Geometry

1104 Submissions

[18] viXra:1104.0079 [pdf] submitted on 19 Apr 2011

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

Authors: Linfan Mao
Comments: 16 pages

A tendering is a negotiating process for a contract through by a tenderer issuing an invitation, bidders submitting bidding documents and the tenderer accepting a bidding by sending out a notification of award. As a useful way of purchasing, there are many norms and rulers for it in the purchasing guides of the World Bank, the Asian Development Bank,..., also in contract conditions of various consultant associations. In China, there is a law and regulation system for tendering and bidding. However, few works on the mathematical model of a tendering and its evaluation can be found in publication. The main purpose of this paper is to construct a Smarandache multi-space model for a tendering, establish an evaluation system for bidding based on those ideas in the references [7] and [8] and analyze its solution by applying the decision approach for multiple objectives and value engineering. Open problems for pseudo-multi-spaces are also presented in the final section.
Category: Geometry

[17] viXra:1104.0078 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(IV)

Authors: Linfan Mao
Comments: 26 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers (part IV)
Category: Geometry

[16] viXra:1104.0077 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(III)

Authors: Linfan Mao
Comments: 74 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers (part III)
Category: Geometry

[15] viXra:1104.0076 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(II)

Authors: Linfan Mao
Comments: 78 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n &t; 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.
Category: Geometry

[14] viXra:1104.0075 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(I)

Authors: Linfan Mao
Comments: 47 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.
Category: Geometry

[13] viXra:1104.0074 [pdf] submitted on 19 Apr 2011

On Multi-Metric Spaces

Authors: Linfan Mao
Comments: 9 pages

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with classical metric spaces, the conception of multi-metric space is introduced. Some characteristics of a multi-metric space are obtained and Banach's fixed-point theorem is generalized in this paper.
Category: Geometry

[12] viXra:1104.0073 [pdf] submitted on 19 Apr 2011

On Algebraic Multi-Vector Spaces

Authors: Linfan Mao
Comments: 7 pages

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with linear vector spaces in classical linear algebra, the conception of multi-vector spaces is introduced. Some characteristics of a multi-vector space are obtained in this paper.
Category: Geometry

[11] viXra:1104.0072 [pdf] submitted on 19 Apr 2011

On Algebraic Multi-Ring Spaces

Authors: Linfan Mao
Comments: 8 pages

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with rings in classical ring theory, the conception of multi-ring spaces is introduced. Some characteristics of a multi-ring space are obtained in this paper
Category: Geometry

[10] viXra:1104.0071 [pdf] submitted on 19 Apr 2011

On Algebraic Multi-Group Spaces

Authors: Linfan Mao
Comments: 8 pages

A Smarandache multi-space is a union of n spaces A1,A2, ... ,An with some additional conditions holding. Combining classical of a group with Smarandache multi-spaces, the conception of a multi-group space is introduced in this paper, which is a generalization of the classical algebraic structures, such as the group, filed, body,..., etc.. Similar to groups, some characteristics of a multi-group space are obtained in this paper.
Category: Geometry

[9] viXra:1104.0070 [pdf] submitted on 19 Apr 2011

A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

Authors: Linfan Mao
Comments: 16 pages

As we known, the Seifert-Van Kampen theorem handles fundamental groups of those topological spaces (see paper)
Category: Geometry

[8] viXra:1104.0069 [pdf] submitted on 19 Apr 2011

A Generalization of Stokes Theorem on Combinatorial Manifolds

Authors: Linfan Mao
Comments: 16 pages

For an integer m > 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for(...) there is a local chart (see paper) where Bnij is an nij -ball for integers 1 < j < s(p) < m. Integral theory on these smoothly combinatorial manifolds are introduced. Some classical results, such as those of Stokes' theorem and Gauss' theorem are generalized to smoothly combinatorial manifolds in this paper.
Category: Geometry

[7] viXra:1104.0068 [pdf] submitted on 19 Apr 2011

Geometrical Theory on Combinatorial Manifolds

Authors: Linfan Mao
Comments: 37 pages

For an integer m ≥ 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for (...), there is a local chart (see paper) where Bnij is an nij -ball for integers 1 ≤ j ≤ s(p) ≤ m. Topological and differential structures such as those of d-pathwise connected, homotopy classes, fundamental d-groups in topology and tangent vector fields, tensor fields, connections, Minkowski norms in differential geometry on these finitely combinatorial manifolds are introduced. Some classical results are generalized to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed and geometrical inclusions in Smarandache geometries for various geometries are also presented by the geometrical theory on finitely combinatorial manifolds in this paper.
Category: Geometry

