A five element basis for the uncountable linear orders

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X, omega_1, omega_1^*, C, C^* where X is any suborder of the reals of cardinality aleph_1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.Comment: 21 page

A finitely presented group of piecewise projective homeomorphisms

In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing the elements of the group by labeled tree diagrams in a manner which closely parallels Richard Thompson's group F.Comment: Formerly "A geometric solution to the von Neumann-Day problem for finitely presented groups". Section added on tree diagrams. Minor revisions elsewher

A Phenomenological Study of Lesbian and Gay People in Leadership Roles: How Perspectives and Priorities Shift in the Workplace as Sexual Orientation Evolves Through Social Constructs

The queer community\u27s presence continues to increase in the United States, while more individuals become visible throughout workplaces. An increasingly diverse workforce leaves organizations struggling to leverage the knowledge and experience lesbian and gay leaders bring to discussions. Limited research informs organizations of the strategies lesbian and gay people in leadership roles use to navigate their ascension into their positions, while managing the disclosure of their sexual orientation and professional identity. This qualitative, phenomenological study involved recruiting 15 participants through purposeful, homogeneous sampling processes and snowball sampling, identifying cisgender lesbian or gay people in leadership or managerial roles, with a minimum of 5 years of experience, who are 50 years of age and older. Data analysis revealed seven key findings: (a) coming out involves disclosure and concealment strategies, (b) identity includes self-care and integrating personal and professional life, (c) social change establishes new norms that cultures adapt, (d) discrimination occurs most frequently through the lavender ceiling and microaggression, (e) the workplace culture is most affected by leaders and policies, (f) activism is critical to the advancement of equity, and (g) Supportive leadership involves mentoring and building a professional network. Through social constructionism theory and queer theory, this dissertation aimed to understand the key findings from the data-collection phase. The conclusions from this study highlight the impact the gay-rights movement of the 1970s and the AIDS crisis of the 1980s had on creating a generation of activist leaders. Further, workplace climate plays a significant role in disclosure decisions. Leaders of the lesbian, gay, bisexual, transgender, and queer (LGBTQ) community aged 50 years and older tend to follow a passive-disclosure process and use a mask of professionalism to conceal their sexual orientation. Organizations have several recommended approaches to create an inclusive and open workplace for the LGBTQ community

The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.Comment: 7 page

Riemann Versus Lebesgue Integrals

The purpose of this project is to examine how integrals work and compare Reimann integrals to Lebesgue integrals, determining their applications and distinctions