On the linear independence of spikes and sines

The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof involves depends on an extrapolation argument of Bourgain and Tzafriri.Comment: 16 pages, 4 figures. Revision with new proof of major theorem

Heterotic Strings in Two Dimensions and New Stringy Phase Transitions

We discuss heterotic string theories in two dimensions with gauge groups Spin(24) and Spin(8) x E_8. After compactification the theories exhibit a rich spectrum of states with both winding and momentum. At special points some of these stringy states become massless, leading to new first order phase transitions. For example, the thermal theories exhibit standard thermodynamics below the phase transition, but novel and peculiar behavior above it. In particular, when the radius of the Euclidean circle is smaller than the phase transition point the torus partition function is not given by the thermal trace over the spacetime Hilbert space. The full moduli space of compactified theories is 13 dimensional, when Wilson lines are included; the Spin(24) and Spin(8) x E_8 theories correspond to distinct decompactification limits.Comment: 32 pages; v2: references added, minor change

User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's inequality has been moved to a separate note; other martingale bounds are described in Caltech ACM Report 2011-0

Improvement of cast nephropathy with plasma exchange depends on the diagnosis and on reduction of serum free light chains

Cast nephropathy is the most common cause of renal disease in multiple myeloma, however, treatment with plasma exchange remains controversial even after 3 randomized controlled studies. We sought to determine the importance of diagnostic confirmation and goal directed therapy in the treatment of cast nephropathy in forty patients with confirmed multiple myeloma and renal failure who underwent plasma exchange. A positive renal response was defined as a decrease by half in the presenting serum creatinine and dialysis independence. No baseline differences were noted between eventual renal responders and non-responders. Three quarters of the patients with biopsy proven cast nephropathy resolved their renal disease when the free light chains present in the serum were reduced by half or more but there was no significant response when the reduction was less. The median time to a response was about 2 months. In patients without cast nephropathy, renal recovery occurred despite reductions in free light chain levels of the serum. No association was found between free light chains in the serum, urinary monoclonal proteins, overall proteinuria and cast nephropathy. We found that the relationship between renal recovery and free light chain reduction was present only in patients with biopsy proven cast nephropathy showing the importance of extracorporeal light chain removal in this disease

Global relationships in tree functional traits

Due to massive energetic investments in woody support structures, trees are subject to unique physiological, mechanical, and ecological pressures not experienced by herbaceous plants. Despite a wealth of studies exploring trait relationships across the entire plant kingdom, the dominant traits underpinning these unique aspects of tree form and function remain unclear. Here, by considering 18 functional traits, encompassing leaf, seed, bark, wood, crown, and root characteristics, we quantify the multidimensional relationships in tree trait expression. We find that nearly half of trait variation is captured by two axes: one reflecting leaf economics, the other reflecting tree size and competition for light. Yet these orthogonal axes reveal strong environmental convergence, exhibiting correlated responses to temperature, moisture, and elevation. By subsequently exploring multidimensional trait relationships, we show that the full dimensionality of trait space is captured by eight distinct clusters, each reflecting a unique aspect of tree form and function. Collectively, this work identifies a core set of traits needed to quantify global patterns in functional biodiversity, and it contributes to our fundamental understanding of the functioning of forests worldwide.Environmental Biolog