arab journal of mathematical sciences number 1 arab

AJMS_27.1._edb_proof 1..1A rab
Jo u
Volume 27 Issue 1 2021
Number 1
1 Editorial advisory board
2 On warped product bi-slant submanifolds of Kenmotsu manifolds Siraj Uddin, Ion Mihai and Adela Mihai
15 On the number of edges of a graph and its complement Jamel Dammak, Gerard Lopez and Hamza Si Kaddour
20 On Hilbert class fi eld tower for some quartic number fi elds Abdelmalek Azizi, Mohamed Talbi and Mohammed Talbi
26 A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces Getahun Bekele Wega and Habtu Zegeye
41 Reconstruction of a homogeneous polynomial from its additive decompositions when identifi ability fails E. Ballico
53 Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications Lamia Saeed Alqahtani
73 Toeplitz and slant Toeplitz operators on the polydisk Munmun Hazarika and Sougata Marik
94 Structure and substructure connectivity of circulant graphs and hypercubes T. Tamizh Chelvam and M. Sivagami
104 Note on a W1, ∞ (L2)-error estimate of a nonlinear fi nite volume scheme for a semi-linear heat equation Ahmed Berkane and Abdallah Bradji
119 The Q 0 -matrix completion problem
Kalyan Sinha
Quarto trim size: 174mm x 240mm
The Arab Journal of Mathematical Sciences is the official science journal of the Saudi Association for Mathematical Sciences. It is dedicated to the publication of original and expository papers in pure and applied mathematics, and is reviewed and edited by an international group of scholars. The Arab Journal of Mathematical Sciences will accept submissions in the mainstream areas of pure and applied mathematics, including algebra, analysis, geometry, diff erential equations, and discrete mathematics.
EDITOR Dr Tariq Alfhadel Department of Mathematics, King Saud University, Saudi Arabia Homepage: https://www.emeraldgrouppublishing.com/ journal/ajms
e-ISSN 2588-9214 p-ISSN 1319-5166 © King Saud University
Guidelines for authors can be found at: https://www.emeraldgrouppublishing.com/journal/ ajms#author-guidelines
Emerald Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, United Kingdom Tel +44 (0) 1274 777700; Fax +44 (0) 1274 785201 E-mail [email protected] For more information about Emerald’s regional offi ces please go to http://www.emeraldgrouppublishing.com/offi ces Customer helpdesk: Tel +44 (0) 1274 785278; Fax +44 (0) 1274 785201 E-mail [email protected] The Publisher and Editors cannot be held responsible for errors or any consequences arising from the use of information contained in this journal; the views and opinions expressed do not necessarily refl ect those of the Publisher and Editors, neither does the publication of advertisements constitute any endorsement by the Publisher and Editors of the products advertised.
Emerald is a trading name of Emerald Publishing Limited
EDITORIAL BOARD
Professor Sharief Deshmukh King Saud University, Saudi Arabia [email protected]
Professor Alberto Ferrero Universita del Piedmonte Oriental, Italy [email protected]
Professor Filippo Gazzola Politecnico di Milano, Italy [email protected]
Professor Mohammed Guediri King Saud University, Saudi Arabia [email protected]
Professor Mourad Ismail University of Central Florida, USA [email protected]
Dr Wissem Jedidi King Saud University, Saudi Arabia [email protected]
Professor Abdellatif Laradji King Saud University, Saudi Arabia [email protected]
Professor Elisabetta Maluta Politecnico di Milano, Italy [email protected]
Professor Nabil Ourimi King Saud University, Saudi Arabia [email protected]
Professor Vicentiu Radulescu University of Craiova, Romania [email protected]
Professor Bassem Samet King Saud University, Saudi Arabia [email protected]
EDITORIAL ADVISORY BOARD
Professor M. Al-Gwaiz King Saud University, Saudi Arabia
Professor V. Anandam Institute of Mathematical Science, India [email protected]
Professor Vieri Benci University of Pisa, Italy [email protected]
Professor Christian Berg University of Copenhagen, Denmark [email protected]
Professor Bang-Yen Chen Michigan State University, USA [email protected]
Professor S. Hedayat University of Illinois, USA [email protected]
Professor S. Kabbaj King Fahad University, Saudi Arabia [email protected]
Professor Mokhtar Kirane Universite de la Rochelle, France [email protected]
Professor Zuhair Nashed University of Central Florida, USA [email protected]
Professor Maurice Pouzet Universite Claude-Bernard, France [email protected]
Professor David Yost Federation University, Australia [email protected]
Editorial boards
Vol. 27 No. 1, 2021 p. 1
Emerald Publishing Limited e-ISSN: 2588-9214 p-ISSN: 1319-5166
On warped product bi-slant submanifolds of
Kenmotsu manifolds Siraj Uddin
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
Ion Mihai Faculty of Mathematics and Computer Science, University of Bucharest,
Bucharest, Romania, and
Adela Mihai Department of Mathematics and Computer Science,
Technical University of Civil Engineering Bucharest, Bucharest, Romania
Abstract Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form).
In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant) of warped product bi-slant submanifolds of Kenmotsu manifolds in terms of the warping function (an intrinsic invariant) and bi-slant angles. The equality case is also considered. Some applications of derived inequality are given.
Keywords Warped products, Semi-slant, Pseudo-slant, Bi-slant submanifolds, Warped product bi-slant
submanifolds, Kenmotsu manifolds
Paper type Original Article
1. Introduction In [18], K. Kenmotsu studied one class of almost contact metric manifolds known as Kenmotsu manifolds. He proved that:
AJMS 27,1
2
JEL Classification — 53C15, 53C40, 53C42, 53B25 © Siraj Uddin, Ion Mihai and Adela Mihai. Published in the Arab Journal of Mathematical Sciences.
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CCBY4.0) license. Anyonemay reproduce, distribute, translate and create derivativeworks of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons. org/licences/by/4.0/legalcode
Declaration of Competing Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The publisher wishes to inform readers that the article “Onwarped product bi-slant submanifolds of Kenmotsu manifolds” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Uddin, S., Mihai, I., Mihai, A. (2019), “On warped product bi-slant submanifolds of Kenmotsu manifolds”, Arab Journal of Mathematical Sciences, Vol. 27 No. 1, pp. 2-14. The original publication date for this paper was 05/07/2019.
The current issue and full text archive of this journal is available on Emerald Insight at:
https://www.emerald.com/insight/1319-5166.htm
Arab Journal of Mathematical Sciences Vol. 27 No. 1, 2021 pp. 2-14 Emerald Publishing Limited e-ISSN: 2588-9214 p-ISSN: 1319-5166 DOI 10.1016/j.ajmsc.2019.06.001
1. Locally a Kenmotsu manifold is a warped product I 3 f M of an interval I and a Kaehler manifold M, with warping function f ¼ cet, where c is a nonzero constant.
2. A Kenmotsu manifold with constant sectional curvature is a space of constant curvature −1, and so it is locally a hyperbolic space.
A ð2mþ 1Þ-dimensional manifold ~M is said to be an almost contact manifold if it admits an endomorphism w of its tangent bundle T ~M, a vector field ξ and a 1-form η, which satisfy:
w2 ¼ −I þ η⊗ ξ;wξ ¼ 0; ηðξÞ ¼ 0; η+w ¼ 0: (1.1)
There exists a compatible metric g, which satisfies
gðwX ;wY Þ ¼ gðX ;Y Þ ηðXÞηðY Þ; ηðXÞ ¼ gðX ; ξÞ; (1.2)
for all vector fields X ;Y on ~M [6]. In addition, an almost contact metric manifold ~M is said to be a Kenmotsu manifold [18] if the relation
ð~∇XwÞY ¼ gðwX ;Y Þξ ηðY ÞwX (1.3)
holds, where ~∇ is the Levi-Civita connection of g. From (1.3), for a Kenmotsu manifold ~M, we also have
~∇Xξ ¼ X ηðXÞξ: (1.4)
As Kenmotsu manifolds are warped product manifolds, therefore it is interesting to investigate the geometry of its warped product submanifolds. The notion of warped submanifolds was first introduced by B.-Y. Chen as a CR-warped product submanifold of Kaehler manifolds in his series of articles [11,12]. He established a general sharp inequality between the main extrinsic invariant (the second fundamental form) and an intrinsic invariant (the warping function) of such submanifolds. Motivated by Chen’s work many geometers studied warped product submanifolds for different spaces (see, e.g., [4,17,19–25,29,30] among many others. For the most up-to-date overview of this subject, see [13–15]).
On the other hand, J.L. Cabrerizro et al. studied in [7] bi-slant submanifolds of almost contact metric manifolds. In [28], the first author and B.-Y. Chen investigated warped product bi-slant submanifolds in Kaehler manifolds. They proved that there do not exist any warped product bi-slant submanifolds of Kaehler manifolds other than hemi-slant warped products and CR-warped products. The non-existence of warped product bi-slant submanifolds is proved in [2] for cosymplectic manifolds.
In this paper, we study warped product bi-slant submanifolds of a Kenmotsu manifold. The geometry of such submanifolds in Kenmotsu manifolds is quite different from Sasakian and cosymplectic case because in case of Kenmotsu manifolds such submanifolds exist while there is no proper warped product bi-slant submanifolds in Sasakain and cosymplectic as well. On their existence, we establish a generalized Chen type sharp inequality for the squared norm of the second fundamental form in terms of the warping function and bi-slant angles. The equality case is considered. Some applications are given in the last section.
2. Preliminaries Let ψ : Mn
→M ðnþdÞ be an isometric immersion of an n-dimensional Riemannian manifold M into an ðnþ dÞ-dimensional Riemannian manifold ~M. We denote by ∇ and ~∇ the Levi- Civita connections onM and ~M, respectively. Then the Gauss and Weingarten formulas are respectively given by [31]
~∇XY ¼ ∇XY þ hðX ;Y Þ (2.1)
Submanifolds of Kenmotsu
XV ; (2.2)
for any X ;Y ∈TM and V ∈T⊥M, where ∇⊥ is the normal connection in the normal bundle T⊥M andAV is the shape operator ofM with respect toV . Moreover, h : TM 3TM →T⊥M is the second fundamental form of M in ~M. Furthermore, AV and h are related by
gðhðX ;Y Þ;V Þ ¼ gðAV X ;Y Þ; (2.3)
for any X ;Y ∈TM and V ∈T⊥M. A submanifold M of a Riemannianmanifold ~M is said to be a totally umbilical submanifold
if hðX ;Y Þ ¼ gðX ;Y ÞH, for any X ;Y ∈TM, where H ¼ 1 n
Pn i¼1 hðei; eiÞ is the mean
curvature vector ofM. A submanifold M is said to be totally geodesic if hðX ;Y Þ ¼ 0. Also, one denotes
khk2 ¼ Xn
gðhðei; ejÞ; hðei; ejÞÞ; hrij ¼ gðhðei; ejÞ; erÞ; (2.4)
with i; j ¼ 1; . . . ; n; r ¼ nþ 1; . . . ; nþ d, where fe1; . . . ; eng is an orthonormal basis of the tangent space TpM and fenþ1; . . . ; enþdg an orthonormal basis of the normal space T⊥
p , for any p∈M.
