A strong triangle inequality in hyperbolic geometry

For a triangle in the hyperbolic plane, let $\alpha,\beta,\gamma$ denote the angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of the altitude to side $c$. Under the assumption that $\alpha,\beta, \gamma$ can be chosen uniformly in the interval $(0,\pi)$ and it is given that $\alpha+\beta+\gamma c + h$ holds approximately 79\% of the time. To accomplish this, we prove a number of theoretical results to make sure that the probability can be computed to an arbitrary precision, and the error can be bounded

Chandrasekhar equations for infinite dimensional systems. Part 2: Unbounded input and output case

A set of equations known as Chandrasekhar equations arising in the linear quadratic optimal control problem is considered. In this paper, we consider the linear time-invariant system defined in Hilbert spaces involving unbounded input and output operators. For a general class of such systems, the Chandrasekhar equations are derived and the existence, uniqueness, and regularity of the results of their solutions established

The Current State of Drug Discovery and a Potential Role for NMR Metabolomics

The pharmaceutical industry has significantly contributed to improving human health. Drugs have been attributed to both increasing life expectancy and decreasing health care costs. Unfortunately, there has been a recent decline in the creativity and productivity of the pharmaceutical industry. This is a complex issue with many contributing factors resulting from the numerous mergers, increase in out-sourcing, and the heavy dependency on highthroughput screening (HTS). While a simple solution to such a complex problem is unrealistic and highly unlikely, the inclusion of metabolomics as a routine component of the drug discovery process may provide some solutions to these problems. Specifically, as the binding affinity of a chemical lead is evolved during the iterative structure-based drug design process, metabolomics can provide feedback on the selectivity and the in vivo mechanism of action. Similarly, metabolomics can be used to evaluate and validate HTS leads. In effect, metabolomics can be used to eliminate compounds with potential efficacy and side effect problems while prioritizing well-behaved leads with druglike characteristics

Advances in Nuclear Magnetic Resonance for Drug Discovery

Background—Drug discovery is a complex and unpredictable endeavor with a high failure rate. Current trends in the pharmaceutical industry have exasperated these challenges and are contributing to the dramatic decline in productivity observed over the last decade. The industrialization of science by forcing the drug discovery process to adhere to assembly-line protocols is imposing unnecessary restrictions, such as short project time-lines. Recent advances in nuclear magnetic resonance are responding to these self-imposed limitations and are providing opportunities to increase the success rate of drug discovery. Objective/Method—A review of recent advancements in NMR technology that have the potential of significantly impacting and benefiting the drug discovery process will be presented. These include fast NMR data collection protocols and high-throughput protein structure determination, rapid protein-ligand co-structure determination, lead discovery using fragment-based NMR affinity screens, NMR metabolomics to monitor in vivo efficacy and toxicity for lead compounds, and the identification of new therapeutic targets through the functional annotation of proteins by FASTNMR. Conclusion—NMR is a critical component of the drug discovery process, where the versatility of the technique enables it to continually expand and evolve its role. NMR is expected to maintain this growth over the next decade with advancements in automation, speed of structure calculation, incell imaging techniques, and the expansion of NMR amenable targets