scholarly journals V. A New Method of resolving Cubic Equations

1805 ◽  
Vol 5 (1) ◽  
pp. 99-116 ◽  
Author(s):  
James Ivory
Keyword(s):  
Real Root ◽  
The Other ◽  
New Method ◽  
Real Roots ◽  
The One ◽  
Image Position

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.

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

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.

Optimal Camera Placement in a Virtual Environment

2021 ◽  
pp. 247-260
Author(s):  
Hocine Chebi
Keyword(s):  
Real Time ◽  
Virtual Environment ◽  
The Other ◽  
New Method ◽  
Camera Placement ◽  
Virtual Camera ◽  
Clear Vision ◽  
Transmission Of Information ◽  
The One ◽  
Visual Properties

Camera placement in a virtual environment consists of positioning and orienting a 3D virtual camera so as to respect a set of visual or cinematographic properties defined by the user. Carrying out this task is difficult in practice. Indeed, the user has a clear vision of the result he wants to obtain in terms of the arrangement of the objects in the image. In this chapter, the authors identify three areas of research that are relatively little covered by the literature dedicated to camera placement and which nevertheless appear essential. On the one hand, existing approaches offer little flexibility in both solving and describing a problem in terms of visual properties, especially when it has no solution. They propose a flexible solution method which computes the set of solutions, maximizing the satisfaction of the properties of the problem, whether it is over constrained or not. On the other hand, the existing methods calculate only one solution, even when the problem has several classes of equivalent solutions in terms of satisfaction of properties. They introduce the method of semantic volumes which computes the set of classes of semantically equivalent solutions and proposes a representative of each of them to the user. Finally, the problem of occlusion, although essential in the transmission of information, is little addressed by the community. Consequently, they present a new method of taking into account occlusion in dynamic real-time environments.

Configuration of Quadrivalent Atoms

1929 ◽  
Vol 25 (2) ◽  
pp. 219-221
Author(s):  
T. M. Lowry
Keyword(s):  
The Other ◽  
Tetrahedral Configuration ◽  
X Ray ◽  
Metallic Atom ◽  
Other Hand ◽  
Function Change ◽  
The One ◽  
Image Position ◽  
Octahedral Configuration ◽  
As If

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).

scholarly journals Beyond NP: Quantifying over Answer Sets

2019 ◽  
Vol 19 (5-6) ◽  
pp. 705-721
Author(s):  
GIOVANNI AMENDOLA ◽  
FRANCESCO RICCA ◽  
MIROSLAW TRUSZCZYNSKI
Keyword(s):  
Answer Set Programming ◽  
The Other ◽  
Polynomial Hierarchy ◽  
Programming Paradigm ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
The One ◽  
Image Position ◽  
Answer Sets ◽  
Computational Properties

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.

Politics, Trade and Communications in East Asia: Thoughts on Anglo-Russian Relations, 1861–1907

1987 ◽  
Vol 21 (4) ◽  
pp. 667-678
Author(s):  
Ian Nish
Keyword(s):  
Nineteenth Century ◽  
East Asia ◽  
The Other ◽  
The Political ◽  
Total Trade ◽  
North East ◽  
Intelligence Operations ◽  
Political Stance ◽  
The One ◽  
Image Position

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.

Notes on the Theory of Equations

1939 ◽  
Vol 23 (256) ◽  
pp. 376-379
Author(s):  
E. P. Lewis
Keyword(s):  
Real Root ◽  
Real Roots ◽  
The Third ◽  
Complex Roots ◽  
Image Position

Multiply throughout by a 2 and write y for ax+ b ; the equation becomes where H ≡ ac − b 2, G ≡ a2d − 3abc + 2b3. Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written Accordingly the two remaining roots are also real if But since α satisfies (i), and so Hence if (i) has three real roots, G2 +4H 3 ≤ 0; and clearly, when G2 +4H 3 = 0, two roots are numerically equal to and the third to .

On the Adjustment of Mortality Tables

1882 ◽  
Vol 23 (5) ◽  
pp. 335-352 ◽  
Author(s):  
John Adams Higham
Keyword(s):  
Sustained Attention ◽  
The Other ◽  
Inclined Plane ◽  
The One ◽  
Image Position ◽  
Central Term

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 .

A New Method and Device for Temperature Measurement Using One-Dimensional Heat Equation

1994 ◽  
Author(s):  
Nicolae A. Damean
Keyword(s):  
Heat Equation ◽  
Temperature Measurement ◽  
Low Temperatures ◽  
Principal Component ◽  
The Other ◽  
New Method ◽  
Gradient Algorithm ◽  
One Dimensional ◽  
The One ◽  
The Mathematical Model

Abstract A new method and device for temperature measurement are presented. The method reduces the measurement of the unknown temperature to the solving of an optimal control problem, using a numerical computer. The device consists of a hardware part including some conventional transducers and a software one. The problem of temperature measurement, according to this method, is mathematically modelled by means of the one-dimensional heat equation, describing the heat transfer through the device. The principal component of the device is a rod. The variation of the temperature which is produced near one end of the rod is determined using some temperature measurements in the other end of the rod, the mathematical model and a type of gradient algorithm. This device works as an attenuator of high temperatures and as an amplifier of low temperatures.

On the independence of Henkin's axioms for fragments of the propositional calculus

1951 ◽  
Vol 16 (1) ◽  
pp. 43-45
Author(s):  
Maurice L'abbé
Keyword(s):  
Propositional Calculus ◽  
General System ◽  
Function Symbol ◽  
The Other ◽  
Axiom Schema ◽  
Other Hand ◽  
Truth Function ◽  
The One ◽  
Image Position ◽  
Made In

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).

The double cusp has five minima

1978 ◽  
Vol 84 (3) ◽  
pp. 537-538 ◽  
Author(s):  
J. Callahan
Keyword(s):  
Critical Points ◽  
Standard Form ◽  
Singularity Theory ◽  
The Other ◽  
Universal Unfolding ◽  
The Real ◽  
Level Curves ◽  
Intersection Points ◽  
The One ◽  
Image Position

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.

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