He was then appointed to serve as a technician, technical adviser, director of construction management department, and then finally, the general engineer in construction project, respectively. He obtained an undergraduate diploma in applied mathematics and Bachelor of Science of Peking University in After then, he attended postgraduate courses in graph theory, combinational mathematics, and other areas.

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Yanpei Liu at Northern Jiaotong University in , and conducted postdoctoral research on automorphism groups of maps and surfaces at the Chinese Academy of Mathematics and System 1 Received. Feng Tian from to In his postdoctoral report, Dr.

Mao pointed out that the motivation for developing mathematics for understanding the reality of things is a combinatorial notion, i. Generally, a thing is complex and hybrid with other things but the understanding of human beings is limitation, which results in the difficult to hold on the true face of things in the world. However, there always exist universal connection between things. We can separate mathematical questions into three ranks: Rank 1 they contribute to all sciences.

Rank 2 they contribute to all or several branches of mathematics. Rank 3 they contribute only to one branch of mathematics, for instance, just to the graph theory or combinatorial theory.

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Classical combinatorics is just a rank 3 mathematics by this view. This conclusion is despair for researchers in combinatorics, also for me 5 years ago. Whether can combinatorics be applied to other mathematics or other sciences?

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## Smarandache Geometries and Map Theory with Applications (I) English and Chinese Languages

Although become a universal genus in science is nearly impossible, our world is a combi- natorial world. A combinatorician should stand on all mathematics and all sciences, not just on classical combinatorics and with a real combinatorial notion, i. This notion requires us answering three questions for solving a combinatorial problem before. What is this problem working for? What is its objective? Af- ter these works be well done, modern combinatorics can applied to all sciences and all sciences are combinatorialization. The metrical combinatorics and mathematics combinatorialization There is a prerequisite for the application of combinatorics to other mathematics and other sciences, i.

The combinatorial conjecture for mathematics, abbreviated to CCM is stated in the fol- lowing. Conjecture 2. Remark 2. Therefore, a branch in mathematics can not be ended if it has not been combinatorialization and all mathematics can not be ended if its combinatorialization has not completed. A typical example for the combinatorialization of classical mathematics is the combinatorial map theory, i. Combinatorially, a surface is topological equivalent to a polygon with even number of edges by identifying each pairs of edges along a given direction on it.

A map M is a connected topological graph cellularly embedded in a surface S. For each vertex of a map M , its valency is defined to be the length of the orbits of P action on a quadricell incident with u. For example, the graph K4 on the tours with one face length 4 and another 8 shown in Fig. Therefore, combinatorial maps are the combinatorialization of surfaces.

Many open problems are motivated by the CCM Conjecture. For example, a Gauss mapping among surfaces is defined as follows. Now the questions are i what is its combinatorial meaning of the Gauss mapping? How to realizes it by combi- natorial maps? By the CCM Conjecture, the following questions should be considered. One can see references [15] and [16] for more open problems for the classical mathematics motivated by this CCM Conjecture, also raise new open problems for his or her research works. The contribution of combinatorial speculation to mathematics 3.

The combinatorialization of algebra By the view of combinatorics, algebra can be seen as a combinatorial mathematics itself. The combinatorial speculation can generalize it by the means of combinatorialization.

## Notes on Differential Geometry (Van Nostrand Reinhold - Database Error Library

For this objective, a Smarandache multi-algebraic system is combinatorially defined in the following definition. Definition 3. An example of multi-algebra systems is constructed by a finite additive group. Some of them are explained in the following. Then we get a generalization of the Lagrange theorem of finite group.

For multi-algebra systems with two or more operations on one set, we introduce the conception of multi-rings and multi-vector spaces in the following. If for any integer e is called a multi-filed. Similar to multi-groups, we can also obtain results for multi-rings and multi-vector spaces to generalize classical results in rings or linear spaces. Certainly, results can be also found in the references [17] and [18]. The combinatorialization of geometries First, we generalize classical metric spaces by the combinatorial speculation. We generalized two well-known results in metric spaces.

We get the Banach fixed-point theorem again. Corollary 3. Then T has just one fixed point. Smarandache geometries were proposed by Smarandache in [29] which are generalization of classical geometries, i. In general, Smarandache geometries are defined in the next. A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom For example, let us consider an euclidean plane R2 and three non-collinear points A, B and C.

## SMARANDACHE GEOMETRIES & MAP THEORY WITH APPLICATIONS(I)

Define s-points as all usual euclidean points on R2 and s-lines as any euclidean line that passes through one and only one of points A, B and C. Then this geometry is a Smarandache geometry because two axioms are Smarandachely denied comparing with an Euclid geometry: i The axiom A5 that through a point exterior to a given line there is only one parallel passing through it is now replaced by two statements: one parallel and no parallel.

Let L be an s-line passing through C and is parallel in the euclidean sense to AB. Notice that through any s-point not lying on AB there is one s-line parallel to L and through any other s-point lying on AB there is no s-lines parallel to L such as those shown in Fig. Notice that through any two distinct s-points D, E collinear with one of A, B and C, there is one s-line passing through them and through any two distinct s-points F, G lying on AB or non-collinear with one of A, B and C, there is no s-line passing through them such as those shown in Fig.

A Smarandache n-manifold is an n-dimensional manifold that supports a Smarandache geometry. Generally, we can ever generalize the ideas in Definitions 3. P Theorem 3. Therefore, we get a relation for Smarandache geometries with Finsler or Riemann geometry.

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Theorem 3. The contribution of combinatorial speculation to theoretical physics The progress of theoretical physics in last twenty years of the 20th century enables human beings to probe the mystic cosmos: where are we came from? Today, these problems still confuse eyes of human beings. Accompanying with research in cosmos, new puzzling problems also arose: Whether are there finite or infinite cosmoses? Are there just one?

### Florentin Smarandache Research

What is the dimension of the Universe? We do not even know what the right degree of freedom in the Universe is, as Witten said [3]. We are used to the idea that our living space has three dimensions: length, breadth and height, with time providing the fourth dimension of spacetime by Einstein. Applying his princi- ple of general relativity, i.

The following diagram describes the developing process of the Universe in different periods after the Big Bang. Here, a brane is an object or subspace which can have various spatial dimensions. We mainly discuss line elements in differential forms in Riemann geometry. By a geomet- rical view, these p-branes in M-theory can be seen as volume elements in spaces.

Definition 4. For example, the expansion factor is 3. According to M-theory, the evolution picture of our cosmos started as a perfect 11 dimen- sional space. However, this 11 dimensional space was unstable. The original 11 dimensional space finally cracked into two pieces, a 4 and a 7 dimensional subspaces. The cosmos made the 7 of the 11 dimensions curled into a tiny ball, allowing the remaining 4 dimensions to inflate at enormous rates, the Universe at the final. Combinatorial Speculations and Combinatorial Conjecture for Mathematics 15 4. The combinatorial cosmos The combinatorial speculation made the following combinatorial cosmos in the reference [17].

In Fig.