The Njiki’s Fundamental Theorem-Definition on Fractions in the Mathematical Set ℚ and by Extension in ℝ and ℂ, for the Purpose of Leading to the Construction of Some Algebraic Structures as Its Theoretical Applications and for the Practical Ones

The purpose of the research in the NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set ℚand by extension in ℝand ℂand in order to construct some algebraic structures is about the proved EXISTENCE and the DEFINITION by NJIKI of two INNOVATIVE, IMPORTANT and TEACHABLE operations of addition or additive operations, in ℚ, marked ⊕and +α,β, and taken as VECTORIAL, TRIANGULAR, of THREE or PROPORTIONAL operations and in order to make THEM not be different from the RATIONAL ONE, +, but to bring much more and new information on fractions, and, by extension in ℝand ℂ. And the very NJIKI’s fundamental THEOREM-DEFINITION having many APPLICATIONS in the everyday life of the HUMAN BEINGS and without talking about computer sciences, henceforth being supplied with very interesting new ALGORITHMS. And as for the work done in the research, it will be waiting for its extension to be done after publication and along with the research results concerned.

AN EXTENSION OF BROSOWSKI-MEINARDUS THEOREM ON INVARIANT APPROXIMATION

We obta(?) gener(?)ization of a fixed point th(?)of Dotson for nan-expansive mapptngs on star-sba p(?)sets and then(?)it to Prooe a unified Brosowski-M(?)us theorem on in(?)ariant approximatton in the setting,(?)fp-normed line(?)spaces.

SOME NEW EXTENSIONS OF EDELSTEIN-SUZUKI-TYPE FIXED POINT THEOREM TO G-METRIC AND G-CONE METRIC SPACES

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 （1962）, 74-79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 （2009）, 5313-5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.

THE EXTENSION THEOREMS OF CONE LINEAR OPERATORS

A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators is given.

AN EXTENSION OF LEVI-CIVITA’S THEOREM TO NONHOLONOMIC DYNAMICAL SYSTEMS

This paper uses Poincaré formalism to extend the Levi-Civita theorem to cope with nonholonomic sys- tems admitting certain invariant relations whose equations of motion involve constraint multipliers.Sufficient condi- tions allowing such extension are obtained and,as an application of the theory a generalization of Routh's motion is presented.

A Theorem for Generalized Extension of Operators and Some Applications

In this note it is shown that every bounded linear operator T∈B(Y, Z) defined on a closed subspace Y of a Banach space X admits a generalized extension T∈B(X, V). Some examples of the applications are given; especially, a character-ization of H.I. spaces is obtained.

Origin of nucleation and growth of extension twins in grains unsuitably oriented for twinning during deformation of Mg-1%Al

A large number of anomalous extension twins,with low or even negative twinning Schmid factors,were found to nucleate and grow in a strongly textured Mg-1Al alloy during tensile deformation along the extruded direction.The deformation mechanisms responsible for this behaviour were investigated through in-situ electron back-scattered diffraction,grain reference orientation deviation,and slip trace-modified lattice rotation.It was found that anomalous extension twins nucleated mainly at the onset of plastic deformation at or near grain boundary triple junctions.They were associated with the severe strain incompatibility between neighbour grains as a result from the differentbasal slip-induced lattice rotations.Moreover,the anomalous twins were able to grow with the applied strain due to the continuous activation ofbasal slip in different neighbour grains,which enhanced the strain incompatibility.These results reveal the complexity of the deformation mechanisms in Mg alloys at the local level when deformed along hard orientations.

