# APM Institute for the Advancement of Physics and Mathematics

apminstitute.org

#### Donators

#### Location of the APM Institute

The APM Institute begun its operations in its new space in March 2014 in **Athens, Greece**. Its premises include research space, a presentation hall and a library.

#### Purpose

Purpose of the APM Institute is the progress and **A**dvancement of **P**hysics and **M**athematics and the support of its members. Fundamental physics, Cosmology, Iterative systems, Chaos, Information physics, History of science, are some research areas pursued by the institute’s members. A main target of the Institute is to keep gifted Greek scientists from leaving Greece. One such promising research topic is the physical **Theory of Selfvariations**, originated by Emmanuil Manoussos in 2007.

MATHEMATICS |

### Record Factorization of Chebyshev Integers

Maintained by Kyriakos Kefalas

Record factorizations are well known for some famous integer sequences like **Fermat numbers** and **Fibonacci numbers**. At the APM Institute’s site we present tables of record factorizations of integer sequences which may be less known to the general public, but are well known to mathematicians. In particular we maintain tables of record factorizations for the following sequences

**Chebyshev integers**(i.e. integer values of Chebyshev polynomials)

A completely factorized integer is placed in our tables according to its 2nd largest principal prime factor (explained in the text), which is exactly the prime factor which is most difficult to find. However, under some conditions, we include in our tables also incompletely factorized integers with large principal prime factors. The most difficult factorization included in our tables (as of 10.4.2019) is the Chebyshev integer:

*7* • 1880678238403581484491261600488335063 {37, P} • 12131078272029453254528593283997361448667051367 {47, P}

contributed by Ilias Kotsireas

Anyone is welcome to contribute to our factorization tables. For this please contact Kyriakos Kefalas through the contact page of the Institute, but make sure that the material you send us deserves to be included in the tables.

PHYSICS |

### Introduction to the Theory of Selfvariations (TSV)

The law of Selfvariations expresses mathematically a slight continuous increase of the rest mass and electric charge of material particles. The mathematical investigation showed that it contains the information of our core knowledge in physics. It can justify, as a common cause, the particle structure of matter, the interactions of material particles, the quantum phenomena and the cosmological data. Moreover it gives a large amount of new knowledge. One of these is the mathematical expression for conserved quantities. We call the set of statements emerging from the mathematical investigation of the law, the theory of Selfariations (TSV).

HISTORY OF SCIENCE |

### The Antikythera Mechanism

Prof. Xenophon Moussas*

…*The origin of all technical achievements is the divine curiosity [of Socrates/Plato] and the play instinct of the working and thinking researcher as well as the constructive fantasy of the inventor*… Albert Einstein, speech on the radio at the opening of the 7 Deutsche Funkausstellung in Berlin, 1930.

Astronomy is the oldest science developed as humans that have been watching the sky for centuries and millennia started attempting to understand all celestial motions, of the stars, the Sun, the Moon and finally the planets. This eventually led them to try to understand their existence in the Cosmos. This was the birth of Philosophy and Humanity. Humans develop calendars form prehistoric times and for this they develop mathematics and astronomy…

The **Antikythera Mechanism** is the oldest known astronomical instrument and astronomical computer that we have in hands, probably made between 150 and 100 BC, by a Greek mechanic and astronomer with excellent knowledge of mathematics.

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* Professor of Space Physics, University of Athens, Greece

MATHEMATICS |

### Solving the Abel Difference Equation

In simple cases, dynamical systems are described by the 1st order autonomous difference equation of the form:

(1)

which we call the **Abel Difference Equation** (ADE)*, where *u*, is an arbitrary function defining the system. Despite its simplicity, it is difficult (or impossible) to find smooth functions *f* (i.e. smooth flows) which are solutions of the ADE*,* for arbitrary non-linear smooth functions *u*. An elegant solution to this problem is presented by **Kyriakos Kefalas** ( On smooth solutions of non linear dynamical systems, *f _{n+1} = u(f_{n})*, part I, Phys. Int., 5: 112-127, 2014) for wide classes of increasing functions

*u.*There it is shown that sufficiently smooth solutions of (1) , are given by:

(2)

where, , is the *k*th iterate of *u*. The solutions depend on a modulator function, , which introduces a (usually remarkably tiny) oscillation into *f*. Central inside the parenthesis is an infinite composition of functions called a continued form, defined as:

where, {*u _{j} *}, is a sequence of functions and the variable s, on top indicates the composition variable. The symbolism stems from the analogy with , with the binary operation replaced by (non-commutative) composition. The conditions for the existence of the overall limit in (2) are investigated as well as the conditions for the convergence of the infinite continued form inside, where the two limits have to be independent from each other. It is also proved that (2) converges to the desired function

*f*, , for a wide class of defining functions

*u*. It is also remarkable that in many cases (for ex., ), the speed of convergence is amazing.