For a differentiable function f on a m-dimensional manifold ~M, the gradient ∇ ! f of f is
defined as gð∇!f ;XÞ ¼ Xðf Þ, for any X tangent to ~M. For any vector field X tangent to M, we write
wX ¼ TX þ FX ; (2.5)
whereTX is the tangential component and FX is the normal component of wX. Thus,T is an endomorphism on the tangent bundle TM and F is a normal bundle valued 1-form of TM. A submanifoldM is called invariant if F is identically zero, that is, wX ∈TM for any X ∈TM; while, M is anti-invariant if T is identically zero, that is, wX ∈T⊥M, for any X ∈TM.
Similarly, for any vector field V normal to M, we put
wV ¼ tV þ fV ; (2.6)
where tV and fV are the tangential and normal components of wV , respectively.
LetM be a submanifold of an almost contact metric manifold ð ~M ;w; ξ; η; gÞ. Hence, if we denote D the orthogonal distribution to ξ in TM, then TM ¼ D⊕ hξi, where hξi is the 1-dimensional distribution spanned by ξ. For any nonzero vector X tangent toM at the point p∈M, such that X is not proportional to ξp, we denote by θðXÞ, the angle between wX and TpM. In fact, sincewξ ¼ 0, θðXÞagrees with the angle betweenwX andDp. A submanifoldM
of an almost contact metric manifold ~M is said to be slant [8], if for any non-zero vector X tangent toM at p such thatX is not proportional to ξp, the angle θðXÞbetweenwX andTpM is constant, i.e., it does not depend on the choice of p∈M andX ∈TpM − hξpi. In this caseD is a slant distribution with slant angle θ.
A slant submanifold is said to be proper slant, if neither θ ¼ 0 nor θ ¼ π 2 [9,10]. We note
that on a slant submanifold if θ ¼ 0, then it is an invariant submanifold and if θ ¼ π 2, then it is
an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant.
A characterization of slant submanifolds was given in [8] as follows:
Theorem 1 ([8]). Let M be a submanifold of an almost contact metric manifold ~M such that ξ∈TM. Then M is slant if and only if there exists a constant λ∈ ½0; 1 such that
AJMS 27,1
T2 ¼ λð−I þ η⊗ ξÞ: (2.7)
Furthermore, in such case, if θ is slant angle, then λ ¼ cos2 θ .
The following relations are straightforward consequences of (2.7)
gðTX ;TY Þ ¼ cos2 θ½gðX ;Y Þ ηðXÞηðY Þ; (2.8)
gðFX ;FY Þ ¼ sin2 θ½gðX ;Y Þ ηðXÞηðY Þ; (2.9)
for any X ;Y tangent to M. The following useful relation is obtained as a consequence of (2.7) in [26].
Theorem2 ([26]). LetM be a proper slant submanifold of an almost contact metric manifold ~M. Then
ðaÞ tFX ¼ sin2 θð−X þ ηðXÞξÞ; ðbÞ fFX ¼ −FTX ; (2.10)
for any X ∈TM.
Another characterization of slant submanifolds was given in [7]:
Theorem3 ([7]). LetD be a distribution onM , orthogonal to ξ . Then,D is slant if and only if there exists a constant λ∈ ½0; 1 such that ðPTÞ2 X ¼ −λX , for any X ∈Dp at p∈M , where P denotes the orthogonal projection on D. Furthermore, in this case, λ ¼ cos2θD.
J.L. Cabrerizo et al. [7] defined bi-slant submanifolds as follows:
Definition1. AsubmanifoldM of an almost contactmetricmanifold ~M is said to be bi-slant if there exists a pair of orthogonal distributions D1 and D2 on M such that:
(i) The tangent bundle TM admits the orthogonal direct decomposition: TM ¼ D1 ⊕D2 ⊕ hξi.
(ii) Each Di ∀i ¼ 1; 2 is a slant distribution with slant angle θi.
Given a bi-slant submanifold M, for any X ∈TM, we put
X ¼ P1X þ P2X þ ηðXÞξ; (2.11)
where PiX denotes the component ofX inDi, for any i ¼ 1; 2. In particular, ifX ∈Di, then we obtain X ¼ PiX. If we define Ti ¼ Pi+T, then we have
wX ¼ T1X þ T2X þ FX ; (2.12)
for any X ∈TM. Given i ¼ 1; 2, from Theorem 3, we get
T2 i X ¼ −cos2θiX ; (2.13)
for any X ∈Di. Non-trivial examples of bi-slant submanifolds are given in [7].
3. Warped product bi-slant submanifolds LetM1 3 f M2 be a warped product manifold of two RiemannianmanifoldsM1 andM2. Then from a result of [5], we have
∇XZ ¼ ∇Z X ¼ 1
f ðXf ÞZ (3.1)
for any vector fields X ; Z tangent to M1;M2, respectively.
manifolds
5
Recently, B.-Y. Chen and the first author introduced the notion of warped product bi-slant submanifolds of Kaehler manifolds [28]. They proved the non-existence of proper warped product bi-slant submanifolds. Then, they introduced the notion of warped product pointwise bi-slant submanifolds and obtained several fundamental results [16]. In this section, we give some useful lemmas for warped product bi-slant submanifolds of Kenmotsu manifolds. First we define these submanifolds as follows:
A warped productM1 3 f M2 of two slant submanifoldsM1 andM2 with slant angles θ1 and θ2 of a Kenmotsu manifold ~M is called a warped product bi-slant submanifold.
A warped product bi-slant submanifold M ¼ M1 3 f M2 is called proper if both M1 and M2 are proper slant submanifolds with slant angles θ1; θ2 ≠ 0; π2 of
~M. A warped product bi- slant submanifold M1 3 f M2 is a contact CR-warped product if θ1 ¼ 0; θ2 ¼ π
2 or θ2 ¼ 0; θ1 ¼ π
2 ; such submanifolds were discussed in [3,27]. Also, it is a warped product pseudo-slant submanifold if θ1 ¼ θ and θ2 ¼ π
2 [1] or θ2 ¼ θ and θ1 ¼ π 2 [23]. The warped
product bi-slant submanifolds with slant angles θ1 ¼ 0; θ2 ¼ θ or θ2 ¼ 0; θ1 ¼ θ were discussed in [25,30].
LetM ¼ M1 3 f M2 be awarped product bi-slant submanifold of a Kenmotsumanifold ~M such that the structure vector field ξ is tangent to M, where M1 and M2 are proper slant submanifolds of ~M. Then, we distinguish 2 cases:
(i) ξ is tangent to M1;
(ii) ξ is tangent to M2.
From (1.4), (2.1) and (3.1), the second case is trivial i.e., there does not exist any proper warped product bi-slant submanifold of a Kenmotsu manifold when the structure vector field is tangent to the fiber.
Now, we start with the case (i). Throughout this paper, we assume that the tangent spaces ofM1 andM2, respectively areD1 andD2. From now on, we use the following conventions: X1;Y1 are vector fields in D1 and X2;Y2 are vector fields in D2.
Lemma 1. Let M ¼ M1 3 f M2 be a warped product bi-slant submanifold of a Kenmotsu manifold ~M such that ξ∈D1 . Then, we have
(i) ξðln f Þ ¼ 1;
(ii) gðhðX1;Y1Þ; FX2Þ ¼ gðhðX1;X2Þ;FY1Þ;
for any X1;Y1 ∈D1 and X2 ∈D2 .
