We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least.87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of 1/2 for MAX CUT and 3/4 or MAX 2SAT. Slight extensions of our analysis lead to a.79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a.758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1/4 and 3/4, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

/pdf/improved-approximation-algorithms-for-maximum-cut-and-3utsb8qos4.pdf

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

https://www.math.ucdavis.edu/~saito/data/graphlap/merris-graphlap-eigvals.pdf

Laplacian matrices of graphs: a survey

The Analysis of Variance.

conditions and an accurate semidefinite programming solver," The Journal of the SDPA family for solving large-scale SDPs," in Handbook on Semidefinite. Semidefinite Programming, Combinatorial. Optimization and Primal semidefinite programming problem Handbook of Semidefinite Programming. Kluwer. Solving Euclidean distance matrix completion problems via semidefinite programming. AY Alfakih, A Khandani, H Wolkowicz. Computational optimization.

/pdf/handbook-of-semidefinite-programming-1hywp52iad.pdf

Handbook of Semidefinite Programming

The combined distribution of heights of men and women has become the canonical illustration of bimodality when teaching introductory statistics. But is this example appropriate? This article investigates the conditions under which a mixture of two normal distributions is bimodal. A simple justification is presented that a mixture of equally weighted normal distributions with common standard deviation σ is bimodal if and only if the difference between the means of the distributions is greater than 2σ. More generally, a mixture of two normal distributions with similar variability cannot be bimodal unless their means differ by more than approximately the sum oftheirstandard deviations. Examination of national survey data on young adults shows that the separation between the distributions of men's and women's heights is not wide enough to produce bimodality. We suggest reasons why histograms of height nevertheless often appear bimodal.

Is Human Height Bimodal

Linear maps that preserve commuting pairs of matrices

Permanental polynomials of graphs

A statistical analysis has been carried out on the distribution and allelism of nearly 500 sex-linked, X-ray-induced, cytologically normal and rearranged lethal mutations in Drosophila melanogaster that were obtained by G. Lefevre. The mutations were induced in four different regions of the X chromosome: (1) 1A1-3E8, (2) 6D1-8A5, (3) 9E1-11A7 and (4) 19A1-20F4, which together comprise more than one-third of the entire chromosome.—The analysis shows that the number of alleles found at different loci does not fit a Poisson distribution, even when the proper procedures are taken to accomodate the truncated nature of the data. However, the allele distribution fits a truncated negative binomial distribution quite well, with cytologically normal mutations fitting better than rearrangement mutations. This indicates that genes are not equimutable, as required for the data to fit a Poisson distribution.—Using the negative binomial parameters to estimate the number of genes that did not produce a detectable lethal mutation in our experiment (n) gave a larger number than that derived from the use of the Poisson parameter. Unfortunately, we cannot estimate the total numbers of nonvital loci, loci with undetectable phenotypes and loci having extremely low mutabilities. In any event, our estimate of the total vital gene number was far short of the total number of bands in the analyzed regions; yet, in several short intervals, we have found more vital genes than bands; in other intervals, fewer. We conclude that the one-band, one-gene hypothesis, in its literal sense, is not true; furthermore, it is difficult to support, even approximately.—The question of the total gene number in Drosophila will, not doubt, eventually be solved by molecular analyses, not by statistical analysis of mutation data or saturation studies.

https://www.genetics.org/content/genetics/113/4/869.full.pdf

The question of the total gene number in drosophila melanogaster

is nonsingular the theorem is false.Any map whose image is contained in the algebra % of the strictly uppertriangular matrices preserves nilpotence. The proof of the theorem de-pends on a result of Gerstenhaber about maximal spaces of nilpotentmatrices. We also use some elementary algebraic geometry and the funda-mental theorem of projective geometry [l,p. 88, Theorem 2.26].

/pdf/linear-transformations-that-preserve-the-nilpotent-matrices-4o7c20sp4l.pdf

William Watkins

Papers

Is Human Height Bimodal

Linear maps that preserve commuting pairs of matrices

Permanental polynomials of graphs

The question of the total gene number in drosophila melanogaster

Linear transformations that preserve the nilpotent matrices.