Geometric combinatorial algebras: cyclohedron and simplex

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra (one-sided) with basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices, with journal correction

Excitonic Funneling in Extended Dendrimers with Non-Linear and Random Potentials

The mean first passage time (MFPT) for photoexcitations diffusion in a funneling potential of artificial tree-like light-harvesting antennae (phenylacetylene dendrimers with generation-dependent segment lengths) is computed. Effects of the non-linearity of the realistic funneling potential and slow random solvent fluctuations considerably slow down the center-bound diffusion beyond a temperature-dependent optimal size. Diffusion on a disordered Cayley tree with a linear potential is investigated analytically. At low temperatures we predict a phase in which the MFPT is dominated by a few paths.Comment: 4 pages, 4 figures, To be published in Phys. Rev. Let

Disorder and Funneling Effects on Exciton Migration in Tree-Like Dendrimers

The center-bound excitonic diffusion on dendrimers subjected to several types of non-homogeneous funneling potentials, is considered. We first study the mean-first passage time (MFPT) for diffusion in a linear potential with different types of correlated and uncorrelated random perturbations. Increasing the funneling force, there is a transition from a phase in which the MFPT grows exponentially with the number of generations $g$, to one in which it does so linearly. Overall the disorder slows down the diffusion, but the effect is much more pronounced in the exponential compared to the linear phase. When the disorder gives rise to uncorrelated random forces there is, in addition, a transition as the temperature $T$ is lowered. This is a transition from a high-$T$ regime in which all paths contribute to the MFPT to a low-$T$ regime in which only a few of them do. We further explore the funneling within a realistic non-linear potential for extended dendrimers in which the dependence of the lowest excitonic energy level on the segment length was derived using the Time-Dependent Hatree-Fock approximation. Under this potential the MFPT grows initially linearly with $g$ but crosses-over, beyond a molecular-specific and $T$-dependent optimal size, to an exponential increase. Finally we consider geometrical disorder in the form of a small concentration of long connections as in the {\it small world} model. Beyond a critical concentration of connections the MFPT decreases significantly and it changes to a power-law or to a logarithmic scaling with $g$, depending on the strength of the funneling force.Comment: 13 pages, 9 figure

Associahedra via spines

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the classical permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction

The fishery, biology and stock assessment of jew fish resources of India

Sciaenids are one of the major component of the demersal trawl The total catch of this resource during 1990-94 period was 1.50,142 t contributing 8.86% to the demersal catch of India. A number of species are found in different states of India. Of which biological and stock assessment studies were made on eleven important species. Crustaceans and fish appear to be the chief food in Juvenile and adult stage respectively. Most of the species have a protracted spawning season. Among all the species studied the largest asymptotic length was estimated for O.ruber from Tuticorin and the smallest for J. sina from Cochin. The highest Z of 7.59 was recorded for K. axillaris from Chennai and the lowest was for O. cuvierifrom Mumbai. The average exploitation rate (E) and the Lc/ Lao was 0.62 and 0.53 respectively. The present yield is 91.222 t and the MSY is 1.42,613 t for all the species taken together. The exploitation rate for almost all the stocks in the states appears to be more than the optimum leve

Demersal finfish resources in certain areas of the EEZ of southwest and southeast coast of India

Bottom trawling data from the cruises of the FORV Sagar Sampada undertaken during 1989-91 in the middle and outer shelf waters of the southern EEZ, at 97 stations within a depth zone of 30 to 130 m, showed catch rates of 3 to 14000 kg/hr. The highest rate of 14 t/hr was recorded at lat. 1 l°N/75° 23.4' E, followed by 5.5t/hr at 8°54'N N/76° 19.6' E off the southwest coast, whereas the abundance was comparatively less off the southeast coast. The average catch rate per hour off the southwest was 724 kg and off the southeast it was 405 kg. The important demersal finfishes available were: threadfin breams (72%), major perches (5%), rays, carangids and goatfish (4% each) off the southwest; and carangids (29%), major perches (23%), rays (19%), threadfin breams (10%) and goatfish (5%) off the southeast. The depth belt of 41-80 m off both the coasts was found to be more productive than deeper regions. The results indicate that the potential yield from the depth zone of 50-100 m off the southwest coast is 2.6 x 10 tonne, which is much higher than the previous estimates of up to 1.5 x 10 tonne. The single largest group (80%) in this zone is threadfin breams, followed by bull's eye, lizard fishes and flatheads

On the stranding of whales along the Tamil nadu and Andhra coasts

Two young female pilot whales Globicephala macrorhyma were caught in the gill nets by a 10 m mechanised boat at Pudukuppam near Cuddalore along the Tamil Nadu coast. The two whales measured 3.02 and 1.40 m in total length respectively. . The fishermen atPomraaiyarpalayam near Pondicherry noticed a whale identified as young female sperm whale Physeter macrocephalus which measured 5.55 m. T he local fishermen at MotupalJy near Ramachandrapuram in Prakasam District of Andhra Pradesh noticed a whale was identified as blue whale Balaenoptera musculu. The various morphometric measurements of these whales listed in detail and tabled