Abstract:
We present a general result giving us families of incomplete and boundedly complete families of discrete distributions. For such families, the classes of unbiased estimators of zero with finite variance and of parametric functions which will have uniformly minimum variance unbiased estimators with finite variance are explicitly characterized. The general result allows us to construct a large number of families of incomplete and boundedly complete families of discrete distributions. Several new examples of such families are described.

Abstract:
Two series of compounds, one comprising of 2-(2′-hydroxyphenyl)benzoxazole (HBO), 2-(2′-hydroxyphenyl)benzimidazole (HBI), 2-(2′-hydroxyphenyl)benzothiazole (HBT), and the other of 2-(2′-hydroxyphenyl)oxazole (HPO), 2-(2′-hydroxyphenyl)imidazole (HPI) and 2-(2′-hydroxyphenyl)thiazole (HPT) are susceptible to ground state rotamerization as well as excited state intramolecular proton transfer (ESIPT) reactions. Some of these compounds show experimental evidence of the existence of two ground state conformers. Out of these two one undergoes ESIPT reaction leading to the formation of the tautomer. The two photophysical processes, in combination, result in the production of a number of fluorescence bands each one of which corresponding to a particular species. Semiempirical AM1-SCI calculations have been performed to rationalize the photophysical behaviour of the compounds. The calculations suggest that for the first series of compounds, two rotational isomers are present in the ground state of HBO and HBI while HBT has a single conformer under similar circumstances. For the molecules of the other series existence of rotamers depends very much on the polarity of the environment. The potential energy curves (PEC) for the ESIPT process in different electronic states of the molecules have been generated theoretically. The simulated PECs reveal that for all these systems the IPT reaction is unfavourable in the ground state but feasible, both kinetically and thermodynamically, in the S1 as well as T1 states.

Abstract:
An investigation was carried out on two ponds of Chatla floodplain, Barak valley, Assam with special reference to aquatic insects. Pond 1 is purely a fish pond where as pond 2 is a community pond too. Present study revealed the status of water quality and in turn diversity, density, dominance and abundance of aquatic insects in both the ponds. Almost all the physico chemical parameters of both the ponds were found within permissible range for aquatic life .However in pond 2 level of phosphate was found little higher than pond 1 due to release of soaps and detergents by human influence. In both the ponds order Hemiptera showed maximum relative abundance ( 98% in pond 1 and 94% in pond 2). The study revealed lower diversity of aquatic insects in pond 2 than that in pond 1.

Abstract:
There is limited available information on the characterization of alkali-stable Bacillus sp. producing lipase and its commercial production. Therefore, thirty bacterial strains were isolated from oil industry soil samples and were screened for lipolytic activity. Bacillus sp. PD-12 was selected for lipase production because of its high lipolytic activity (4.2 IU mL-1). Bacillus sp. PD-12 lipase was optimally active at pH 8.0 and at 40C temperature. The lipase is capable of hydrolyzing vegetable oils and synthetic triglycerides. Maximum lipase production by Bacillus sp. PD-12 was obtained when grown under shaking conditions (250 rpm) at 30C for 24 h. In order to increase lipase production, optimization of carbon and nitrogen sources was studied. Maximum lipolytic activity (4.250.020 IU mL-1) was obtained with olive oil as a carbon source followed by coconut oil (2.50.030 IU mL-1). Among nitrogen sources, ammonium nitrate resulted in maximum lipolytic activity (15.60.036 IU mL-1). Lipase production by Bacillus sp. PD-12 was studied in a 3 liter fermentor with a working volume of 1.8 liter under optimized conditions resulted in lipolytic activity of 22 IU mL-1 after 21 h. Thus, short fermentation time (21 h) makes this fermentation system a promising one in terms of lipase productivity and alkali-stable Bacillus sp. can be used in detergent industry.

Abstract:
Entanglement spectrum of finite-size correlated electron systems are investigated using the Gutzwiller projection technique. The product of largest eigenvalue and rank of the block reduced density matrix, which is a measure of distance of the state from the maximally entangled state of the corresponding rank, is seen to characterise the insulator to metal crossover in the state. The fraction of distinct eigenvalues exhibits a `chaotic' behaviour in the crossover region, and it shows a `integrable' behaviour at both insulating and metallic ends. The integrated entanglement spectrum obeys conformal field theory (CFT) prediction at the metal and insulator ends, but shows a noticeable deviation from CFT prediction in the crossover regime, thus it can also track a metal-insulator crossover. A modification of the CFT result for the entanglement spectrum for finite size is proposed which holds in the crossover regime also. The adjacent level spacing distribution of unfolded non-zero eigenvalues for intermediate values of Gutzwiller projection parameter $g$ is the same as that of an ensemble of random matrices obtained by replacing each block of reduced density matrix by a random real symmetric Toeplitz matrix. It is strongly peaked at zero, with an exponential tail proportional to $e^{-(n/R)s}$, where $s$ is the adjacent level spacing, $n$ is number of distinct eigenvalues and $R$ is the rank of the reduced density matrix.

Abstract:
The block entanglement entropy and fluctuations are investigated in one dimension in finite size correlated electron systems using the Gutzwiller wave function as a prototype correlated electron state. Entanglement entropy shows logarithmic divergence for all values of the correlation projection parameter $g$, as predicted by conformal field theories for critical systems, but the central charge requires finite size corrections. There is an infinite correlation length corresponding to correlation between same kinds of spins, for all values of $g$. A scaling form for the block entropy, as a function of $g$ and the system size $N$, is proposed which predicts a metal-insulator crossover at $N^{1/3} g\approx 0.24$. Bipartite fluctuations in the number of particles in a block, and the spin fluctuations also obey an approximate scaling. A relation is found between the block entropy and the bipartite spin fluctuations. Our results show some correspondence with an experiment on Ni nanochains.

Abstract:
We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the "ordered metric" characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.

Abstract:
We derive new estimates of the size of codes and orthogonal arrays in the ordered Hamming space (the Niederreiter-Rosenbloom-Tsfasman space). We also show that the eigenvalues of the ordered Hamming scheme, the association scheme that describes the combinatorics of the space, are given by the multivariable Krawtchouk polynomials, and establish some of their properties.