pp. 389-399 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Stable Polynomials ◽  
Necessary And Sufficient

Existence and determination of the set of Metzler matrices for given stable polynomialsThe problem of the existence and determination of the set of Metzler matrices for given stable polynomials is formulated and solved. Necessary and sufficient conditions are established for the existence of the set of Metzler matrices for given stable polynomials. A procedure for finding the set of Metzler matrices for given stable polynomials is proposed and illustrated with numerical examples.

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