XIV.—Note on the Construction of an Orthogonant

1919 ◽  
Vol 38 ◽  
pp. 146-153
Author(s):  
Thomas Muir

(1) The essence of Cayley's mode* of arriving at the formation of an orthogonal substitution lies, as is known, in the observation that if two sets of variablesx, y, z, w, and ξ, η, ζ, ωbe taken linear functions of a third setin such a way that the determinant of the coefficients is in the one caseand in the other the conjugate of this, then the first and second set of variables are orthogonally related.

Author(s):  
B. Choudhary

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.


1929 ◽  
Vol 25 (2) ◽  
pp. 219-221
Author(s):  
T. M. Lowry

Two alternative views have been expressed in regard to the configuration of quadrivalent atoms. On the one hand le Bel and van't Hoff assigned to quadrivalent carbon a tetrahedral configuration, which has since been confirmed by the X-ray analysis of the diamond. On the other hand, Werner in 1893 adopted an octahedral configuration for radicals of the type MA6, e.g. inand then suggested that “the molecules [MA4]X2 are incomplete molecules [MA6]X2. The radicals [MA4] result from the octahedrally-conceived radicals [MA6] by loss of two groups A, but with no function-change of the acid residue…. They behave as if the bivalent metallic atom in the centre of the octahedron could no longer bind all six of the groups A and lost two of them leaving behind the fragment [MA4]” (p. 303).


Author(s):  
Marco Carricato ◽  
Joseph Duffy ◽  
Vincenzo Parenti-Castelli

Abstract In this article the inverse static analysis of a two degrees of freedom planar mechanism equipped with spiral springs is presented. Such analysis aims to detect the entire set of equilibrium configurations of the mechanism once the external load is assigned. While on the one hand the presence of flexural pivots represents a novelty, on the other it extremely complicates the problem, since it brings the two state variables in the solving equations to appear as arguments of both trigonometric and linear functions. The proposed procedure eliminates one variable and leads to write two equations in one unknown only. The union of the root sets of such equations constitutes the global set of solutions of the problem. Particular attention has been reserved to the analysis of the “reliability” of the final equations: it has been sought the existence of critical situations, in which the solving equations hide solutions or yield false ones. A numerical example is provided. Also, in Appendix it is shown a particular design of the mechanism that offers computational advantages.


2019 ◽  
Vol 19 (5-6) ◽  
pp. 705-721
Author(s):  
GIOVANNI AMENDOLA ◽  
FRANCESCO RICCA ◽  
MIROSLAW TRUSZCZYNSKI

AbstractAnswer Set Programming (ASP) is a logic programming paradigm featuring a purely declarative language with comparatively high modeling capabilities. Indeed, ASP can model problems in NP in a compact and elegant way. However, modeling problems beyond NP with ASP is known to be complicated, on the one hand, and limited to problems in $\[\Sigma _2^P\]$ on the other. Inspired by the way Quantified Boolean Formulas extend SAT formulas to model problems beyond NP, we propose an extension of ASP that introduces quantifiers over stable models of programs. We name the new language ASP with Quantifiers (ASP(Q)). In the paper we identify computational properties of ASP(Q); we highlight its modeling capabilities by reporting natural encodings of several complex problems with applications in artificial intelligence and number theory; and we compare ASP(Q) with related languages. Arguably, ASP(Q) allows one to model problems in the Polynomial Hierarchy in a direct way, providing an elegant expansion of ASP beyond the class NP.


1987 ◽  
Vol 21 (4) ◽  
pp. 667-678
Author(s):  
Ian Nish

As Britain saw it, trade was not the prime motivating force for Russian expansion in east Asia or, put another way, the Russian frontiersmen were not driven by the actual amount of their trade there or its future potentialities. While Russia was primarily concerned with the tea trade over land frontiers, Britain was concerned with the seaborne commerce of China. The customs revenue paid to China in the year 1894 worked out as follows:Judging from the returns of the Chinese Imperial Maritime Customs Organization, British ships carried 83.5% of China's total trade. But Britain's commercial dominance affected her political stance because she wanted to preserve China's stability for most of the second half of the nineteenth century. This was at the root of the political tensions between Britain and Russia which emerged in China after 1860 and especially those which derived from the spate of railway building which took place from 1890 onwards. It would be foolish to deny that intense rivalry did exist in the area from time to time or that detailed observations of the actions of the one were regularly conducted by the other—what we should now call ‘intelligence operations’. But what I shall suggest in this paper is that, despite all the admitted antagonism and suspicion between Britain and Russia in east Asia, Britain regularly made efforts to reach accommodations with Russia in north-east Asia.


1882 ◽  
Vol 23 (5) ◽  
pp. 335-352 ◽  
Author(s):  
John Adams Higham

Mr. John Finlaison's method of graduation (Journal, vol. xxi, page 50) deals with an irregularity, which, for convenience, may be assumed to be unity, in the following manner:—It diminishes the irregularity to a small fraction on its first appearance at one end of the formula; conducts it along the formula by easy stages, as over a double inclined plane; and dismisses it gently at the other end. The method has the advantage of simplicity in working and check, but it is correct to first differences only.Mr. Woolhouse's method is correct to third differences, and the irregularity glides over a curve; but the application of the formula requires sustained attention.The object of this paper is to combine the facility of the one method with the smoothness and correctness of the other.Let S be the sum of nine numbers, u0 to u8, increasing by third differences,the central term, whence .


1951 ◽  
Vol 16 (1) ◽  
pp. 43-45
Author(s):  
Maurice L'abbé

A general system of axioms has been given by Henkin for a fragment of the propositional calculus having as primitive symbols, in addition to the usual parentheses, variables, and implication sign ⊃, an arbitrarily given truth function symbol ϕ. This system of axioms, which we shall denote by S(⊃, ϕ), contains the following three axiom schemataplus the 2m further axiom schemata involving the symbol ϕwhere ϕ is an m-placed function symbol. We refer to Henkin's paper, p. 43, for the detailed description of the axiom schemata (4).The remark was made in the above mentioned paper that each of the 2m axiom schemata of (4) is trivially independent of the rest of the axioms of S(⊃, ϕ), and it was conjectured that the axiom schemata (1), (2) and (3) are also independent. In this note, we prove the general independence of the axiom schemata (1) and (2). As for (3), we show on the one hand its independence in the systems S(⊃) and S(⊃, f), and, on the other hand, its dependence in the system S(⊃, ∼). The net result is, therefore, that in any of these systems of axioms S(⊃, ϕ) all the axiom schemata are independent, except possibly the axiom schema (3).


1978 ◽  
Vol 84 (3) ◽  
pp. 537-538 ◽  
Author(s):  
J. Callahan

The double cusp is the real, compact, unimodal singularitysee (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.


1986 ◽  
Vol 33 (2) ◽  
pp. 207-218 ◽  
Author(s):  
S. J. Goodenough

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).


1805 ◽  
Vol 5 (1) ◽  
pp. 99-116 ◽  
Author(s):  
James Ivory

1. I Divide cubic equations into two varieties or species: the one, comprehending all cubic equations with three real roots; the other, all those with only one real root.2. Let φ denote any angle whatever, and let τ = tan φ, the radius being unity: let also : then from the doctrine of angular sections we havewhich being reduced to the form of an equation, isZ3-3τZ2-3Z+τ=0.


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