From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.     

A Highly Selective and Sensitive Chemiluminescent Probe for Real-Time Monitoring of Hydrogen Peroxide in Cells and Animals

Selective and sensitive molecular probes for hydrogen peroxide (H₂O₂), which plays diverse roles in oxidative stress and redox signaling, are urgently needed to investigate the physiological and pathological effects of H₂O₂. A lack of reliable tools for in vivo imaging has hampered the development of H₂O₂ mediated therapeutics. By combining a specific tandem Payne/Dakin reaction with a chemiluminescent scaffold, H₂O₂-CL-510 was developed as a highly selective and sensitive probe for detection of H₂O₂ both in vitro and in vivo. A rapid 430-fold enhancement of chemiluminescence was triggered directly by H₂O₂ without any laser excitation. Arsenic trioxide induced oxidative damage in leukemia was successfully detected. In particular, cerebral ischemia-reperfusion injury-induced H₂O₂ fluxes were visualized in rat brains using H₂O₂-CL-510 , providing a new chemical tool for real-time monitoring of H₂O₂ dynamics in living animals.     

Estimating the distance to a monotone function

In standard property testing, the task is to distinguish between objects that have a property 𝒫 and those that are ε‐far from 𝒫, for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy 𝒫. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1,…,n} ↦ R is at distance ε_f from being monotone if it can (and must) be modified at ε_fn places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 − δ)ε,ε] that encloses ε_f. The running time of our algorithm is O(ε_f⁻¹ log log ε_f^{− 1} log n), which is optimal within a factor of loglog ε_f⁻¹ and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(ε_f⁻¹ log nlog log log n). Finally, we extend our results to multivariate functions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

ON THE RELATIONSHIPS BETWEEN NATURAL AND HARVEST MORTALITIES OF THE MALLARD

A discrete optimal control approach is used to show that the intensity of harvest of mallard populations in North America, at steady state, is sensitive to the strength of compensation between natural mortality and harvest mortality. Optimal harvest is less sensitive to changes in the natural mortality rate than to the interaction between natural and harvest mortalities. The management implications are: (1) if compensatory mortality operates, then more animals may be harvested without endangering the population, and (2) managers should pay particular attention to this mechanism in studying population dynamics.     

In vivo effects of porphyrins on bacterial DNA.

The DNA damage in intact Staphylococcus aureus and E. coli cells induced by photosensitized deuteroporphyrin or hemin is described. Treatment of S. aureus cultures with hemin or photosensitized deuteroporphyrin (Dp) caused time-dependent changes in the plasmidial DNA profiles. The major observation was the disappearance of the plasmid supercoiled fraction. The chromosomal DNA was also affected by hemin and by photosensitized Dp, since its degradation products were detected after exposing the bacterial cells to the porphyrin drugs. Photosensitization of E. coli cells, pretreated with Dp and polymyxin B nonapeptide (PMBNP), also resulted in plasmidial damage. No such damage occurred when E. coli cultures were treated with hemin and PMBNP. The above results can be tightly correlated with the antimicrobial action of porphyrins. Their damage to the bacterial DNA seems to reflect one of the in vivo effects of these porphyrins.     

Functional polymers. XVI. Synthesis and polymerization of 2(2-hydroxy-5-isopropenylphenyl)-2H-benzotriazole and a new synthesis of 2(2-hydroxy-5-vinylphenyl)2H-benzotriazole

2(2-Hydroxy-5-isopropenylphenyl)2H-benzotriazole was synthesized in 40% overall yield starting from o-nitroaniline. Diazotization in aqueous hydrochloric acid gave o-nitrophenyl diazonium chloride which was condensed with p-hydroxyacetophenone; the azo compound was reduced to 2(2-hydroxy-5-acetylphenyl) 2H-benzotriazole with zinc powder in sodium hydroxide solution and the 2-hydroxy group of the compound was acetylated. Treatment of the acetyl compound with methyl Grignard reagent resulted in the methylation of the 5-acetyl group to 2[2-acetoxy-5(2-hydroxy-2-propyl)phenyl]2H-benzotriazole which was then dehydrated with potassium hydrogen sulfate to the desired 2(2-hydroxy-5-isopropenylphenyl)2H-benzotriazole. This monomer did not homopolymerize, but was copolymerized readily with styrene, methyl methacrylate, and n-butyl acrylate with azobisisobutyronitrile as the initiator. 2(2-Acetoxy-5-acetylphenyl)2H-benzotriazole was also reduced with sodium borohydride to form 2[2-acetoxy-5-(1-hydroxyethyl)phenyl]2H-benzotriazole which was dehydrated and hydrolyzed to the known 2(2-hydroxy-5-vinylphenyl)-2H-benzotriazole. This route provides a novel and simpler synthesis of 2(2-hydroxy-5-vinylphenyl)2H-benzotriazole.     

Bounds on multiplicities of spherical spaces over finite fields

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number C such that the multiplicity $\dim \mathrm{Hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by C, for any finite field F and any irreducible representation ρ of $G(F)$. We give an explicit bound for C. We conjecture that this result is true for any reductive group scheme and when F ranges (in addition) over all local fields of characteristic 0.Different aspects of this conjecture were studied in [3], [11], [6], [7].