Desargues' theorem in n-space

1960 ◽  
Vol 1 (3) ◽  
pp. 311-318 ◽  
Author(s):  
Sahib Ram Mandan
Keyword(s):  
Projective Space ◽  
Algebraic Approach ◽  
The Other ◽  
Desargues Theorem

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

scholarly journals Deformation of finite morphisms and smoothing of ropes

2008 ◽  
Vol 144 (3) ◽  
pp. 673-688 ◽  
Author(s):  
Francisco Javier Gallego ◽  
Miguel González ◽  
Bangere P. Purnaprajna
Keyword(s):  
General Theory ◽  
Projective Space ◽  
General Result ◽  
The Other ◽  
Independent Interest ◽  
Higher Dimension ◽  
Smooth Curves ◽  
Other Hand ◽  
Finite Covers ◽  
The One

AbstractIn this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

scholarly journals Alexander- and Markov-type theorems for virtual trivalent braids

2019 ◽  
Vol 28 (01) ◽  
pp. 1950003 ◽  
Author(s):  
Carmen Caprau ◽  
Abigayle Dirdak ◽  
Rita Post ◽  
Erica Sawyer
Keyword(s):  
Type Theorem ◽  
Algebraic Approach ◽  
The Other ◽  
Markov Type ◽  
Trivalent Graphs

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on [Formula: see text]-moves.

scholarly journals Algebraic approach for the one-dimensional Dirac–Dunkl oscillator

2020 ◽  
Vol 35 (31) ◽  
pp. 2050255
Author(s):  
D. Ojeda-Guillén ◽  
R. D. Mota ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Energy Spectrum ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Representation Theory ◽  
Algebraic Approach ◽  
The Other ◽  
One Dimensional ◽  
The One

We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.

scholarly journals ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

2010 ◽  
Vol 21 (07) ◽  
pp. 939-949 ◽  
Author(s):  
KEIJI MATSUMOTO ◽  
TOMOHIDE TERASOMA
Keyword(s):  
Projective Space ◽  
Hyperelliptic Curve ◽  
Geometric Mean ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
The Other ◽  
Geometric Means ◽  
Dimensional Projective Space ◽  
Period Integral ◽  
Generalized Arithmetic

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomae's formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

A characterization of sets of n points which determine n hyperplanes

1968 ◽  
Vol 64 (3) ◽  
pp. 585-588 ◽  
Author(s):  
J. G. Basterfield ◽  
L. M. Kelly
Keyword(s):  
Projective Space ◽  
The Other ◽  
Finite Projective Space ◽  
Other Hand ◽  
Finite Set ◽  
Set Of Points

Suppose N is a set of points of a d-dimensional incidence space S and {Ha}, a ∈ I, a set of hyperplanes of S such that Hi ∈ {Ha} if and only if Hi ∩ N spans Hi. N is then said to determine {Ha}. We are interested here in the case in which N is a finite set of n points in S and I = {1, 2,…, n}; that is to say when a set of n points determines precisely n hyperplanes. Such a situation occurs in E3, for example, when N spans E3 and is a subset of two (skew) lines, or in E2 if N spans the space and n − 1 of the points are on a line. On the other hand, the n points of a finite projective space determine precisely n hyperplanes so that the structure of a set of n points determining n hyperplanes is not at once transparent.

scholarly journals A quadratic transformation of the three-dimensional space of all conics through two points of a projective plane

2009 ◽  
Vol 42 (3) ◽  
Author(s):  
Giorgio Donati
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Quadratic Transformation ◽  
Pencil Of Conics ◽  
Dimensional Projective Space ◽  
Three Dimensional Space

AbstractUsing the Steiner’s method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projective mappings are induced by quadratic transformations of the three-dimensional projective space of all conics through two distinct points of a projective plane.

scholarly journals Algebraic Approach to Casimir Force Between Two $$\delta $$-like Potentials

2021 ◽  
Author(s):  
Kamil Ziemian
Keyword(s):  
Asymptotic Expansion ◽  
Casimir Force ◽  
Casimir Effect ◽  
Scaling Limit ◽  
Algebraic Approach ◽  
Casimir Energy ◽  
Universal Function ◽  
The Other ◽  
Three Space ◽  
Physical Quantities

AbstractWe analyse the Casimir effect of two nonsingular centers of interaction in three space dimensions, using the framework developed by Herdegen. Our model is mathematically well-defined and all physical quantities are finite. We also consider a scaling limit, in which the problem tends to that with two Dirac $$\delta $$ δ ’s. In this limit the global Casimir energy diverges, but we obtain its asymptotic expansion, which turns out to be model dependent. On the other hand, outside singular supports of $$\delta $$ δ ’s the limit of energy density is a finite universal function (independent of the details of the nonsingular model before scaling). These facts confirm the conclusions obtained earlier for other systems within the approach adopted here: the form of the global Casimir force is usually dominated by the modification of the quantum state in the vicinity of macroscopic bodies.

Implementing The Standards: Multiple Approaches to Geometry: Teaching Similarity

1990 ◽  
Vol 83 (4) ◽  
pp. 274-280
Author(s):  
Sharon L. Senk ◽  
Daniel B. Hirschhorn
Keyword(s):  
Mathematics Education ◽  
Secondary School ◽  
Algebraic Approach ◽  
Physical World ◽  
School Mathematics ◽  
The Other ◽  
Synthetic Approach ◽  
Evaluation Standards

Geometry as a subject uniquely furnishes a language for describing our physical world. It also gives a way visually to represent concepts and relations in other branches of mathematics. Although debate might always ensue on whether geometry should be a full-year secondary school course, the importance of geometry throughout a student's mathematics education seems to have broad acceptance. Consequently, it is not surprising that in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) we find an explicit standard on geometry for all levels K–4, 5–8, and 9–12. In fact, for grades 9–12, two standards on geometry are included—one focusing on a synthetic approach, the other on an algebraic approach.

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.

scholarly journals GENERATING COMPLEX POTENTIALS WITH REAL EIGENVALUES IN SUPERSYMMETRIC QUANTUM MECHANICS

2001 ◽  
Vol 16 (16) ◽  
pp. 2859-2872 ◽  
Author(s):  
B. BAGCHI ◽  
S. MALLIK ◽  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebra ◽  
Algebraic Approach ◽  
Factorization Method ◽  
Bound State ◽  
Complex Potentials ◽  
The Other ◽  
Darboux Transformations ◽  
Weierstrass Function ◽  
First Time

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra [Formula: see text] . This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass ℘ type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT nonsymmetric and its known eigenvalue corresponds to an unbound state.

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