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.     

Selectivities in 1,2,3-triazolide displacements of halides and additions to alkynes

4,5-Dicarbomethoxy-1,2,3-triazolide or 4-phenyl-1,2,3-triazolide displace chloride from ethyl chloroacetate or β-chloropropionate to give both 1-N and 2-N alkylated products. Our highest 2-N to 1-N selectivity was ca 5/1 and was found with the base triethylamine in DMF. The same triazolides and others add to alkynes, e.g. ethyl propiolate, methyl acetylenedicarboxylate, phenylpropiolaldehyde, ethyl phenylpropiolate, etc, to give Michael adducts at the 2-N position exclusively. Here the usual preference holds, i.e., the anti adduct is favored, but anti to syn isomerization usually sets in. On the basis of the available data for nucleophilic substitutions and additions, a limited directioselectivity pattern emerges for H-1,2,3-triazoles (T) and their anions (T^?): neutral T almost invariably leads with 1-N; T^t-- usually adds to unsaturates at 2-N; unsubstituted, 4-substituted and 4,5-disubstituted T^? attack organic halides at both 1-N and 2-N. Compared to phenyl, 2-triazolyl exerts a greater deshielding effect on proton chemical shifts; these and other patterns in the PMR spectra of the Michael adducts are discussed. CNDO calculations indicate that the 1-H is more stable than the 2-H-1,2,3-triazole and that in both neutral triazole and in triazolide, the 1-nitrogen position should lead nucleophilic attacks-this directioselectivity prediction is only partly (and probably fortuitously) correct.     

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.     

Preparative uv-laser photochemistry of the azoalkane spiro(2,3-diazabicyclo [2.2.1]hept-2-ene-7′,1-cyclopropane): Trapping of the 1,4-Diradical 2-(3-Cyclopentenyl) ethyl by Molecular Oxygen

Photo-extrusion of nitrogen from the azoalkane 1 in the presence of molecular oxygen gave besides the hydrocarbons 3 and 5, the endoperoxide 10 and hydroperoxide 11, the former via trapping of the 1,4-diradical 4 by triplet oxygen, the latter by ene-reaction-6f hydrocarbon 5 with singlet oxygen.     

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.     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

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.     

Error bounds for continued fractionsK(1/b n)

Summary A priori truncation error bounds are obtained for continued fractions of the formK(1/b _n),b _n complex. The error bounds are easily applied to the case whenb _n0 asn. A numerical example involving the complex error function is given.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].     

A 19F NMR study of the transmission of electronic effects in triarylantimony and triarylbismuth compounds. A comparison with the nitrogen- and carbon-containing analogues

A ¹⁹F NMR study of the transmission of electronic effects has been made for the systems Ar₂EC₆H₄F-4 (E = Sb, Bi, CH, N). The fluorine chemical shifts obtained are correlated with the polar constants (Σσ^o and Σσ) of the substituents, suggesting that electronic effects are transmitted through the SbC_ar, BiC_ar and CC_ar bonds predominantly by an inductive mechanism, whereas the transmission through the NC_ar bonds is contributed to significantly by classical resonance effects due to competitive conjugation of the lone pair with the aromatic rings, and the substituents therein. A dual parameter correlation of the fluorine chemical shifts with the inductive (σ_I) and resonance (σ^o_R and σ_R) parameters of the substituents in the aromatic rings has led to similar conclusions. The inductive transmission through the bridging Sb and Bi atoms has been assigned to the absence of conjugation of lone pair and vacant d-orbitals of the metals with π-electron systems of the aromatic rings. On the basis of the values of the ? coefficients for the correlation equations obtained it has been established that the transmitting ability of the BiC_ar bonds is close to that of the C_alC_ar bonds and considerably lower than the transmitting ability of the NC_ar bonds.