Synthesis of natural compounds in Echinacea

In this dissertation, we have investigated the direct and efficient synthetic route to biologically active natural products. Chapter 1 describes the synthesis of some of the natural compounds in Echinacea. Several main constituents of the plant Echinacea have been synthesized for the first time. We have developed a direct and flexible route to amides and ketones in Echinacea. The synthesized natural products have been used for standard samples for the biological studies. Chapter 2 describes the new tandem strategy to construct bicyclic systems by a Diels-Alder/ene reaction and its application for the synthesis of Isoligularone. During this tandem reaction, as many as five stereogenic centers can be created in a single reaction. The efficient synthesis of Isoligularone nicely demonstrates the utility of this tandem Diels-Alder/ene reaction sequence for the synthesis of natural products

Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces

In this paper, we consider a symmetric pure jump Markov process $X$ on a general metric measure space that satisfies volume doubling conditions. We study estimates of the transition density $p(t,x,y)$ of $X$ and their stabilities when the jumping kernel for $X$ has general mixed polynomial growths. Unlike [24], in our setting, the rate function which gives growth of jumps of $X$ may not be comparable to the scale function which provides the borderline for $p(t,x,y)$ to have either near-diagonal estimates or off-diagonal estimates. Under the assumption that the lower scaling index of scale function is strictly bigger than $1$, we establish stabilities of heat kernel estimates. If underlying metric measure space admits a conservative diffusion process which has a transition density satisfying a general sub-Gaussian bounds, we obtain heat kernel estimates which generalize [2, Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the rate function and the function $F$ related to walk dimension of underlying space. As an application, we proved that the finite moment condition in terms of $F$ on such symmetric Markov process is equivalent to a generalized version of Khintchine-type law of iterated logarithm at the infinity

Synthesis and Natural Distribution of Anti-inflammatory Alkamides from Echinacea

The synthesis of the alkamides 2Z,4E-undeca-2,4-dien-8,10-diynoic acid isobutyl amide (1) and 2Z,4E-undeca-2,4-dien-8,10-diynoic acid isobutyl amide (5) was accomplished by organometallic coupling followed by introduction of the doubly unsaturated amide moiety. The distribution of these two amides in accessions of the nine species of Echinacea was determined

Inscribing or Circumscribing a Histogon to a Convex Polygon

We consider two optimization problems of approximating a convex polygon, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation ? is a copy of an axis-aligned histogon rotated by ? in counterclockwise direction. The goal is to find a largest inscribed histogon and a smallest circumscribed histogon over all orientations in [0,?). Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems

Beyond the Standard Model B-parameters with improved staggered fermions in Nf=2+1N_f=2+1 QCD

We calculate the kaon mixing B-parameters for operators arising generically in theories of physics beyond the standard model. We use HYP-smeared improved staggered fermions on the $N_f = 2+1$ MILC asqtad lattices. Operator matching is done perturbatively at one-loop order. Chiral extrapolations are done using "golden combinations" in which one-loop chiral logarithms are absent. For the combined sea-quark mass and continuum extrapolation, we use three lattice spacings: $a \approx 0.045, 0.06$ and $0.09 \text{fm}$. Our results have a total error of 5-6%, which is dominated by the systematic error from matching and continuum extrapolation. For two of the BSM $B$-parameters, we agree with results obtained using domain-wall and twisted-mass dynamical fermions, but we disagree by $(4-5)\sigma$ for the other two.Comment: 7 pages, 5 figures, Lattice 2013 Proceedin