Notes on the Quadratic Integers and Real Quadratic Number Fields

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_d$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_d \gg (\log d)^2$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $\mu = -1$, the construction gives all $d$'s for which the negative Pell's equation $X^2 - d Y^2 = -1$ (or more generally $X^2 - D Y^2 = -4$) is soluble. When $\mu$ is a prime, it gives all of the real quadratic fields in which the prime ideals lying over $\mu$ are principal

Spatial spectral analysis of elastic waves in a three-dimensional truss structure

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1996.Includes bibliographical references (leaves 132-133).by Jeongho Park.M.S

Gut Microbiota-Derived Short-Chain Fatty Acids, T Cells, and Inflammation

T cells are central players in the regulation of adaptive immunity and immune tolerance. In the periphery, T cell differentiation for maturation and effector function is regulated by a number of factors. Various factors such as antigens, co-stimulation signals, and cytokines regulate T cell differentiation into functionally specialized effector and regulatory T cells. Other factors such as nutrients, micronutrients, nuclear hormones and microbial products provide important environmental cues for T cell differentiation. A mounting body of evidence indicates that the microbial metabolites short-chain fatty acids (SCFAs) have profound effects on T cells and directly and indirectly regulate their differentiation. We review the current status of our understanding of SCFA functions in regulation of peripheral T cell activity and discuss their impact on tissue inflammation