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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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