Proof. First part is trivial and can be obtained by using (1.4), (2.1) and (3.1). For the second part, we have
gð~∇X1 Y1;X2Þ ¼ −gð∇X1
X2;Y1Þ ¼ −X1ðln f Þ g ðY1;X2Þ ¼ 0; (3.2)
for anyX1;Y1 ∈D1 andX2 ∈D2. Also, from (1.2) and the fact that ξ is tangent toM1, we have
gð~∇X1 Y1;X2Þ ¼ gðw~∇X1
Y1;wX2Þ ¼ gð~∇X1 wY1;wX2Þ g
ð~∇X1 wÞY1;wX2Þ: (3.3)
wY1;wX2Þ ¼ gð~∇X1 T1Y1;T2X2Þ
þ gð~∇X1 T1Y1;FX2Þ þ gð~∇X1
FY1;wX2Þ: Then using the covariant derivative property of Riemannian connection and (1.2), (1.3), (2.1) and (3.1), we derive
AJMS 27,1
6
ηðY1Þ gðT1X1;T2X2Þ ¼ −X1ðln f Þ gðT1Y1;T2X2Þ þ gðhðX1;T1Y1Þ;FX2Þ þ gðð~∇X1
wÞFY1;X2Þ gð~∇X1 wFY1;X2Þ:
By the orthogonality of vector fields, the left hand side and the first term in the right hand side vanish identically. Then using (1.3), (2.6) and (2.10), we find
0 ¼ gðhðX1;T1Y1Þ;FX2Þ þ sin2 θ1 g ð~∇X1 Y1;X2Þ sin2 θ1 g ð~∇X1
ξ;X2Þ þ gð~∇X1
FT1Y1;X2Þ: Using (1.4), (2.2), (2.3), (3.1) and the orthogonality of vector fields, we derive
gðhðX1;X2Þ;FT1Y1Þ ¼ gðhðX1;T1Y1Þ;FX2Þ: (3.4)
Interchanging Y1 by T1Y1 in (3.4), we obtain
cos2 θ1 gðhðX1;X2Þ;FY1Þ ¼ cos2 θ1gðhðX1;Y1Þ;FX2Þ cos2 θ1ηðY1Þ g ðhðX1; ξÞ;X2Þ:
(3.5)
Since for a submanifold M of a Kenmotsu manifold ~M, hðX ; ξÞ ¼ 0; ∀X ∈TM, then the second part of the lemma follows from above relation. Hence, the proof is complete. -
Lemma 2. Let M ¼ M1 3 f M2 be a warped product bi-slant submanifold of a Kenmotsu manifold ~M such that ξ∈D1 , whereM1 andM2 are proper slant submanifolds of ~M with slant angles θ1 and θ2 , respectively. Then, we have
gðhðX2;Y2Þ;FT1X1Þ gðhðX2;T1X1Þ;FY2Þ ¼ T1X1ðln f Þ gðX2;T2Y2Þ cos2 θ1ðX1ðln f Þ ηðX1ÞÞ gðX2;Y2Þ
(3.6)
Proof. For any X1 ∈D1 and X2;Y2 ∈D2, we have
gð~∇X2 X1;Y2Þ ¼ X1ðln f Þ gðX2;Y2Þ: (3.7)
On the other hand, we also have
gð~∇X2 X1;Y2Þ ¼ gðw~∇X2
X1;wY2Þ ¼ gð~∇X2 wX1;wY2Þ g
ð~∇X2 wÞX1;wY2Þ:
gð~∇X2 X1;Y2Þ ¼ gð~∇X2
T1X1;T2Y2Þ þ gð~∇X2 T1X1;FY2Þ
þ gð~∇X2 FX1;wY2Þ þ ηðX1Þ g ðwX2;wY2Þ:
From (1.2), (2.1) and (3.1), the above equation takes the form
gð~∇X2 X1;Y2Þ ¼ T1X1ðln f ÞgðX2;T2Y2Þ þ gðhðX2;T1X1Þ;FY2Þ gð~∇X2
wFX1;Y2Þ þ gðð~∇X2
wÞFX1;Y2Þ þ ηðX1Þ g ðX2;Y2Þ: Again, using (1.3) and (2.6), we derive
gð~∇X2 X1;Y2Þ ¼ T1X1ðln f Þ g ðX2;T2Y2Þ þ gðhðX2;T1X1Þ;FY2Þ gð~∇X2
t FX1;Y2Þ gð~∇X2
gð~∇X2 X1;Y2Þ ¼ T1X1ðln f Þ gðX2;T2Y2Þ þ gðhðX2;T1X1Þ;FY2Þ
þ sin2 θ1 X1ðln f Þ gðX2;Y2Þ sin2 θ1 ηðX1Þ gðX2;Y2Þ þ gð~∇X2
FT1X1;Y2Þ þ ηðX1Þ gðX2;Y2Þ ¼ T1X1ðln f Þ gðX2;T2Y2Þ þ gðhðX2;T1X1Þ;FY2Þ þ sin2 θ1X1ðln f Þ gðX2;Y2Þ þ cos2 θ1ηðX1Þ gðX2;Y2Þ gðhðX2;Y2Þ;FT1X1Þ:
(3.8)
Thus, the result follows from (3.7) and (3.8), which proves the lemma completely. -
The following useful relations are easily derived by interchangingX1 byT1X1,X2 byT2X2
and Y2 by T2Y2 in Lemma 2.
gðhðX2;Y2Þ;FX1Þ gðhðX1;X2Þ;FY2Þ ¼ T1X1ðln f Þ gðX2;Y2Þ ðX1ðln f Þ ηðX1ÞÞ gðT2X2;Y2Þ;
(3.9)
gðhðT2X2;Y2Þ;FT1X1Þ gðhðT2X2;T1X1Þ;FY2Þ ¼ cos2 θ2T1X1ðln f Þ gðX2;Y2Þ cos2 θ1 ðX1ðln f Þ ηðX1ÞÞ gðT2X2;Y2Þ;
(3.10)
gðhðX2;T2Y2Þ;FT1X1Þ gðhðX2;T1X1Þ;FT2Y2Þ ¼ − cos2 θ2T1X1ðln f Þ gðX2;Y2Þ þ cos2 θ1ðX1ðln f Þ ηðX1ÞÞ gðT2X2;Y2Þ;
(3.11)
cos2 θ1 cos 2 θ2 ðX1ðln f Þ
ηðX1ÞÞ gðX2;Y2Þ;
(3.12)
gðhðT2X2;Y2Þ;FX1Þ gðhðT2X2;X1Þ;FY2Þ ¼ T1X1ðln f Þ gðT2X2;Y2Þ þ cos2 θ2 ðX1ðln f Þ ηðX1ÞÞ gðX2;Y2Þ;
(3.13)
gðhðX2;T2Y2Þ;FX1Þ gðhðX2;X1Þ;FT2Y2Þ ¼ T1X1ðln f Þ gðX2;T2Y2Þ cos2 θ2 ðX1ðln f Þ ηðX1ÞÞ gðX2;Y2Þ
(3.14)
and gðhðT2X2;T2Y2Þ;FX1Þ gðhðT2X2;X1Þ;FT2Y2Þ ¼ cos2 θ2T1X1ðln f Þ g ðX2;Y2Þ
cos2 θ2 ðX1ðln f Þ ηðX1ÞÞ gðT2X2;Y2Þ:
(3.15)
8
4. An inequality for warped product bi-slant submanifolds LetM ¼ M1 3 f M2 be a warped product bi-slant submanifolds of a Kenmotsu manifold ~M; we decompose the normal bundle of M as follows
T⊥M ¼ FD1 ⊕FD2 ⊕ μ; μ ⊥ FD1 ⊕FD2; (4.1)
where μ is a w-invariant normal subbundle of T⊥M. A warped product bi-slant submanifold M ¼ M1 3 f M2 of a Kenmotsu manifold ~M is
said to bemixed totally geodesic, if hðX ; ZÞ ¼ 0, for anyX ∈D1 and Z ∈D2, whereD1 andD2
are the tangent bundles of M1 and M2, respectively. Now, we set the following frame fields for an n-dimensional warped product bi-slant
submanifoldM ¼ M1 3 f M2 of a ð2mþ 1Þ-dimensional Kenmotsumanifold ~M such that ξ is
tangent toM1, whereM1 andM2 are proper slant submanifolds of ~M with slant angles θ1 and θ2 respectively. Let us consider the dimensions dimðM1Þ ¼ 2pþ 1 and dimM2 ¼ 2q, i.e., n ¼ 2pþ 1þ 2q. Then the orthonormal frames of the corresponding tangent spacesD1
andD2, respectively are given by fe1; . . . ; ep; epþ1 ¼ sec θ1T1e1; . . . ; e2p ¼ sec θ1T1ep; e2pþ1 ¼ ξg and fe2pþ2 ¼ e1; . . . ; e2pþ1þq ¼ eq; e2pþqþ2 ¼ eqþ1 ¼ sec θ2T2e1; . . . ; en ¼ e2q ¼ sec θ2T2eqg. Thus, the orthonormal frame fields of the normal subbundles of FD1;FD2 and μ, respectively are fenþ1 ¼ ~e1 ¼ csc θ1Fe1; . . . ; enþp ¼ ~ep ¼ csc θ1Fep; enþpþ1 ¼ ~epþ1 ¼ csc θ1 sec θ1FT1e1; . . . ; enþ2p ¼ ~e2p ¼ csc θ1 sec θ1FT1epg; fenþ2pþ1 ¼ ~e2pþ1 ¼ be1 ¼ csc θ2 Fe1; . . . ; enþ2pþq ¼ ~e2pþq ¼ beq ¼ csc θ2 Feq; enþ2pþqþ1 ¼ ~e2pþqþ1 ¼ beqþ1 ¼ csc θ2 sec θ2 FT2e1; . . . ; enþ2pþ2q ¼ ~e2pþ2q ¼ be2q ¼ csc θ2 sec θ2 FT2eqg and fe2n ¼ ~en; . . . ; e2mþ1 ¼ ~e2ðm−nþ1Þg.
Now, using the results of Section 3 and the above frame fields, we give following main result of this paper.
Theorem 4. Let M ¼ M1 3 f M2 be a mixed totally geodesic warped product bi-slant submanifold of a Kenmotsu manifold ~M such that ξ is tangent to M1 , where M1 and M2 are proper slant submanifolds of ~M with slant angles θ1 and θ2 , respectively. Then
(i) The second fundamental form h of M satisfies the following inequality
khk2 ≥ 2q csc2 θ1 cos2 θ1 þ cos2 θ2
k∇!ðln f Þk2 1 ; (4.2)
where 2q ¼ dimM2 and ∇ !ðln f Þ is the gradient of ln f along M1.
(ii) If equality sign in (i) holds identically, then:
(a) M1 is a totally geodesic submanifold of ~M;
(b) M2 is a totally umbilical submanifold of ~M.
