scholarly journals On Certain Mappings of Riemannian Manifolds

1965 ◽  
Vol 25 ◽  
pp. 121-142
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Conformal Mapping ◽  
Riemannian Manifolds ◽  
Stereographic Projection ◽  
Point Of View ◽  
The Other ◽  
Central Projection ◽  
Special Cases ◽  
Special Case

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

scholarly journals The annealing of copper: with special reference to dilatation

1907 ◽  
Vol 80 (535) ◽  
pp. 1-12
Keyword(s):  
Common Knowledge ◽  
Point Of View ◽  
The Other ◽  
Annealed Condition ◽  
Practical Point ◽  
Special Cases ◽  
Soft State ◽  
Rate Of Cooling ◽  
The Difference ◽  
Annealed Copper

It is well known that copper is met with in two distinct forms, viz., the soft state as in cast or annealed metal, and the hard variety which is the result ofmechanical work. There is more difference between the mechanical properties of hard and of soft copper than is observed in the case of two distinct metals; such, for example, as nickel and cobalt. For instance, one of the most important of these differences, from a practical point of view, is in connection with the tensile strength of the material, which is only about 10 to 14 tons per square inch in cast or in annealed copper, while in hardened copper the tenacity is about twice as great, and usually runs from about 20 to 28 tons per square inch, or even more in special cases. The difference between hard and soft copper can be also readily illustrated by bending two rods about ¼ inch square section, one in the hardened and the other in the annealed condition. The latter can be easily bent in the hands or even tied in a knot, while the mechanically worked bar is rigid and elastic, and can only be bent by the application of considerable force. It is common knowledge that hard copper becomes perfectly annealed by heating to 500° C,; that the heating need not be for any lengthened period, and the rate of cooling afterwards is unimportant.

scholarly journals XXVI.—On a Group of Linear Differential Equations of the 2nd Order, including Professor Chrystal's Seiche-equations

1906 ◽  
Vol 41 (3) ◽  
pp. 651-676 ◽  
Author(s):  
J. Halm
Keyword(s):  
Differential Equations ◽  
Stokes Equation ◽  
Mathematical Theory ◽  
General Type ◽  
The Other ◽  
Special Cases ◽  
Linear Differential ◽  
Image Position ◽  
Special Case ◽  
The Stokes Equation

It is readily seen that the two differential equationswhich play an important rôle in Professor Chrystal's mathematical theory of the Seiches, are special cases of the more general typeWith regard to the first, the Seiche-equation, this becomes at once apparent by writing a= − ½. Equation (2), on the other hand, which we may briefly call the Stokes equation [see Professor Chrystal's paper on “Some further Results in the Mathematical Theory of Seiches,” Proc. Roy. Soc. Edin., vol. xxv.] will be recognised as a special case (a = + 1) of the equationwhich is transformed into (3) by the substitution .

scholarly journals THE ADHM CONSTRUCTION OF INSTANTONS ON NONCOMMUTATIVE SPACES

2011 ◽  
Vol 23 (03) ◽  
pp. 261-307 ◽  
Author(s):  
SIMON BRAIN ◽  
WALTER D. VAN SUIJLEKOM
Keyword(s):  
Euclidean Space ◽  
Noncommutative Geometry ◽  
Space Time ◽  
Point Of View ◽  
Noncommutative Space ◽  
Special Cases ◽  
Noncommutative Spaces ◽  
Coordinate Algebra ◽  
Classical Construction

We present an account of the ADHM construction of instantons on Euclidean space-time ℝ4 from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parametrized by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal–Groenewold plane [Formula: see text] and the Connes–Landi plane [Formula: see text].

scholarly journals On the Group Structure in Homotopy Groups

1954 ◽  
Vol 7 ◽  
pp. 133-144
Author(s):  
Masatake Kuranishi
Keyword(s):  
Homotopy Group ◽  
Group Structure ◽  
Algebraic Approach ◽  
Point Of View ◽  
The Other ◽  
Homotopy Groups ◽  
Other Hand ◽  
Special Case

Usually the group structure in a homotopy group is defined directly and explicitly. But the algebraic approach to the topology, now common, seems to raise the following question : is that the only group sturcture which is natural from the algebraic topological point of view? On the other hand, several algebraists have begun to feel a necessity to construct a “homotopy or cohomotopy theory of groups,” and it may be allowed to say that one of the first steps to the problem is the axiomatization of homotopy groups. Our first question is of course a special case of the latter problem.

