Algebraic geometry over semi-structures and hyper-structures of characteristic one

In this thesis, we study algebraic geometry in characteristic one from the perspective of semirings and hyperrings. The thesis largely consists of three parts: (1) We develop the basic notions and several methods of algebraic geometry over semirings. We first construct a semi-scheme by directly generalizing the classical construction of a scheme, and prove that any semiring can be canonically realized as a semiring of global functions on an affine semischeme. We then develop Cech cohomology theory for semi-schemes, and show that the classical isomorphism is still valid for a semi-scheme. Finally, we introduce the notion of a valuation on a semiring, and prove that an analogue of an abstract curve by using the (suitably defined) function field Qmax(T) is homeomorphic to the projective line over the field with one element. (2) We develop algebraic geometry over hyperrings. The first motivation for this study arises from the following problem posed in [9]: if one follows the classical construction to define the hyper-scheme (X = SpecR,O_X), where R is a hyperring, then a canonical isomorphism R ≃ O_X(X) does not hold in general. By investigating algebraic properties of hyperrings (which include a construction of a quotient hyperring and Hilbert Nullstellensatz), we give a partial answer for their problem as follows: when R does not have a (multiplicative) zero-divisor, the canonical isomorphism R ≃ O_X(X) holds for a hyper-scheme (X = SpecR,OX). In other words, R can be realized as a hyperring of global functions on an affine hyper-scheme. We also give a (partial) affirmative answer to the following speculation posed by Connes and Consani in [7]: let A = k[T] or k[T, 1/T ], where k = Q or Fp. When k = Fp, the topological space SpecA is a hypergroup with a canonical hyper-operation ∗ induced from a coproduct of A. The similar statement holds with k = Q and SpecA\{δ}, where δ is the generic point (cf. [7, Theorems 7.1 and 7.13]). Connes and Consani expected that the similar result would be true for Chevalley group schemes. We prove that when X = SpecA is an affine algebraic group scheme over arbitrary field, then, together with a canonical hyper-operation ∗ on X introduced in [7], (X, ∗) becomes a slightly general (in a precise sense) object than a hypergroup. (3) We give a (partial) converse of S.Henry’s symmetrization procedure which produces a hypergroup from a semigroup in a canonical way (cf. [21]). Furthermore, via the symmetrization process, we connect the notions of (1) and (2), and prove that such a link is closely related with the notion of real prime ideals

The Combination of Ephedrae herba and Coicis semen in Gambihwan Attenuates Obesity and Metabolic Syndrome in High-Fat Diet–Induced Obese Mice

Gambihwan is a herbal prescription used in Korean medicine to treat obesity. The authors evaluated the effects and mechanisms of two types of Gambihwan (GBH1 and 2) administered to high-fat diet– (HFD-) induced obese mice. Four-week-old C57BL/6 mice were fed a HFD for 8 weeks with or without GBH1 or 2 (100-200 mg/kg/day by oral gavage). All mice were subjected to glucose tolerance testing after the 8-week treatment period and then euthanized. Serum insulin, lipids, and inflammatory cytokine levels were analyzed using commercial kits. Hepatic enzyme levels and lipid profiles were also investigated. Liver section slides were stained with Oil Red O (ORO) or hematoxylin and eosin (H&E) to assess lipid accumulation. GBH1 and 2 both significantly decreased body, liver, or adipose tissue weights in HFD-fed mice and significantly improved glucose tolerance (p<0.05 in all groups). Cholesterol levels in both sera and liver homogenates were significantly decreased by GBH1 and 2 (p<0.05 in all groups). In addition, serum inflammatory cytokines (p<0.05 in 200 mg/kg/day groups) and hepatic enzyme levels were significantly diminished by GBH administration at 200mg/kg/day (p<0.05 in all groups). Furthermore, histologic analyses of liver sections revealed GBH suppressed lipid accumulation. Both GBH types suppressed HFD-induced increases in body weight and obesity-related markers in HFD-fed mice despite the difference in constituents between GBH1 and 2. It is strongly assumed that the combination of Ephedrae herba and Coicis semen exerted the antiobesity effect. The results obtained show that the antiobesity effects of GBH warrant further investigation