Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

     

Hearing the shape of a general doubly-connected domain inR 3 with mixed boundary conditions

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR ³ with mixed boundary conditions, from the complete knowledge of the eigenvalues { _n } _n=1 for the three-dimensional Laplacian, using the asymptotic expansion of the spectral function ast0.     

Spectrophotometric study of the reaction mechanism between DDQ Pi- and iodine sigma-acceptors and chloroquine and pyrimethamine drugs and their determination in pure and in dosage forms

Two simple and accurate spectrophotometric methods are presented for the determination of anti-malarial drugs, chloroquine phosphate (CQP) and pyrimethamine (PYM), in pure and in different pharmaceutical preparations. The charge transphere (CT) reactions between CQP and PYM as electron donors and 2,3-dichloro-5,6-dicyano-p-benzoquinone (DDQ) pi-acceptor and iodine sigma-acceptor reagents to give highly coloured complex species have been spectrophotometrically studied. The optimum experimental conditions have been studied carefully. Beer' law is obeyed over the concentration range of 1.0-15 microg ml(-1) for CQP and 1.0-40 microg ml(-1) for PYM using I(2) and at 5.0-53 microg ml(-1) for CQP and 1.0-46 microg ml(-1) for PYM using DDQ reagents, respectively. For more accurate results, Ringbom optimum concentration range is calculated and found to be 10-53 and 8-46 microg ml(-1) for CQP and PYM using DDQ, respectively and 5-15 and 8-40 microg ml(-1) for CQP and PYM using iodine, respectively. The Sandell sensitivity is found to be 0.038 and 0.046 g cm(-2) for DDQ method and 0.0078 and 0.056 g cm(-2) for I(2) method for CQP and PYM, respectively which indicates the high sensitivity of both methods. Standard deviation (S.D.=0.012-0.014 and 0.013-0.015) and relative standard deviation (R.S.D.=0.09-1.4 and 1.3-1.5%) (n=5) for DDQ and I(2) methods respectively, refer to the high accuracy and precision of the proposed methods. These results are also confirmed by between day precision of percent recovery of 99-100.6%, and 98-101% for CQP and PYM by DDQ method and 99-102% and 99.2-101.4% for CQP and PYM by I(2) method respectively. These data are comparable to those obtained by British and American pharmacopoeias assay for the determination of CQP and PYM in raw materials and in pharmaceutical preparations.     

Regioselectivity in Reactions of Bis‐Hydrazonoyl Halides with Some Bifunctional Heterocycles

Bis‐hydrazonoyl chloride 1 reacts regioselectively with 3‐mercapto‐1,2,4‐triazole 2a , 2,3‐dihydro‐3‐thioxo‐1,2,4‐triazin‐5(4H)‐one 2b and 2‐mercaptobenzimidazole 2c to give the hitherto unknown annelated 2,3‐bis‐(phenylhydrazono)thiazoles 6a‐c , respectively. Reactions of 1 with the methylthio derivatives of such heterocycles afforded the annelated 3,3′‐bis‐(1,2,4‐triazoles) 11a‐c , respectively. Similar reaction of 1 with 2‐phenylamino‐4(3H)‐pyrimidinones 4 gave 2,3‐bis(phenylhydrazono)imidazo[1,2‐a]pyrimidin‐5(1H)‐ones 16 . Oxidation of 6c yielded the corresponding bis(phenylazo) derivative 15 . The regiochemistry of the studied reactions is discussed.     

Mass spectra of gliclazide drug at various ion sources temperature

Gliclazide (GL, C₁₅H₂₁N₃O₃S) drug is used as non-insulin-dependant diabetes mellitus. The drug was investigated using thermal analysis (TA) measurements (TG/DTG) and electron impact mass spectral (EI–MS) fragmentation at 70 eV techniques. The mass spectra of GL at different values of ion source temperatures (400, 416, 425, and 440 K) are recorded and investigated. Semiempirical MO calculation, using PM3 procedure, has been carried out on neutral molecule and positively charged species. These calculations included bond length, bond order, bond strain, partial charge distribution, ionization energy, and heats of formation (ΔH _f). PM3 procedure provides a basis for fine distinction among sites of initial bond cleavage, which is crucial to the rationalization of subsequent fragmentation of the molecule. The primary fragmentation pathway in both TA and MS (at different values of ion source temperature) is initiated by S–N bond rupture. TA and DTG show one main weight loss at 250.38 °C and four peaks at 271.6, 360.99, 427.93 and 479.17 °C in DTA, which may be attributed to various fragments. Also, the rate constant (K′) of thermal degradation has been tested isothermally at 210 and 600 °C. The calculated rate values are 9.6 × 10⁻³ and 0.33 × 10⁻³ s⁻¹, respectively, and discussed. In MS, the effect of ion source temperature on mass spectral fragmentation processes is discussed on the basis of energy considerations using quasi equilibrium theory.     