THE EXTENSION THEOREM OF LINEAR RANDOM FUNCTIONALS AND ITS APPLICATIONS

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Diagnostic Study of an Extreme Explosive Cyclone over the Kuroshio/Kuroshio Extension Region

Explosive cyclones(ECs)occur frequently over the Kuroshio/Kuroshio Extension region.The most rapidly intensified EC over the Kuroshio/Kuroshio Extension region during the 42 years(1979-2020)of cold seasons(October-April)was studied to reveal the variations of the key factors at different explosive-developing stages.This EC had weak low-level baroclinicity,mid-level cyclonic-vorticity advection,and strong low-level water vapor convergence at the initial explosive-developing stage.The low-level baroclinicity and mid-level cyclonic-vorticity advection increased substantially during the maximum-deepening-rate stage.The diagnostic analyses using the Zwack-Okossi equation showed that diabatic heating was the main contributor to the initial rapid intensification of this EC.The cyclonic-vorticity advection and warm-air advection enhanced rapidly in the middle and upper troposphere and contributed to the maximum rapid intensification,whereas the diabatic heating weakened slightly in the mid-low troposphere.The relative contribution of the diabatic heating decreased from the initial explosive-developing stage to the maximum-deepening-rate stage due to the enhancement of other factors(the cyclonic-vorticity advection and warm-air advection).Furthermore,the physical factors contributing to this EC varied with the explosive-developing stage.The non-key factors at the initial explosive-developing stage need attention to forecast the rapid intensification.

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Estimation of Displacement and Extension due to Reverse Drag of Normal Faults: Forward Method

In the case of reverse drag of normal faulting, the displacement and horizontal extension are determined based on the established equations for the three mechanisms: rigid body, vertical shear and inclined shear. There are three sub-cases of basal detachment for the rigid body model: horizontal detachment, antithetic detachment and synthetic detachment. For the rigid body model, the established equations indicate that the total displacement on the synthetic base (Dt2) is the largest, that on the horizontal base (Dt1) is moderate, and that on the antithetic base (Dt3) is the smallest. On the other hand, the value of (Dt1) is larger than the displacement for the vertical shear (Dt4). The value of (Dt1) is larger than or less than the displacement for the inclined shear (Dt5) depending on the original fault dip δ0, bedding angle θ, and the angle of shear direction β. For all original parameters, the value of Dt5 is less than the value of Dt4. Also, by comparing three rotation mechanisms, we find that the inclined shear produces largest extension, the rigid body model with horizontal detachment produces the smallest extension, and the vertical shear model produces moderate extension.

SOME LIOUVILLE-TYPE THEOREMS FOR THE STATIONARY 3D MAGNETO-MICROPOLAR FLUIDS

In this paper,we prove some Liouville-type theorems for the stationary magnetomicropolar fluids under suitable conditions in three space dimensions.We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials.And then we show similar results assuming that the solutions are contained in L^(p)(R^(3))with p∈[2,9/2).Finally,we show the same result for lower values of p∈[1,9/4)with the further assumption that the solutions vanish at infinity.

Analysis of the Digital Skill Gap of Agricultural Extension Personnel in South-East, Nigeria

The study analysed the digital skill gap of agricultural extension personnel on the use of digital technologies for extension services delivery in South-East, Nigeria. The specific objectives were to describe the socioeconomic characteristics of agricultural extension personnel in South-East Nigeria and identify the digital skill gaps among agricultural extension personnel in the area of study. Purposive sampling technique was used to select 364 Agricultural Extension personnel for the study. Data were collected through the use of structured questionnaire and were analysed using simple descriptive statistical tools such as percentages, mean score, and standard deviation. Findings indicated that most of the personnel were male (57.8%), within the age bracket of 38 - 47 years (62.9%), had B.Sc./HND as their highest educational qualification (74.7%), married (86.3%), and had a household size of 6 - 10 Persons (57.7%). It was further revealed that the majority (70.1%) were members of professional organization, earned a monthly income of N50,001.00 - N100,000.00 (65.7%), had a work experience of 11 - 15 years (51.1%), and owned a smartphone/ iPad/laptop (91.5%). Findings further indicated that they had moderate skill gap in Basic Computer skills (Mean = 4.32), and digital communication and collaboration skills (Mean = 4.26). Findings also showed that they had a high skill gap in digital technical skills (Mean = 2.46), digital data analysis skills (Mean = 2.09), digital content creation skills (Mean = 2.43), digital ethical skills (Mean = 2.79), multimedia production skills (Mean = 2.81), and video library management skills (Mean = 2.39). The study concluded that though there exists a high digital skill gap among agricultural extension personnel in South-East, Nigeria, their socioeconomic characteristics are capable of supporting the implementation of digital extension services in the area. The study recommended that the management of Agricultural Development Programs in South-East, Nigeria, should provide digital training for extension personnel to close the digital skill gap that currently exists among the personnel.

