Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations

Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .

一个关于Smarandache Quotients函数序列的均值

主要应用初等方法研究了Smarandache Quotients函数序列的均值问题,得到了两个重要的等式.

Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12

Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .

MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.

Binomial coefficients,Catalan numbers and Lucas quotients

Let p be an odd prime and let a,m ∈ Z with a > 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0<k<pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.

Contamination assessment,source apportionment and associated health risks of PTEs in agricultural soil under five land-use patterns in Sanya,China

To understand the levels of potentially toxic elements(PTEs)contamination in soils and their effects on human health from different agricultural land use in Sanya,China.128 soil samples(64 topsoil samples and corresponding subsoil samples)were collected from the five representative land-use patterns.Inductively coupled plasma mass spectrometry(ICP-MS),Atomic fluorescence spectrometry(AFS),and Inductively coupled plasma optical emission spectrometry(ICP-OES)were used to determine the content of PTEs(As,Cd,Hg,Cu,Cr,Ni,Pb,Zn,Co,Mo,Sb,and V).Correlation analysis and factor analysis were used to determine the source of PTEs.Geo-accumulation index(I_(geo)),hazard quotient(HQ),and total carcinogenic risk index(TR)were used to measure the PTEs contamination and its relative health impacts.Results showed that the average values of 12 PTEs in topsoil were higher than the Hainan soil geochemical baseline,showing different degrees of PTEs accumulation effect.The concentration of PTEs in the topsoil was lower than those in the subsoil except for Cd and Hg.The I_(geo)revealed that the major accumulated element in soils was As followed by Mo.Source apportionment suggested that parent materials and agricultural practices were the dominant factors for PTEs accumulation in the topsoil.Noncarcinogenic risks of soil samples from five land-use patterns presented a trend of paddy field>dry field>woodland>orchard>garden plot.However,the HQ values of 12 PTEs were less than the recommended limit of HQ=1,representing that there are no non-carcinogenic risks of PTEs for children and adults in the study area.The TR values are within 6.95×10^(-6)-1.38×10^(-5),which corresponds to the low level.Therefore the PTEs in the agricultural soil of the study area show little influence on the health status of the local population.

Rational and Continuous Measurement of the Emotional Decision Making in Visual Recognition of Facial Emotional Expressions with M.A.R.I.E.: First Half