Proof. From (2.4), we have
khk2 ¼ Xn
r¼nþ1
gðhðei; ejÞ; erÞ2:
Splitting the above expression for the tangent bundles of M1 and M2, we derive
khk2 ¼ X2mþ1
r¼1
r¼1
gðhðei; ejÞ; erÞ2: (4.3)
manifolds
9
Since M is mixed totally geodesic, then the second term in the right hand side of (4.3) is identically zero. Thus, we find
khk2 ¼ X2p r¼1
X2pþ1
X2pþ1
gðhðei; ejÞ;~erÞ2 þ X2q r¼1
X2q i; j¼1
gðhðei; ejÞ;berÞ2
gðhðei; ejÞ;~erÞ2:
(4.4)
The third and sixth terms have μ-components and we could not find any relation for warped products in terms of μ-components, therefore we shall leave these two terms. Also, we could not find any relations for gðhðei; ejÞ;~erÞ; i; j ¼ 1; . . . ; 2pþ 1; r ¼ 1; . . . ; 2p and gðhðei; ejÞ;berÞ; i; j; r ¼ 1; . . . ; 2q. Therefore, we also leave the first and fifth terms. By using Lemma 1(ii) for a mixed totally geodesic warped product the second term in the right hand side is also zero. Thus, the evaluated term is only fourth term which can be expressed by using the constructed frame fields as follows
khk2 ≥ csc2 θ1 sec 2 θ2
Xp
Xp
2 θ2 Xp
r¼1
2 θ2 Xp
r¼1
Xp
Xp
4 θ2 Xp
r¼1
gðhðT2ei;T2ejÞ;FT1erÞ2:
(4.5)
Using Lemma 2 and the relations (3.9)–(3.15) for a mixed totally geodesic warped product submanifold, we derive
AJMS 27,1
Xp
2 θ2 Xp
r¼1
ðT1erðln f ÞÞ2
r¼1
ðT1erðln f ÞÞ2 þ 2q csc2 θ1 cos 2 θ1
Xp
Xp
2 θ1 Xp
r¼1
ðT1erðln f ÞÞ2:
(4.6)
Since ηðerÞ ¼ 0; r ¼ 1; . . . ; 2p, then above expression will be
Xp
2 θ1 X2pþ1
r¼1
ðT1erðln f ÞÞ2
2 θ1 X2p r¼pþ1
gðer;T1∇ !ðln f ÞÞ2
2 θ1 ðT1e2pþ1ðln f ÞÞ2:
(4.7)
Since T1e2pþ1 ¼ T1ξ ¼ 0, (4.7) can be written as
Xp
2 θ1 kT1∇
!ðln f Þk2
1þ cos2 θ2 sec
kT1∇ !ðln f Þk2 ¼ gðT1∇
!ðln f Þ;T1∇ !ðln f ÞÞ:
Using (2.8), we get
kT1∇ !ðln f Þk2 ¼ cos2 θ1
:
kT1∇ !ðln f Þk2 ¼ cos2 θ1
k∇!ðln f Þk2 1 : (4.9)
Xp
k∇!ðln f Þk2 1
2q csc2 θ1 1þ cos2 θ2 sec
2 θ1 csc2 θ1
Xp
k∇!ðln f Þk2 1
2q csc2 θ1 cos2 θ1 þ cos2 θ2
Xp
k∇!ðln f Þk2 1 ;
which is inequality (4.2). For the equality case, since M is mixed totally geodesic, then
hðD1;D2Þ ¼ f0g: (4.10)
From the leaving third and sixth terms in the right hand side of (4.4), we have
hðX ;Y Þ ⊥ μ; (4.11)
for any X ;Y ∈TM. Also, from the leaving first term of (4.4), we find
hðD1;D1Þ ⊥ FD1: (4.12)
hðD1;D1Þ ⊂ FD2: (4.13)
But for a mixed totally geodesic submanifold, from Lemma 1(ii), we get
hðD1;D1Þ ⊥ FD2: (4.14)
hðD1;D1Þ ¼ f0g: (4.15)
SinceM1 is totally geodesic inM [5,11], then using this fact with (4.10) and (4.15) we conclude
that M1 is a totally geodesic submanifold of ~M, which is the first relation of inequality. Similarly, from the leaving fifth term in the right hand side of (4.4), we get
hðD2;D2Þ ⊥ FD2: (4.16)
hðD2;D2Þ ⊂ FD1: (4.17)
SinceM2 is totally umbilical inM [5,11], using this fact with (4.10) and (4.17), we conclude that
M2 is a totally umbilical submanifold of ~M, which proves the statement ðbÞ. Hence, the proof is complete. -
AJMS 27,1
12
5. Some applications of derived inequality In this section, we give some applications of the derived inequality (4.2).
1. In Theorem 4, if θ1 ¼ π 2 and θ2 ¼ 0, then the warped product bi-slantM1 3 f M2 takes
the formM⊥3 f MT i.e., M is a contact CR-warped product studied in [27]. In this case, the inequality (4.2) will be khk2 ≥ 2qðk∇!ðln f Þk2 − 1Þ, which is the main Theorem 3.4 of [27].
2. If we consider θ1 ¼ θ and θ2 ¼ 0 in a warped product bi-slant submanifold M ¼ M1 3 f M2, then M is a warped product semi-slant submanifold of the form Mθ 3 f MT studied in [30]. Then, the inequality (4.2) change into khk2 ≥ 2qðcsc2 θþ cot2 θÞðk∇!ðln f Þk2 − 1Þ, which is Theorem 4.2 of [30]. Thus, Theorem 4.2 of [30] is a special
case of Theorem 4.
3. Also, if we consider θ1 ¼ θ and θ2 ¼ π 2 in a warped product bi-slant submanifold
M ¼ M1 3 f M2 of a Kenmotsu manifold ~M, thenM turns into a warped product pseudo- slant submanifold of the formM ¼ Mθ 3 f M⊥, whereMθ andM⊥ are proper slant and anti- invariant submanifolds of ~M, respectively. In this case, if we put θ1 ¼ θ and θ2 ¼ π
2 in Theorem 4, then inequality (4.2) will be khk2 ≥ 2q cot2 θðk∇!ðln f Þk2 − 1Þ, which the inequality (5.1) of [1]. Hence, Theorem 5.1 of [1] is a special case of Theorem 4.
4. Similarly, if we assume that θ1 ¼ π 2 and θ2 ¼ θ in a warped product bi-slant
submanifold M ¼ M1 3 f M2 of a Kenmotsu manifold ~M, then M is a warped product pseudo-slant submanifold M ¼ M⊥ 3 f Mθ such that M⊥ is an anti-invariant submanifold and Mθ is a proper slant submanifold of ~M. Thus inequality (4.2) takes the
form khk2 ≥ 2q cos2 θðk∇!ðln f Þk2 – 1Þ, which the inequality (4.1) of [23]. Hence, Theorem
4.1 of [23] is a special case of Theorem 4.
References
[1] F.R. Al-Solamy, M.F. Naghi, S. Uddin, Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds, Quaest. Math. 42 (3) (2019) 373–389, http://dx.doi.org/10.2989/16073606. 2018.1452800.
[2] L.S. Alqahtani, M.S. Stankovic, S. Uddin, Warped product bi-slant submanifolds of cosymplectic manifolds, Filomat 31 (16) (2017) 5065–5071, http://dx.doi.org/10.2298/FIL1716065A.
[3] K. Arslan, R. Ezentas, I. Mihai, C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc. 42 (2005) 1101–1110.
[4] M. Atceken, Warped product semi-slant submanifolds in Kenmotsu manifolds, Turk. J. Math. 36 (2012) 319–330.
[5] R.L. Bishop, B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 1–49.
[6] D.E. Blair, Contact Manifolds in Riemannian Geometry, in Lecture Notes in Mathematics, vol. 509, Springer-Verlag, New York, 1976.
[7] J.L. Cabrerizo, A. Carriazo, L.M. Fernandez, M. Fernandez, Semi-slant submanifolds of a Sasakian manifold, Geom. Dedicata 78 (2) (1999) 183–199.
[8] J.L. Cabrerizo, A. Carriazo, L.M. Fernandez, M. Fernandez, Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000) 125–138.
[9] B.-Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, 1990.
[10] B.-Y. Chen, Slant immersions, Bull. Aust. Math. Soc. 41 (1990) 135–147.
[11] B.-Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math. 133 (2001) 177–195.
manifolds
13
[12] B.-Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II, Monatsh. Math. 134 (2001) 103–119.
[13] B.-Y. Chen, Pseudo-Riemannian Geometry δ-Invariants and Applications, World Scientific, Hackensack, NJ, 2011.
[14] B.-Y. Chen, Geometry of warped product submanifolds: a survey, J. Adv. Math. Stud. 6 (2) (2013) 1–43.
[15] B.-Y. Chen, Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific, Hackensack, NJ, 2017.
[16] B.-Y. Chen, S. Uddin, Warped product pointwise bi-slant submanifolds of Kaehler manifolds, Publ. Math. Debr. 92 (1–2) (2018) 183–199.
[17] I. Hasegawa, I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata 102 (2003) 143–150.
[18] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24 (1972) 93–103.
[19] K. Matsumoto, I. Mihai, Warped product submanifolds in Sasakian space forms, SUT J. Math. 38 (2) (2002) 135–144.
[20] A. Mihai, Warped product submanifolds in complex space forms, Acta Sci. Math. Szeged 70 (2004) 419–427.
[21] A. Mihai, Warped product submanifolds in quaternion space forms, Rev. Roum. Math. Pures Appl. 50 (2005) 283–291.
[22] C. Murathan, K. Arslan, R. Ezentas, I. Mihai, Warped product submanifolds in Kenmotsu space forms, Taiwanese J. Math. 10 (6) (2006) 1431–1441.
[23] M.F. Naghi, S. Uddin, F.R. Al-Solamy, Warped product submanifolds of Kenmotsu manifolds with slant fiber, Filomat 32 (6) (2018) 2115–2126.
[24] B. Sahin, Warped product pointwise semi-slant submanifolds of Kaehler manifolds, Port. Math. 70 (2013) 252–268.
[25] S. Uddin, Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds, Bull. Math. Sci. 8 (2018) 435–451, http://dx.doi.org/10.1007/s13373-017-0106-9.
[26] S. Uddin, F.R. Al-Solamy, Warped product pseudo-slant immersions in Sasakian manifolds, Publ. Math. Debr. 91 (3–4) (2017) 331–348, http://dx.doi.org/10.5486/PMD.2017.7640.
[27] S. Uddin, A. Alghanemi, M.F. Naghi, F.R. Al-Solamy, Another class of warped product CR- submanifolds in Kenmotsu manifolds, J. Math. Comput. Sci. 17 (2017) 148–157.
[28] S. Uddin, B.-Y. Chen, F.R. Al-Solamy, Warped product bi-slant immersions in Kaehler manifolds, Mediterr. J. Math. 14 (2) (2017) http://dx.doi.org/10.1007/s00009-017-0896-8, Art. 95, p. 11.
[29] S. Uddin, V.A. Khan, K.A. Khan, Warped product submanifolds of a Kenmotsu manifold, Turk. J. Math. 36 (2012) 319–330.
[30] S. Uddin, M.F. Naghi, F.R. Al-Solamy, Another class of warped product submanifolds of Kenmotsu manifolds, RACSAM 112 (4) (2018) 1141–1155, http://dx.doi.org/10.1007/s13398-017-0415-6.
[31] K. Yano, M. Kon, Structures on Manifolds, in: Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984.
Corresponding author Siraj Uddin can be contacted at: [email protected]
For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: [email protected]
AJMS 27,1
On the number of edges of a graph and its complement
Jamel Dammak Department of Mathematics, Faculty of Sciences of Sfax, Sfax, Tunisia
Gerard Lopez Institut de Mathematiques de Luminy, CNRS-UPR 9016, Marseille, France, and
Hamza Si Kaddour Institut Camille Jordan, CNRS UMR 5208, Universite Claude Bernard Lyon 1,
Univ Lyon, Villeurbanne, France
Abstract
Let G ¼ ðV ;EÞbe a graph. The complement of G is the graph G :¼ ðV ; ½V 2 nEÞwhere ½V 2 is the set of pairs fx; ygof distinct elements ofV . IfK is a subset ofV , the restriction ofG toK is the graphG
K :¼ ðK; ½K2 ∩EÞ. We prove that ifG ¼ ðV ;EÞ is a graph and k is an integer, 2 ≤ k ≤ v− 2, then there is a k -element subsetK ofV
such that eðG KÞ≠ eðG
KÞ, moreover the condition k ≤ v− 2 is optimal. We also study the case
eðG KÞ u eðG
KÞðmod pÞ where p is a prime number. Following a question from M.Pouzet, we show this:
LetG ¼ ðV ;EÞbe a graphwith vvertices. If eðGÞ ≠ eðGÞ (resp. eðGÞ ¼ eðGÞ) then there is an increasing family
ðHnÞ2≤n≤v (resp. ðHnÞ2≤n≤v−2) of n -element subsets Hn of V such that eðG Hn
Þ≠ eðG Hn
Þ for all n. Similarly if
eðGÞ u eðGÞ ðmod pÞwhere p is a prime number, p > v− 2, then there is an increasing family ðHnÞ2≤n≤v of n -element subsets Hn of V such that eðG
Hn Þ u eðG
Hn Þðmod pÞ for all integer n∈ f2; 3; . . . ; vg.
