Fundamental Weights, Permutation Weights and Weyl Character Formula

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(\Lambda^{++})$ of strictly dominant weights $\Lambda^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(\Lambda^{++})$, there is a very peculiar subset $\wp(\Lambda^{++})$ for which we always have $$dim\wp(\Lambda^{++})=dimW(G_N)/dimW(A_{N-1}) .$$ For any dominant weight $\Lambda^+$, the elements of $\wp(\Lambda^+)$ are called {\bf Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $\wp(\Lambda^{++})$ and $\wp(\rho)$ where $\rho$ is the Weyl vector of $G_N$. The concept of signature factor which enters in Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant $\Lambda^+$, calculation of the character $ChR(\Lambda^+)$ for irreducible representation $R(\Lambda^+)$ will then be provided by $A_N$ multiplicity rules governing generalized Schur functions. The main idea is again to express everything in terms of the so-called {\bf Fundamental Weights} with which we obtain a quite relevant specialization in applications of Weyl character formula.Comment: 6 pages, no figures, TeX, as will appear in Journal of Physics A:Mathematical and Genera

An Explicit Construction of Casimir Operators and Eigenvalues : I

We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients $g^{A_1,A_2,.. A_p}$. It is seen that these coefficients can be descibed by some rational polinomials of rank N. These polinomials are also multilinear in Cartan sub-algebra indices taking values from the set $I_0 = {1,2,.. N}$. The crucial point here is that for each degree one needs, in general, more than one polinomials. This in fact is related with an observation that the whole set of symmetric coefficients $g^{A_1,A_2,.. A_p}$ is decomposed into sum subsets which are in one to one correspondence with these polinomials. We call these subsets clusters and introduce some indicators with which we specify different clusters. These indicators determine all the clusters whatever the numerical values of coefficients $g^{A_1,A_2,.. A_p}$ are. For any degree p, the number of clusters is independent of rank N. This hence allows us to generalize our results to any value of rank N. To specify the general framework explicit constructions of 4th and 5th order Casimir operators of $A_N$ Lie algebras are studied and all the polinomials which specify the numerical value of their coefficients are given explicitly.Comment: 14 pages, no figures, revised version, to appear in Jour.Math.Phy

Laparoscopic versus conventional appendectomy - a meta-analysis of randomized controlled trials

<p>Abstract</p> <p>Background</p> <p>Although laparoscopic surgery has been available for a long time and laparoscopic cholecystectomy has been performed universally, it is still not clear whether open appendectomy (OA) or laparoscopic appendectomy (LA) is the most appropriate surgical approach to acute appendicitis. The purpose of this work is to compare the therapeutic effects and safety of laparoscopic and conventional "open" appendectomy by means of a meta-analysis.</p> <p>Methods</p> <p>A meta-analysis was performed of all randomized controlled trials published in English that compared LA and OA in adults and children between 1990 and 2009. Calculations were made of the effect sizes of: operating time, postoperative length of hospital stay, postoperative pain, return to normal activity, resumption of diet, complications rates, and conversion to open surgery. The effect sizes were then pooled by a fixed or random-effects model.</p> <p>Results</p> <p>Forty-four randomized controlled trials with 5292 patients were included in the meta-analysis. Operating time was 12.35 min longer for LA (95% CI: 7.99 to 16.72, p < 0.00001). Hospital stay after LA was 0.60 days shorter (95% CI: -0.85 to -0.36, p < 0.00001). Patients returned to their normal activity 4.52 days earlier after LA (95% CI: -5.95 to -3.10, p < 0.00001), and resumed their diet 0.34 days earlier(95% CI: -0.46 to -0.21, p < 0.00001). Pain after LA on the first postoperative day was significantly less (p = 0.008). The overall conversion rate from LA to OA was 9.51%. With regard to the rate of complications, wound infection after LA was definitely reduced (OR = 0.45, 95% CI: 0.34 to 0.59, p < 0.00001), while postoperative ileus was not significantly reduced(OR = 0.91, 95% CI: 0.57 to 1.47, p = 0.71). However, intra-abdominal abscess (IAA), intraoperative bleeding and urinary tract infection (UIT) after LA, occurred slightly more frequently(OR = 1.56, 95% CI: 1.01 to 2.43, p = 0.05; OR = 1.56, 95% CI: 0.54 to 4.48, p = 0.41; OR = 1.76, 95% CI: 0.58 to 5.29, p = 0.32).</p> <p>Conclusion</p> <p>LA provides considerable benefits over OA, including a shorter length of hospital stay, less postoperative pain, earlier postoperative recovery, and a lower complication rate. Furthermore, over the study period it was obvious that there had been a trend toward fewer differences in operating time for the two procedures. Although LA was associated with a slight increase in the incidence of IAA, intraoperative bleeding and UIT, it is a safe procedure. It may be that the widespread use of LA is due to its better therapeutic effect.</p