Introducing Quaternions to Integer Factorisation

The key purpose of this paper is to open up the concepts of the sum of four squares and the algebra of quaternions into the attempts of factoring semiprimes, the product of two prime numbers. However, the application of these concepts here has been clumsy, and would be better explored by those with a more rigorous mathematical background. There may be real immediate implications on some RSA numbers that are slightly larger than a perfect square.

Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a4 + b4 = c4. Part I

In this paper we prove in a new way, the well known result, that Fermat’s equation a4 + b4 = c4, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F0) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−10) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F0): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=0 ( b 2 , b ′ 2 , b ″ 2 ) if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. .

Passive robust control for uncertain Hamiltonian systems by using operator theory

In this study,the passivity-based robust control and tracking for Hamiltonian systems with unknown perturbations by using the operator-based robust right coprime factorisation method is concerned.For the system with unknown perturbations,a design scheme is proposed to guarantee the uncertain non-linear systems to be robustly stable while the equivalent non-linear systems is passive,meanwhile the asymptotic tracking property of the plant output is discussed.Moreover,the design scheme can be also used into the general Hamiltonian systems while the simulation is used to further demonstrate the effectiveness of the proposed method.

Introducing “Arithmetic Calculus” with Some Applications: New Terms, Definitions, Notations and Operators

New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements;and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned;and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.

A Provably Secure and Efficient Remote Password Authentication Scheme Using Smart Cards

Communication technology has advanced dramatically amid the 21st century,increasing the security risk in safeguarding sensitive information.The remote password authentication(RPA)scheme is the simplest cryptosystem that serves as the first line of defence against unauthorised entity attacks.Although the literature contains numerous RPA schemes,to the best of the authors’knowledge,only few schemes based on the integer factorisation problem(IFP)and the discrete logarithm problem(DLP)that provided a provision for session key agreement to ensure proper mutual authentication.Furthermore,none of the previous schemes provided formal security proof using the random oracle model.Therefore,this study proposed an improved RPA scheme with session key establishment between user and server.The design of the proposed RPA scheme is based on the widely established Dolev-Yao adversary model.Moreover,as the main contribution,a novel formal security analysis based on formal definitions of IFP and DLP under the random oracle model was presented.The proposed scheme’s performance was compared to that of other similar competitive schemes in terms of the transmission/computational cost and time complexity.The findings revealed that the proposed scheme required higher memory storage costs in smart cards.Nonetheless,the proposed scheme is more efficient regarding the transmission cost of login and response messages and the total time complexity compared to other scheme of similar security attributes.Overall,the proposed scheme outperformed the other RPA schemes based on IFP and DLP.Finally,the potential application of converting the RPA scheme to a user identification(UI)scheme is considered for future work.Since RPA and UI schemes are similar,the proposed approach can be expanded to develop a provably secure and efficientUI scheme based on IFP and DLP.

Factorisable Quasi-adequate Semigroups

In this paper,we investigate a class of factorisable IC quasi-adequate semigroups,so-called,factorisable IC quasi-adequate semigroups of type-（H,I）.Some characterizations of factorisable IC quasi-adequate semigroups of type-（H,I） are obtained.In particular,we prove that any IC quasi-adequate semigroup has a factorisable IC quasi-adequate subsemigroups of type-（H,I） and a band of cancellative monoids.

Flood susceptibility mapping using a novel integration of multi-temporal sentinel-1 data and eXtreme deep learning model

Flash floods(FFs)are amongst the most devastating hazards in arid regions in response to climate change and can cause the loss of agricultural land,human lives and infrastructure.One of the major challenges is the high-intensity rainfall events affecting low-lying areas that are vulnerable to FF.Several works in this field have been conducted using ensemble machine learning models and geohydrological models.However,the current advancement of eXtreme deep learning,which is named eXtreme deep factorisation machine(xDeepFM),for FF susceptibility mapping(FSM)is lacking in the literature.The current study introduces a new model and employs a previously unapplied approach to enhance FSM for capturing the severity of floods.The proposed approach has three main objectives:(i)During-and after-flood effects are assessed through flood detection techniques using Sentinel-1 data.(ii)Flood inventory is updated using remote sensing-based methods.The derived flood effects are implemented in the next step.(iii)An FSM map is generated using an xDeepFM model.Therefore,this study aims to apply xDeepFM to estimate susceptible areas using 13 factors in the emirates of Fujairah,UAE.The performance metrics show a recall of 0.9488),an F1-score of 0.9107),precision of(0.8756)and an overall accuracy of 90.41%.The accuracy of the applied xDeepFM model is compared with that of traditional machine learning models,specifically the deep neural network(78%),support vector machine(85.4%)and random forest(88.75%).Random forest achieves high accuracy,which is due to its strong performance that depends on factors contribution,dataset size and quality,and available computational resources.Comparatively,the xDeepFM model works efficiently for complicated prediction problems having high non-collinearity and huge datasets.The obtained map denotes that the narrow basins,lowland coastal areas and riverbank areas up to 5 km(Fujairah)are highly prone to FF,whilst the alluvial plains in Al Dhaid and hilly regions in Fujairah show low probability.The coastal city areas are bounded by high-rise steep hills and the Gulf of Oman,which can elevate the water levels during heavy rainfall.Four major synchronised influencing factors,namely,rainfall,elevation,drainage density,distance from drainage and geomorphology,account for nearly 50%of the total factors contributing to a very high flood susceptibility.This study offers a platform for planners and decision makers to take timely actions on potential areas in mitigating the effects of FF.

Source apportionment of PM_(2.5)light extinction in an urban atmosphere in China

Haze in China is primarily caused by high pollution of atmospheric fine particulates（PM2.5）.However, the detailed source structures of PM2.5 light extinction have not been well established, especially for the roles of various organic aerosols, which makes haze management lack specified targets. This study obtained the mass concentrations of the chemical compositions and the light extinction coefficients of fine particles in the winter in Dongguan, Guangdong Province, using high time resolution aerosol observation instruments. We combined the positive matrix factor（PMF） analysis model of organic aerosols and the multiple linear regression method to establish a quantitative relationship model between the main chemical components, in particular the different sources of organic aerosols and the extinction coefficients of fine particles with a high goodness of fit（R^2= 0.953）. The results show that the contribution rates of ammonium sulphate,ammonium nitrate, biomass burning organic aerosol（BBOA）, secondary organic aerosol（SOA） and black carbon（BC） were 48.1%, 20.7%, 15.0%, 10.6%, and 5.6%, respectively. It can be seen that the contribution of the secondary aerosols is much higher than that of the primary aerosols（79.4% versus 20.6%） and are a major factor in the visibility decline. BBOA is found to have a high visibility destroying potential, with a high mass extinction coefficient, and was the largest contributor during some high pollution periods. A more detailed analysis indicates that the contribution of the enhanced absorption caused by BC mixing state was approximately 37.7% of the total particle absorption and should not be neglected.

Triply Factorised Groups and the Structure of Skew Left Braces

The algebraic structure of skew left brace has proved to be useful as a source of settheoretic solutions of theYang-Baxter equation.We study in this paper the connections between left and right π-nilpotency and the structure of finite skew left braces.We also study factorisations of skew left braces and their impact on the skew left brace structure.As a consequence of our study,we define a Fitting-like ideal of a left brace.Our approach depends strongly on a description of a skew left brace in terms of a triply factorised group obtained from the action of the multiplicative group of the skew left brace on its additive group.