Context: The advent of Artificial Intelligence (AI) requires modeling prior to its implementation in algorithms for most human skills. This observation requires us to have a detailed and precise understanding of the interfaces of verbal and emotional communications. The progress of AI is significant on the verbal level but modest in terms of the recognition of facial emotions even if this functionality is one of the oldest in humans and is omnipresent in our daily lives. Dysfunction in the ability for facial emotional expressions is present in many brain pathologies encountered by psychiatrists, neurologists, psychotherapists, mental health professionals including social workers. It cannot be objectively verified and measured due to a lack of reliable tools that are valid and consistently sensitive. Indeed, the articles in the scientific literature dealing with Visual-Facial-Emotions-Recognition (ViFaEmRe), suffer from the absence of 1) consensual and rational tools for continuous quantified measurement, 2) operational concepts. We have invented a software that can use computer-morphing attempting to respond to these two obstacles. It is identified as the Method of Analysis and Research of the Integration of Emotions (M.A.R.I.E.). Our primary goal is to use M.A.R.I.E. to understand the physiology of ViFaEmRe in normal healthy subjects by standardizing the measurements. Then, it will allow us to focus on subjects manifesting abnormalities in this ability. Our second goal is to make our contribution to the progress of AI hoping to add the dimension of recognition of facial emotional expressions. Objective: To study: 1) categorical vs dimensional aspects of recognition of ViFaEmRe, 2) universality vs idiosyncrasy, 3) immediate vs ambivalent Emotional-Decision-Making, 4) the Emotional-Fingerprint of a face and 5) creation of population references data. Methods: With M.A.R.I.E. enable a rational quantified measurement of Emotional-Visual-Acuity (EVA) of 1) a) an individual observer, b) in a population aged 20 to 70 years old, 2) measure the range and intensity of expressed emotions by 3 Face-Tests, 3) quantify the performance of a sample of 204 observers with hyper normal measures of cognition, “thymia,” (ibid. defined elsewhere) and low levels of anxiety 4) analysis of the 6 primary emotions. Results: We have individualized the following continuous parameters: 1) “Emotional-Visual-Acuity”, 2) “Visual-Emotional-Feeling”, 3) “Emotional-Quotient”, 4) “Emotional-Deci-sion-Making”, 5) “Emotional-Decision-Making Graph” or “Individual-Gun-Trigger”6) “Emotional-Fingerprint” or “Key-graph”, 7) “Emotional-Finger-print-Graph”, 8) detecting “misunderstanding” and 9) detecting “error”. This allowed us a taxonomy with coding of the face-emotion pair. Each face has specific measurements and graphics. The EVA improves from ages of 20 to 55 years, then decreases. It does not depend on the sex of the observer, nor the face studied. In addition, 1% of people endowed with normal intelligence do not recognize emotions. The categorical dimension is a variable for everyone. The range and intensity of ViFaEmRe is idiosyncratic and not universally uniform. The recognition of emotions is purely categorical for a single individual. It is dimensional for a population sample. Conclusions: Firstly, M.A.R.I.E. has made possible to bring out new concepts and new continuous measurements variables. The comparison between healthy and abnormal individuals makes it possible to take into consideration the significance of this line of study. From now on, these new functional parameters will allow us to identify and name “emotional” disorders or illnesses which can give additional dimension to behavioral disorders in all pathologies that affect the brain. Secondly, the ViFaEmRe is idiosyncratic, categorical, and a function of the identity of the observer and of the observed face. These findings stack up against Artificial Intelligence, which cannot have a globalist or regionalist algorithm that can be programmed into a robot, nor can AI compete with human abilities and judgment in this domain. *Here “Emotional disorders” refers to disorders of emotional expressions and recognition.

Rational and Continuous Measurement of Emotional-Fingerprint, Emotional-Quotient and Categorical vs Proportional Recognition of Facial Emotions with M.A.R.I.E., Second Half

Context: The advent of Artificial Intelligence (AI) requires modeling prior to its implementation in algorithms for most human skills. This observation requires us to have a detailed and precise understanding of the interfaces of verbal and emotional communications. The progress of AI is significant on the verbal level but modest in terms of the recognition of facial emotions even if this functionality is one of the oldest in humans and is omnipresent in our daily lives. Dysfunction in the ability for facial emotional expressions is present in many brain pathologies encountered by psychiatrists, neurologists, psychotherapists, mental health professionals including social workers. It cannot be objectively verified and measured due to a lack of reliable tools that are valid and consistently sensitive. Indeed, the articles in the scientific literature dealing with Visual-Facial-Emotions-Recognition (ViFaEmRe), suffer from the absence of 1) consensual and rational tools for continuous quantified measurement, 2) operational concepts. We have invented a software that can use computer-morphing attempting to respond to these two obstacles. It is identified as the Method of Analysis and Research of the Integration of Emotions (M.A.R.I.E.). Our primary goal is to use M.A.R.I.E. to understand the physiology of ViFaEmRe in normal healthy subjects by standardizing the measurements. Then, it will allow us to focus on subjects manifesting abnormalities in this ability. Our second goal is to make our contribution to the progress of AI hoping to add the dimension of recognition of facial emotional expressions. Objective: To study: 1) categorical vs dimensional aspects of recognition of ViFaEmRe, 2) universality vs idiosyncrasy, 3) immediate vs ambivalent Emotional-Decision-Making, 4) the Emotional-Fingerprint of a face and 5) creation of population references data. Methods: M.A.R.I.E. enables the rational, quantified measurement of Emotional Visual Acuity (EVA) in an individual observer and a population aged 20 to 70 years. Meanwhile, it can measure the range and intensity of expressed emotions through three Face- Tests, quantify the performance of a sample of 204 observers with hypernormal measures of cognition, “thymia” (defined elsewhere), and low levels of anxiety, and perform analysis of the six primary emotions. Results: We have individualized the following continuous parameters: 1) “Emotional-Visual- Acuity”, 2) “Visual-Emotional-Feeling”, 3) “Emotional-Quotient”, 4) “Emotional-Decision-Making”, 5) “Emotional-Decision-Making Graph” or “Individual-Gun-Trigger”, 6) “Emotional-Fingerprint” or “Key-graph”, 7) “Emotional-Fingerprint-Graph”, 8) detecting “misunderstanding” and 9) detecting “error”. This allowed us a taxonomy with coding of the face-emotion pair. Each face has specific measurements and graphics. The EVA improves from ages of 20 to 55 years, then decreases. It does not depend on the sex of the observer, nor the face studied. In addition, 1% of people endowed with normal intelligence do not recognize emotions. The categorical dimension is a variable for everyone. The range and intensity of ViFaEmRe is idiosyncratic and not universally uniform. The recognition of emotions is purely categorical for a single individual. It is dimensional for a population sample. Conclusions: Firstly, M.A.R.I.E. has made possible to bring out new concepts and new continuous measurements variables. The comparison between healthy and abnormal individuals makes it possible to take into consideration the significance of this line of study. From now on, these new functional parameters will allow us to identify and name “emotional” disorders or illnesses which can give additional dimension to behavioral disorders in all pathologies that affect the brain. Secondly, the ViFaEmRe is idiosyncratic, categorical, and a function of the identity of the observer and of the observed face. These findings stack up against Artificial Intelligence, which cannot have a globalist or regionalist algorithm that can be programmed into a robot, nor can AI compete with human abilities and judgment in this domain. *Here “Emotional disorders” refers to disorders of emotional expressions and recognition.

