[7] **viXra:1501.0229 [pdf]**
*submitted on 2015-01-25 14:47:30*

**Authors:** Bolonkin A.A.

**Comments:** 111 Pages.

Это краткий конспект лекций по курсу "Теория оптимальных систем", прочитанных автором для студентов старших курсов, аспирантов, инженеров и преподавателей в 1967-1969гг в Московском авиационном технологическом институте и в 1969-1971гг в МВТУ им. Баумана.
Автор излагает принципиально новые методы оптимизации, поиска глобального минимума и применяет их в технических задачах автоматики, динамики полета, авиации, космонавтике, комбинаторике, в теории игр, задачах с противодействием и т.п.
This is a brief summary of lectures for the course "Theory of optimal systems", read by the author for senior students, graduate students, engineers and lecturers in 1967-1969 at the Moscow Aviation Institute of Technology and in 1969-1971 at MVTU. named Bauman.
The author presents the fundamentally new optimization techniques, search for the global minimum and applies them to the technical problems of automation, flight dynamics, aviation, aerospace, combinatorics, game theory, problems with resistance, etc.

**Category:** General Mathematics

[6] **viXra:1501.0228 [pdf]**
*submitted on 2015-01-25 12:08:37*

**Authors:** Bolonkin A.A.

**Comments:** 111 Pages.

Это краткий конспект лекций по курсу "Теория оптимальных систем", прочитанных автором для студентов старших курсов, аспирантов, инженеров и преподавателей в 1967-1969гг в Московском авиационном технологическом институте и в 1969-1971гг в МВТУ им. Баумана.
Автор излагает принципиально новые методы оптимизации, поиска глобального минимума и применяет их в технических задачах автоматики, динамики полета, авиации, космонавтике, комбинаторике, в теории игр, задачах с противодействием и т.п.

**Category:** General Mathematics

[5] **viXra:1501.0223 [pdf]**
*replaced on 2015-01-27 09:02:02*

**Authors:** Koji KOBAYASHI

**Comments:** 2 Pages.

This article provide new approach to solve P vs NP problem by using cardinality of bases function. About NP-Complete problems, we can divide to infinite disjunction of P-Complete problems. These P-Complete problems are independent of each other in disjunction. That is, NP-Complete problem is in infinite dimension function space that bases are P-Complete. The other hand, any P-Complete problem have at most a finite number of P-Complete basis. The reason is that each P problems have at most finite number of Least fixed point operator. Therefore, we cannot describe NP-Complete problems in P. We can also prove this result from incompleteness of P.

**Category:** General Mathematics

[4] **viXra:1501.0153 [pdf]**
*submitted on 2015-01-14 06:34:12*

**Authors:** Adriaan van der Walt

**Comments:** 33 Pages. This monograph is of a fundamental nature and its ideas should be accessible to any reader with a post graduate background in Mathematics and/or Physics (hence no references are given).

The Leibnizean cosmology, where both space and discrete objects (including particles) are assumed to be formed from continuous entities called infinitesimals, replaces the Euclidean Cosmology, where continuous space is assumed to be formed from discrete entities called points. Infinitesimals as well as infinitesimal numbers are defined in this monograph and an Arithmetic for non-standard analysis is developed by generalising the concept of real number to a wider class of numbers, called here the Cauchy numbers, and by rearranging Number Theory somewhat.
This is motivated in part one by showing that Cantor’s famous diagonal proof, which is an algebraic formulation of Euclidean Cosmology, rests on the fallacy that an infinite decimal fraction can be identified by specifying its finite digits. The argument of this part includes a proof that the set of equivalence classes of Cauchy sequences is countable.
In part two the Leibnizean model for the infinite divisibility of space is developed by introducing the concepts of infinitesimal and infinitesimal number from the context of Calculus. An Arithmetic for Cauchy numbers is developed, including an interpretation of L’Hospital’s rules and a description of the Real Continuum. The concept of cascades of infinitesimals as directed sets is introduced and the Fundamental Theorem of the Calculus is studied as a Net defined on such a cascade.
In part three the Leibnizean Cosmology is argued to be in line with the ideas of Parmenides and that space, in this cosmology, is ‘fuzzy’ - thus clarifying the paradox of the arrow. It is also pointed out that, with particles suitably defined as infinitesimals, Parmenides’ ideas are vindicated because everything is one, and motion is only an illusion because nothing comes into being where it did not exist before. Furthermore, some intractable problems of Physics, like the particle/wave duality and action at a distance, are pointed out to be direct consequences of the Euclidean Cosmology and that they all but disappear in the Leibnizean Cosmology.
In part four it is pointed out that when the role of points as building blocks of space is discarded, points can be used comfortably in the Leibnizean model as indicators of locations in space. Thus Mathematics can once more become a canonical model, but without the parts that depend on the Euclidean properties of points; e.g. open and closed sets on the real line.

**Category:** General Mathematics

[3] **viXra:1501.0074 [pdf]**
*submitted on 2015-01-05 19:59:45*

**Authors:** Samuel C. Hsieh

**Comments:** 6 Pages.

A common question from students on the usual diagonalization proof for the uncountability of the set of real numbers is: when a representation of real numbers, such as the decimal expansions of real numbers, allows us to use the diagonalization argument to prove that the set of real numbers is uncountable, why can't we similarly apply the diagonalization argument to rational numbers in the same representation? why doesn't the argument similarly prove that the set of rational numbers is uncountable too? We consider two answers to this question. We first discuss an answer that is based on the familiar decimal expansions. We then present an unconventional answer that is based on continued fractions.

**Category:** General Mathematics

[2] **viXra:1501.0062 [pdf]**
*replaced on 2015-02-14 11:41:14*

**Authors:** Arm Boris Nima

**Comments:** 15 Pages.

We generalized the concept of hyperbolic\\ and trigonometric functions to the\\ third and the fourth case which \\ gives rise to the parametrization \\ of the orbifold defined by\\ $x^3+y^3+z^3-3xyz=1$\\ in the third case.

**Category:** General Mathematics

[1] **viXra:1501.0004 [pdf]**
*replaced on 2015-02-05 15:51:41*

**Authors:** Elemer E Rosinger

**Comments:** 7 Pages.

Hilbert's Fifth Problem, in English translation, [1], is as follows :
``How far Lie's concept of continuous groups of transformations is approachable in our investigations without the assumption of the differentiability of the functions ?"
followed by :
``In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumptions ?"
Lately, in the American mathematical literature, due to unclear reasons, it has often been distorted and truncated as follows, [3] :
``Hilbert's fifth problem, like many of Hilbert's problems, does not have a unique interpretation, but one of the most commonly accepted accepted interpretations ..."
A recent letter in this regard, sent to Terence Tao, and the editors of [3], Dan Abramovich, Daniel S Freed, Rafe Mazzeo and Gigliola Staffilani can be found below.

**Category:** General Mathematics