Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

Cone complementarity problems with finite solution sets

We introduce the notion of a complementary cone and a nondegenerate linear transformation and characterize the finiteness of the solution set of a linear complementarity problem over a closed convex cone in a finite dimensional real inner product space. In addition to the above, other geometrical properties of complementary cones have been explored.     

Study of fiber optic sugar sensor

Over the last two decades, the fiber optic technology has passed through many analytical stages. Some commercially available fiber optic sensors, though in a small way, are being used for automation in mechanical and industrial environments. They are also used for instrumentation and controls. In the present work, an intensity-modulated intrinsic fiber optic sugar sensor is presented. This type of sensor, with slight modification, can be used for on-line determination of the concentration of sugar content in sugarcane juice in sugar industry. In the present set-up, a plastic fiber made of polymethylmethacrylate is used. A portion of the cladding (1 cm, 2 cm, 3 cm) at the mid-point along the length of the fiber is removed. This portion is immersed in sugar solution of known concentration and refractive index. At one end of the fiber an 850 nm source is used and at the other end a power meter is connected. By varying the concentration of sugar solution, the output power is noted. These studies are made due to the change in refractive index of the fluid. The device was found to be very sensitive which is free from EMI and shock hazards, stable and repeatable and they can be remotely interfaced with a computer to give on-line measurements and thus become useful for application in sugar industries.     

Catalytic conversion of 2-propanol on La2MnZnO6

The perovskite La₂MnZnO₆ has been synthesized by a ceramic technique and its catalytic activity has been tested for 2-propanol decomposition. The catalyst is totally selective to dehydrogenation of the alcohol. A possible mechanism for the reaction is proposed.

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La₂MnZnO₆ 2-. . .

     

Bismuth compounds in organic synthesis. A one-pot synthesis of homoallyl ethers and homoallyl acetates from aldehydes catalyzed by bismuth triflate

[reaction: see text] Three one-pot methods for the conversion of aldehydes to homoallyl ethers catalyzed by Bi(OTf)(3).xH(2)O (1 < x < 4) have been developed. The one-pot synthesis of homoallyl ethers can be achieved either by in situ generation of the acetal followed by its reaction with allyltrialkylsilane or by a three-component synthesis in which the aldehyde, trimethylorthoformate or an alkoxytrimethylsilane and allyltrimethylsilane are mixed together in the presence of bismuth triflate (0.1-1.0 mol %). In addition, a three-component synthesis of homoallyl acetates, which is achieved by reacting the aldehyde, acetic anhydride, and allyltrimethylsilane in the presence of bismuth triflate (3.0-5.0 mol %), has been developed. The use of a relatively nontoxic, easy to handle, and inexpensive catalyst adds to the versatility of these methods.     

Embeddings of anisotropic Sobolev spaces on unbounded domains

Summary Integral representations of junctions in the anisotropic Sobolev spaces on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain,