Functional and Local Renormalization Groups

We discuss the relation between functional renormalization group (FRG) and local renormalization group (LRG), focussing on the two dimensional case as an example. We show that away from criticality the Wess-Zumino action is described by a derivative expansion with coefficients naturally related to RG quantities. We then demonstrate that the Weyl consistency conditions derived in the LRG approach are equivalent to the RG equation for the $c$-function available in the FRG scheme. This allows us to give an explicit FRG representation of the Zamolodchikov-Osborn metric, which in principle can be used for computations.Comment: 19 pages, 1 figur

Synthesis of biodegradable polyesteramides with pendant functional groups

Morpholine-2,5-dione derivatives having substituents with benzyl-protected carboxylic acid, benzyloxycarbonyl-protected amine and p-methoxy-protected thiol groups, respectively, were prepared in 29-58% yield by cyclization of the corresponding N-[(2RS)-bromopropionyl]-L-amino acids. Polyesteramides with protected pendant functional groups were obtained by ring-opening copolymerization of either ε-caprolactone or DL-lactide with morpholine-2,5-dione derivatives having protected functional substituents. The copolymerizations were carried out in the bulk at 130°C using stannous octoate as an initiator and using low mole fractions (0,05, 0,10 and 0,20) of morpholine-2,5-dione derivatives in the feed. The molecular weight of the resulting copolymers ranged from 1,4 to 8,3 · 104. The ring-opening homopolymerization of morpho-line-2,5-dione derivatives with protected functional substituents was not successful. Polyesteramides with either pendant carboxylic acid groups or pendant amine groups were prepared by catalytic hydrogenation of the corresponding protected copolymers. Treatment of copolymers having pendant p-methoxybenzyl-protected thiol groups with trifluoromethanesulfonic acid resulted not only in the removal of the p-methoxybenzyl group but also in severe degradation of the copolymers, due to acidolysis of main-chain ester bonds

Understanding β-Hydride Eliminations from Heteroatom Functional Groups

Using density functional theory, we investigated detailed aspects of the quintessential organometallic process, β-hydride elimination (BHE). In general, we find that most BHE processes from alkyl functional group β-atoms are facile, while BHE processes from heteroatom functional groups (N and O) are prohibitively high in energy. We present calculated molecular orbitals and atomic NBO charges obtained from snapshots along reaction profiles to present a qualitative overview for how heteroatoms adversely affect these processes. We discuss these results to provide an illustration for how these processes proceed, clarifying a sometimes oversimplified model for these reactions

Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $\mathcal{L}$ on a connected and simply connected nilpotent Lie group $G$ with the aid of the corresponding \emph{Rockland} operator $\mathcal{L}_0$ on the `local' contraction $G_0$ of $G$, as well as of the corresponding Rockland operator $\mathcal{L}_\infty$ on the `global' contraction $G_\infty$ of $G$. We provide asymptotic estimates of the Riesz potentials associated with $\mathcal{L}$ at $0$ and at $\infty$, as well as of the kernels associated with functions of $\mathcal{L}$ satisfying Mihlin conditions of every order. We also prove some Mihlin-H\"ormander multiplier theorems for $\mathcal{L}$ which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherel measure' associated with $\mathcal{L}$ from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.Comment: 42 pages, no figure

Functional central limit theorems on Lie groups: A survey

The general solution of the functional central limit problems for triangular arrays of random variables with values in a Lie group is described. The role of processes of finite variation is clarified. The special case of processes with independent increments having Markov generator is treated. Connections with Hille–Yosida theory for two–parameter evolution families of operators and with the martingale problem are explained

Global functional calculus for operators on compact Lie groups

In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex powers of such operators. As an application, we give a constructive symbolic proof of the G\r{a}rding inequality for operators in $(\rho,\delta)$-classes in the setting of compact Lie groups.Comment: 23 pages; minor correction

Functional analytic background for a theory of infinite-dimensional reductive Lie groups

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.Comment: 17 page