Some Extensions of Askey-Wilson's Q-Beta Integral and the Corresponding Orthogonal Systems

1988 ◽  
Vol 31 (4) ◽  
pp. 467-476 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Polynomial System ◽  
Biorthogonal System ◽  
The Other ◽  
Symmetric Form ◽  
Ultraspherical Polynomials ◽  
First Case ◽  
Special Cases ◽  
Beta Integral ◽  
Orthogonal Systems ◽  
Special Case

AbstractA seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

scholarly journals The Euler–Lagrange Cohomology and General Volume-Preserving Systems

2003 ◽  
Vol 18 (27) ◽  
pp. 1911-1923 ◽  
Author(s):  
Bin Zhou ◽  
Han-Ying Guo ◽  
Jianzhong Pan ◽  
Ke Wu
Keyword(s):  
Symplectic Manifold ◽  
Point Of View ◽  
The Other ◽  
Cohomology Groups ◽  
Nambu Mechanics ◽  
Special Case ◽  
Volume Preserving

We briefly introduce the concept of Euler–Lagrange cohomology groups on a symplectic manifold (ℳ2n, ω) and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important two-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian H are included as a special case with the two-form [Formula: see text]. The other volume preserving systems on (ℳ2n, ω) are studied. The relations between our approach and Feng–Shang's volume-preserving systems as well as the Nambu mechanics are also explored.

scholarly journals Geometry of k-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 222
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Pişcoran Laurian-Ioan ◽  
Nadia Alluhaibi
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Point Of View ◽  
Euclidean Spaces ◽  
Field Point ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.

Presupposition and Givenness

2016 ◽  
Author(s):  
Bart Geurts
Keyword(s):  
The Other ◽  
Local Context ◽  
Special Cases ◽  
Different Types ◽  
Special Case

Presuppositions are items of information triggered by certain words and constructions that exhibit ‘projection behaviour’, which is to say that, except in special cases, they will escape from any level of embedding. Presupposed information is given, or at least presented as such, and there are two main theories of what it means for presuppositions to be given. On one account, a presupposition must be entailed in the local context in which it is triggered; on the other, presuppositions require that certain discourse entities be available in the context. On the latter account, but not on the former, anaphora is a special case of presupposition. It might be that both accounts are correct, though for different types of presupposition.

The Linearization of the Product of Continuous q-Jacobi Polynomials

1981 ◽  
Vol 33 (4) ◽  
pp. 961-987 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Hypergeometric Series ◽  
Jacobi Polynomials ◽  
Sufficient Conditions ◽  
Exact Formula ◽  
Point Of View ◽  
Integral Expression ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Linearization Coefficients ◽  
Special Case

The problem of linearizing the product of two Jacobi polynomials, Pm(α, β)(x)Pn(α, β)(x), and to establish the conditions for the non-negativity of the coefficients has been of considerable interest for many years. Explicit non-negative representations were sought and found by many authors [7, 8, 13, 14], but only in the special case α = β, although Hylleraas [14] succeeded in finding a formula in another case α = β + 1. Gasper [9, 10] found the necessary and sufficient conditions for the non-negativity of the linearization coefficients by exploiting a recurrence relation obtained by Hylleraas for the above-mentioned product. Koornwinder [16] approached the same problem from a different point of view and managed to find a non-negative integral expression to these coefficients when . However, an exact formula in a hypergeometric series form for general α, β has been very elusive so far, in spite of the fact that all computation of special cases seemed to indicate that such a formula should exist.

Can the Academic Self-Concept be Positive and Realistic?

2005 ◽  
Vol 19 (3) ◽  
pp. 129-132 ◽  
Author(s):  
Reimer Kornmann
Keyword(s):  
Academic Performance ◽  
Opposite Effect ◽  
Self Esteem ◽  
Point Of View ◽  
The Other ◽  
Self Concept ◽  
High Ability ◽  
Self Image ◽  
Other Hand

Summary: My comment is basically restricted to the situation in which less-able students find themselves and refers only to literature in German. From this point of view I am basically able to confirm Marsh's results. It must, however, be said that with less-able pupils the opposite effect can be found: Levels of self-esteem in these pupils are raised, at least temporarily, by separate instruction, academic performance however drops; combined instruction, on the other hand, leads to improved academic performance, while levels of self-esteem drop. Apparently, the positive self-image of less-able pupils who receive separate instruction does not bring about the potential enhancement of academic performance one might expect from high-ability pupils receiving separate instruction. To resolve the dilemma, it is proposed that individual progress in learning be accentuated, and that comparisons with others be dispensed with. This fosters a self-image that can in equal measure be realistic and optimistic.

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