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We introduce partially ordered sets and relate them to random Cantor sets of E-infinity theory. Subsequently we derive the dimensionality of Cantorian-fractal spacetime using posets and E-infinity transfinite Cantor sets.

The article attempts to give a short historical overview of the discovery of fractal-Cantorian space-time starting from the 17th century up to the present. In the last 25 years a great number of scientists worked on fractal space-time notably Garnet Ord in Canada, Laurent Nottale in France and Mohamed El Naschie in England who gave an exact mathematical procedure for the derivation of the dimensionality and curvature of fractal space-time fuzzy manifold.

The interrelation between pure mathematics and physics is very deep and intricate. It is similar to the duality between pure reason and reality. However, the difference between number theory and numerology when used in physics should be clear and easily sorted out. Confusion in this respect is not possible and while numerology must be deplored, number theoretical argument should be an integral part of theoretical physics.

In this work we introduce Kobayashi hyperbolic manifolds with their automorphism groups which are Lie groups. Bounded Stein manifolds are Kobayashi hyperbolic with Lie groups as automorphism groups. One and two-Stein spaces are in close connection with dimensions of El Naschie’s exceptional Lie groups hierarchies. The sum of seventeen one and two-Stein spaces can be expressed in terms of the dimensions of certain Lie groups. The number of high energy elementary particles in an extended standard model and their fuzzy values are found.

In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E -infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals.

In this work we present different mathematical derivations of the inverse electromagnetic fine structure constant from the dimension of the compactified Klein modular curve, the dimension of the exceptional Lie group E8 ⊗ E8 and from the number of states 548, 576 and 672. We show the close connection of the number of particle-like states 548, 576 and 672 with the dimensions of Lie groups, specially with E71/2 and other exceptional Lie groups.

In this work and following some major developments in our understanding of the connection between the exceptional Lie symmetry groups hierarchy and physics, we present finite and infinite dimensional Lie groups and their connections to the orthogonal and unitary groups. The dimension of the Lie group E 7 1 2 is included in the derivation of the number of elementary particle-like super states 576 and 672. Connections to the dimensions of Lie groups and the 72 and 84 elementary particles recently considered by the Author are discussed.

In the present work, different derivations of the 548 isometries of E-infinity symmetry group are presented. The connection between the dimensions of exceptional Lie groups, orthogonal, unitary groups and the 548 is found. The work gives some arguments for deriving the inverse electromagnetic fine structure constant from 1152 bosons and an equal number of fermions following the light cone quantization of the GS action of a super Maxwell theory.

The aim of this article is to present spherical, Euclidean and hyperbolic polyhedra and find some connections of the order of their reflection groups and simple groups such as PGL(2, 7), PGL(2, 8), PGL(2, 7) × C 2 , PSL(2, 31) × C 2 to the number of elementary particles. In the present work we show that a larger number of 72 or 84 elementary particles is consistent with super string theory, M -theory and heterotic string theory. The philosophy of the work is based on El Naschie’s E -infinity interpretation of Emmy Nother’s theorem.

The essay gives arguments for deriving the electromagnetic fine structure constant from maximally symmetric spaces. A connection between the order of some subgroups of the monster simple group, the ratio of the proton mass to the electron mass and the fine structure constant is found. A derivation of the fine structure constant from the number of elements in the Cristoffel symbol and the order of the reflection group F4 is given.

Symmetric positive systems of first-order linear partial differential equations were introduced by Kurt Otto Friedrichs (1958) while treating equations that change their type, like the equations modelling transonic fluid flow. This system, today also known as a Friedrichs system, appeared to be convenient for the numerical treatment of various boundary value problems which also inspired Ern, Guermond and Caplain (2007) to express the theory in terms of operators acting in abstract Hilbert spaces obtaining well-posedness result in this abstract setting. Although some evolution (non- stationary) problems can be treated within this framework, their theory is not suitable for evolution problems like the initial-boundary value problem for the non-stationary Maxwell system, or the Cauchy problem for the symmetric hyperbolic system. Some numerical treatment of such non-stationary problems was done by Burman, Ern and Fernandez (2010), while the existence and uniqueness result was recently provided by Burazin and Erceg. Most classical papers deal with Friedrichs systems in real space setting. We extend both stationary and non-stationary theory to the complex Hilbert space setting and, more general, to complex Banach spaces. We also prove the existence and uniqueness result taking the pivot space H s ( R d ; C r ) Hs(Rd ; Cr) instead of L 2 ( R d ; C r ) L2(Rd ; Cr). Besides linear, semilinear problems can also be treated in complex case. We apply these results to some tipical examples of complex systems of partial differential equation, such as Dirac, Dirac-Klein-Gordon and Dirac-Maxwell system.

