A symmetry-adapted force-constant lattice-dynamical model for single-walled carbon nanotubes

We develop a lattice-dynamical model based on the screw symmetry of single-walled carbon nanotubes that allows for reducing the size of the dynamical matrix to six, for all tube chiralities. The model uses force constants derived from fitting to the phonon dispersion of 2D graphite. We present the calculation procedure in a clear and transparent way, making the model easier to follow. We calculate the phonon dispersions of a number of nanotubes of different chiralities. The splitting of two highest Raman active modes and the radial breathing mode frequency are studied by changing the tube diameter and chirality.     

Conservation Laws of K(m, n) and mK(m, n) Equations

Based on the rank analysis method, algorithmization idea, and symbolic computation, in this paper we have presented a method to construct the conservation laws for nonlinear evolution equations. The polynomial conservation laws for K (n 2, n) equations and mnK(m, n) equations are found by using of this approach and some new results have been obtained.     

Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

Thermodynamic properties of the spin-Peierls transition in CuGeO3

We present an investigation of the spin-Peierls transition atT _SP=14.5 K in polycrystalline CuGeO₃ through specific-heat and thermal-expansion measurements. Clear second-order phase-transition anomalies are found in both properties atT _SP, although only a small entropy of S0.1 Rn2 is released at the transition. Most of the entropy is released atT _SP<T<150 K, where the temperature dependence of the magnetic contribution to the specific heat as well as the thermal expansion exhibit extrema atT ^*40 K. These are caused by one-dimensional antiferromagnetic fluctuations along the Cu chains, possibly accompanied by structural fluctuations. Using Ehrenfest's relation, a hydrostatic pressure coefficient (T _SP/_p)_p0 (0.45±0.06) K/kbar is derived.     

Perturbation analysis of the orthogonal procrustes problem

Given two arbitrary real matricesA andB of the same size, the orthogonal Procrustes problem is to find an orthogonal matrixM such that the Frobenius norm MA – B is minimized. This paper treats the common case when the orthogonal matrixM is required to have a positive determinant. The stability of the problem is studied and supremum results for the perturbation bounds are derived.     

About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let . The numbers and as well as all partially ordered sets for which these maxima are attained are determined.     

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

Organoruthenaborane Chemistry. X. The SM2-catalysed formation of [ l-(η6-C6Me6)-isocloso-RuB9H9], and some B-phenylamino derivatives. NMR and X-ray diffraction studies

The action of SMe₂ on the ten-vertex nido-ruthenaborane [6-(η⁶-C₆Me₆)RuB₉H_l3] ( 1 ) provides a high-yield route to the unsubstituted isocloso-ruthenaborane [1-(η⁶-C₆Me₆)RuB₉H₉] (2). The benzene analogue [1-(η⁶-C₆Me₆)RuB₉H₉] is prepared similarly. By contrast, reaction of (1) with PhNH₂ gives a variety of B-phenylamino isocloso derivatives, including orange crystals of [1-(η⁶-C₆Me₆)-2-(PhNH)-isocloso-1-RuB 9 H₈] ( 3 ), red-orange [1-(η⁶-C₆Me₆)-2,3-(PhNH)₂-isocloso-1-RuB₉H₇] ( 4 ) and dark-red [1-(η⁶-C₆Me₆)-5,6,7-(PhNH)₃-isocloso-1-RuB₉H₆] ( 5 ). Detailed ¹H and ¹¹B nmr properties of these various compounds are described. The structure of ( 3 ) has been established by a single-crystal X-ray diffraction study of the solvate [1-(η⁶-C₆Me₆)-2-(PhNH)-isocloso-1-RuB₉H₈] · 1/2 CH₂Cl₂; the crystals were monoclinic, space group C2/c, with a = 1895.1(3), b = 1556.6(3), c = 1716.4(3) pm, β = 104.37(1)° and z = 8.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

On the affinity of cytosine towards electrophiles

Ab initio SCF computations on the intrinsic preferences of the H⁺, CH ₃ ⁺ and C₂H ₅ ⁺ cations towards the two principal sites of protonation or alkylation on cytosine, N₃ or O₂, show that this preference undergoes a continuous modification with the increase in size and complexity of the cation. N₃ is the preferred site of fixation of H⁺, O₂ the preferred site of C₂H ₅ ⁺ , while CH ₃ ⁺ has no marked preference. The exchange repulsion term of the binding energy appears responsible for the preference of C₂H ₅ ⁺ for O₂.This work was supported by the Ligue Francaise contre le Cancer and the National Foundation for Cancer Research (USA)