Finsler space is differentiable manifold for which Minkowski space is the
fiber of the tangent bundle. To understand structure of the reference frame in
Finsler space, we need to understand the structure of orthonormal basis in
Minkowski space.

In this short paper we will show, via elementary arguments, the equivalence
of the Twin Prime Conjecture to a problem which might be simpler to prove. Some
conclusions are drawn, and it is shown that proving the Twin Prime Conjecture
is equivalent to proving that there cannot be an infinite string of consecutive
natural numbers satisfying some specified equations.

We propose a demonstration of Fermat's Last Theorem using a technique
certainly within the reach of Fermat himself, and then we infer that this is
the marvelous proof that Fermat claimed to have. An original procedure to find
all the infinite Pythagorean triples will emerge spontaneously, too.

We give a simple finitary proof that every Goodstein Sequence G(m) over the
natural numbers must terminate finitely. We note first that if G(p) terminates
finitely then so does G(m) for all natural numbers m<p. We then show that for
every natural number k there is a natural number u such that the (k-1)th term
of G(u) is k^k, and that G(u) terminates finitely.

The main results of this paper are the construction, both rigourous and
intuitive, of "the" intrinsic extension of the set of non negative integers N
and the smallest over-field of R set which is continue (according to
R.Dedekind). The aim of this article is to provide a new Non Standard Analysis,
very simple, which is compared in the Introduction with A.Robinson's and
J.H.Conway's

In this article we use the Desargues' theorem and its reciprocal to solve two
problems.

In a previous paper we have introduced the ortho-homological triangles, which
are triangles that are orthological and homological simultaneously.

In this article we call attention to two remarkable ortho-homological
triangles (the given triangle and its first Brocard's triangle), and using the
Sondat's theorem relative to orthological triangles, we emphasize on four
important collinear points in the geometry of the triangle.

The new classes of super special codes are constructed in this book using the
specially constructed super special vector spaces. These codes mainly use the
super matrices. These codes can be realized as a special type of concatenated
codes. This book has four chapters. In chapter one basic properties of codes
and super matrices are given. A new type of super special vector space is
constructed in chapter two of this book. Three new classes of super special
codes namely, super special row code, super special column code and super
special codes are introduced in chapter three.

We give a simple proof of the Poincar\'e conjecture by using the contact
Ricci flow associated with the Reeb vector field.

Mathematical theories are classified in two distinct classes : {\it rigid},
and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory,
topology, category theory, etc., have a basic concept - given for instance by a
set of axioms - from which all the other concepts are defined in a unique way.
Non-rigid theories, like ring theory, certain general enough pseudo-topologies,
etc., have a number of their concepts defined in a more free or relatively
independent manner of one another, namely, with {\it compatibility} conditions
between them only.

In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle
$ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal
triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the
lines $AA_1, BB_1, CC_1$ are concurrent), then the triangles $ABC$ and
$A_2B_2C_2$ are also homological.

It is well known that the distribution of the prime numbers plays a central
role in number theory. It has been known, since Riemann's memoir in 1860, that
the distribution of prime numbers can be described by the zero-free region of
the Riemann zeta function $\zeta(s)$. This function has infinitely many zeros
and a unique pole at $s = 1$. Those zeros at $s = -2, -4, -6, ...$ are known as
trivial zeros. The nontrivial zeros of $\zeta(s)$ are all located in the
so-called critical strip $0<\Re(s) <1$.

It is shown that there is no non-trivial cycle and divergent cycle in the
Collatz sequences, or frequently called 3x + 1 mapping . This proves that the
conjecture by Collatz is true.

We obtained the probabilities for the values of the M\"obius function for
arbitrary numbers and found that the asymptotic densities of the squarefree
integers among the odd and even numbers are $8/\pi^2$ and $4/\pi^2$,
respectively. It is determined that statistics of successive outcomes of the
M\"obius function for very large squarefree odd and even numbers behaves
similar to statistics of heads and tails of two flipping coins. These
preliminary results are giving arguments supporting the Riemann Hypothesis. Its
plausibility is based on statistical phenomena for integers.