Keywords Set, Matrix, Graph, Edge, Prime number
1. Introduction Our notations and terminology follow [2]. A graph is an ordered pair G :¼ ðV ;EÞ (or
ðV ðGÞ;EðGÞÞ), where E is a subset of ½V 2, the set of pairs fx; yg of distinct elements of V . Elements of V are the vertices of G and elements of E are its edges. An edge fx; yg is also noted by x y. The cardinality jV j of V is called the order of G. Two distinct vertices x and y are adjacent if x y∈EðGÞ, otherwise x and y are non-adjacent. We denote by eðGÞ :¼ jEðGÞj the number of edges of G. The degree of a vertex x of G, denoted by dGðxÞ, is the number of edges which contain x. The graph G is δ -regular (or regular) if dGðxÞ ¼ δ for all x∈V ;
Edges of a graph and its complement
15
JEL Classification — 05C50, 05C60 © Jamel Dammak, Gerard Lopez and Hamza Si Kaddour. Published in the Arab Journal of
Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http:// creativecommons.org/licences/by/4.0/legalcode
The publisher wishes to inform readers that the article “On the number of edges of a graph and its complement” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Dammak, J., Lopez, G., Kaddour, H. S. (2019), “On the number of edges of a graph and its complement”,Arab Journal of Mathematical Sciences, Vol. 27 No. 1, pp. 15-19. The original publication date for this paper was 29/05/2019.
Accepted 26 May 2019
Vol. 27 No. 1, 2021 pp. 15-19
Emerald Publishing Limited e-ISSN: 2588-9214 p-ISSN: 1319-5166
DOI 10.1016/j.ajmsc.2019.05.006
δ is called the degree of the regular graph G. The complement of G is the graph
G :¼ ðV ; ½V 2 nEÞ. If K is a subset of V , the restriction of G to K, also called the induced
subgraph of G on K, is the graph G K :¼ ðK; ½K2 ∩EÞ. For instance, given a set V , ðV ; 0=Þ is
the empty graph on V whereas ðV ; fxy : x ≠ y∈VgÞ is the complete graph. Our first result is Theorem 1.1, we prove that: given a graph G ¼ ðV ;EÞ and k be an
integer, 2 ≤ k ≤ v− 2, we cannot have eðG KÞ ¼ eðG
KÞ for all k -element subsets K of V , moreover the condition k ≤ v− 2 is optimal, indeed for k ¼ v− 1 a counterexample is given by (2) of Theorem 1.1.
Theorem 1.1. Let G ¼ ðV ;EÞ be a graph of order v.
(1) If v ≥ 4 , then for each integer kwith 2 ≤ k ≤ v− 2 , there is a k-element subset K of V
such that eðG KÞ ≠ eðG
KÞ. (2) If G is regular with degree v− 1
2 then eðGÞ ¼ eðGÞ and eðG− xÞ ¼ eðG− xÞ for all vertex x.
(3) Let p be a prime number with p ≥ 3 such that 2dGðxÞ ≡ v− 1ðmod pÞ for all x∈V .
Then eðGÞ ≡ eðGÞðmod pÞ and eðG− xÞ ≡ eðG− xÞðmod pÞ for all vertex x. Our second result is Theorem 1.2. Given a graph G ¼ ðV ;EÞ, p a prime number, and k an
integer, 2 ≤ k ≤ v− 2, under some conditions on k, we cannot have eðG KÞ ≡ eðG
KÞ ðmod pÞ for all k-element subsets K of V .
Theorem 1.2. Let G ¼ ðV ;EÞ be a graph with v≥ 4 vertices and let p be a prime number. Let k be an integer, 2 ≤ k ≤ v− 2.
(1) If ( p ¼ 2 and k ≡ 2ðmod 4Þ ) or ( p ≥ 3 and k u 0; 1ðmod pÞ ), then there is a k-element subset K of V such that eðG
KÞ u eðG KÞðmod pÞ.
(2) If p ≥ 3 and k ≡ 0ðmod pÞ then there is a k-element subset K of V such that
eðG KÞ u eðG
KÞðmod pÞ if and only if G is neither the complete graph nor the empty graph.
Our third result is Theorem 1.3. It is related to a question thatM.Pouzet asked us about the existence, in a graph G ¼ ðV ;EÞ, of an increasing family ðHnÞn of n -element subsetsHn ofV
such that eðG Hn
Þ ≠ eðG Hn
Þ. Theorem 1.3. Let G ¼ ðV ;EÞ be a graph of v vertices with v≥ 3.
(1) If eðGÞ ≠ eðGÞ then there is a vertex x such that eðG− xÞ ≠ eðG− xÞ. (2) If eðGÞ ≠ eðGÞ then there is an increasing family ðHnÞ2≤n≤vof n-element subsets Hn of V
such that eðG Hn
Þ ≠ eðG Hn
Þ for all integer n∈ f2; 3; . . . ; vg. (3) If eðGÞ ¼ eðGÞ and v ≥ 4 then there is an increasing family ðHnÞ2≤n≤v−2 of n-element
subsets Hn of V such that eðG Hn
Þ ≠ eðG Hn
Þ for all integer n∈ f2; 3; . . . ; v− 2g. (4) Let p be a prime number, p > v− 2 . If eðGÞu eðGÞðmod pÞ then there is an increasing
family ðHnÞ2≤n≤v of n-element subsets Hn of V such that eðG Hn
Þu eðG Hn
Þðmod pÞ for all integer n∈ f2; 3; . . . ; vg.
2. Incidence matrices We consider the matrix Wt k defined as follows: Let V be a finite set, with v elements. Given
non-negative integers t; k, letWt k be the
v
t
by
v
k
matrix of 0’s and 1’s, the rows ofwhich
AJMS 27,1
16
are indexed by the t-element subsets T of V , the columns are indexed by the k-element subsets K of V , and where the entry WtkðT;KÞ is 1. T ⊆K and is 0 otherwise. The matrix transpose of Wt k is denoted
tWt k. A fundamental result, due to D.H. Gottlieb [4], and independently W. Kantor [5], is this:
Theorem 2.1 (D.H. Gottlieb [4], W. Kantor [5]). For t ≤ minðk; v− kÞ , Wt k has full row rank over the field of rational numbers.
It is clear that t ≤ minðk; v− kÞ implies v
t
following result.
Corollary 2.2. For t ≤ minðk; v− kÞ , the rank of Wt k over the field of rational numbers
is v
and thus KerðtWt kÞ ¼ f0g.
Corollary 2.2 and the following theorem are important tools in the proof of our main results. In fact, Theorem 2.3 has made to establish a version modulo a prime of Kelly’s combinatorial lemma [6]; it also allows to obtain a version modulo a prime of the particular version of Pouzet’s combinatorial lemma [7].
Let n; p be positive integers, the decomposition of n ¼ PnðpÞ i¼0 nip
i in the basis p is also denoted by ½n0; n1; . . . ; nnðpÞp where nnðpÞ ≠ 0 if and only if n≠ 0.
Theorem 2.3 [1]. Let p be a prime number. Let v; t and k be non-negative integers, k ¼ ½k0; k1; . . . ; kkðpÞp, t ¼ ½t0; t1; . . . ; ttðpÞp, t ≤ minðk; v− kÞ . We have:
(1) kj ¼ tj for all j < tðpÞ and ktðpÞ ≥ ttðpÞ if and only if KerðtWtkÞ ¼ f0g (mod p ).
(2) t ¼ ttðpÞptðpÞ and k ¼ PkðpÞ i¼tðpÞþ1kip
i if and only if dKerðtWtkÞ ¼ 1 (mod p ) and
fð1; 1; . . . ; 1Þg is a basis of KerðtWt kÞ.
The notation a jb (resp. a b) means a divide b (resp. a not divide b).
Theorem 2.4 (Lucas’ Theorem [3] ). Let p be a prime number, t; k be positive integers, t ≤ k, t ¼ ½t0; t1; . . . ; ttðpÞp and k ¼ ½k0; k1; . . . ; kkðpÞp . Then
k
t
¼
The following result is a consequence of Lucas’ theorem.
Corollary 2.5 ([1]). Let p be a prime number, t; k be positive integers, t ≤ k, t ¼ ½t0; t1; . . . ; ttðpÞp and k ¼ ½k0; k1; . . . ; kkðpÞp . Then:
pj k
t
if and only if there is i∈ f0; 1; . . . ; tðpÞg such that ti > ki:
3. Proofs of main results
Let G ¼ ðV ;EÞ be a graph of order v. Let T1;T2; . . . ;T v 2
be an enumeration of the
2-element subsets ofV , letK1;K2; . . . ;K v k
be an enumeration of the k-element subsets ofV .
Let wG be the row matrix ðg1; g2; . . . ; g v 2
Þ where gi ¼ 1 if Ti is an edge of G, 0 otherwise.
We have wGW2 k ¼ ðeðG K1
Þ; eðG K2
Þ; . . . ; eðG K
17
2
Þ with gi ¼ 0 if Ti is an edge of G, 1 otherwise. We
have w G W2 k ¼ ðeðG
K1 Þ; eðG
K2 Þ; . . . ; eðG
K v k
3.1 Proof of Theorem 1.1 (1) Assume that eðG
KÞ ¼ eðG KÞ for all k-element subsetsK ofV , thenwGW2 k ¼ w
G W2 k.
By Corollary 2.2,KerðtWt kÞ ¼ f0g. ThenwG ¼ w G , soG ¼ G, which is impossible. So there is
a k-element subset K of V such that eðG KÞ ≠ eðG
KÞ. (2) For x∈V , dGðxÞ þ d
G ðxÞ ¼ v− 1. Since dGðxÞ ¼ v− 1
2 then dGðxÞ ¼ d G ðxÞ. We haveP
x∈V dGðxÞ ¼ 2eðGÞ and P
x∈V d G ðxÞ ¼ 2eðGÞ, then eðGÞ ¼ eðGÞ. Now from eðGÞ ¼
eðG− xÞþ d G ðxÞ and eðGÞ ¼ eðG− xÞ þ dGðxÞ, we deduce that eðG− xÞ ¼ eðG− xÞ.
