Integer Subsets with High Volume and Low Perimeter

We consider a certain variation of the 'isoperimetric problem' adopted for subsets of nonnegative integers. More specifically, we explore the sequence P(n) as described in OEIS A186053. We provide the first exact formulas for P(n) including multiple recursive relations involving auxiliary functions as well as concise and satisfying representations and even quasi-explicit formulas. We also discuss some of the intricate fractal-like symmetry of the sequence as well as the development of algorithms for computing P(n). We conclude with open questions for further research. Note this is a more developed, but more concise version of a previous arXiv paper arXiv:1107.2954 by the name "Sets with High Volume and Low Perimeter".Comment: 14 pages, 2 figures. Note this is a more developed, but more concise version of a previous arXiv paper arXiv:1107.2954 by the name "Sets with High Volume and Low Perimeter

Some integer formula-encodings and related algorithms

We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such formulas, we describe two essen- tially distinct families of canonical formula-encodings for integers, respectively deduced from the decimal encoding and the fundamental theorem of arithmetic. Our main contribution is the de- tailed description of two algorithms which efficiently determine the canonical formula-encodings associated with relatively large sets of consecutive integers

A Strange Family of Calabi-Yau 3-folds

We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds $\tilde{X}$ with hodge numbers $h^{11}=31,h^{21}=1$ constructed in \cite{BN}. We calculate the Picard-Fuchs differential equation associated to this family, and use it to predict the instanton numbers on the hypothetical mirror. These exhibit a strange vanishing in odd degrees. We also calculate the monodromy action on H^3(\tilde{X},\QQ) and find that it strangely predicts a positive Euler characteristic for its mirror. From a degenerate fiber of our family we construct a new rigid Calabi-Yau 3-fold. In an appendix we prove the expansion of the conifold period conjectured in \cite{ES} to hold for all 1-parameter families.Comment: 23 page

Inflammatory Response Following Hemorrhagic Stroke: The Role of Cytokines

Hemorrhagic stroke is a dangerous form of stroke resulting from the rupturing of weakened blood vessels, releasing blood that increases intracranial pressure and causes the death of surrounding tissue. Treatment options are improving but remain limited, as evidenced by this condition being characterized by high rates of mortality as well as long-term morbidity. Unregulated inflammatory responses that occur following injury may be partially to blame for these poor outcomes. In response to any injury, the immune system releases cytokines to recruit immune cell activation and promote inflammation. But after hemorrhagic stroke, whether aneurysmal subarachnoid hemorrhage (aSAH) or intracerebral hemorrhage (ICH), the rapid increase in inflammation and an imbalance of inflammatory cytokines can lead to the development of secondary ischemic injury. Several animal models have been developed to investigate these forms of stroke, and despite their shortcomings, these models have been crudely applied for the study of neuroinflammation post-stroke. There is an imperative need to establish a clear and robust animal model that accurately represents the condition taking place in human patients. This dissertation work began with a systematic review of the literature to more clearly understand the work that has been done in this area. Next, the endovascular puncture model of aSAH was utilized in rats. Tissue samples were collected and compared to human blood and cerebrospinal fluid samples to assess for cytokine changes. Lastly, collagenase mouse models for ICH were utilized to understand the changes that take place in the TXNIP-NLRP3 inflammasome following stroke and the therapeutic effects of verapamil on inflammatory, functional, and behavioral outcomes. This dissertation work adds to the body of literature on the relationship between inflammation, cytokine release, and outcomes, which will ultimately allow for the development of improved treatment protocols

Equity, Due Process and the Seventh Amendment: A Commentary on the Zenith Case

The seventh amendment to the United States Constitution requires that [i]n Suits at common law . . . the right of trial by jury shall be preserved. What exactly is a suit at common law? When the amendment was enacted in 1791, there was no law that was common to all the states. In 1812 Supreme Court Justice Story, in a Circuit Court ruling, held that the common law alluded to was the common law of England, the grand reservoir of all of our jurisprudence. This means that when today an American judge has to decide whether in any set of proceedings trial by jury is constitutionally required, he has to ask himself whether in England in 1791 the case would have been tried at common law. That is the test: it has come to be called the historical test