[6] viXra:1104.0062 [pdf] submitted on 20 Apr 2011

Pseudo-Manifold Geometries with Applications

Authors: Linfan Mao
Comments: 15 pages.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969), i.e., an axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways and a Smarandache n-manifold is a n-manifold that support a Smarandache geometry. Iseri provided a construction for Smarandache 2-manifolds by equilateral triangular disks on a plane and a more general way for Smarandache 2-manifolds on surfaces, called map geometries was presented by the author in [9]-[10] and [12]. However, few observations for cases of n ≥ 3 are found on the journals. As a kind of Smarandache geometries, a general way for constructing dimensional n pseudo-manifolds are presented for any integer n ≥ 2 in this paper. Connection and principal fiber bundles are also defined on these manifolds. Following these constructions, nearly all existent geometries, such as those of Euclid geometry, Lobachevshy-Bolyai geometry, Riemann geometry, Weyl geometry, Kähler geometry and Finsler geometry, ...,etc., are their sub-geometries.
Category: Geometry

[5] viXra:1104.0061 [pdf] submitted on 20 Apr 2011

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

Authors: Linfan Mao
Comments: 19 pages.

Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an more important work for mathematician is to apply combinatorics to other mathematics and other sciences not merely to find combinatorial behavior for objectives. Recently, such research works appeared on journals for mathematics and theoretical physics on cosmos. The main purpose of this paper is to survey these thinking and ideas for mathematics and cosmological physics, such as those of multi-spaces, map geometries and combinatorial cosmoses, also the combinatorial conjecture for mathematics proposed by myself in 2005. Some open problems are included for the 21th mathematics by a combinatorial speculation.
Category: Geometry

[4] viXra:1104.0060 [pdf] submitted on 20 Apr 2011

Parallel Bundles in Planar Map Geometries

Authors: Linfan Mao
Comments: 16 pages.

Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one's intuition. But in a planar map geometry, a kind of the Smarandache geometries, the situation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behaviors of parallel bundles in planar map geometries, a generalization of parallel lines in plane geometry and obtains characteristics for parallel bundles.
Category: Geometry

[3] viXra:1104.0059 [pdf] submitted on 20 Apr 2011

A New View of Combinatorial Maps by Smarandache's Notion

Authors: Linfan Mao
Comments: 19 pages.

On a geometrical view, the conception of map geometries is introduced, which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surfaces. Some open problems related combinatorial maps with the Riemann geometry and Smarandache geometries are presented.
Category: Geometry

[2] viXra:1104.0054 [pdf] submitted on 18 Apr 2011

Microscopes and Telescopes for Theoretical Physics : How Rich Locally and Large Globally is the Geometric Straight Line ?

Authors: Elemér E Rosinger
Comments: 31 pages.

One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is but one of the possible mathematical models chosen upon certain reasons. And as is known, there are a a variety of mathematical models of GSL, among them given by nonstandard analysis, reduced power algebras, the topological long line, or the surreal numbers, among others. As shown in this paper, the GSL can allow coordinatisations which are arbitrarily more rich locally and also more large globally, being given by corresponding linearly ordered sets of no matter how large cardinal. Thus one can obtain in relatively simple ways structures which are more rich locally and large globally than in nonstandard analysis, or in various reduced power algebras. Furthermore, vector space structures can be defined in such coordinatisations. Consequently, one can define an extension of the usual Differential Calculus. This fact can have a major importance in physics, since such locally more rich and globally more large coordinatisations of the GSL do allow new physical insights, just as the introduction of various microscopes and telescopes have done. Among others, it and general can reassess special relativity with respect to its independence of the mathematical models used for the GSL. Also, it can allow the more appropriate modelling of certain physical phenomena. One of the long vexing issue of so called "infinities in physics" can obtain a clarifying reconsideration. It indeed all comes down to looking at the GSL with suitably constructed microscopes and telescopes, and apply the resulted new modelling possibilities in theoretical physics. One may as well consider that in string theory, for instance, where several dimensions are supposed to be compact to the extent of not being observable on classical scales, their mathematical modelling may benefit from the presence of infinitesimals in the mathematical models of the GSL presented here. However, beyond all such particular considerations, and not unlikely also above them, is the following one : theories of physics should be not only background independent, but quite likely, should also be independent of the specific mathematical models used when representing geometry, numbers, and in particular, the GSL. One of the consequences of considering the essential difference between the GSL and its various mathematical models is that what appears to be the definitive answer is given to the intriguing question raised by Penrose : "Why is it that physics never uses spaces with a cardinal larger than that of the continuum ?".
Category: Geometry

[1] viXra:1104.0053 [pdf] submitted on 17 Apr 2011

A New Proof of Menelaus's Theorem of Hyperbolic Quadrilaterals in the Poincaré Model of Hyperbolic Geometry

Authors: Catalin Barbu, Florentin Smarandache
Comments: 6 pages.

In this study, we present a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, and a proof for the transversal theorem for triangles.
Category: Geometry