In this work, we show the connection between the coupling constants and the dimensions of special orthogonal and unitary groups. A derivation of the exact theoretical value of the fine structure constant from groups SU(3), SO(9) and the Betti number is found.

Quarks are the elementary units of matter and similarly the Betti numbers are the elementary units of the invariant topology of space–time. In the present work we use the Betti numbers of the first and second fuzzy Kahler manifold to write down the coupling constants α ¯ g , α ¯ gs and the inverse electromagnetic fine structure constant α ¯ 0 . The masses of the average pentaquark and the Higgs particle are written in terms of α ¯ g , α ¯ gs and also in terms of the Betti numbers of the first fuzzy Kahler manifold. The paper should help in demystifying the meaning of the value of the inverse electromagnetic constant α ¯ 0 = 137 .

In a series of papers over the last few years El Naschie addressed the question of the minimum and maximum number of elementary particles which a mathematically consistent and a physically meaning full extended standard model should contain. El Naschie’s minimum is 62 particles namely 60 believed to have been already discovered in addition to one Higgs boson and one graviton which are theoretically needed but are not jet experimentally conformed. By contrast the maximum number of 69 particles is although consistent with many quantum field theories based models as well as a classical result by Dyson may not be the only possibility. In the present work we show that a larger number of 72 or even 84 particles are easily shown to be consistent with super string theory and super symmetry. Our work consists of two parts. The first part is a reappraisal of El Naschie’s results and the second is a derivation of the proposed possibility of an upper bound of 72 or 84 elementary particles.

First we give some definitions and theorems about subfactors and Jones’ index. Subsequently the connection between Jones’ index and the geometry of four manifolds is outlined. It is shown that the braid relation can be satisfied only when the parameters τ and q ∈ e 2 π i n , n = 3 , 4 , … ∪ ( 0 , ∞ ) are related by Jones’ equation τ −1 = q + q −1 + 2. The invariant V L of oriented links is shown to describe knots in E-infinity Cantorian space–time. In other words E-infinity may be constructed using the mathematics of subfactors and knot theory.

In this short work, we present a summary about Platonic solids specially about the different dodecahedra. Kepler’s Conjecture and the Dodecahedral Conjecture are of a great importance in circle searching specially in the Poincare dodecahedron space as a cosmological model. Some properties of the 120-cell Coxeter polytop are presented. E-infinity Cantorian space-time may be regarded as a fuzzy version of a hyperbolic manifold based on the 120-cell Coxeter polytop. The topology of fuzzy 4-dimensional polytops are used to give information about the number of the elementary particles in the standard model of high energy physics.

In the present work we show the connections between the topology of four-manifolds, conformal field theory, the mathematical probability theory and Cantorian space-time. In all these different mathematical fields, we find as the main connection the appearance of the golden mean.

In this short paper we discuss some interesting Higgs models. It is concluded that the most likely scheme for the Higgs particles consists of five physical Higgs particles. These are two charged H + , H − and three neutrals h 0 , H 0 , A 0 . Further more the most probably total number of elementary particles for each model is calculated [El Naschie MS. Experimental and theoretical arguments for the number of the mass of the Higgs particles. Chaos, Solitons & Fractals 2005;23:1091–8; El Naschie MS. Determining the mass of the Higgs and the electroweak bosons. Chaos, Solitons & Fractals 2005;24:899–905; El Naschie MS. On 366 kissing spheres in 10 dimensions, 528 P-Brane states in 11 dimensions and the 60 elementary particles of the standard model. Chaos, Solitons & Fractals 2005;24:447–57].

In the present work, first we give some theorems on continued fraction expansions. As shown in [Chaos, Solitons & Fractals 21 (2004) 9–19] the limit set of the Kleinian transformation acting on the E-infinity Cantorian space-time is a set of periodic continued fractions. We expand all possible quark’s mass ratios as periodic continued fractions and following [Chaos, Solitons & Fractals 22 (2004) 749–752], we establish that their vibrations are like fractal polylines.

