Class Number Two for Real Quadratic Fields of Richaud-Degert Type

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the Euler-Rabinowitsch polynomial for real quadratic fields. As well, we complete the list of Richaud-Degert types given in [16] and show how the behaviour of the Euler-Rabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination of the class number 2 problem for such types. For some values the determination is unconditional, and for others, the wide Richaud-Degert types, the determination is conditional on the generalized Riemann hypothesis (GRH)

Sign Changes of the Liouville function on quadratics

Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \vspace{1mm} \noindent {\bf Conjecture (Chowla).} {\em \begin{equation} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x) \end{equation} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. } \vspace{1mm} \noindent The prime number theorem is equivalent to \eqref{a.1} when $f(x)=x$. Chowla's conjecture is proved for linear functions but for the degree greater than 1, the conjecture seems to be extremely hard and still remains wide open. One can consider a weaker form of Chowla's conjecture, namely, \vspace{1mm} \noindent {\bf Conjecture 1 (Cassaigne, et al).} {\em If $f(x) \in \Z [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \Z[x]$, then $\lambda (f(n))$ changes sign infinitely often.} Clearly, Chowla's conjecture implies Conjecture 1. Although it is weaker, Conjecture 1 is still wide open for polynomials of degree $>1$. In this article, we study Conjecture 1 for the quadratic polynomials. One of our main theorems is {\bf Theorem 1.} {\em Let $f(x) = ax^2+bx +c$ with $a>0$ and $l$ be a positive integer such that $al$ is not a perfect square. Then if the equation $f(n)=lm^2$ has one solution $(n_0,m_0) \in \Z^2$, then it has infinitely many positive solutions $(n,m) \in \N^2$.} As a direct consequence of Theorem 1, we prove some partial results of Conjecture 1 for quadratic polynomials are also proved by using Theorem 1

Girl Child Education for Inclusive Sustainable Development

The socio-economic marginalization of females derived from gender disparities, lack of access and quality education leads to subsequent youth unemployment, working poverty, poor standards of living, diminished career choices, unequal salaries and economic exclusion. It is therefore imperative for leaders and governments to invest in girl-child education for Africa\u27s development by castigating gender-based violence, sexual discrimination, gender stereotypes and obliterating early-child marriage. The blatant repudiation of girl\u27s fundamental rights to access equal and quality education is the indictment of a continent that has failed young women and their broader continuous struggle for emancipation, equality and power. This paper lays out some of the problems that girls face and presents an outline of how girl child education is critical to fostering inclusive sustainable development in Africa

Writers’ habits

Transcription of performance talk jointly written and performed at the Kunstraum Riehen, Switzerland

Oktaedrische Ruthenium(II)- und Rhodium(III)-Komplexe mit mehrzähnigen Liganden als Kinaseinhibitoren

