Class Numbers of CM-Fields with Solvable Normal Closure

Stark conjectured that given a positive integer h, there are only a finite number of CM-fields L with class number equal to h. We prove this conjecture for fields L of degree 6 whose normal closure is solvable.     

Combined forced and free convective flow of water at 4°C past a semi-infinite vertical plate

Effects of buoyancy forces on forced and free convective flow of water at 4°C past a semi-infinite vertical plate at constant temperature are studied. Flow is assumed to be vertically upwards. Similarity solutions are derived and the resulting equations are solved numerically on a computer. Velocity and temperature profiles are shown graphically and numerical values of the skin friction and the rate of heat transfer are entered in tables. It is observed that the skin friction and the Nusselt number increase with increasing Gr/Re², where Gr is the Grashof number and Re is the Reynolds number     

Fourier coefficients of forms of CM-type

Let f be a cuspidal normalized eigenform of weight ≥ 2 for Г₀(N),with Fourier expansion $f(z) = \sum\limits_{n = 1}^\infty {a_f (n)e^{2\pi inz} } $ While the Galois representations associated to f can be used effectively to study the divisibility properties of the Fourier coefficients, it is very difficult to analyze the condition a _f (p) = 0 (mod p). In this paper, we show that the problem is accessible in the case that f has complex multiplication. Under some mild conditions on f, we show that for p sufficiently large, a _f (p) = 0 (mod p) in fact implies that a _f (p) = 0.     

Magnetically Recyclable Nano-Fe2O3-Catalyzed Chemoselective Synthesis and Antioxidant Activity of Diethyl (3-((5-Aryl-1H-1,2,4-triazol-3-yl)thio)propyl)phosphonates

An efficient, green, and chemoselective S-alkylation of 5-aryl-1H-1,2,4-triazole-3-thiones with diethyl (3-bromopropyl)phosphonate in water, catalyzed by nano-Fe₂O₃ under ligand- and base-free conditions, is reported. Clean reaction, less expensive catalyst, excellent yields, and easy workup are the advantages of the present method. The catalyst can be easily collected by a magnet and recycled without significant loss in catalytic activity. The newly synthesized compounds were screened for their antioxidant property by using 1,1-diphenyl-2-picrylhydrazyl (DPPH) scavenging assay. The majority of the compounds exhibited good antioxidant activity.     

Synthesis and Biological Activity of a Novel Pentacyclic Heterocycle

A novel pentacyclic heterocycle has been synthesized starting from D ‐glucose involving two crucial conversions: the intramolecular cycloaddition of O‐allyloxy furanaldoxime derived from D ‐glucose to furanopyran‐2‐isoxazoline and its diastereoselective reductive cleavage to the corresponding cis‐1,3‐amino alcohol. The antibacterial and antifungal activities of the synthesized heterocycle have been examined.     

On amorphization and nanocomposite formation in Al-Ni-Ti system by mechanical alloying

Amorphous structure generated by mechanical alloying (MA) is often used as a precursor for generating nanocomposites through controlled devitrification. The amorphous forming composition range of ternary Al-Ni-Ti system was calculated using the extended Miedema’s semi-empirical model. Eleven compositions of this system showing a wide range of negative enthalpy of mixing (−ΔH ^mix) and amorphization (−ΔH ^amor) of the constituent elements were selected for synthesis by MA. The Al₈₈Ni₆Ti₆ alloy with relatively small negative ΔH ^mix (−0.4 kJ/mol) and ΔH^amor (−14.8 kJ/mol) became completely amorphous after 120 h of milling, which is possibly the first report of complete amorphization of an Al-based rare earth element free Al-TM-TM system (TM = transition metal) by MA. The alloys of other compositions selected had much more negative ΔH^mix and H^amor; but they yielded either nanocomposites of partial amorphous and crystalline structure or no amorphous phase at all in the as-milled condition, evidencing a high degree of stability of the intermetallic phases under the MA environment. Hence, the negative ΔH ^mix and ΔH ^amor are not so reliable for predicting the amorphization in the present system by MA     

Holographic grating based Ebert spectrograph

In this paper we discuss the aberration properties and design procedure of in-plane Ebert spectrograph using conventional as well as holographic diffraction gratings. In both cases the gratings are situated at the well-known √3-position so that the spectrum can be recorded on a flat surface. It has been found that the holographic grating system has better resolution than the conventional grating system. The design parameters of a medium sized holographic grating spectrograph in which a concave spherical mirror is mounted in the off-axis configuration have been specified. The performance of the spectrograph has been evaluated by plotting spot diagram.     

Crystalline constituents from euphorbiaceae—XIII : The structure of a new triterpene from Euphorbia nerifolia L.

From the leaves and stems of Euphorbia nerifolia Linn., a new triterpene was isolated besides friedelan-3α- andβ-ols, and taraxerol. From a study of its chemical reactions and spectral data, the new triterpene was shown to be glut-5(10)-en-1-one(3).     

Computational complexity of parametric linear programming

We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646.