Simultaneous determination of cinnarizine and nicergoline in a binary mixture using first derivative spectra,first derivative of ratio spectra and multivariate calibration techniques

Four methods are described for the simultaneous determination of cinnarizine (Cinn) and nicergoline (Nic). The first method is based on the first derivative (¹ D) ultraviolet spectrophotometry, with zero crossing and peak to base measurement. The first derivative amplitudes at 216.1 and 235.0 nm were selected for the assay of cinnarizine and nicergoline, respectively. The second method is based on the first derivative of the ratio-spectra (¹ DD) obtained by measurement of the amplitudes at 256 and 290.6 nm for cinnarizine and nicergoline, respectively. The other two chemometric methods applied were classical least squares (CLS) and inverse least squares (ILS). The calibration of these chemometric methods that involves absorbance and concentration data matrices is used for the prediction of unknown concentrations of Cinn and Nic. The numerical values were calculated by using Matlab R12 version 6.0 and Origin 5.0 software. The linear ranges are 1–40 and 1–46 μg/mL for cinnarizine and nicergoline, respectively, for all methods. The four methods were successfully applied to assaying the pharmaceutical formulations (Cinibral® tablets).     

Two charge-transfer complex spectrophotometric methods for the determination of sulpiride in pharmaceutical formulations

Two simple, rapid, accurate and sensitive spectrophotometric methods are described for the determination of sulpiride. They are based on charge transfer complexation between the drug as n-electron donor and p-chloranilic acid as π acceptor or iodine as σ-acceptor. These give highly coloured complexes with absorption maxima at 518 and 363, 294 nm, respectively. Beer’s law linear ranges were 13.7–341.4 and 1.7–20.5 µg mL^?1 for the p-chloranilic acid and iodine methods. The methods were successfully applied to the determination of the drug in Dogmatil® Fort tablets and the results compared with the official method. The complex association constants and standard free energy changes were calculated using Benesi-Hildebrand plots.      

Flow injection potentiometric determination of clobutinol hydrochloride

New clobutinol (Clob) ion-selective polyvinyl chloride (PVC) membrane electrodes, based on the ion-associates of Clob with phosphotungstic acid or phosphomolybdic acid were prepared using dibutyl phthalate as plasticizing solvent. The electrodes were characterized in terms of membrane composition, temperature and pH. The sensors showed a near-Nernstian response over the concentration ranges (6.31 × 10⁻⁶)-(1.00 × 10⁻²) and (5.01 × 10⁻⁵)-(1.00 × 10⁻²) M in the case of clobutinol-phosphotungstate ((Clob)₃-PT) applying batch and flow injection (FI) analysis, respectively, and (1.58 × 10⁻⁵)-(1.00 × 10⁻²) and (5.01 × 10⁻⁵)-(1.00 × 10⁻²) M in case of clobutinol-phosphomolybdate ((Clob)₃-PM) for batch and FI analysis systems, respectively. The electrodes were successfully applied for the potentiometric determination of ClobCl in pharmaceutical preparation and urine in steady state and flow injection conditions. The electrodes exhibit good selectivity for Clob with respect to a large number of inorganic cations, sugars and amino acids.     

A -analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem plays an important role not only in harmonic analysis and approximation theory, but also in communication engineering since it enables engineers to reconstruct analog signals from their samples at a discrete set of data points. The main aim of this paper is to derive a -analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem. The proof uses recent results in the theory of -orthogonal polynomials and basic hypergeometric functions, in particular, new results on the addition theorems for -exponential functions.