Aufgabenstellung (gekürzt): Während klinisch zugelassene bioaktive Substanzen vornehmlich organische Moleküle sind, werden anorganische Verbindungen hauptsächlich wegen der Reaktivität des Metallzentrums wie in DNA-bindenden Cytostatika (z.B. Cisplatin) oder in Kontrastmitteln (z.B. Gd(III)-Komplexe) eingesetzt. Eine weitere Anwendung wurde von Meggers et al. im Jahre 2004 etabliert, in der erstmals stabile und kinetisch inerte Metallkomplexe als Kinaseinhibitoren publiziert wurden. In diesen erfüllt das Metallzentrum ausschließlich strukturelle Funktionen und bildet mit verschiedenen organischen Liganden rigide und komplexe Strukturen. Abgeleitet vom natürlichen Kinaseinhibitor Staurosporin dient ein Pyridocarbazol als Pharmakophor-Ligand, während das Metallzentrum mit den vier verbliebenen Koordinationsstellen Raum für Modifikationen bietet. In den vergangenen Jahren wurden mit dieser Strategie mehrere potente und selektive Kinaseinhibitoren dargestellt. Die ersten metallorganischen Kinaseinhibitoren waren Halbsandwich-Komplexe mit einem Cyclopentadienyl und einem einzähnigen Liganden. Diese Komplexe ließen sich in wenigen Schritten herstellen, boten jedoch nur begrenzt viele Strukturmöglichkeiten. In weiteren Arbeiten wurden oktaedrische Komplexe mit vier variablen Koordinationsstellen A–D hervorgebracht. Mit diesen wurde die Darstellung von sehr komplexen Strukturen ermöglicht, welche mit denen von größeren Naturstoffen vergleichbar sind. Allerdings wurden auch gleichzeitig neue synthetische Herausforderungen geschaffen. Während von Halbsandwich-Komplexen stets ein Enantiomerenpaar gebildet wird, steigt die Zahl an möglichen Stereoisomeren in oktaedrischen Komplexen mit sechs einzähnigen Liganden auf bis zu 30. Mit einem zweizähnigen Pyridocarbazol und vier einzähnigen Liganden bleiben immer noch bis zu 24 mögliche Stereoisomere. Diese hohe Zahl an verschiedenen Konfigurationen ist natürlich nicht attraktiv für die gezielte Synthese von bioaktiven Verbindungen. Aufgabe dieser Arbeit war die Entwicklung von neuen mehrzähnigen Liganden und deren möglichst selektive Einführung in Metallkomplexen. So sind z.B. mit acyclischen dreizähnigen Liganden bei fac-Koordination nur noch maximal zwölf Stereoisomere (sechs Diastereomere als Enantiomerenpaare) zu erwarten. Dreizähnige Liganden in mer-Koordination führen zu maximal vier Stereoisomeren (zwei Diastereomere als Enantiomerenpaare), vierzähnige Liganden zu höchstens zwölf verschiedenen Stereoisomeren (sechs Diastereomere als Enantiomerenpaare). Etwaige strukturelle Bedingungen sowie kinetische oder thermodynamische Effekte lassen zudem eine Bevorzugung von bestimmten Konfigurationen erwarten. Schließlich sollten die erhaltenen Komplexe auf ihre chemischen Eigenschaften und biologischen Aktivitäten gegenüber Proteinkinasen untersucht werden. Mit den neuen mehrzähnigen Liganden sollten neben Proteinkinasen auch Lipidkinasen inhibiert werden. Hauptmerkmal der Lipidkinaseinhibitoren und gleichzeitig verantwortlich für die Selektivität gegenüber Proteinkinasen ist eine Methylgruppe an der Imidfunktion des Pyridocarbazols. Seit Beginn der Arbeiten von Meggers et al. besteht das Problem, dass die Kinaseinhibitoren stets als racemische Gemische erhalten werden, wobei sich die Enantiomere in ihrer Aktivität stark unterscheiden können. Enantiomerentrennungen gelten allerdings als zeit- und kostenintensiv und halbieren zudem nochmals die Ausbeute. Erste Lösungsansätze zur asymmetrischen Synthese von einfachen Ru(II)-Komplexen wurden bereits in der Arbeitsgruppe etabliert. Jedoch war es bislang nicht möglich, diese Auxiliar-vermittelten Methoden auf eine enantioselektive Synthese von Kinaseinhibitoren umzusetzen. Aus diesen Gründen sollte versucht werden, mit Hilfe von chiralen mehrzähnigen Liganden ein zweites Stereozentrum in die Metallkomplexe einzuführen. Von den resultierenden Komplexen wären dann Diastereomere zu erwarten, welche sich kinetisch und thermodynamisch unterscheiden. Im Idealfall würden diese so begünstigt bzw. diskriminiert, dass nur ein enantiomerenreines Produkt entsteht. Abschließend sollte untersucht werden, inwieweit sich die hier erworbenen Erkenntnisse und Methoden auf andere Projekte innerhalb der Arbeitsgruppe übertragen ließen. Von besonderem Interesse waren dabei Kinaseinhibitoren mit neuartigen Pharmakophorliganden, in die verschiedene mehrzähnige Liganden einzuführen waren. Auch solche Metallkomplexe sollten auf ihre chemischen Eigenschaften und biologischen Aktivitäten evaluiert werden