The properties of the modulus and arguments of the Riemann zeta function are
examined to establish the behavior of the zero level curves. These properties
are used to locate the non trivial zeros of the zeta function. It is shown that
a pair of zeros off the critical line is inconsistent with the Laplacian of the
zeta function, thus establishing the Riemann Hypothesis.

We propose a Relational Calculus based on the concept of unary relation. In
this Relational Calculus different axiomatic systems converge to a model called
Dynamic Generative System with Symmetry (DGSS). In DGSS we define the concepts
of relational set and function and prove that extensionality and the
substitution property of equality are theorems of DGSS. As a first
exemplification of DGSS, we construct a model of natural numbers without
relying on Peano's Axioms. Eventually, some new clarifications regarding the
nature of the number zero are given.

Four new probability models are derived which generalize the common
univariate continuous distributions. Classical distributional measures are
derived from Hoel, et al., Introduction to Probability Theory, 1971. Measures
include probability density function, moments generating function, cumulative
distribution function, inverse distributions, skewness, kurtosis, change of
variable distributions, log distributions. Maximum likelihood estimation
technique is briefly outlined.

Cantor's diagonal argument makes use of an hypothetical table T containing
all real numbers within the real interval (0, 1). That table can be easily
redefined in order to ensure it contains at least all rational numbers within
(0, 1). In these conditions, could the rows of T be reordered so that the
resulting diagonal and antidiagonal were rational numbers? In that case not
only the set of real numbers but also, and for the same reason, the set of
rational numbers would be non denumerable.

Arguments on the need, and usefulness, of going beyond the usual
Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are
presented. The motivation comes, among others, from well known {\it topological
type processes}, or in short TTP-s, in the theories of Measure, Integration and
Ordered Spaces. These TTP-s, as shown by the classical characterization given
by the {\it four Moore-Smith conditions}, can {\it no longer} be incorporated
within the usual HKB topologies.

Recently uncovered second derivative discontinuous solutions of the simplest
linear ordinary differential equation define not only an nonstandard extension
of the framework of the ordinary calculus, but also provide a dynamical
representation of the ordinary real number system. Every real number can be
visualized as a living cell -like structure, endowed with a definite
evolutionary arrow. We discuss the relevance of this extended calculus in the
study of living systems. We also present an intelligent version of the Newton's
first law of motion.

The formulation of a new analysis on a zero measure Cantor set $C (\subset
I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$
exploiting the concept of {\em relative} infinitesimals and a scale invariant
ultrametric valuation of the form $\log_{\varepsilon^{-1}} (\varepsilon/x) $
for a given scale $\varepsilon>0$ and infinitesimals $0<x<\varepsilon, x\in
I\backslash C$. Using this new absolute value, a valued (metric) measure is
defined on $C $ and is shown to be equal to the finite Hausdorff measure of the
set, if it exists.

The origin of complex structures, randomness, and irreversibility are
analyzed in the scale free SL(2,R) analysis, which is an extension of the
ordinary analysis based on the recently uncovered scale free $C^{2^n-1}$
solutions to linear ordinary differential equations. The role of an intelligent
decision making is discussed. We offer an explanation of the recently observed
universal renormalization group dynamics at the edge of chaos in logistic maps.
The present formalism is also applied to give a first principle explanation of
1/$f$ noise in electrical circuits and solid state devices.

We propose a new algorithm to obtain max flow for the multicommodity flow.
This algorithm utilizes the max-flow min-cut theorem and the well known
labeling algorithm due to Ford and Fulkerson [1]. We proceed as follows: We
select one source/sink pair among the n distinguished source/sink pairs at a
time and treat the given multicommodity network as a single commodity network
for such chosen source/sink pair.

The paper suggests a slightly more rigor justification to Wang et al.'s work
from 2007, introduces the Slanted Integral (via a combination of Riemann's
integral and Lebesgue's integral), suggests that it is assentially a dual,
integral calculus-based form of the Line Integral, outlines uncountably many
novel integration methods, and outlines uncountably many novel proofs to all
the known theorems of integral calculus. The author suggests that a better
comprehension to the ideas in this paper is achievable via first reading Wang
et al.'s extraordinary paper from 2007.