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*To avoid confusion with the related Abel Functional Equation (AFE), .

PHYSICS |

### Dark Matter Cosmology and Astrophysics

Vladimir S. Netchitailo*

Hypersphere World-Universe Model (WUM) envisions Matter carried from Universe into World from fourth spatial dimension by Dark Matter Particles (DMPs). Luminous Matter is byproduct of Dark Matter (DM) annihilation. WUM introduces Dark Epoch (spanning from Beginning of World for 0.4 billion years) when only DMPs existed, and Luminous Epoch (ever since for 13.8 billion years). Big Bang discussed in standard cosmological model is, in our view, transition from Dark Epoch to Luminous Epoch due to Rotational Fission of Overspinning DM Supercluster’s Cores and annihilation of DMPs. WUM solves a number of physical problems in contemporary Cosmology and Astrophysics through DMPs and their interactions: **Angular Momentum problem** in birth and subsequent evolution of Galaxies and Extrasolar systems—how do they obtain it; **Fermi Bubbles**—two large structures in gamma-rays and X-rays above and below Galactic center; **Mysterious Star KIC 8462852** with irregular dimmings; **Coronal Heating problem** in solar physics—temperature of Sun’s corona exceeding that of photosphere by millions of degrees; **Cores of Sun and Earth** rotating faster than their surfaces; Diversity of **Gravitationally-Rounded Objects** in Solar system and their **Internal Heat****; Lightning Initiation problem**—electric fields observed inside thunderstorms are not sufficient to initiate sparks; **Terrestrial Gamma-Ray Flashes**—bursts of high energy X-rays and gamma rays emanating from Earth. Model makes predictions pertaining to **Masses of DMPs**, proposes **New Types of their Interactions**. WUM reveals **Inter-Connectivity of Primary Cosmological Parameters** and calculates their values, which are in good agreement with the latest results of their measurements.

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### 5-Dim General Relativity

Late Prof. Paul S. Wesson*

Our group is using an extension of general relativity from four to five dimensions where the extra dimension is interpreted as giving rise to matter. Hence the name, **Space-Time-Matter theory**. Mathematically, STM theory is based on an old embedding theorem of Campbell, which ensures that the 5D Ricci-flat field equations contain the 4D Einstein field equations with matter. Physically, the forms of 4D matter are determined by the dependence of the 5D metric on the extra coordinate. Vacuum is especially simple, being the result of a 5D metric whose 4D part is quadratically factorized by the extra coordinate.

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* Late Professor in Astrophysics and Theoretical Physics, University of Waterloo, Canada

### Shannon’s idea of information in biological respiratory sedimentation

Dr. Ioannis Haranas*, Dr. Ioannis Gkigkitzis**

The concept of “information” is widely used in modern science and means “data” or “message”. In spite its use by various scientists no physical essence can be still attached to it. The first appearance of the original concept of information theory took place during the development of various communications systems, which had to ensure information transfer or exchange. The operational principles of various systems obey strictly the laws of physics, where optimization of the same systems operation requires the dealing with the amount of information, which can be transmitted through communication channels. H. Nyquist, R. Hartlee and C. Shannon, and Landauer all employees of Bell Laboratories were pioneers in the field. Their works resulted to Shannon’s theory of information which further resulted in the optimization of the operation of the technical systems. Furthermore, a collection of these notions, called by Shannon himself “*Mathematical Theory of Communication*”, became the basis of classic Information Theory (Shannon, 1948).

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*Adjunct Professor, Department of Physics and Computer Science, Wilfrid Laurier University, Canada

**Ass. Professor, Department of Mathematics, East Carolina University, USA

**More Topics**

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#### Donations

*We kindly thank our Donators who have contributed to the Research Departments and the Scientific Library of the APM Institute.*

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