We investigate the death of entanglement and the purity loss of a two qubits–field system in the dispersive regime with a reservoir. For an alternative entanglement measure, we calculate the negativity of the eigenvalues of a partially transposed density matrix and compare it with the mutual entropy. A new measure related to the mutual entropy, namely, the index of entropy, is proposed to measure the degree of entanglement, and this agrees well with the negativity. We found that the entanglement has a strong sensitivity to the phase damping. The asymptotic behavior of the field states, the two qubits, and the total system fall into a mixed state. We treat the phenomena of death of entanglement and purity as they arise from the effect of phase damping.

Quantumness of the correlations between two qubits, coupled to a cavity field whenever the system is open or closed, is investigated when sudden death (or birth) of the entanglement occurs. It is found that quantum correlation not exists only in the entangled state. It’s found that the dephasing parameters and the purity of the initial states play an important role in the dynamics behaviors of the quantum correlations, including entanglement. Quantumness of the correlations and entanglement, due to dephasing of the cavity, are damped until the same their values which are contained in their initial states.

The analytical description of a superconducting qubit strongly coupled to a resonator with a thermal reservoir is given. Link between nonlocality and entanglement is investigated, they can be manipulated and controlled by any change in the thermal reservoir parameters. According to the Peres conjecture: if the state has negative partial transposition, then the violation of Bell's inequality occurs. Under the effect of thermal noise, the negative partial transposition is transformed to positive. Therefore, the state shows the phenomenon of sudden death of both entanglement and nonlocality.

The quantum correlation (QC) of two qubits coupled to a thermal field is investigated based on measurement-induced disturbance. It is found that QC does not exist only in the entangled state. Due to the thermal parameter, QC is inhibited from vanishing and the phenomenon of the death and birth of entanglement occurs. Due to the field-mode structure, QC is equal to zero for separated qubits, whereas for a maximally entangled state, it has a periodic asymptotic behavior which does not vanish. QC and entanglement differ when the qubit state is initially in the separated or entangled state.

This paper mainly studies the singularities of Gauss Map of pedal hypersurface in R n +1 . It contains the geometry of pedal hypersurfaces in R n +1 and their Gauss maps. The singularity of Gauss map of the pedal hypersurface using the rank of jacobian matrix of Gauss map is given and classified. The sets of singularities and its graphs under the Gauss map are plotted.

This paper mainly studies the Singularities of smooth mapping. The singularities of the families of Gauss maps corresponding to the family of mappings are studied and the shape of these families and their singularities using mathematica program are illustrated and plotted.

We present algorithms for computing the differential geometry properties of Frenet apparatus (t,n,b1,b2,κ1,κ2,κ3) of intersectioncurves of implicit–parametric–parametric and implicit–implicit–parametric hypersurfaces in R4, for transversal intersection. Some examples are given and plotted.

A brachiopod fauna of late Bathonian age recovered from the Kehailia Formation from Gebel Engabashi in northern Sinai consists of six species (two rhynchonellids and four terebratulids) referred to six genera, of which one genus and two species are new: Globirhynchia sphaerica (Cooper, 1989) new combination, Daghanirhynchia angulocostata Cooper, 1989, Ectyphoria sinaiensis new species, Cooperithyris circularis new genus and species, and new material: Avonothyris species A, and Ptyctothyris species A. The brachiopods described herein comprise a fauna located at the northern part of the Indo-African Faunal Realm within the Jurassic Ethiopian Province. They extend the The geographic distribution of those taxa that show great affinity with the Jurassic brachiopod fauna of Saudi Arabia described by cooper (1989). Differentiation of the endemic faunas that is so characteristic of many of these Ethiopian province faunas is becoming more well-defined

Thermolysis of N-arylnicotinamide oximes 1a–c (R = H, CH3 and Cl) under nitrogen gives rise to benzimidazoles 3a–c as the major products (60–62%), in addition to nicotinonitrile 4, arylamines 5a–c, nicotinic acid 6, phenols 7a–c and 8a–c, nicotinanilides 9a–c, 2-(pyridin-3-yl) benzoxazoles 10a and carbazoles 11a–c. In the presence of naphthalene, thermolysis of 1a gave - and -naphthols 12 and 13 beside the previous products. Also pyrolysis of 1a in boiling tetralin lead to the formation of 1-hydroxytetralin 14, -tetralone 15 and 1,1-bitetralyl 16 as the major products. The isolated products have been interpreted in the terms of a free radical mechanism involving the homolysis of N–O and/or C–N bonds.

Thermolysis of N-phenylbenzamide oximes I, II and III (R = Cl, NO2 and OCH3) under nitrogen gives rise to benzimidazoles as the major products (45–52%), in addition to benzonitrile, arylamines, benzoic acid, phenols, benzanilides, 2-phenyl benzoxazoles and carbazoles. In the presence of naphthalene, I gave - and -naphthols beside the previous products. Also heating of I under reflux boiling tetralin lead to the formation of 1-hydroxytetralin, -tetralone and 1,1-bitetralyl as the major products. The isolated products have been interpreted in the terms of a free radical mechanism involving the homolysis of N–O and/or C–N bonds.