We prove Cramer's conjecture that $p_{n+1} - p_n = O(\ln(p_n)^2)$, where
$p_n$ is the $n^{th}$ prime and $\ln(x)$ is the natural logarithm of $x$. Also,
Legendre's conjecture follows from this, that is, there exists at least one
prime between two successive square numbers.

Withdraw by the author due to errors

In analogy with the Poisson algebra of the quadratic forms on the symplectic
plane, and the notion of duality in the projective plane introduced by Arnold,
where the concurrence of the triangle altitudes is deduced from the Jacobi
identity, we consider the Poisson algebras of the first degree harmonics on the
sphere, the pseudo-sphere and on the hyperboloid, to obtain analogous duality
notions and similar results for the spherical, pseudo-spherical and hyperbolic
geometry.

Migration has various dimensions; urbanization due to migration is one of
them. In Rajasthan State, district level analysis of urbanization due to
migrants shows trend invariably for all districts of the state, though the
contribution in urbanization by migrants varies from district to district. In
some districts the share of migrants moving to urban areas is very impressive,
in others it is not that much high. The migrants' contribution is on the
raising over the decades.

We shall show that for positive integers n >= 2, the Riemann Zeta Function
zeta(n) is irrational. We shall deduce that from an integral based on
fractional parts and then use the inequality | x - u/v | < v^(-2) to show
irrationality.

In [1], we give Dickson's conjecture on $N^n$. In this paper, we further give
Dickson's conjecture on $Z^n$ and obtain an equivalent form of Green-Tao's
conjecture [2]. Based on our work, it is possible to establish a general theory
that several multivariable integral polynomials on $Z^n$ represent
simultaneously prime numbers for infinitely many integral points and generalize
the analogy of Chinese Remainder Theorem in [3].

The famous four color theorem states that for all planar graphs, every vertex
can be assigned one of 4 colors such that no two adjacent vertices receive the
same color. Since Francis Guthrie first conjectured it in 1852, it is until
1976 with electronic computer that Appel and Haken first gave a proof by
finding and verifying 1936 reducible unavoidable sets, and a simplified proof
of Robertson, Sanders, Seymour and Thomas in 1997 only involved 633 reducible
unavoidable sets, both proofs could not be realized effectively by hand.

The aim of this paper is to try to establish a generic model for the problem
that several multivariable number-theoretic functions represent simultaneously
primes for infinitely many integral points. More concretely, we introduced
briefly the research background-the history and current situation-from Euclid's
second theorem to Green-Tao theorem.

This paper proves a generalization of the Butterfly Theorem, a classical
Euclidean result, which is valid in the complex projective plane.

To factor an integer N, given that it is equal to the product of two primes,
it suffices to find an integer d satisfying a certain simple numerical test. In
this approach, the factorization problem equates to the problem of designing an
optimal data base of values d to be tested.

In this article we give, for the fist time the solution of the general
difference equation of 2-degree. We also give as application the expansion of a
continued fraction into series, which was first proved, found in the past by
the author.

In this paper we propose a method of solving a Nonlinear Diophantine Equation
by converting it into a System of Diophantine Linear Equations.

In this note, we present an elementary proof of the uniqueness of the
solutions of the initial value problems of linear ordinary differential
equations (odes).

Consider two circles, externally tangential,and with integer radii R1, R2;
and with R1>R2.The two circles have three tangent lines in common, one of them
being T1T2. If M is the midpoint of T1T2, and K the point of intersection of
the lines C1C2 and T1T2;then 16 right triangles are formed(C1 and C2 are the
two circle centers), see Figure 1.In Section 6 of this paper, we find the
precice form the two integers R1 and R2 must have, in order that the sixteen
aforementioned right triangles be Pythagorean.

The presence of infinitesimals is traced back to some of the most general
algebraic structures, namely, semigroups, and in fact, magmas, [1], in which
none of the structures of linear order, field, or the Archimedean property need
to be present. Such a clarification of the basic structures from where
infinitesimals can in fact emerge may prove to have a special importance in
Physics, as seen in [4-16].