(3) For x∈V , dGðxÞ þ d G ðxÞ ¼ v− 1. Since 2dGðxÞ ≡ v− 1ðmod pÞ then dGðxÞ ≡ d
G ðxÞ
ðmod pÞ. We conclude using similar arguments to those in item (2). ,
3.2 Proof of Theorem 1.2 We set t:¼ 2. We recall the notation k ¼ ½k0; k1; . . . ; kkðpÞp. (1) Case 1. p ¼ 2 and k ≡ 2ðmod 4Þ. We have k0 ¼ 0, k1 ¼ 1, t ¼ ½0; 12. Since k0 ¼ t0 and k1 ≥ t1 ¼ ttðpÞ then, by Theorem 2.3,
KerðtWt kÞ ¼ f0g (mod p). Case 2. p ≥ 3 and k u 0; 1ðmod pÞ. We have k0 ≥ 2, t ¼ t0 ¼ 2. Since k0 ≥ 2 ¼ ttðpÞ then, by Theorem 2.3, KerðtWt kÞ ¼ f0g
(mod p).
In the two cases,KerðtWt kÞ ¼ f0g (mod p). Assume that eðG KÞ ≡ eðG
KÞðmod pÞ for all k-element subsets K of V . Then wGW2k ¼ w
G W2 kðmod pÞ. As KerðtWt kÞ ¼ f0g (mod p),
then wG ¼ w G ðmod pÞ, so G ¼ G, which is impossible. Then there is a k-element subset K
of V such that eðG KÞ u eðG
KÞðmod pÞ. (2) If p ≥ 3 then t ¼ t0 ¼ 2 ¼ ttðpÞ. Since k ≡ 0 ðmod pÞ then k0 ¼ 0, and thus
k ¼ PkðpÞ i¼tðpÞþ1 ki p
i. By Theorem 2.3, fð1; 1; . . . ; 1Þg is a basis of KerðtWt kÞ. IfG is the complete graph or the empty graph then eðG
KÞ ≡ eðG KÞðmod pÞ. Indeed, ifG
is the complete graph then eðG KÞ ¼
k
2
, eðG
KÞ ¼ 0. Since t0 ¼ 2 > k0 ¼ 0 then, by
Corollary 2.5, pj k
. So eðG
KÞ ≡ 0 ≡ eðG KÞðmod pÞ. If G is the empty graph,
then eðG KÞ ¼ 0 ≡ eðG
KÞ ¼ k
ðmod pÞ.
Conversely if G is neither the complete graph nor the empty graph, assume that
eðG KÞ ≡ eðG
KÞðmod pÞ for all k-element subsetsK ofV . ThenwGW2 k ¼ w G W2 kðmod pÞ.
So wG −w G ¼ λð1; . . . ; 1Þðmod pÞ with λ∈ f0; 1; − 1g. As G is neither the complete graph
nor the empty graph, there are i; j such that gi ¼ 0and gj ¼ 1, so gi − gi ¼ −1 and gj − gj ¼ 1.
Then λ ≠ 1 and λ ≠ − 1. Thus λ ¼ 0, and wG ¼ w G ðmod pÞ, so G ¼ G, which is impossible.
Then there is a k-element subset K of V such that eðG KÞ u eðG
KÞðmod pÞ. ,
3.3 Proof of Theorem 1.3 We need the following lemma.
AJMS 27,1
18
Lemma 3.1. Let G ¼ ðV ;EÞ be a graph of order v and let p be a prime number, p ≥ 3.
(1) Let k be an integer, 2 ≤ k ≤ v− 2. If
v− 2
k− 2
eðGÞ u
v− 2
k− 2
eðGÞ ðmod pÞ then there
is a k-element subset K of V such that eðG KÞ u eðG
KÞðmod pÞ . (2) If ðv− 2ÞeðGÞ u ðv− 2ÞeðGÞ ðmod pÞ then there is x∈V such that eðG− xÞu
eðG− xÞ ðmod pÞ . Proof. (1) It is an immediate consequence of the following formula
v 2 k 2
eðG kÞ
(2) Follows from (1) by taking k ¼ v− 1. ,
Now we prove Theorem 1.3.
(1) Assume that eðG− xÞ ¼ eðG− xÞ for all x∈V . From eðGÞ ¼ eðG− xÞ þ dGðxÞ, we obtain
P x∈V eðGÞ ¼ P
x∈V eðG− xÞ þP x∈V dGðxÞ. Since
P x∈V dGðxÞ ¼ 2eðGÞ then ðv− 2Þ
eðGÞ ¼ P x∈V eðG− xÞ. Similarly, ðv− 2ÞeðGÞ ¼ P
x∈V eðG− xÞ. Then eðGÞ ¼ eðGÞ. (2) We make the proof by induction on v. We set Hv :¼ V . We assume that Hi is defined
for all k ≤ i ≤ v. Let us defineHk−1. As eðGHk Þ ≠ eðG
Hk Þ then by (1), there is x∈Hk such that
eðG Hk
− xÞ ≠ eðG Hk
− xÞ. We setHk−1 :¼ Hk n fxg. SoHk−1 ⊂Hk and eðGHk− 1 Þ ≠ eðG
Hk− 1 Þ.
(3) By applying (1) of Theorem 1.1 for the graphG and k ¼ v− 2we obtain x ≠ y∈V such
that eðG− fx; ygÞ ≠ eðG− fx; ygÞ. If v ¼ 4 we are done. If v ≥ 5, then v− 2 ≥ 3 and here we conclude using (2).
(4) By induction on v. Since eðGÞ u eðGÞðmod pÞ and p > v− 2 then ðv− 2ÞeðGÞ u ðv− 2ÞeðGÞ ðmod pÞ. By (2) of Lemma 3.1, there is x ∈ V such that eðG− xÞu eðG− xÞ ðmod pÞ. We conclude by using the induction hypothesis. ,
References
[1] A. Ben Amira, J. Dammak, H. Si Kaddour, On a generalization of Kelly’s combinatorial lemma, Turk. J. Math. 36 (2014) 949–964.
[2] J.A. Bondy, Basic graph theory: paths and circuits, in: R.L. Graham, M. Gr€otschel, L. Lovasz (Eds.), HandBook of Combinatorics, Vol. 1, North-Holland, 1995, pp. 3–110.
[3] N.J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 ((10) Part 1) (1947) 589–592.
[4] D.H. Gottlieb, A certain class of incidence matrices, Proc. Amer. Math. Soc. 17 (1966) 1233–1237.
[5] W.M. Kantor, On incidence matrices of finite projective and affine spaces, Math. Z. 124 (1972) 315–318.
[6] P.J. Kelly, A congruence theorem for trees, Pac. J. Math. 7 (1957) 961–968.
[7] M. Pouzet, Application d’une propriete combinatoire des parties d’un ensemble aux groupes et aux relations, Math. Z. 150 (1976) 117–134.
Corresponding author Hamza Si Kaddour can be contacted at: [email protected]
Edges of a graph and its complement
19
On Hilbert class field tower for some quartic number fields
Abdelmalek Azizi Department of Mathematics, Faculty of Sciences, Mohammed First University,
Oujda, Morocco, and
Mohamed Talbi and Mohammed Talbi Regional center of Education and Training, Oujda, Morocco
Abstract We determine the Hilbert 2-class field tower for some quartic number fields k whose 2-class group Ck;2 is isomorphic to =2 3 =2. Keywords Cyclic quartic field, Class group, Hilbert class field
1. Introduction Let k be a number field, Ck;2 its 2-class group, it is the 2-Sylow subgroup of the class group
(in the wide sense) of k; kð0Þ ¼ k ⊆ kð1Þ ⊆ kð2Þ . . . kðiÞ . . ., the tower of Hilbert 2-class field of k which means that kð1Þ is the Hilbert 2-class field of k (this is the maximal abelian unramified
extension of k of degree a power of 2) and kðiþ1Þ is the Hilbert 2-class field of kðiÞ for i≥ 1.
Lemma 1. If G is a 2-group of order 2m , m ≥ 2 , such that G=G0 ’ =23=2 , then G is isomorphic to Qm ðrespectively Dm; Sm; ð2; 2ÞÞ the quaternion (respectively dihedral, semidihedral, Klein) group of order 2m . In particular G0, the commutator subgroup of G , is cyclic.
Proof. See [3]. ,
Let G ¼ Galðkð2Þ=kÞ, be the Galois group of kð2Þ=k, then, with class field theory we have
that G0 ¼ Galðkð2Þ=kð1ÞÞ ’ C kð1Þ;2 and G=G
0 ¼ Galðkð1Þ=kÞ ’ Ck;2. According to [3] and [8], if
Ck;2 is an elementary group of rank 2, then Ckð1Þ;2 is cyclic, which implies that the Hilbert
2-class field tower stops at kð2Þ.
AJMS 27,1
JEL Classification — 11R27, 11R29, 11R37 © Abdelmalek Azizi, Mohamed Talbi and Mohammed Talbi. Published in the Arab Journal of
Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http:// creativecommons.org/licences/by/4.0/legalcode
Declaration of Competing Interest: Authors do not have any conflict of interest. The publisher wishes to inform readers that the article “On Hilbert class field tower for some quartic
number fields”was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Azizi, A., Talbi, M., Talbi, M. (2019), “OnHilbert class field tower for some quartic number fields”, Arab Journal of Mathematical Sciences, Vol. 27 No. 1, pp. 20-25. The original publication date for this paper was 22/05/2019.
Received 15 March 2019 Revised 19 May 2019 Accepted 19 May 2019
Definition 1 (Taussky’s Conditions). Let F=k be a cyclic unramified extension and j ¼ jF=k ðrespectively NF=kÞ denote the conorm (respectively the norm) of F=k.
F=k is of type ðAÞ if and only if kerðjÞ ∩ NF=kðCFÞ
> 1,
F=k is of type ðBÞ if and only if kerðjÞ ∩ NF=kðCFÞ
¼ 1.
Note that kerðjÞ is the set of all the class ideals of k which capitulate in F.
