John Peebles in a Senior Voice Recital

This is the program for the senior voice recital of tenor John Peebles. Pianist Debbie Theobalt assisted the performance. This recital took place on April 4, 1978, in the Mabee Fine Arts Center Recital Hall

Sampling Random Spanning Trees Faster than Matrix Multiplication

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in $\tilde{O}(n^{4/3}m^{1/2}+n^{2})$ time (The $\tilde{O}(\cdot)$ notation hides $\operatorname{polylog}(n)$ factors). The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, $O(n^\omega)$. For the special case of unweighted graphs, this improves upon the best previously known running time of $\tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\})$ for $m \gg n^{5/3}$ (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute $\epsilon$-approximate effective resistances for a set $S$ of vertex pairs via approximate Schur complements in $\tilde{O}(m+(n + |S|)\epsilon^{-2})$ time, without using the Johnson-Lindenstrauss lemma which requires $\tilde{O}( \min\{(m + |S|)\epsilon^{-2}, m+n\epsilon^{-4} +|S|\epsilon^{-2}\})$ time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate

Hypergraph Capacity with Applications to Matrix Multiplication

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics. We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such general purpose bound). We then elaborate on some attempted generalizations/modifications of the Lovász number to undirected hypergraphs that we have tried. It is not currently known whether these attempted generalizations/modifications upper bound the capacity of arbitrary hypergraphs. An important method for proving lower bounds on hypergraph capacity is to exhibit a large independent set in a strong power of the hypergraph. We examine methods for this and show a barrier to attempts to usefully generalize certain of these methods to hypergraphs. We then look at cap sets: independent sets in powers of a certain hypergraph. We examine certain structural properties of them with the hope of finding ones that allow us to prove upper bounds on their size. Finally, we consider two interesting generalizations of capacity and use one of them to formulate several conjectures about connections between cap sets and sunflower-free sets

Sample-Optimal Identity Testing with High Probability

We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1-delta, whether the distributions are identical versus epsilon-far in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via black-box amplification, which multiplies the required number of samples by Theta(log(1/delta)). We show that black-box amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta. An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worst-case failure probability for a broad class of uniformity testing statistics over all possible input distributions - including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well

Staphylococcus alpha toxin

The present thesis describes the general properties mode of action in vivo and in vitro, and a method of purification, of staphylococcal alpha toxin. The optimum pH and temperature of the haemolytic reaction were found to be 5.5 and 31 C. respectively. In proteolytic digestion and heat sensitivity, the toxin behaved as a typical protein. Contrary to current views alpha toxin was found intrinsically heat sensitive. A revised explanation of the paradoxical "Arrhenius phenomenon" is suggested and disscused in light of this. Alpha toxin was not used up in haemolysis; it acts as a catalyst. Kinetic experiments showed that at low concentrations of toxin the relationship between the rate of haemolysis and concentration was compatible with an enzymic reaction. At high concentrations there was a marked falling off; possible reasons for this are suggested and discussed , and the limitations or haemolysis as an indicator system are pointed out. Also the velocity of haemolysi was dependent of the concentration of rabbit red blood cells. Considering the talk of evidence it is concluded that alpha toxin belongs to that group of bacterial haemolysins which are thought to be enzymes. Divalent ions are not required for haemolysis. Inhibitors of bacterial protenses had no effect. Heavy metal ions such as Hg 2+ were found to inhibit at concentrations of 10-3M; below this they became haemolytic by themselves . Some organic sulphydryl inhibitors also inhibited alpha toxin; it seems therefore that free SH groups may play a role in haemolysis. The trypanocidal drug Suramin and related substances were powerful inhibitors of both the haemolytic and the lethal activity of alpha toxin. Phospholipids were tested as possible competitive inhibitors of alpha toxin in an attempt to gain information concerning the point of attack. Apart from a crude preparation of sheep brain cephalin none of them inhibited. The possible significance of this is discussed. Death from alpha toxin was found to be dose dependent: it was either very rapid, or occurred after considerable delay. Dose dependence and the pattern of death was largely the same for the four species examined, viz., rabbits, mice, fowl and frogs. Rapid death in seconds or minutes, without histological lesions is most likely explained by an action on heart or central nervous system. Slow death in several hours or days may result from lesions in a multiplicity of organs. Large doses( which killed rapidly intravenously) when administered subcutaneously or intraperitoneally killed much slower, probably because of the time required for diffusion of toxin to vital organs. Alpha toxin caused local flaccid paralysis of voluntary muscle when injected into the dorsal sac of 6 week old mice; at high doses this occurred before any detectable histological lesion. Muscles of the paralysed limbs not respond to electrical stimulation IN SITU. In presence of alpha toxin the reactivity of excised voluntary muscles of mice and frogs to acetyl choline and electrical stimulation was abolished IN VITRO. Muscles of curarised mice beghaved in the same way, and it concluded that alpha toxin has a direct myotoxic action. Heart muscle appeared less sensitive. Hearts of mice killed with alpha toxin continued to beat for a few minutes after death and hearts of frogs for up to several hours. In tissue culturos mouse heart explants were less affected than the whole animal. Whereas 128 MHD is an L.D. 50 for 20g. mice,explants of 20-1,000 cells were only moderately affected by 2,000 MHD/ml: some explants continued to beat, some stopped after 30 min., but recovered overnight. The possible significance of this is discussed. Alpha toxin, purified by combination of gel filtration on G.75 and DEAE gamma 50 Sephadex and fractional methanol precipitation, behaved as serologically and physically homogencous protein. The sedimentation constant was 3.1 S at 0,13% protein, which is close to that of 3.0 S suggested recently by Bernheimer and Schwartz (1963). The potency was 119,000 MH/mg. protein. The product was unstable and evidence was obtained which suggested that on standing or dialysis alpha toxin tends to polymerise with the appearance of heavy component