Bidirectional Controlled Teleportation by Using 5-Qubit States: A Generalized View

Recently bidirectional controlled perfect teleportation using 5-qubit states are reported in Int. J. Theor. Phys. (2013), doi:10.1007/s10773-013-1484-8 and ibid (2012), doi:10.1007/s10773-012-1208-5. In this paper we have shown that there exists a class of 5-qubit quantum states that can be used for bidirectional controlled teleportation. Two out of the three reported cases are the special cases of the proposed class of 5-qubit quantum states and one of them is not strictly a case of controlled bidirectional quantum teleportation. Further, we have shown that one can in principle, construct infinitely many 5-qubit quantum states for this purpose. We have also shown that the idea can be extended to bidirectional controlled probabilistic teleportation. Some potential applications of the proposed scheme and its modified versions are also discussed in relation with the implementation of quantum remote control and quantum cryptography.     

The Pseudo-differential Operator hμ,a on Hankel Invariant Spaces

Using Hankel transform the symbol 'a' is defined and the pseudo-differential operator (p.d.o.) h_μ,a associated with the Bessel operator d ²/dx ² + (1 ? 4μ ²)/4x ² in terms of this symbol is defined. It is shown that the operator h_μ,a is a continuous linear map of a Hankel invariant space into itself. A special pseudo-differential operator called the Hankel potential is defined and some of its properties are investigated.     

Pseudo-differential operators involving Watson transform

A class of pseudo-differential operators (p.d.o.'s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class T ^m is introduced. It is shown that the p.d.o.'s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.'s and their products are bounded in certain Sobolev type space.     

Geometrical Model for Phenylthallium Dihalides

The geometry of the phenylthallium dihalides, PhTlX₂(X=F, Cl or Br) which has heitherto been unknown, has been proposed on the basis of several qualitative aspects and quantitative arguments. These molecules have a planar structure with C_2v symmetry and the ring geometry has been assumed to be the same as that of benzene. The ring-substituent bond length has been taken to be 2.218 A° in all the three molecules, and the Tl-F, Tl-Cl and Tl-Br bond lengths in respective molecules have been estimated to be 1.878, 2.366 and 2.520 A°. The remaining parameter inter-bond angle, F-Tl-F has been taken to be 116° in PhTlF₂, whereas, angles Cl-Tl-Cl and Br-Tl-Br in respective molecules have been taken to be 120°, as expected in the case of a pure sp² hybridization.     

Subweakly Biased Pairs and Jungck Contractions with Applications

The existence of common fixed point results for a weakly biased pair under Jungck contraction is established. We define a new class of subweakly biased pair which properly contains the classes of C _q -commuting, pointwise R-subweakly commuting and R-subweakly commuting maps and establish common fixed point results for this class of maps. As applications several invariant best approximation results are proved which unify, extend, and complement the well-known results.     

Coordination and Hydroboration of Ru(II)-Borate Complexes: Dihydridoborate vs. Bis(dihydridoborate)

Treatment of [Cp*RuCl₂]₂, 1 , [(COD)IrCl]₂, 2 or [(p-cymene)RuCl₂]_2, 3 (Cp*=η⁵-C₅Me_5, COD= 1,5-cyclooctadiene and p-cymene=η⁶-ⁱPrC₆H₄Me) with heterocyclic borate ligands [Na[(H₃B)L], L₁ and L₂ ( L₁ : L=amt, L₂ : L=mp; amt=2-amino-5-mercapto-1,3,4-thiadiazole, mp=2-mercaptopyridine) led to the formation of borate complexes having uncommon coordination. For example, complexes 1 and 2 on reaction with L₁ and L₂ afforded dihydridoborate species [L^AM(μ-H)₂BHL] 4 – 6 ( 4 : L^A=Cp*, M=Ru, L=amt; 5 : L^A=Cp*, M=Ru, L=mp; 6 : L^A=COD, M=Ir, L=mp). On the other hand, treatment of 3 with L₂ yielded cis- and trans-bis(dihydridoborate) species, [Ru{(μ-H)₂BH(mp)}₂], cis- 7 and trans- 7 . The isolation and structural characterization of fac- and mer-[Ru{(μ-H)₂BH(mp)}{(μ-H)BH(mp)₂}], 8 from the same reaction offered an insight into the behaviour of these dihydridoborate species in solution. Fascinatingly, despite having reduced natural charges on Ru centres both at cis-and trans- 7 , they underwent hydroboration reaction with alkynes that yielded both Markovnikov and anti-Markovnikov addition products, 10 a – d .