On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists

Recently, Segal constructed a derived equivalence for an interesting 5-fold flop that was provided by Abuaf. The aim of this article is to add some results for the derived equivalence for Abuaf's flop. Concretely, we study the equivalence for Abuaf's flop by using Toda-Uehara's tilting bundles and Iyama-Wemyss's mutation functors. In addition, we observe a "flop-flop=twist" result and a "multi-mutation=twist" result for Abuaf's flop.Comment: 38 pages, v2;major revision. improved the readability of proofs, Appendix C adde

Quest for potentials in the quintessence scenario

The time variation of the equation of state $w$ for quintessence scenario with a scalar field as dark energy is studied up to the third derivative ($d^3w/da^3$) with respect to the scale factor $a$, in order to predict the future observations and specify the scalar potential parameters with the observables. The third derivative of $w$ for general potential $V$ is derived and applied to several types of potentials. They are the inverse power-law ($V=M^{4+\alpha}/Q^{\alpha}$), the exponential ($V=M^4\exp{(\beta M/Q)}$), the cosine ($V=M^4(\cos (Q/f)+1)$) and the Gaussian types ($V=M^4\exp(-Q^2/\sigma^2)$), which are prototypical potentials for the freezing and thawing models. If the parameter number for a potential form is $n$, it is necessary to find at least for $n+2$ independent observations to identify the potential form and the evolution of the scalar field ($Q$ and $\dot{Q}$). Such observations would be the values of $\Omega_Q, w, dw/da. \cdots$, and $dw^n/da^n$. Since four of the above mentioned potentials have two parameters, it is necessary to calculate the third derivative of $w$ for them to estimate the predict values. If they are tested observationally, it will be understood whether the dark energy could be described by the scalar field with this potential. Numerical analysis for $d^3w/da^3$ are made under some specified parameters in the investigated potentials. It becomes possible to distinguish the freezing and thawing modes by the accurate observing $dw/da$ and $d^2w/da^2$ in some parameters.Comment: 6 pages, 2 figures. arXiv admin note: text overlap with arXiv:1503.0367

A New Approach for Gay and Lesbian Domestic Partners: Legal Acceptance Through Relational Property Theory

In the past twenty years, the number of couples living together on a long-term basis without marrying has dramatically increased. 1 With the increase in domestic partnerships 2 has come new litigation in which one scorned domestic partner sues the other, claiming a right to property procured during the relationship. Often, one partner brought more present or future financial resources to the relationship than the other. The wealthier partner may have placed some or all of the property acquired during the relationship in her name only for a variety of seemingly innocuous reasons (e.g., business expertise, tax benefits, facilitation of financial management); nonetheless, the result is that one partner has title to the assets while the other partner has nothing. In fact, such a situation is similar to marriage, where one spouse typically enters the union with more assets, business expertise, or earning potential than the other. As this Note argues, the key difference between spouses and domestic partners, however, is that marital dissolution statutes protect spouses. Domestic partners must rely on scattered judicial decisions which often do not agree on the proper theory a domestic partner should advance in order to state a claim. The problems a domestic partner faces are compounded when the plaintiff making the property claim is gay or lesbian. Homosexuals are not a protected class, which means that they can be discriminated against as long as there is a rational basis for the statutory distinction and a legitimate government interest.

Strong full exceptional collections on certain toric varieties with Picard number three via mutations

In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional collections by using Orlov's blow-up formula and mutations.Comment: 17 pages, To apper in Le Matematich