The paper present a way to show that the subset sum problem cannot be solved
in polynomial time.

Existence results for Hilbert's problem 13th prove that arbitrary continue
function of many variables can be represented as a superposition of continue
functions of one variable or of continue functions of two variables.
Constructive results for discrete functions are given in this paper. So any
equation constructed by discrete functions can be given solution represented as
a superposition of discrete functions of one variable or of two variables.

Previous derivations of the sum and product rules of probability theory
relied on the algebraic properties of Boolean logic. Here they are derived
within a more general framework based on lattice theory. The result is a new
foundation of probability theory that encompasses and generalizes both the Cox
and Kolmogorov formulations. In this picture probability is a bi-valuation
defined on a lattice of statements that quantifies the degree to which one
statement implies another.

We present an elementary proof that the qualitative picture of a linear
hyperbolic flow is insensitive to slight measurements errors in its associated
vector field.

Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy
sequences are common notations for number systems. In this note, first, it is
proven that both simple continued fractions and base-b expansions fail to
denote real numbers and thus lack logic; second, it is shown that Dedekind cuts
and Cauchy sequences fail to join in algebraical operations and thus lack
intuition; third, we construct a logical calculus and deduce numbers to
intuitively and logically denote number systems.

Based on Euclid's algorithm, we find a kind of special sequences which play
an interesting role in the study of primes. We call them W Sequences. They not
only ties up the distribution of primes in short interval but also enables us
to give new weakened forms of many classical problems in Number Theory. The
object of this paper is to provide a brief introduction and preliminary
analysis on this kind of special sequences.

The object of this paper is to generalize a theorem on the binomial
coefficient [4] to the case in an arithmetic progression. We will also give a
slightly stronger result than Langevin's [2].

In this note, we generalize an ancient Greek inequality about the sequence of
primes to the cases of arithmetic progressions even multivariable polynomials
with integral coefficients. We also refine Bouniakowsky's conjecture [16] and
Conjecture 2 in [22]. Moreover, we give two remarks on conjectures in [22]

In this book the authors introduce three new types of fuzzy model called the
super column Fuzzy Relational Model using super column matrices, super row
fuzzy relational model using super row matrices and super mixed fuzzy
relational model using supermatrices. These new models are used to study the
role of media on 27 percent reservation for the other backward classes in the
educational institutions run by the Indian central Government. This book has
four chapters. Chapter one introduces the new notion of super fuzzy relational
models using supermatrices.

In this paper I explore the set of quaternion algebras over field. In
contrast to quaternion algebra H=E(R,-1,-1), linear function of quaternion
algebra E(C,-1,-1) over complex field satisfies to the Cauchy--Riemann
equations.

We define a sequence ${a_n}$ by $a_1=a$ and $a_{n+1}=P(a_n)$, where $P(x)$ is
a polynomial with real coefficients. We then find out for which values $a$ and
for which polynomials $P(x)$ this sequence will be constant after a certain
rank.

In this short note we prove that certain generalized Mertens conjectures are
still false, and as a consequence, the Riemann hypothesis is also false.

In 1998, in the winter issue of the journal Mathematics and Computer
education (see [1]), Monte Zerger posed the following problem. He had noticed
the Pythagorean triple (216,630,666);(216)^2+(630)^2=(666)^2. Note that 216=6^3
and 666 is the hypotenuse length. The question was then, whether there existed
a digit d and a positive integer k(other than the above); such that d^k is the
leglength of a Pythagorean triangle whose hypotenuse length has exactly k
digits, each being equal to d. In 1999, F.Luca and P.Bruckman, answered the
above question in the negative.

Step by step a substitution of the well known Brouwer choice sequence will be
constructed. It begins with an establishing of quasi alternating prime number
series followed by a construction of a virtual sequence in sense of the virtual
set definition. The last step gives reasons for why this virtual sequence
substitutes the choice sequence created by L. E. J. Brouwer.