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 erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

Posets isomorphic to their extensions

A standard extension for a poset P is a system Q of lower ends (descending subsets) of P containing all principal ideals of P. An isomorphism between P and Q is called recycling if [Y]Q for all YQ. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.     

苯胺气相N－烷基化反应研究Ⅱ．苯胺和乙醇常压催化合成N－乙基苯胺的工业应用

苯胺气相Ｎ－烷基化反应研究Ⅱ．苯胺和乙醇常压催化合成Ｎ－乙基苯胺的工业应用陈骏如，李瑞祥，韩银仙，李贤均（四川大学化学系成都６１００６４）Ｎ－乙基苯胺是制造染料、塑料、炸药和医药等许多化工产品的重要中间体，其合成方法过去多采用无机酸催化剂，如Ｈ＿２Ｓ...     

σ,π-acetylido complexes via P---C bond cleavage of phosphinoacetylenes: X-ray structure of Fe2(CO)6(CCPh)(PPh2)

Solutions of Co^II salts of organic acids in primary amines absorb molecular oxygen rapidly and irreversibly forming μ-dioxygen—cobalt complexes. Thermolysis leads to a homopolar cleavage of the OO bond with subsequent radical reactions involving ligand amine. After thermolysis the capacity of oxygen uptake is reestablished (catalysis).     

Concerted cycloaddition of chlorosulfonyl isocyanate to α-pinene stepwise rearrangement of the β-lactam cycloadduct to a γ-lactam

Chlorosulfonyl isocyanate adds to α-pinene in concerted fashion to give the unrearranged N-chlorosulfonyl-β-lactam 2 (75%). Thermolysis of cycloadduct 2 affords the rearranged N-chlorosulfonyl-γ-lactam 6 (60%) via a sequence of transient carbonium ions (45). Reductive hydrolysis of 2 and 6 gave, respectively, the NH-β-lactam 3 (70%) and the NH-γ-lactam 7 (70%). Proof of structure of the rearranged γ-lactams 6 and 7 are provided, respectively, by X-ray crystallographic analysis and the application of the Eu(dpm)₃ NMR shift reagent.     

Reaction of η5-cyclopentadienylbis(triphenylphosphine)cobalt(I) with alkadiynes. Preparation and reactions of bicyclic cobalt metallocycles and a stable trimethylsilyl-substituted mono-complexed alkadiyne. Comments on the mechanism of cobalt-catalyzed alkyne cooligomerization

Reactions of η⁵-cyclopentadienylbis(triphenylphosphine)cobalt(I) (5) with several 2,n-alkadiynes (2) were investigated. Each of these reactions leads initially to a material in which one of the acetylene functional groups is π-coordinated to cobalt; this complex then undergoes conversion to a metallocycle. In cases where the two acetylene functions are connected by three- and four-carbon bridges (2b, 2c), metallocycles formed by intramolecular reaction of two acetylene functions in the same molecule may be isolated. In cases where the acetylene functions are joined by larger or smaller bridges, the reactions are more complex, and both inter- and intramolecular metallocycles are formed. Reactions of 5 with 1,8-bis(trimethylsilyl)-1,7-octadiyne (16) gives an isolable crystalline mono-acetylene complex (17), this material is stable in the solid state but undergoes conversion to metallocycle (18) in benzene solution. The relationship of these results to the mechanism of the CpCo(CO)₂-catalyzed benzocycloalkene synthesis is discussed; it is suggested that intramolecular metallocycles are intermediates in reactions leading to benzocyclopentanes and -cyclohexenes, but intermolecular metallocycles are probably involved in reactions leading to benzocyclobutenes.     

A Laplace-like method for solving linear difference equations

This paper is concerned with the problem of solving linear difference equations of ordern with constant coefficients and with given initial conditions in which the variable runs not only through the integers but over . The main idea is the introduction of a suitable commutative ring of functions with discrete convolution as multiplication rule which works, although it is not a field. The existence of inverses is studied and, after the introduction of suitable functions, the problem is reduced by means of a Laplace-like relation to an algebraic equation. Examples of application are given. Finally some remarks make the connection with the Operational Calculus of Mikusinski and with the Operational Calculus of Fenyö. The advantages of this method lie in the fact that it is applicable to functions others than the step functions, in its simplicity from the theoretical point of view, in its usefulness even when computation is required and in its formal similarity to the classical treatment of linear differential equations with constant coefficients.     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper Chromatic Sums for Rooted Planar Triangulations, V: Special Equations. (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy ₂ of two variablesy ₁ andz. A parameter is involved. The main result is that for = 2 cos (2/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.The present work is basically an expository paper on the theory given in Chromatic Sums, V, but it includes some new results and many simplifications.