Remark on factorials that are products of factorials

In a paper published in 1993, Erdos proved that if n! = a! b!, where 1 < a a parts per thousand currency sign b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + epsilon)/ log 2) log log n and generalize it to the equation a (1)!a (2)! ... a (k) ! = n!. In a recent paper, F. Luca proved that n - b = 1 for large enough n provided that the ABC-hypothesis holds.

Influence of methylamine and N,N'-dimethylurea, the hydrolysis products of methyl isocyanate, on its systemic toxicity

Subcutaneous administration of the LD50 dose of methyl isocyanate (MIC) to rats induced severe hyperglycaemia, lactic acidosis and uraemia in rats. Neither methylamine (MA) nor N,N′-dimethylurea (DMU), the hydrolysis products of MIC, administered in equimolar doses had any influence on these parameters except for a marginal transient increase in plasma urea by DMU. Methyl isocyanate administration led to haemoconcentration, resulting in an increase in the plasma concentration of total proteins and a decrease in both the plasma concentration of albumin and the plasma cholinesterase activity. The hydrolysis products of MIC had no influence on any of these parameters. Thus, it seems reasonable to suggest that the systemic effects of MIC are caused by MIC per se, in spite of its high hydrolytic instability.

Thermal degradation products of poly(styrenesulfides) investigated by direct pyrolysis�mass spectrometry and flash pyrolysis�gas chromatography/mass spectrometry

The thermal degradation products of two sulfur polymers, poly(styrenedisulfide) (PSD) and poly(styrenetetrasulfide) (PST), were investigated in parallel by direct pyrolysis-mass spectrometry (DPMS) and by flash pyrolysis-GC/MS (Py-GC/MS). The time-scale of the two pyrolysis techniques is quite different, and therefore they were able to detect significantly different products in the pyrolysis of PSD and PST because of the thermal lability of sulfur-containing compounds. However, the results obtained are not contradictory, and satisfactory mechanisms for the thermal degradation of PSD and PST have been derived from the overall evidence available. Pyrolysis compounds containing sulfur, styrene, and a number of cyclic styrene sulfides and diphenyldithianes have been observed by DPMS. However, in flash pyrolysis-GC/MS, styrene, sulfur, only one cyclic styrene sulfide, and two isomers of diphenylthiophene have been detected. These thiophene derivatives were indeed absent among the compounds obtained by DPMS because they were the terminal (most thermally stable) species arising from further decomposition of the cyclic styrene sulfides formed in the primary thermal degradation processes of PSD and PST.

Primary pyrolysis products of thiokol liquid polysulfide

An analysis of the primary degradation products of the widely used commercial polysulfide polymer Thiokol LP-33 by direct pyrolysis-mass spectrometry (DP-MS) is reported. The mechanism of degradation is through a radical process involving the random cleavage of a formal C-O bond followed by backbiting to form the cyclic products.

Evaluation of Precipitation Retrievals From Orbital Data Products of TRMM Over a Subtropical Basin in India

The spatial error structure of daily precipitation derived from the latest version 7 (v7) tropical rainfall measuring mission (TRMM) level 2 data products are studied through comparison with the Asian precipitation highly resolved observational data integration toward evaluation of the water resources (APHRODITE) data over a subtropical region of the Indian subcontinent for the seasonal rainfall over 6 years from June 2002 to September 2007. The data products examined include v7 data from the TRMM radiometer Microwave Imager (TMI) and radar precipitation radar (PR), namely, 2A12, 2A25, and 2B31 (combined data from PR and TMI). The spatial distribution of uncertainty from these data products were quantified based on performance metrics derived from the contingency table. For the seasonal daily precipitation over a subtropical basin in India, the data product of 2A12 showed greater skill in detecting and quantifying the volume of rainfall when compared with the 2A25 and 2B31 data products. Error characterization using various error models revealed that random errors from multiplicative error models were homoscedastic and that they better represented rainfall estimates from 2A12 algorithm. Error decomposition techniques performed to disentangle systematic and random errors verify that the multiplicative error model representing rainfall from 2A12 algorithm successfully estimated a greater percentage of systematic error than 2A25 or 2B31 algorithms. Results verify that although the radiometer derived 2A12 rainfall data is known to suffer from many sources of uncertainties, spatial analysis over the case study region of India testifies that the 2A12 rainfall estimates are in a very good agreement with the reference estimates for the data period considered.

Determinantal point processes in the plane from products of random matrices

We show that the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of k independent n x n matrices with i.i.d. complex Gaussian entries with a few of matrices being inverted. In second example we calculate the same for (compatible) product of rectangular matrices with i.i.d. Gaussian entries and in last example we calculate for product of independent truncated unitary random matrices. We derive exact expressions for limiting expected empirical spectral distributions of above mentioned ensembles.

On a rational criterion for assigning batch number to processed products of shrimps

Investigations were conducted to find the variability of processed shrimps with respect to two quality characteristics, namely, the numbers of deteriorated and discoloured pieces. Samples were collected for three days from two arbitrarily selected processing factories from Cochin at the pre-freezing stages. Results show that both the quality characteristics vary significantly between different size-grades, but while the variation in the number of deteriorated pieces between days is not significant, the variation in the number of discoloured pieces between days is significant.

Quality aspects of some exportable dried fishery products of Bangladesh

A study was conducted to evaluate the quality aspects of eight exportable dried fishery products of Bangladesh. The products were evaluated by examining organoleptic properties, water reconstitution behavior, microbiological and biochemical aspects. The water reconstitution rate was faster in ribbon fish and Bombay duck, and slower in other fishes and air bladder and shark fin products. Organoleptic and physical characteristics in respect of color, odor, texture, insect infestation and broken pieces of the products revealed that all of the products were either in excellent or acceptable condition. Total viable bacterial load ranged from 0.95 x 104 CFU/g to 1.8 x 104 CFU/g in dried products. Coliform bacteria was absent in all the dried fishery products. The results of moisture, crude protein, lipid and ash content of the products ranged from 16.20 to 23.51%, 58.37 to 82.86%, 0.17 to 14.44% and 0.58 to 9.23%, respectively. Non-protein nitrogen (NPN) contents were in the range of 0.104 to 2.372% and the pepsin digestibility values were of 88.41 to 94.23%. The results of the study suggested that the exportable dried products were of good quality and hygienically safe.

Flavour components of some processed. fish and fishery products of Japan

A study was conducted to examine the flavour components of some processed fish and fishery products of Japan by gas chromatography-mass spectrometry (GC-MS). In brief the method was to absorb the headspace volatiles at 70°C into the fused silica fibre of needle of the solid phase micro extraction fibre. The absorbed components were injected to the GC-MS. The components were identified by computer matching with library database as well as by authentic standard components. In general the number of flavour components were higher in the processed fish and fishery products (except frozen prawn) than that of the raw fish and prawn. The concentration (quantity) of the f1avour components in processed fish and fishery products was much higher than that of the raw fish and prawn. Smoked salmon and baked salmon possessed double number of flavour components than that of the raw salmon. Smoking resulted the highest number of flavour components followed by baking (grilling) and canning, surimi products (kamaboko and chikuwa), drying and lastly salting. However, freezing and frozen storage resulted loss of flavour components in prawn.