Congruence Criteria for Finite Subsets of Quaternionic Elliptic and Quaternionic Hyperbolic Spaces

We investigate congruence classes of m-tuples of points in the quaternionic elliptic space ?P ⁿ . We establish a canonical bijection between the set of congruence classes of m-tuples of points in ?P ⁿ and the set of equivalence classes of positive semidefinite Hermitian m×m matrices of rank at most n+1 with the 1's on the diagonal. We show that with each m-tuple of points in ?P ⁿ is associated a tuple of points on the real unit sphere S ². Then we get that the congruence class of an m-tuple of points in ?P ⁿ is determined by the congruence classes of all its triangles and by the direct congruence class of the associated tuple on the sphere S ² provided that no pair of points of the m-tuple has distance π/2. Finally we carry out the same kind of investigation for the quaternionic hyperbolic space ?H ⁿ . Most of the results are completely analogous, although there are also some interesting differences.     

Stabilization of the rotating disk-beam system with a delay term in boundary feedback

In this article, the stabilization problem of a rotating disk-beam system is addressed. It is assumed that the flexible beam is free at one end, whereas the other end is attached to the center of the rotating disk whose angular velocity is time-varying. The proposed feedback law consists of a torque control which acts on the disk, whereas a delayed boundary force control is exerted at the free end of the beam. Thereafter, it is proved that the presence of such controls in the system guarantees the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk as well as the feedback gain in the delay term. This result is illustrated by numerical examples.     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.     

Synthesis of palladium-bidentate complex and its application in Sonogashira and Suzuki coupling reactions

A new palladium-bidentate complex [m-C₆H₄(CH₂ImMe)₂(PdCl₂)] (III) was prepared in two steps. In the first step, [m-C₆H₄(CH₂ImMeAgCl)₂] (II) (Im = imidazole moiety) was obtained by reacting imidazolium salt [m-C₆H₄(CH₂ImMe)₂]Cl₂ (I) (prepared by quaternisation of Nmethylimidazole with 1,3-bis(chloromethyl)benzene) and Ag₂O in CH₂Cl₂. In the next step, treatment of (II) with Pd(CH₃CN)₂Cl₂ afforded complex III which was evaluated for its catalytic activity for C-C bond-formation reactions by examining the coupling reaction of 3-iodoanisole with phenylacetylene in the Sonogashira reaction. In addition, 3-methoxybiphenyls were obtained with good to excellent yields by Suzuki coupling reactions of 3-iodoanisole with phenylboronic acids or phenylborates salts in the presence of this complex.     

Production of γ by 194 GeV/c negative pions on tungsten

On the basis of 2000 ?, ?′ and ?″ events obtained in π^?-W interactions at 194 GeV/c we extract a value for the cross section times the branching ratio of B_μμσ = (0.96±0.04±0.17) pb nucleon^?1. The $\text{(?′ + ?″)}\text{?}$ ratio is found to be 0.53 ± 0.19. The observed differential x_F and P_T distributions are compared with those of the Drell-Yan continuum, and the cross section and the x_F distribution are compared with theoretical predictions.     

On p-tuples of equi-isoclinic 3-spaces in the Euclidean space

In this paper we show on the one hand that ?⁸ contains two congruence classes of sextuples of equi-isoclinic 3-subspaces, whereas no such 7-tuple exists and on the other hand, we give the list of all regular p-tuples of equi-isoclinic 3-subspaces.     

Optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a Hardy potential

We deal with the following parabolic problem