On the cyclotomic polynomial

AbstractFor a given positive integer m and an algebraic number field K necessary and sufficient conditions for the mth cyclotomic polynomial to have K-integral solutions modulo a given integer of K are given. Among applications thereof are: that the solvability of the cyclotomic polynomial mod an integer yields information about the class number of related number fields; and about representation of integers by binary quadratic forms. The latter extends previous work of the author. Moreover some information is obtained pertaining to when an integer of K is the norm of an integer in a given quadratic extension of K. Finally an explicit determination of the pqth cyclotomic polynomial for distinct primes p and q is provided, and known results in the literature as well as generalizations thereof are obtained

Gender oppression and possibilities of empowerment: images of women in African literature with specific reference to Mariama Ba's So long a letter, Buchi Emecheta's The Joys of motherhood and Tsitsi Dangarembga's Nervous conditions

This study consists of a comparative analysis of three novels by three prominent African women writers which cast light on the ways in which women are oppressed by traditional and cultural norms in three different African countries. These three primary texts also explore the ways in which African women's lives are affected by other issues, such as colonialism and economic factors, and this study discusses this. An analysis of these novels reveals that the inter-connectedness of racial, class and gender issues exacerbates the oppression of many African women, thereby lessening the opportunities for them to attain self-realization. This study goes on to investigate whether there are possibilities of empowerment for the women in the primary texts, and examining the reasons why some women fail to transcend their situations of oppression. The primary novels will be discussed in different chapters, which explore the problems with which various women are beset, and discuss the extent to which the various women in the novels manage to attain empowerment. In conclusion, this study compares and contrasts the ways in which the women in the primary texts are oppressed and highlights the reasons why some women are able to attain empowerment, whilst others are unable to do so. It also shows that many women are beset with comparable forms of oppression, but they may choose to react to these situations differently. Over and above these issues, the study seeks to draw attention to the fact that women need to come together and contribute to the ways in which they can attain various forms of empowerment

