Ammonium hexanitratocerate(IV) as an oxidizing agent-V Titrimetric determination of arsenic(III) at room temperature

Conditions have been established for the accurate titration of arsenic(III) with ammonium hexanitratocerate(IV) at room temperature in sulphuric, nitric and hydrochloric acid media, using suitable catalysts, and ferroin as indicator.     

Synthesis of some fused triazoloquinolines

The reaction of 3 -amino-1,2,4-triazolo[4,3-a]quinoline ( II ) with diethyl ethoxymethylenemalonate and ethyl acetoacetate/ethyl trifluoroacetoacetate afforded 10-carboethoxy-9-oxo-9H-pyrimido[1′,2′:1,5][1,2,4]triazolo-[4,3-a]quinoline ( III ) and 11-methyl/trifluoromethyl-9-oxo-9H-pyrimido[1′,2′:1,5][1,2,4]triazolo[4,3-a]quinoline ( IV/V ) respectively. 2-Chloropyridine-3-carboxylic acid chloride reacted with II to yield 5-oxo-5H-pyrido-[3″,2″:5′,6′]pyrimido[1′,2′:1,5][1,2,4]triazolo[4,3-a]quinoline ( VII ), a new ring system.     

Non-linear dynamic analysis of rotors by finite element method

Non-linear natural vibration characteristics and the dynamic response of hingeless and fully articulated rotors of rectangular cross-section are studied by using the finite element method. In the formulation of response problems, the global variables are augmented with appropriate additional variables, facilitating direct determination of sub-harmonic response. Numerical results are given showing the effect of the geometric non-linearity on the first three natural frequencies. Response analysis of typical rotors indicates a possibility of substantial sub-harmonic response especially in the fully articulated rotors widely adopted in helicopters.     

A catadioptric magnifier with two magnifications for microfiche reading

We have considered the possibility of using four internal reflections in a single lens to reduce its effective focal length by a very large factor. Hence, by using very shallow curvatures on the lens surfaces, it is possible to obtain very high visual magnification for viewing transmission objects such as microfiche. Some details of such a system are presented here, where a lower magnification is obtained for two internal reflections and a higher magnification for four internal reflections.     

Convergence of the steepest descent method for minimizing quasiconvex functions

To minimize a continuously differentiable quasiconvex functionf: ⁿ , Armijo's steepest descent method generates a sequencex ^k+1 =x ^k –t _k f(x ^k ), wheret _k >0. We establish strong convergence properties of this classic method: either , s.t. ; or arg minf = , x ^k andf(x ^k ) inff. We also discuss extensions to other line searches.The research of the first author was supported by the Polish Academy of Sciences. The second author acknowledges the support of the Department of Industrial Engineering, Hong Kong University of Science and Technology.We wish to thank two anonymous referees for their valuable comments. In particular, one referee has suggested the use of quasiconvexity instead of convexity off.     

First order difference system- existence and uniqueness

In this paper, the general solution of the homogeneous matrix difference system is constructed in terms of two fundamental matrix solutions. The general solution of the inhomogeneous matrix difference system is established by the variation of parameters formula. A unique solution of the two-point boundary value problem associated with the matrix difference system is constructed by applying the QR-algorithm and the Bartels-Stewart algorithm.

     

Segments in enumerating faces

We introduce the concept of a segment of a degenerate convex polytope specified by a system of linear constraints, and explain its importance in developing algorithms for enumerating the faces. Using segments, we describe an algorithm that enumerates all the faces, in time polynomial in their number. The role of segments in the unsolved problem of enumerating the extreme points of a convex polytope specified by a degenerate system of linear constraints, in time polynomial in the number of extreme points, is discussed.Work carried out while on sabbatical leave in the Industrial and Operations Engineering Department at the University of Michigan in Ann Arbor, USA.     

On the number of groups of a given order

Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)^c on a set of positive density. Letting F_k(x) denote the number of n ≤ x for which G(n) = k, it is shown that $\text{F}{}_2\text{(x) =}\text{O(x(}\text{log}{}_4\text{x)}\text{(}\text{log}{}_3\text{x)}{}^2\text{)}$, where log_rx denotes the r-fold iterated logarithm.