Theorem 1. Let k be a number field such that Ck;2 is isomorphic to =23=2;F1;F2;F3
the three unramified quadratic extensions of k within kð1Þ and G be the Galois group of
kð2Þ=k, then
(1) G is abelian if and only if the four classes of Ck;2 capitulate in each extension Fi=k ;
(2) G ’ Q3 if and only if the three extensions Fi=k are of type ðAÞ and in each extension Fi=k only two classes of Ck;2 capitulate;
(3) G ’ Qm avec m > 3 if and only if uniquely one extension Fi=k is of type ðAÞ and in each extension Fi=k two classes of Ck;2 capitulate;
(4) G ’ Sm if and only if the three extensions Fi=k are of type ðBÞ and in each extension Fi=k two classes of Ck;2 capitulate;
(5) G ’ Dm if and only if the four classes of Ck;2 capitulate uniquely in one extension Fi=k .
Proof. See [8]. ,
2. Units of some number fieldss In the remainder of this paper, let ‘ be a prime number congruent to 1 modulo 8, ε the
fundamental unit of ð ffiffi ‘
p Þ and L ¼ ð ffiffiffiffiffiffiffiffiffi ε
ffiffi ‘
pp Þ.
The extension L= is real cyclic of degree 4, of Galois group H ¼ hσi and quadratic
subfieldð ffiffi ‘
p Þ. Since L is of conductor ‘, then L is a subfield of the ‘th cyclotomic fieldðζ‘Þ and there is a character χ0 of Galððζ‘Þ=Þ ’ ð=‘Þ*, where its kernel is Galððζ‘Þ=LÞ, we will call the character of L.
Let χ ¼ χ0 þ χ0−1, then χ is a rational character of ðζ‘Þ and L is fixed by the Common
kernel of χ0 and χ−1. Let EL be the group of units of L, Eχ ¼ fω ∈ ELjω1þσ2 ¼ ±1g the group of χ-relatives units of L (according to definition of H. W. Leopoldt in [10]), jELj (respectively jEχ j) the group of the absolute values of EL, (respectively of Eχ), jELj ¼ jELj⊕ jEχ j and εχ a generator of Eχ, then:
Theorem 2. Let L ¼ ð ffiffiffiffiffiffiffiffiffi ε
ffiffi ‘
pp Þwhere ‘ be a prime number congruent to 1modulo 8 and ε
the fundamental unit ofð ffiffi ‘
p Þ , then there exists ξ in EL , such as ξ 2 ¼ ±εε1−σχ and fξ; ξσ ; ξσ2g
is a fundamental system of units of L
Proof. See [4]. ,
(1) ξ1þσ ¼ ±εχ;
(2) ξ1þσ2 ¼ ±ε;
21
(4) Nð ffiffi ‘
p Þ=ðεÞ ¼ NL=ð ffiffi ‘
p ÞðεχÞ ¼ NL=ðξÞ ¼ −1.
Lemma 2. With the same notation of Theorem 2 , fξ; ξσ ; ξσ2g is also a fundamental system of units of F ¼ Lð ffiffiffiffiffiffi
−n p Þ where n is a positive integer prime to ‘ and squarefree.
Proof. If Lð ffiffiffiffiffiffi −n
p Þ ≠ Lð ffiffiffiffiffiffi −1
p Þ, then, according to [1, Proposition 3], to show that fξ; ξσ ; ξσ2g is a fundamental system of units of F it suffices to show that nμ is not a square in L, for
μ ¼ ξ01 j1 ξ02
j2ξ03 where fξ01; ξ02; ξ03g ¼ fξ; ξσ ; ξσ2g and j1; j2 ∈ f0; 1g. Indeed, if μ ¼ ξ, then if nξ ¼ x2 in L, by calculating the norm in L=k, we find that
ξ1þσ2 ¼ ±ε is a square in k, which is impossible. If μ ¼ ξσ, then if nξσ ¼ ±n εχ
ξ ¼ x2 in L, by calculating the norm in L=k, we find that
±ε1þσ
ε ¼ ±1 ε is a square in k, which is absurd.
If μ ¼ ξσ 2
, then if nξσ 2 ¼ ±n ε
ξ ¼ x2 in L, by calculating the norm in L=k, we find that
±ε2 ε ¼ ±ε is a square in k, which is not the case.
If μ ¼ ξ1þσ ¼ ±εχ, then if ±nεχ ¼ x2 in L, by calculating the norm in L=k, we find that
ε1þσ2 χ ¼ ε1þσ ¼ −1 is a square in k, which is absurd.
If μ ¼ ξ1þσ2 ¼ ±ε, then ±nε cannot be a square in L.
If μ ¼ ξσþσ2, then if nξσþσ2 ¼ x2 in L, by calculating the norm in L=k, we find that
ξ1þσþσ2þσ3 ¼ −1 is a square in k, which is not the case.
If μ ¼ ξ1þσþσ2, then if nξ1þσþσ2 ¼ x2 in L, by calculating the norm in L=k, we find that
ξ1þσþσ2þσ3ξ1þσ2 ¼ ±ε is a square in k, which is impossible.
If F ¼ Lð ffiffiffiffiffiffi −1
p Þ, then we have m ¼ 2 is the largest integer such that ζm ∈F, so, from
[1, Proposition 2], to show that fξ; ξσ ; ξσ2g is a fundamental system of units of F it suffices to
show that 2μ is not a square in L, for μ ¼ ξ01 j1 ξ02
j2ξ03 where fξ01; ξ02; ξ03g ¼ fξ; ξσ; ξσ2g and j1; j2 ∈ f0; 1g. Using the same reasoning as above we find the result, which completes the proof of the lemma. ,
3. Hilbert 2-class field tower of ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ε
ffiffi ‘
ffiffi ‘
pp Þ where ‘ is a prime number such that ‘ ≡ 1ðmod 8Þ and ð2‘Þ4 ¼ −1,
according to [2], Ck;2 is isomorphic to =2 3 =2, thus kð1Þ=k has three intermediate
subfields, F1; F2; F3 and letG be the Galois group of kð2Þ=k. Using [6, Theorem 4, p. 48–49],
it is easy to show that the genus field of k is kð*Þ ¼ F3 ¼ kð ffiffiffiffiffiffi −2
p Þ, this is the maximal extension of k of the form kMwhich is unramified for all prime ideals of k, finite and infinite, and such that M is abelian over .
As ð2‘Þ ¼ 1, then there exist two prime ideals
B1 ¼ 2; aþ b
ffiffi ‘
p
2
ffiffi ‘
p
with a and b are integers, but Bhð‘Þ 1 and Bhð‘Þ
2 are principal ideals of ð ffiffi ‘
p Þwhere hð‘Þ is the class number of ð ffiffi
‘ p Þ, so we can choose c and d positive integers such as
Bhð‘Þ 1 ¼ ðαÞ and Bhð‘Þ
2 ¼ ðαÞ in ffiffi
‘ p
p 2
p 2
are positive. Let H1 and H2 be prime ideals of kaboveB1 and B2 respectively and H ¼ H1H2.
Proposition 1. Let k ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ε
ffiffi ‘
pp Þ where ‘ is a prime number such that ‘ ≡ 1ðmod 8Þ
and ð2‘Þ4 ¼ −1 . Then the class ½H is of order 2 in k . Moreover H capitulates in kð ffiffiffiffiffiffi −2
p Þ. Proof. The class ofH is of order 2, indeed, we haveH2 ¼ ð2Þ in k. Suppose thatH ¼ ðλÞ for some λ in k, which is equivalent to ðλ2Þ ¼ ð2Þ in k, therefore, there exists a unit ηof k such that
2η ¼ λ2, although there exist x and y in ð ffiffi ‘
p Þ such that λ ¼ xþ y ffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ε
ffiffi ‘
p y2 þ 2xy
ffiffi ‘
pq :
Since ε is a fundamental unit of k and i ¼ ffiffiffiffiffiffi −1
p ∉ k, then 2η ∈ ð ffiffi
‘ p Þ, therefore x or y is
equal to 0. If y ¼ 0, then 2η ¼ x2, so 2 ¼ x02 or 2ε ¼ x002 in ð ffiffi ‘
p Þ, which gives thatffiffiffi 2
p ∈ ð ffiffi
−1 p
∈ ð ffiffi ‘
p Þ in the second case, which is impossible. Similarly, if x ¼ 0 we find that ±‘ is a square in , consequently ½H is of order 2.
To show thatH is capitulated in F3 ¼ kð ffiffiffiffiffiffi −2
p Þ, it suffices to remark that ffiffiffiffiffiffi −2
p is in F3 and
ð ffiffiffiffiffiffi −2
p 2Þ ¼ ð2Þ in F3, thus H capitulates in F3 ,.
Theorem 3. Let k ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ε
ffiffi ‘
pp Þwhere ‘ is a prime number such that ‘ ≡ 1ðmod 8Þ and
ð2‘Þ4 ¼ −1 and F3 ¼ kð ffiffiffiffiffiffi −2
p Þ. Then CF3;2, the 2-part of the class group of F3, is cyclic of
order h2ðF3Þ ¼ 4.
Proof. Let α, α, B1, B2,H1,H2 andH ¼ H1H2 defined as above and let L ¼ ð ffiffiffiffiffiffiffiffiffi ε
ffiffi ‘
we have kL ¼ F3,Nk=ð ffiffi ‘
p ÞðH1Þ ¼ B1 andB1 is unramified in L=ð ffiffi ‘
p Þ. So, to prove thatH1
is inert in F3=k it suffices to show that B1 is inert in L=ð ffiffi ‘
p Þ (Translation Theorem), and for
this, we calculate the following norm residue symbol ðα;ε ffiffi ‘
p B1
ffiffi ‘
p
B1
¼
2
soB1 is inert in L=ð ffiffi ‘
p Þ, which givesH1 remains inert in F3=kand in the samewaywe have shown thatH2 remains inert in F3=kand sinceH ¼ H1H2 capitulates in F3, then by the Artin reciprocity law, we find that F3=K is of type ðAÞ, therefore, according to [8], we find that CF3 ;2
is cyclic.