The nexus between Community Engagement and Academic Language Development

Community engagement (CE) is now widely considered a core function of higher education worldwide. In South African higher education institutions (HEIs), there is an increasing focus on CE as a means of transforming the role of the university in society, though the forms and shapes of CE vary by institution. CE is positioned as part of the means of addressing the challenges within the South African education system, such as ensuring equity in academic access in the face of diversity and making sure higher education institutions are responsive to the needs of society. Community Engagement is increasingly being afforded the same status as teaching and learning and research in higher education. The idea that higher education should function as a public good is central to this. This study reflects on how CE can be expansively viewed as places of learning for students to achieve epistemic access with epistemic justice, particularly in increasing diverse and changing contexts. Despite the growing research on CE in HEIs, there is comparatively limited focus on the intersection between CE and language use and potential linkages with identity and epistemic access and success. Given that one of the major challenges in South African HEIs relate to difficulties experienced by students whose home language is not English, the experiences of students learning within CE contexts within those institutions warrants investigation. The main aim of this study was to explore second language English speaking students’ experiences of language in the Engaged Citizen Programme, a Community Engagement programme at Rhodes University aimed at offering students the opportunity of enhanced learning, giving students the opportunity to evaluate the theories and ideas taught in the university against the realities of the South African context. The programme is also intended to offer students the opportunity to learn with and from communities and thereby enable personal growth (ECP Handbook: 2020; p.3). Using an in-depth phenomenological approach, this study explored diverse students’ experiences of language use in both community engaged programmes and in the classroom as a basis for understanding the role language plays in such spaces and the impact of these programmes on epistemic access, justice and success for students in HEIs. The study explored the role that CE plays for second language English students as they navigate complex questions of identity and belonging in HEIs. In CE activities, such as the Engaged Citizen Programme, unlike traditional classroom learning, English is often not the medium of instruction, as learning takes place in community sites, where multiple other languages are spoken. In the traditional classrooms, English is the dominant medium of instruction which can bring challenges for students whose home language is not English. Students are faced with various challenges including failure to communicate effectively and understand content knowledge. Significantly this study found that this often related to a sense of self-worth and belonging and constrained their participation and engagement in class. It was evident from the students’ reflections on their experiences in the Engaged Citizen Programme that CE provided a more flexible space generally more comfortable to these participants; a space that promotes engaged learning without rigid rules. The students’ reflections affirmed the contribution of CE in promoting engagement of students outside the formal classes and enhancing the ways in which they use language freely. It was also evident from the students’ reflections that CE provided a space in which students can identify who they are and have a sense of belonging. In the context of diversity, the majority of the students said they come to the university feeling a level of under preparedness and cannot identity with dominant groups. The reflections from the students’ experiences therefore offer some insights into ways in which we can actively promote CE in supporting student access and addressing issues of epistemic justice in higher education. The findings suggest that many of the benefits of CE, such as higher levels of interaction and significant amounts of translanguaging, need to be brought into the formal classroom spaces because they enhanced student engagement. While CE was also seen to be challenging and there were calls for more support, the essence of the experience was as a space of personal development and awareness of social responsibility. The explicit normative value of CE was in contrast to the absence of such considerations in the formal HE curriculum and the student experiences suggest that much could be learned from this. The use of English, both on campus and in CE activities, was found to be value-laden and politically charged. The participants, black students who spoke English as an additional language, all related experiences of English being positioned as a ‘superior’ language. The students who were highly fluent in English experienced being positioned as ‘showing off’ and seen to have ‘forgotten their roots’. Students who