The output of a quartz transducer oscillating in its fundamental and higher harmonics

The output from a quartz crystal oscillating in three liquids, at its fundamental, 3rd harmonic and 5th harmonic have been measured by the calorimetric method keeping the crystal amplitude the same. The output is found to be much less than that one can expect from theoretical considerations.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

Pressure and temperature dependence of the electrical resistivity of bulk Al23 Te77 glass

Electrical resistivity of bulk amorphous Al₂₃T₇₇ samples has been studied as a function of pressure (up to 80 kbar) and temperature (down to 77 K). At atmospheric pressure the temperature dependence of resistivity obeys the relation ? = π₀ exp(δE/RT) with two activation energies. In the temperature range 300 K ? T > 234 K the activation energy is 0.58 eV and for 234 >T ? 185 K the value is δE = 0.30 ev. The activation energy has been measured as a function of pressure. The electrical resistivity decreases exponentially with the increase of pressure and at 70 kbar pressure the electrical behaviour of the sample shows a metallic nature with a positive temperature coefficient. The high pressure phase of the sample is found to be a crystalline hexagonal phase.     

Crystal structure,absolute configuration and stereochemistry of nupharidine hydrobromide

An X-ray diffraction study of nupharidine hydrobromide has been carried out in order to obtain an unequivocal solution to the question of the N-oxide configuration. Nupharidine hydrobromide (C₁₅H₂₄BrNO₂) crystals are orthorhombic, space groupP2₁2₁2₁ withz = 4. The unit cell constants at 22 ± 3 ° area = 13·419(2),b = 13·668(4),c = 8·731(2) Å (CuK₁, = 1·54051 Å). Three-dimensional intensity data, to the limit 2 = 145 ° for CuK₁, were measured on a GE XRD-490 automatic diffractometer equipped with Ni-Co balanced filters for monochromatization. The crystal structure was solved by the heavy-atom method and refined by least squares to anR of 0·06 for 1960 reflections.X-ray analysis clearly established that (a) the quinolizidine N-oxide system is built of twocis-fused chairs, (b) the methyl and the furan substituents occupy the equatorial position and (c) the absolute configuration indicated by the chemical methods is in agreement with the X-ray work. The stereochemical aspects of the Polonovski elimination reaction in the conversion of (+) nupharidine to ⁶-dehydrodeoxynupharidine has been studied in the light of these X-ray results.     

Spectrophotometric,Thermodynamic and Density Functional Studies of Charge Transfer Complex Between Benzhydryl Piperazine and 2,3-Dichloro-5,6-dicyano-1,4-benzoquinone

The charge transfer complex of benzhydryl piperazine as donor with the π-acceptor 2,3-dichloro-5,6-dicyano-p-benzoquinone has been studied spectrophotometrically in acetonitrile medium at different temperatures. On mixing the donor with acceptor, a reddish brown colored charge transfer complex is formed. Electronic absorption spectra of the complex show charge transfer bands at 587, 546 and 457 nm. The molecular composition of the complex was studied by applying Job’s continuous variation and spectrophotometric titration methods. These results support the formation of the complex in a 1:2 ratio. The Benesi–Hildebrand equation has been applied to compute the formation constant and molecular extinction coefficient. Thermodynamic parameters of the charge transfer complexation reaction (standard entropy, standard enthalpy and standard Gibbs free energy) have been calculated. The results of the spectrophotometric study demonstrated that the charge transfer complex formation is endothermic. The computational studies of the charge transfer complex were performed by using the Gaussian 09 W package of programs. The bond lengths, bond angles, dihedral angles, Mulliken atomic charges, molecular electrostatic potential maps and characterization of the highest occupied molecular orbital and lowest unoccupied molecular orbital surfaces of charge transfer complex were computed.     

Fully copositive matrices

The class of fully copositive (C ₀ ^f ) matrices introduced in [G.S.R. Murthy, T. Parthasarathy, SIAM Journal on Matrix Analysis and Applications 16 (4) (1995) 1268–1286] is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class ofQ ₀-matrices areP ₀-matrices. As a corollary of this main result, we establish that a bisymmetricQ ₀-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization ofQ ₀-matrices within the class ofC ₀ ^f . While establishing this characterization, it will be shown that Graves's principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable toC ₀ ^f Q ₀ class. As a byproduct of this characterization, we observe that aC ₀ ^f -matrix is inQ ₀ if, and only if, it is completelyQ ₀. Also, from Aganagic and Cottle's [M. Aganagic, R.W. Cottle, Mathematical Programming 37 (1987) 223–231] result, it is observed that LCPs arising fromC ₀ ^f Q ₀ class can be processed by Lemke's algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

A new method for constructing factorisable representations for current groups and current algebras

LetC _e (R ⁿ ,G) denote the group of infinitely differentiable maps fromn-dimensional Euclidean space into a simply connected and connected Lie group, which have compact support. This paper introduces a class of factorisable unitary representations ofC _e (R ⁿ ,G) with the property that the unitary operatorU _f corresponding to a functionf inC _e (R ⁿ ,G) depends not only onf, but also on the derivatives off up to a certain order. In particular these representations can not be extended to the group of all continuous functions fromR ⁿ toG with compact support.     

Exact transient solutions of kinetics of first‐order reactions with end effects

The finite set of rate equations C _m,n ^' =α _n,n-1 C _m,n-1 (t)+α _n,n C _m,n (t)+α _n,n+1 C _m,n+1 (t), $$0 \leqslant m \leqslant N,0 \leqslant n \leqslant N,$$ where $$\alpha _{i,j}$$ are $\alpha _{j,j - 1} = A,\alpha _{j,j} = - \left( {A + B} \right),\alpha _{j,j + 1} = B$ , with $\alpha _{0,0} = - \alpha _{1,0} = - \alpha$ and $\alpha _{N,N} = - \alpha _{N - 1,N} = - b,\alpha _{0, - 1} = \alpha _{N,N + 1} = 0$ , subject to the initial condition $C_{m,n} \left( 0 \right) = \delta _{n,m}$ (Kronecker delta) for some $m$ , arises in a number of applications of mathematics and mathematical physics. We show that there are five sets of values of $a$ and $b$ for which the above system admits exact transient solutions.