New Insights into Physical Science Vol.3

We attempt to obtain Pythagorean triples by a simple method consisting in transforming the relation between integers  into an equation in by introduction of a parameter . By this way we obtain easily Pythagorean triples for each choice of  . Following this example we introduce also a suitable parameter  totransform the relation  into an equation in  which must have only one multiple root, i.e. must have coefficients alternated in signs. Observing that this happens only for  and not at all for  , we arrive to conclude that the equation has roots only for   and no root for  thus prove the Fermat’s last theorem.

From a holistic perspective of a physical space of any given size^[1], it is invariably necessary to consider its energy content, since no physical means exists of making a physical space completely devoid of energy. Such a space would therefore only be a fictive “geometric space” – that can be intellectually conceived and treated according to the rules of the appropriate geometry – although not existing in reality in the cosmos. Cosmic space always contains energy in one form or another, limited by the space under consideration. Therefore each space possesses an energy density – no matter how low, which never becomes zero. 

Because of the mass-energy equivalence relationship . c², cosmic space also possesses a mass equivalent and is therefore “materialistic” in nature. If this is considered in association with Einstein’s space-time, what is obtained instead is an “energy-time”, i.e. an energy effect”, which is based on Planck’s action quantum h. Cosmic space as a whole and every part of it is always - up from its very beginning - an energy-effect. Under this condition, a close relationship would appear to exist between the General Theory of Relativity and Quantum Physics. Furthermore, it will be shown that the physical conditions of space are such that a natural = physical quantisation of space and time exists, a necessary prerequisite for the Unification of Quantum Physics with General Theory of Relativity, thus obviating the need for any artificial or arbitrary quantisation.

By comparing the formula giving odd powers sums of integers from Bernoulli numbers and the Faulhaber conjecture form of them, we obtain two recurrence relations for calculating the Faulhaber coefficients. Parallelly we search for and obtain the differential operator which transform a powers sum into a Bernoulli polynomial. From this and by changing arguments from z,n into Z=z(z-1), λ=zn+n(n-1/2) we obtain a formula giving powers sums on arithmetic progressions directly from the  powers sums on integers.

The general expressions, based on the Fermi distribution of the free electrons, are applied for calculation of the kinetic coefficients in donor-doped silicon at arbitrary degree of the degeneracy of electron gas under equilibrium conditions. The classical statistics leads to large errors in estimation of the transport parameters for the materials where Fermi level is located high above the conduction band bottom unless the effective density of randomly moving electrons is introduced. The obtained results for the diffusion coefficient and drift mobility are discussed together with practical approximations applicable for non-degenerate electron gas and materials with arbitrary degree of degeneracy. In particular, the drift mobility of randomly moving electrons is found depend on the degree of degeneracy and can exceed the Hall mobility considerably. When the effective density is introduced, the traditional Einstein relation between the diffusion coefficient and the drift mobility of randomly moving electrons is conserved at any level of degeneracy. The main conclusions and formulae can be applicable for holes in acceptor-doped silicon as well.

In this short review some aspects of applications of free electron theory on the ground of the Fermi statistics will be analysed. There it is an intention to attempt somebody’s attention to problems in widespread literature of interpretation of conductivity of metals, superconductor in the normal state and semiconductors with degenerate electron gas. In literature there are many cases when to these materials the classical statistics is applied. It is well known that the electron heat capacity and thermal noise (and as a consequence the electrical conductivity) are determined by randomly moving electrons, which energy is close to the Fermi energy level, and the other part of electrons, which energy is well below the Fermi level cannot be scattered and change its energy. Therefore, there was tried as simple as possible on the ground of Fermi distribution, and on random motion of charge carriers, and on the well-known experimental results to take general expressions for various kinetic parameters which are applicable for materials both without and with degenerate electron gas. It is shown, that drift mobility of randomly moving charge carriers, depending on the degree degeneracy, can considerably exceed the Hall mobility. It also is shown that the Einstein relation between the diffusion coefficient and the drift mobility of charge carriers is valid at any degree of degeneracy. There are presented the main kinetic parameter values for elemental metals.

The general expressions based on the Fermi distribution of the free charge carriers are applied for estimation of the transport characteristics in superconductors at the temperature well above the superconducting phase transition temperature T_C. The Hall-effect experimental results in the normal state of the superconductor YBa₂Cu₃O_7-d are not finally explained. On the ground of the randomly moving charge carriers there are presented the transport characteristics for both single type and two types of the charge carriers. The particular attention has been pointed to the Hall-effect measurement results of the high-T_C superconductor YBa₂Cu₃O_7-d. It is at the first time derived the Hall coefficient expression for two type of highly degenerate charge carriers (electrons and holes) on the ground of the randomly moving charge carriers at the Fermi surface. It is shown that the Hall coefficient and other transport characteristics are determined by the ratio between the electron-like and hole-like density-of-states at the Fermi surface.

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.

By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials.

In this paper is proven that all relations between a couple of dual operators  i.e. operators obeying the commutation relation  are invariant under substitution of  with any another dual couple.

From this property are obtained many differential operators realizing transformations in space and phase space such as translation, dilatation, hyperbolic, … , fractional order Fourier transforms and Fourier transform itself. Transforms of arbitrary functions and operators and geometric forms by these differential operators are given.

The kernel of the integral transform associated with a differential transform is found. As case study the differential Fourier transform is highlighted in order to see how it is possible to get in a concise manner the known properties of the Fourier transform without doing integrations. 

Finslerian extension of the theory of relativity implies that space-time can be not only in an amorphous state which is described by Riemann geometry but also in ordered, i.e. crystalline states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics. It is shown that there are only two types of curved Finslerian spaces endowed with local relativistic symmetry. However the metric of only one of them satisfies the correspondence principle with Riemannian metric of the general theory of relativity and therefore underlies viable Finslerian extension of the GR. Since the existing purely geometric approaches to a Finslerian generalization of Einstein’s equations do not allow one to obtain such generalized equations which would provide a local relativistic symmetry of their solutions, special attention is paid to the property of the specific invariance of viable Finslerian metric under local conformal transformations of those fields on which it explicitly depends. It is this property that makes it possible to use the well-known methods of conventional field theory and thereby to circumvent the above-mentioned difficulties arising within the framework of purely geometric approaches to a Finslerian generalization of Einstein’s equations.

A formula changing the operator  where  into a sum of operators  is proved. Thank to this relation between operators a new and rapid method for resolutions of differential equations is exposed in details. It is seen to be useful also for obtaining the differential operators that transform monomials into Hermite, Laguerre, associated Laguerre, Gegenbauer, Chebyshev polynomials and for getting quasi all their main properties in a very concise manner. Is proposed also the differential representation of the Laplace transform permitting the differential calculus to prove consicely its properties.