In the present work, first we give some definitions and theorems on hyperbolic maps, structurally stability and deterministic chaos. The limit set of the Kleinian transformation acting on the E-infinity Cantorian space–time turned out to be a set of periodic continued fractions as shown in [Chaos, Solitons & Fractals, 21 (2004) 9]. That set has a hyperbolic structure and is structurally stable. Subsequently, we show that the appearance of transversal homoclinic points induces a chaotic behavior in that set.

In the present work we establish a connection between El Naschie's E-infinity theory and Stieltjes solution of the problem on the distribution of mass along a line and KAM theory following Gantmacher and Krein mechanical interpretation of Stieltjes' classical research on the subject.

In the present work we first give a general representation of the derivatives of the irrational number φ , for instance 1 φ , 1 φ 2 , 1 φ 3 etc., as periodic continued fractions. Any irrational number can then be expanded in an infinite continued fraction. The limit set of the Kleinian transformation acting on the E -infinity Cantorian spacetime turned out to be this set of periodic continued fractions, consequently the vacuum of the E -infinity is described by this limit set. As discussed by El Naschie, every particle can be interpreted geometrically as a scaling of another. This is done using the topology of hyperbolic Kleinian space of VAK, which is nothing but our limit set. Here we will present the ratios of the theoretical masses of certain elementary particles to that of some chosen particles in term of φ . Many of these masses are quite close to integer multiples of the mass of a chosen particle. Finally we discuss the possibility of new transfinite, axion-like particles as discussed recently by Krauss and El Naschie [Quintessence, Vintage, London, 1999].

Quantum space time as given by topology and geometry of El Naschie’s e(∞) theory must be regarded as fundamentally fuzzy. It’s geometry and topology belong to the mathematical category of fuzzy logic and fuzzy set theory. All lines are fuzzy fractal lines in fuzzy spaces and all exact values are exact fuzzy expectation values. That way we remove many paradoxes and contradictions in the standard model of high energy particle physics.

In the present work we give an introduction to the e(∞) Cantorian space–time theory. In this theory every particle can be interpreted as a scaling of another particle. Some particles are a scaling of the proton and are expressed in terms of φ and α 0 . Following the VAK suggestion of El Naschie, the limit sets of Kleinian groups are Cantor sets with Hausdorff dimension φ or a derivative of φ such as 1/φ, 1/φ2, 1/φ3, etc. Consequently and using e(∞) theory, the mass spectrum of elementary particles may be found from the limit set of the Mobius–Klein geometry of quantum space–time as a function of the golden mean φ=( 5 −1)/2=0.618033989 as discussed recently by Datta (see Chaos, Solitons & Fractals 17 (2003) 621–630).

In the present work we give a classical nested mechanical model and corresponding expressions for the theoretical masses of elementary particles, including the masses of quarks as being the joint eigenvalues of combined vibrating sets using the Southwell and the Dunkerly theorems. The role played by the golden mean in KAM theory and consequently our present model is also discussed.

In this note we will give an expression of masses of the elementary particles of the standard model in terms of the golden mean. This is the value φ=( 5 −1)/2 , which corresponds to the Hausdorff dimension of a single random Cantor set. El Naschie’s transfinite field theory which we will call the golden field theory is based on this fact and is the basis of the present paper.

We estimate the dimensionality of the mean sphere in infinite dimensional space using the so called Leech–Conway lattices. The numerical results improve previous calculations by El Naschie [Chaos, Solitons & Fractals 9 (8) (1998) 1445–1471] and confirms the essential four dimensionality of the expectation sphere.

Using a Feynman path integral-like summing procedure we can establish two exact statistical averages for the topological and the Hausdorff dimension of a fractal-Cantorian spacetime. By equating the two expressions we find the Hausdorff and the topological dimension to be 4 + ϕ3 and 4, respectively, where ϕ = ( 5 - 1 ) / 2 .

In various recent formal and informal lectures and discussions Prof. M.S. El Naschie expressed the sentiment that the 496 of E8E8 being a perfect number is not a coincidence and there is deep number theoretical connections to physics in this case. In this short work, we first introduce the perfect numbers. We show the close connection between the fourth perfect number and the sum of the first four perfect numbers to the number of particles in the standard model. As conjectured by El Naschie, it seems very unlikely that 496 being one of only four perfect numbers in the first 33 million numbers is a coincidence.

The present work gives first a review of the mathematical theory of finite and infinite dimensional topological spaces. Subsequently we connect the discussion with E-infinity theory and the theory of partially ordered sets. Finally, we contemplate the relevance of abstract results for quantum gravity.