On the cardinality of positively linearly independent sets

Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in \(\mathbb {R}^n\) has been established to be between \(n+1\) and 2n (with both extremes existing). The lower bound is immediate from being a positive spanning set, while the upper bound uses both positive spanning and positively linearly independent. In this note, we provide details proving that a positively linearly independent set in \(\mathbb {R}^n\) for \(n \in \{1, 2\}\) has at most 2n elements, but a positively linearly independent set in \(\mathbb {R}^n\) for \(n\ge 3\) can have an arbitrary number of elements.     

Hydrated electron extinction coefficient revisited

The extinction coefficient of the hydrated electron (e(-))aq generated by pulse radiolysis is evaluated relative to the methyl viologen radical cation (*)MV(+), whose extinction coefficient at 605 nm has been carefully measured in the past. We find that the room temperature (e(-))aq extinction coefficients reported in the literature are underestimated by 10-20%. We obtain = 22,700 M(-1) cm(-1) for the 20 degrees C hydrated electron at 720 nm, assuming the (*)MV(+) extinction is 13,700 M(-1) cm(-1) at 605 nm. This has implications both for second-order reaction rate measurements of (e(-))aq and for the estimate of its integrated oscillator strength.     

Individually addressable microelectrode array for monitoring oxygen and nitric oxide release

We have fabricated a six individual addressable gold working electrode microarray. The device is wirebonded to an eight-pin DIL package that can be easily interconnected to an external multi-channel potentiostat. A polyion complex film coating on the electrode surface provides a suitable coating for the growth of cells. The responses of oxygen and nitric oxide were assessed on uncoated and coated devices using electroanalytical methods. The film coating reduced the diffusion current by approximately 20% in both cases. No changes in the electrochemical mechanism were observed. Simultaneous recordings were obtained for 2 h in the presence of the cells, thus the device is stable for the duration of the bioanalytical measurements. Measurements were conducted to study the simultaneous changes in oxygen and nitric oxide levels in cultured fibroblast cells in the presence of growth hormones that cause cell proliferation. Increases in oxygen consumption of the cells were coupled with increases in nitric oxide levels when in the presence of the growth hormones. Use of a biological detergent to cause an oxidative burst resulted in a large increase in the current for potentials set to detect nitric oxide and oxygen.     

On univoque Pisot numbers

We study Pisot numbers which are univoque, i.e., such that there exists only one representation of as , with . We prove in particular that there exists a smallest univoque Pisot number, which has degree . Furthermore we give the smallest limit point of the set of univoque Pisot numbers.

     

New techniques for bounds on the total number of prime factors of an odd perfect number

Let denote the sum of the positive divisors of . We say that is perfect if . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This was extended by the author to show that . Using an idea of Carl Pomerance this paper extends these results. The current new bound is .

     

The cluster deletion problem for cographs

The min-edge clique partition problem asks to find a partition of the vertices of a graph into a set of cliques with the fewest edges between cliques. This is a known NP-complete problem and has been studied extensively in the scope of fixed-parameter tractability (FPT) where it is commonly known as the Cluster Deletion problem. Many of the recently-developed FPT algorithms rely on being able to solve Cluster Deletion in polynomial time on restricted graph structures.     

Sporadic Reinhardt Polygons

Let $n$ be a positive integer, not a power of two. A Reinhardt polygon is a convex $n$ -gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all $n$ , there are many Reinhardt polygons with $n$ sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of $n$ , additional Reinhardt polygons exist, which do not possess this structured form. We call these polygons sporadic. We completely characterize the integers $n$ for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when $n=pqr$ with $p$ and $q$ distinct odd primes and $r\ge 2$ . We also prove that a positive proportion of the Reinhardt polygons with $n$ sides is sporadic for almost all integers $n$ , and we investigate the precise number of sporadic Reinhardt polygons that are produced for several values of $n$ by a construction that we introduce.     

Computing proximal points of nonconvex functions

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower- ${\mathcal{C}}^2$ and encouraging preliminary numerical testing is reported.     

Incorporating H chemical shift determination into C-direct detected spectroscopy of intrinsically disordered proteins in solution

Exclusively heteronuclear ¹³C-detected NMR spectroscopy of proteins in solution has seen resurgence in the past several years. For disordered or unfolded proteins, which tend to have poor ¹H-amide chemical shift dispersion, these experiments offer enhanced resolution and the possibility of complete heteronuclear resonance assignment at the cost of leaving the ¹H resonances unassigned. Here we report two novel ¹³C-detected NMR experiments which incorporate a ¹H chemical shift evolution period followed by ¹³C-TOCSY mixing for aliphatic ¹H resonance assignment without reliance on ¹H detection.     

Search for Lorentz and CPT violation effects in Muon spin precession

The spin precession frequency of muons stored in the (g-2) storage ring has been analyzed for evidence of Lorentz and CPT violation. Two Lorentz and CPT violation signatures were searched for a nonzero delta omega a(=omega a mu+ - omega a mu-) and a sidereal variation of omega a mu+/-). No significant effect is found, and the following limits on the standard-model extension parameters are obtained: bZ = -(1.0+/-1.1) x 10(-23) GeV; (m mu dZ0 + HXY)=(1.8+/-6.0) x 10(-23) GeV; and the 95% confidence level limits b perpendicular mu+ <1.4 x 10(-24) GeV and b perpendicular mu- <2.6 x 10(-24) GeV.