Comparative Study of the Socioeconomic Characteristics and Digital Literacy Level of Agricultural Extension Personnel in Imo and Ebonyi States, South-East, Nigeria

The study comparatively analysed the socioeconomic characteristics and digital literacy level of Agricultural Extension personnel (AEP) in Ebonyi and Imo States, South-East, Nigeria. The specific objectives were to describe the socioeconomic characteristics of agricultural extension personnel in Ebonyi and Imo States, and to ascertain the digital literacy level of AEP in the studied states. Purposive sampling technique was used to select 312 Agricultural Extension personnel (132 from Ebonyi State Agricultural Development Program and 180 from Imo State Agricultural Development Program) for the study. Data were collected through the use of validated and structured questionnaire, and administered through the help of well-trained enumerators. Data were analysed using simple descriptive statistical tools such as percentages mean score, standard deviation and weighted mean. Findings indicated that they were more male in the both States (55.3% and 57.8%) for Ebonyi and Imo State respectively and that the average age of AEP in Ebonyi and Imo States were 44.7 years and 49.2 years respectively. It was further revealed that the majority (77.3% and 82.8%) had B.Sc./HND as their highest academic qualifications, belonged to professional organisations (62.1% and 75%), and were earning an average monthly income of N58,798 and N62,648 for Ebonyi and Imo State respectively. Also, it was revealed that their mean years of service were 12.4 years and 13.4 years for Ebonyi and Imo State respectively. Almost all of them (87.9% and 95.0%) own a smartphone, had access to the internet (80.3% and 90.0%), but do not own a laptop/ipad (82.6% and 72.8%) for Ebon-yi and Imo State respectively. Results further revealed that Agricultural extension personnel in both Ebonyi and Imo State respectively had low digital literacy level ( = 2.41 and 2.32). The study concluded that AEP in Ebonyi and Imo State respectively had similar socioeconomic characteristics and low level of digital literacy. The study recommended that the management of ADPs in both Ebonyi and Imo State should ensure the training of AEP in digital skills to enhance their digital literacy level to enable them use digital technologies in their work.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

AAK Theorem and a Design of Multidimensional BIBO Stable Filters

Rational approximation theory occupies a significant place in signal processing and systems theory. This research paper proposes an optimal design of BIBO stable multidimensional Infinite Impulse Response filters with a realizable (rational) transfer function thanks to the Adamjan, Arov and Krein (AAK) theorem. It is well known that the one dimensional AAK results give the best approximation of a polynomial as a rational function in the Hankel semi norm. We suppose that the Hankel matrix associated to the transfer function has a finite rank.

Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

Wellbore Cleaning Degree and Hydraulic Extension in Shale Oil Horizontal Wells

The efficient development and exploitation of shale oil depends on long-distance horizontal wells. As the degreeof cleaning of the wellbore plays a key role in these processes, in this study, this problem is investigated experimentallyby focusing on the dimensionless cuttings bed height. A method is proposed to calculate the horizontalwellhydraulic extension taking into account the influence of the wellbore cleaning degree on the wellborepressure distribution and assess the effect of a variety of factors such as the bottom hole pressure, the circulatingpressure drop, the drilling pump performance and the formation properties. The analysis shows that the hydraulicextension of horizontal wells decreases with an increase in the cuttings bed height, and the higher the displacementof drilling fluid, the faster the hydraulic extension declines. The annular pressure drop of the horizontalsection increases with the increase of the cuttings bed height, resulting in a higher bottom-hole pressure. Severalarguments are provided to guide the safe drilling of shale oil horizontal wells and overcome the limits of currenttechnological approaches.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.