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.     

The importance of nonconventional structures in the binding of Ni<Superscript>+</Superscript> to ethynylsilanes and ethynylgermanes

The influence of the nature of the heteroatom on the Ni⁺ gas-phase binding energies of HCC–XH₃ (X is C, Si, or Ge) compounds has been investigated through the use of high-level density functional theory methods. The structures of the corresponding Ni⁺ complexes were optimized at the B3LYP/6-311G(d,p) level of theory. Final energies were obtained in single-point B3LYP/6-311+G(2df,2p) calculations. Nonconventional complexes, in which the metal cation interacts simultaneously with the CC system and with one of the X–H bonds of the substituent XH₃ group, play a significant role in the binding of Ni⁺ to HCC–XH₃ (X is Si or Ge) derivatives. Conversely, such nonconventional complexes are not local minima of the propyne–Ni⁺ potential-energy surface. This establishes a clear distinction between unsaturated carbon derivatives and the Si- and Ge-containing analogues as far as bonding to transition-metal monocations is concerned. Actually, the attachment of Ni⁺ to HCC–XH₃ (X is Si or Ge) compounds in the gas phase yields a mixture of conventional and nonconventional complexes. These agostic-type interactions can be viewed as a dative bond from the X–H bonding orbital toward the empty s orbital of the metal, and a back-donation from the valence electron pairs of the metal into the X–H antibonding orbital of the neutral species.Proceedings of the 11th International Congress of Quantum Chemistry satellite meeting in honor of Jean-Louis Rivail     

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.     

Electrochemical Determination of Ascorbic Acid at γ-MnO<Subscript>2</Subscript> Modified Carbon Black Microelectrodes

A carbon black microelectrode modified by -MnO₂ has been prepared. The electrocatalytical oxidation of ascorbic acid (AA) at this microelectrode was investigated. The 2nd-order linear scan voltammograms of AA are recorded from –0.5 to 0.5V (vs. SCE). The relationship between the oxidation peak current of AA and its concentration in the range of 1.0×10^–64.0×10^–3molL^–1 is linear. The detection limit (3) was found to be 6.0×10^–7molL^–1. Also, the determination of AA in samples is evaluated, and the results are satisfactory.     

Computations in choice groupoids

Defining achoice as a mapping of the subsets of a setX into their respective subsets, a one-to-one (and naturally) corresponding binary operation,sequential choice, is identified under which the power set ofX is closed as achoice groupoid. A complete logical diagram is given, exhibiting all the implications between conjunctions of the seven conditions: (1) idempotence, (2) consistency, (3) absorbence, and (4) homomorphism of a choice, and (5) commutativity, (6) associativity, and (7) path-independence of the corresponding sequential choice.     

Concerning the solution set ofAx=b whereP≦A≦Q andp≦b≦q

Summary LetPQ ben×n real matrices so that ifPAQ for some matrixA, thenA is nonsingular. Letp andq ben-dimensional real column vectors. This paper determines the set of all solutionsx to the equationAx=b for allA andb so thatPAQ andpbq.     

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.)     

Generalized reinhardt domains

In this paper we investigate a class of Lie group actions on , the so-calledpolar actions, that naturally generalize the standard actions. For a domain invariant under such an action (i.e., a generalized Reinhardt domain) we characterize the invariant plurisubharmonic functions and determine the envelope of holomorphy in geometric terms. For a generalized Reinhardt domain containing the origin of we also compute its automorphism group. Supported in part by NSF Grant 8602020     

On the implementation of Sμ U(2) action in the irreducible representations of twisted canonical commutation relations

The natural S U(2) action on the creation and annihilation operators satisfying twisted canonical commutation relations (TCCR) is investigated. It is shown that the Fock representation is the only covariant irreducible representation of TCCR.Partially supported by the program RP.1.10.     

Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.