Vector Fields, Flows and Lie Groups of Diffeomorphisms

The freedom in choosing finite renormalizations in quantum field theories (QFT) is characterized by a set of parameters $\{c_i \}, i = 1 ..., n >...$, which specify the renormalization prescriptions used for the calculation of physical quantities. For the sake of simplicity, the case of a single $c$ is selected and chosen mass-independent if masslessness is not realized, this with the aim of expressing the effect of an infinitesimal change in $c$ on the computed observables. This change is found to be expressible in terms of an equation involving a vector field $V$ on the action's space $M$ (coordinates x). This equation is often referred to as ``evolution equation'' in physics. This vector field generates a one-parameter (here $c$) group of diffeomorphisms on $M$. Its flow $\sigma_c (x)$ can indeed be shown to satisfy the functional equation $$\sigma_{c+t} (x) = \sigma_c (\sigma_t (x)) \equiv \sigma_c \circ \sigma_t$$ $$\sigma_0 (x) = x,$$ so that the very appearance of $V$ in the evolution equation implies at once the Gell-Mann-Low functional equation. The latter appears therefore as a trivial consequence of the existence of a vector field on the action's space of renormalized QFT.Comment: (8 pages, latex

The so-called renormalization group method applied to the specific prime number logarithmic decrease

A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in $\Bbb N^*$ decreases like the single power of the inverse neperian logarithm.Comment: 6 pages, late

Evidence—Credibility Impeachment and the Drug-Using Witness—State v. Renneberg, 83 Wn. 2d 735, 522 P.2d 835 (1974)

In chambers before the grand larceny trial of Milton and Virginia LaVanway, the court apparently ruled that testimony about the defendants\u27 prior drug use\u27 was inadmissible in the state\u27s case. When the defendants subsequently testified to their good character, however, the trial court admitted such evidence for purposes of impeachment. On appeal of their convictions defendants challenged the admission of the evidence. The Washington Supreme Court affirmed the convictions, holding that once a defendant\u27s character has been placed in issue, evidence of drug use is admissible to attack his or her character on cross-examination. The plurality opinion also stated in dictum that such evidence is inadmissible to attack a defendant\u27s capacity, absent medical or scientific proof connecting addiction to a lack of veracity. State v. Renneberg, 83 Wn. 2d 735, 522 P.2d 835 (1974)

Restructuring the Legislature: A Proposal for Unicameralism in Washington

The tenability of such a unicameral legislature in Washington will be analyzed by examining: 1) the background and history of unicameralism; 2) the effect of the reapportionment cases on the need for a bicameral legislature; 3) the efficiency of a unicameral as opposed to a bicameral legislature; and 4) means of implementing a change from bicameralism to unicameralism. In addition, results of the authors\u27 poll of state legislators and a sample constitutional amendment are presented