were not highly fluent in English, on the other hand, often constrained their participation in class because they experienced concern that their mispronunciations and accents may be mocked. The essence of the experience of language use in both formal classroom settings and in CE activities is that this is tightly bound to identity and is ideologically fraught. This requires more explicit conversation in all learning spaces.Kubatirana pamwe nenharaunda (KPN) (CE) iko zvino kwave kutariswa zvakanyanya sembiru yebasa redzidzo yepamusoro pasirese. MuSouth Africa muzvikoro zvedzidzo yepamusoro (ZZY) (HEIs), kune kuwedzera kwekupa nguva kuKPN kunyangwe mamiriro uye maumbirwo eKPN achisiyana zvichienderana nechikoro chacho. Nematambudziko ari mukati memunezvedzidzo muSouth Africa uye nekumwewo, sekuenzanisira mukuwanikwa kwedzidzo mukusiyana kwevanhu nemaitiro uye kuona kuti zvikoro zvepamusoro zvinoteerera zvinodiwa munharaunda, KPN yaakupihwa kukosha kumwechete nekudzidzisa nekudzidza uyewonetsvagiridzo mudzidzo yepamusoro nechinangwa chekushandura nzira idzo ruzivo rwunoshandurwa mukushanda kwedzidzo yepamusoro sechinhu chakanakira munhu wese (ona Bhagwan: 2017). Shanduko yemaonero ekugadzirwa kwezivo nebasa remayunivhesiti mumagariro, kubva kunzira dzakare dzekuzvionera pamusoro kuenda kumayunivhesiti anobatikana nenharaunda zvinoonekwa senzira chaiyo yekuzadzisa chinangwa chekuti mayunivhesiti ave anodavira kune zvakapoteredza, aine mutoro nazvo uye achiunza shanduko. Pamwongo wemakakatanwa anechekuita neKPN panenyaya yekuti KPN inogona kutariswa zvakanyanya sevanze dzekudzidza dzevadzidzi uye kuwana kupinda munezveruzivo nezvekururamisira, kunyanya mukuwedzera kwokusiyana nekushanduka kwemamiriro ezvinhu. Zvakadaro, kunyangwe paine kukura mukufarirwa kweKPN muZZY, kune kushomeka kwekutarisa panosangana KPN nekushandiswa kwemutauro uye hukama nezvekuti unozviti uri ani uye kuwana kupinda munezvezivo nekubudirira. Muchiitiko chekudzidza kuburikidza nezviitwa zveKPN, zvichisiyana nekudzidza muimba yekudzidzira (kirasi) kwagara kuripo muZZY, Chirungu hachisiriicho nzira yekuraira nayo, sezvo kudzidza kuchiitika munzvimbo dzirimunharaunda. Mumakirasi ekudzidza kwagara kuripo, Chirungu ndomutauro unonyanyoshandiswa pakurairidza izvo zvinogona kuita kuti vadzidzi vanotaura mutauro usiri Chirungu kumba vatarisane nematambudziko anosanganisira kukundikana kutaura zvavarikuda, kuvenehukama nezvirikuitika chaizvo uye kunzwisisa ruzivo rwezvinodzidzwa zvinogona kukanganisa kutora chikamu kwavo nokubatirana kwavo nezvinengezvichiitika mukirasi. Nokuda kwokuti rimwe rematambudziko makuru muZZY zvemuSouth Africa rinechekuita nezvinetso zvinosangana nevadzidzi vane mutauro wekumba usiri Chirungu, panefaniro yekuoongorora zvinosanganikwa nazvo nevadzidzi mukati meKPN. Wongororo iyi idavidzo kumukaha uwu. Chinangwa chikuru chewongororo iyi chachiri chekuongorora zvinosanganikwa nazvo nevadzidzi vanotaura Chirungu semutauro wechipiri muEngaged Citizen Programme, chirongwa cheCommunity Engagement chine chinangwa chekupa vadzidzi mukana wekusimudzira madzidziro, kupa vadzidzi mukana wekuyera/kuongorora pfungwa dzirimukudzidza nemaonero munezvinodzidziswa muyunivhesiti zvichiyenzaniswa nezviri kuitika muSouth Africa pamwe nekupa vadzidzi mukana wekudzidza kubva kune nepamwe nenharaunda zvichitungamirira kukukura semunhu mumwe nemumwe (ECP Handbook: 2020; p.3) paRhodes University, inovayunivhesiti inoshandisaChirungu pakurairidza. Ichishandisa nemaitiro akadzama nzira yekuongorora inonzi phenomenological approach wongororo iyi yakatarisa zvinosanganikwa nazvo zvevadzidzi vakasiyana-siyana mukushandiswa kwemutauro muzvirongwa zvokubatirana pamwe nenharaunda uye mukirasi sehwaro hwekunzwisisa zvibereko zvezvirongwa izvi pakuwanikwa kwezvezivo, kururamisira uye kubudirira kwevadzidzi muZZY. Wongororo iyi yangayakanangawo zvakare kuongorora basa rinoitwa neKPN kuvadzidzi avo mutauro wechipiri urichiRungu pavanenge vachiedza kupindura mibvunzo yakaoma inechekuita nezvekuti ndivanaani uye nekuva kwavo chikamu chemuZZY. Zvakava pachena kubva mukufungisisa kwevadzidzi pamusoro pekuvevangavarimuchirongwa Engaged Citizen Programme kuti KPN kwakavapa nzvimbo yakasununguka uye yakagadzikana, inosimudzira kudzidza kunobata pasina mitemo yakaoma/isingashandurwi. Kufungisisa kwevadzidzi kwakasimbisa zvinounzwa neKPN mukusimudzira kuvanechekuita kwevadzidzi munezvinoitika kunze kwemakirasi nekusimudzira nzira dzavanoshandisa mutauro nadzo vakasununguka. Zvaivawo pachena kubva mukufungisisa kwevadzidzi kuti CE yakapa nzvimbo yekuti vadzidzi vazive kuti ndivanaani uye kuti vanzwe kuti ndevepi. Mumamiriro ekusiyana kwezvinhu, vazhinji vevadzidzi vakati vanouya kuyunivhesiti vachinzwa vaine chiyero chekugadzirira chepasi uye vasingagoni kufambidzana nevemapoka aripamusoro/anemukundo. Zvakabuda mukufungisisa kwevadzidzi pamusoro pezvavakapinda mazviri/zvavakasangana nazvo zvinopa mamwe manzwisisiro atingashandisa enzira dzatingashingairira nadzo kusimudzira CE mukutsigira vadzidzi kuwana kupinda nekugadzirisa nyaya dzekururamisira munezvezivo mudzidzo yepamusoro.Thesis (PhD) -- Faculty of Education, Centre for Higher Education Research, Teaching and Learning, 202