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.     

Theory of Differential Inequalities for Two-Point Boundary Value Problems and their Applications to Three-Point B.V.Ps Associated with n th Order Non-Linear System of Differential Equations

This paper presents a theory of differential inequalities for two-point boundary value problems (B.V.Ps) associated with the system of n th order non-linear differential equations. Using these inequalities as a tool we establish the existence and uniqueness of solutions to three-point B.V.Ps associated with the system of n th order non-linear differential equations by using the idea of matching solutions.     

Electrospray ionization mass spectral studies of N,N-dialkylaminoethane-2-sulphonic acids

Oxidative reactions of VX type compounds and N,N-dialkylaminoethane-2-thiols that are precursors for VX compounds produce N,N-dialkylaminoethane-2-sulphonic acids, N(R(1))(R(2))-CH(2)-CH(2)SO(3)H (where R(1) and R(2) = methyl, ethyl, n-propyl and isopropyl, 1-10), as the degradation products, and these degradation products are considered as markers for the detection of chemicals listed in the schedules of Chemical Weapons Convention (CWC) chemicals. Off-site detection of such degradation products in aqueous samples is an important task in the verification of CWC-related chemicals. Here we report a simple method involving the direct analysis of aqueous samples using positive and/or negative ion electrospray ionization (ESI) for the screening, detection and identification of N,N-dialkylaminoethane-2-sulphonic acids, avoiding sample preparation and chromatographic steps. The positive ion ESI mass spectra of all the compounds result in abundant [M+Na](+) ions, and the negative ion spectra show abundant [M-H](-) ions to confirm their molecular weight. The collision-induced dissociation spectra of [M+Na](+) and [M-H](-) give characteristic product ions by which it is easy to detect and identify all the studied N,N-dialkylaminoethane-2-sulphonic acids including those of isomeric compounds. The method is successfully applied to detect the spiked chemical, N,N-diisopropylaminoethane-2-sulphonic acid, present in a water sample received in a proficiency test.     

Counting integral ideals in a number field

Let K   be an algebraic number field. We discuss the problem of counting the number of integral ideals below a given norm and obtain effective error estimates. The approach is elementary and follows a classical line of argument of Dedekind and Weber. The novelty here is that explicit error estimates can be obtained by fine tuning this classical argument without too much difficulty. The error estimate is sufficiently strong to give the analytic continuation of the Dedekind zeta function to the left of the line R(s)=1$\mathfrak{R}(s)=1$ as well as explicit bounds for the residue of the zeta function at s=1$s=1$.     

Phase holdups in three-phase fluidized beds in the presence of disc promoter

Three-phase fluidized beds are found to have wide applications in process industries. The present investigation essentially comprises of the studies on gas holdup, liquid holdup and bed porosity in three-phase fluidized beds with coaxially placed disc promoter. Holdup data were obtained from bed expansion and pressure drop measurements. Analysis of the data was done to elucidate the effects of dynamic and geometric parameters on gas holdup, liquid holdup and bed porosity. Data were correlated and useful equations were obtained from empirical modeling.     

Spectroscopic studies on Fe and Mn doped SrB4O7 glasses

EPR and optical absorption studies on Fe³⁺ and Mn²⁺ doped strontium tetraborate (SrB₄O₇) glasses are carried out at room temperature. The EPR spectrum of the Fe³⁺ doped glass consists of signals with g-values 9.04, 4.22 and 2.04, whereas the EPR spectrum of Mn²⁺ doped glass exhibits a characteristic hyperfine sextet around g=2.0. The spectroscopic analyses of the obtained results confirmed distorted octahedral site symmetry for the Fe³⁺ and Mn²⁺ impurity ions. Crystal field and Racah parameters evaluated from optical absorption spectra are: Dq=790, B=700 and C=3000 cm⁻¹ for Fe³⁺doped glass and Dq=880, B=700 and C=2975 cm⁻¹ for Mn²⁺ doped glass.     

An efficient synthesis of 2-benzoxepines from Morita-Baylis-Hillman adducts using heterogeneous recyclable catalysts

2-Benzoxepines have efficiently been synthesized from Morita-Baylis-Hillman adducts, alkyl 3-aryl-3-hydroxy-2-methylenepropanoates by treatment with HCHO catalyzed by silica supported perchloric acid (HClO4.SiO2) or Amberlyst-15 in CH2Cl2 under reflux for a short period of time (1.5-2.5 h). The catalyst can be recovered and recycled. The antibacterial properties of the new 2-benzoxepines were studied but no activity was found.