Since F3=ð ffiffi ‘
p Þ is a normal biquadratic extension with Galois group of type ð2; 2Þ and where k;L; k0 ¼ ð ffiffi
‘ p
; ffiffiffiffiffiffi −2
p Þ are these intermediate subfields, then, by [9], we have
h2ðF3Þ ¼ 22−1−2−0qðF3
. ð ffiffi
:
But h2ð‘Þ ¼ 1; h2ðkÞ ¼ 4; h2ðLÞ ¼ 1 (see [12]), and by [7,11] we have h2ðk0Þ ¼ 2, so
h2ðF3Þ ¼ 4qðF3=ð ffiffi ‘
p ÞÞand we have ε is a fundamental unit of kand since 2ε is not a square in ð ffiffi
‘ p Þ, then, by [1, Proposition 3], fεg is also a fundamental system of units of k0 and
according to Lemma 2, L and F3 have the same fundamental system of units which is
fξ; ξσ; ξσ2g, thus qðF3=ð ffiffi ‘
p ÞÞ, the unit index of F3=ð ffiffi ‘
p Þ, is equal to 1, which gives that h2ðF3Þ ¼ 4. ,
Hilbert class field tower for number fields
23
ffiffi ‘
pp Þwhere ‘ is a prime number such that ‘ ≡ 1 ðmod 8Þ and
ð2‘Þ4 ¼ −1 and let G be the Galois group of kð2Þ=k, then G is of order 8.
Indeed, we have kð1Þ=F3 is an unramified extension and the 2-class group of F3 is cyclic,
thus F3 and kð1Þ have the same Hilbert 2-class field, namely kð2Þ, so # G ¼ 2h2ðF3Þ ¼ 8.
Theorem 4. Let k ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ε
ffiffi ‘
pp Þwhere ‘ is a prime number such that ‘ ≡ 1ðmod 8Þ and
ð2‘Þ4 ¼ −1 , F1; F2; F3 the quadratic subfields of kð1Þ=k . Then, in each extension Fi=k ,
i∈ f1; 2; 3g , there exist exactly two classes of Ck;2 which capitulates and G is the quaternion group of order 8.
Proof. According to Lemma 2, we have fξ; ξσ ; ξσ2g is a fundamental system of units of F3. Since
NF3=kðξÞ ¼ NF3=k
ξσ
2 ¼ ±ε and NF3=kðξσÞ ¼ ±ε−1;
then NF3=kðEF3Þ ¼ Ek, consequently, by [5], we have only two classes of Ck;2 which capitulates in F3, namely ½H and its square and since the extension F3=k is of type ðAÞ, then, by Theorem 1, we find that G is a quaternion group, and by Remark 2, the group G is of order 8, therefore G ’ Q3. ,
Remark 3. As G ’ Q3, then, by [8],
CF1 ;2 ’ CF2 ;2 ’ CF3 ;2 ’ 4:
Example 1. Let k ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ð17þ 4
p Þ q
Þ, we have 17 ≡ 1ðmod 8Þ and ð 217Þ4 ¼ −1, then
the Galois group of kð2Þ=k is the quaternion group of order 8, there exist exactly two classes of
Ck;2 which capitulates in F3 ¼ kð ffiffiffiffiffiffi −2
p Þ, as well for F1 and for F2 and the Hilbert 2-class field
tower of k stops at kð2Þ.
References
[1] A. Azizi, Unites de certains corps de nombres imaginaires et abeliens Sur , Ann. Sci. Math. Quebec 23 (2) (1999) 15–21.
[2] E. Brown, C.J. Parry, The 2-class group of certain biquadratic number fields II, Pac. J. Math. 78 (1) (1978) 11–26, Pacific Journal of Mathematics.
[3] D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.
[4] M.N. Gras, Table numerique du nombre de classes et des unites des extensions cycliques reelles de degre 4 de ,, Publ. Math. Fac. Sci. Besançon, Theor. Nombres 2 (1977-78) 1–79.
[5] F.P. Heider, B. Schmithals, Zur kapitulation der idealklassen in unverzweigten primzyklischen erweiterungen, J. Reine Angew. Math. 366 (1982) 1–25.
[6] M. Ishida, The Genus Fields of Algebraic Number Fields, in: Lecture Notes in Mathematics, vol. 555, Springer-Verlag, London., 1976.
[7] P. Kaplan, Sur le 2-groupe des classes d’ideaux des corps quadratiques, J. Reine Angew. Math. 283/284 (1976) 313–363.
[8] H. Kisilevsky, Number fields with class number congruent to 4 modulo 8 and Hilbert’s theorem 94, J. Number Theory 8 (1976) 271–279.
[9] F. Lemmermeyer, Kuroda’s class number formula, Acta Arith. 66 (3) (1994) 245–260.
AJMS 27,1
[10] H.W. Leopoldt, Uber einheitengruppe und klassenzahl reeller abeslscher Zahlk€orper, Abh. Deutsche Akad. Wiss. Berlin, Math. 2 (1953) 1–48.
[11] T.M. McCall, C.J. Parry, R.R. Ranalli, Imaginary bicyclic biquadratic fields with cyclic 2-class group, J. Number Theory 53 (1) (1995) 88–99.
[12] L.C. Washington, Introduction To Cyclotomic Fields, (Book 83), second ed., in: Graduate Texts in Mathematics, Springer–Verlag, New York, 1997.
Corresponding author Mohammed Talbi can be contacted at: [email protected]
Hilbert class field tower for number fields
25
A Method of approximation for a zero of the sum of maximally
monotone mappings in Hilbert spaces
Getahun Bekele Wega and Habtu Zegeye Department of Mathematics and Statistical Sciences,
Botswana International University of Science and Technology, Palapye, Botswana
Abstract Our purpose of this study is to construct an algorithm for finding a zero of the sum of twomaximally monotone mappings in Hilbert spaces and discus its convergence. The assumption that one of the mappings is α-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unifymost of the results that have been proved for this important class of nonlinear mappings.
Keywords Firmly nonexpansive, Hilbert spaces, Maximally monotone mapping, Strong convergence,
Zero points
1. Introduction Let H be a real Hilbert space with inner product h:; :i and induced norm k:k. Let A : H → 2H
be a nonlinear mapping. The domain, range, zero, and graph of A are respectively the sets DomðAÞ ¼ fx ∈ H : Ax ≠ 0=g; RðAÞ ¼ fAx : x ∈ DomðAÞg; ZeroðAÞ ¼ fx ∈ H : 0 ∈ Axg and GphðAÞ ¼ fðx; yÞ ∈ H 3H : y ∈ Axg. A mappingA : H → 2H is called monotone if for any x; y ∈ H and u ∈ Ax; v ∈ Aywe have
AJMS 27,1
JEL Classification — 47H05, 47J25, 49M27, 90C25 ©Getahun BekeleWega and Habtu Zegeye. Published in theArab Journal of Mathematical Sciences.
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CCBY4.0) license. Anyonemay reproduce, distribute, translate and create derivativeworks of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons. org/licences/by/4.0/legalcode
The authors thank the anonymous referees for useful suggestions which improved the contents of this paper.
Funding: The first author gratefully acknowledges the funding received from Simons Foundation based at Botswana International University of Science and Technology (BIUST), Palapye, Botswana.
Declaration of Competing Interest: None. The publisher wishes to inform readers that the article “AMethod of approximation for a zero of the
sum of maximally monotone mappings in Hilbert spaces” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please useWega, G. B., Zegeye, H. (2019), “AMethod of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces”, Arab Journal of Mathematical Sciences, Vol. 27 No. 1, pp. 26-40. The original publication date for this paper was 22/05/2019.
Received 4 April 2019 Revised 16 May 2019 Accepted 17 May 2019
hu v; x yi≥ 0: (1.1)
A monotone mapping A : H → 2H is called maximally monotone if GphðAÞ is not properly contained in the graph of any other monotone operator. The resolvent of A is given by JλA ¼ ðI þ λAÞ−1, where I is the identity mapping on H and λ > 0.
Let A; B : H → 2H be maximally monotone mappings. Consider the problem of finding z∈H such that
0 ∈ Azþ Bz: (1.2)
We denote the solution set of (1.2) by ðAþ BÞ−1ð0Þ. This problem includes, as special cases, convex programming, variational inequalities, split feasibility problem and minimization problem. For solving problem (1.2), we remark that several authors have studied different iterative schemes (see, for example, [3,4,10,11,14,21] and the references therein).
In 1979, Passty [11] introduced Forward–backward splitting method which defines a sequence fxng by
xnþ1 ¼ ðI þ rnBÞ−1ðI rnAÞxn; n ≥ 1; (1.3)
where frng is a sequence of positive numbers, A and B are maximal monotone mappings with DomðAÞ ⊂ DomðBÞ, and A is single valued. This method is essentially a generalization of the classical gradient method for constrained convex optimization and monotone variational inequalities, and inherit restrictions similar to those methods such as A is single valued. In general, this method provides weak convergence even with the restriction that A is single-valued. It fails to provide weak convergence results to the zero of the sum of the general maximal monotone mappings.
In 1979, Lions and Mercier [7] introduced Peaceman–Rachford splitting method whose iteration method is given by
yn ¼ 2ðI þ λBÞ−1 IÞxn;
zn ¼ 2ðI þ λAÞ−1 IÞyn;
xnþ1 ¼ ð1 ρnÞxn þ ρn zn; n ≥ 1;
(1.4)
where λ > 0 is a fixed scalar, and fρng⊂ ð0; 1 is a sequence of relaxation parameters. They proved weak convergence of this sequence to the solution of problem (1.2) under certain conditions.
In 2008, Eckstein and Svaiter [5] constructed new approach splitting algorithms which starts by reformulating (1.2) as the problem of locating a point in a certain extended solution set SeðA; BÞ ⊂ H 3H and proved weak convergence results provided that H has finite dimension or Aþ B is maximal monotone mapping. The extended solution set for the problem (1.2), which is the subset of H 3H is defined by:
SeðA; BÞ ¼ fðz; wÞ∈ H 3 H : w ∈ BðzÞ;w ∈ AðzÞg: (1.5)
We treat H 3 H as a Hilbert space by endowing it with the canonical inner product
hðx; x0 Þ; ðy; y0 Þi ¼ hx; yi þ hx0 ; y
0 i: More recently, in order to solve problem (1.2), Svaiter [17] studied the following Algorithm: for any λ > 0; x0; b0 ∈H,
Step 1. Compute; yn; an such that an ∈ AðynÞ; λan þ yn ¼ xn−1 − λbn−1:
Step 2. Compute; xn; bn such that bn ∈ BðxnÞ; λbn þ xn ¼ yn þ λbn−1; n ≥ 1:
Approximation in Hilbert
spaces
27
They proved that fðxn; bnÞg and fðyn; − anÞg converge weakly to a point ðz; wÞ in SeðA; BÞ, where z ∈ ðAþ BÞ−1ð0Þ. We remark that the convergence is still weak convergence.
With regard to a strong convergence, several authors have studied different iterative schemes (see for example, [12,18–20,22] and the references therein) for a zero of the sum of monotone m

arab journal of mathematical sciences number 1 arab

Documents