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.

Abelian quotients and orbit sizes of linear groups

Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order pr,i.e.,a finite homocyclic abelian group.LetΔn （G） denote the n-th power of the augmentation idealΔ（G） of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Qn（G）=Δn（G）/Δn+1（G） for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.

Augmentation quotients for complex representation rings of dihedral groups

Denote by Dm the dihedral group of order 2m. Let R（Dm） be its complex representation ring, and let △（Dm） be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient △^n（Dm）/△^n＋1（Dm） for each positive integer n.

Predicting non-carcinogenic hazard quotients of heavy metals in pepper (Capsicum annum L.) utilizing electromagnetic waves

Given the tendency of heavy metals to accumulate in soil and plants,the purpose of this study was to determine the contamination.levels of Cd,Ni,Pb,and Zn on peppers(leaves.and fruit),grown in contaminated soils in industral centers.For this purpose,we measured the uptake of the four heavy metals by peppers grown in the heavy metal contaminated soils throughout the four growth stages:two-leaf,growth,fiowering,and fruiting;and calculated various vegetation indices to evaluate the heavy metal contamination potentials.Electromagnetic waves were also applied for analyzing the responses of the target plants to various heavy metals.Based on the relevant spectral bands identified by principal component analysis(PCA)and random search methods,a regression method was then employed.to determine the most optimal.spectral bands for estimating,the target hazard quotient(THQ).The THQ was found to be the highest in the plants contaminated by Pb(THQ=62)and Zn(THQ=5.07).The results of PCA and random search indicated that the spectra at the bands of b570,b650,and b760 for Pb,b400 and b880 for Ni,b560 and b880 for Cd,and b560 b910,and b1050 for Zn were the most optimal spectra for assessing THQ.Therefore,in future studies,instead of examining the amount of heavy metals in plants by chemical analysis in the laboratory,the responses of the plants to the electromagnetic waves in the identified bands can be redily investigated in the field based on the established correlations.

Infinite-dimensional separable quotients of Banach space

In functional analysis, the following problem is fundamental: does every infinite-dimensional Banach space have an infinite-dimensional quotient space? It has remainedunsolved for a long time.

Erratum to:Foliations associated to harmonic maps on some complex two ball quotients

In[Y,p.1146,1.35],it is stated that"PR#T2 has a two-fold cover that is T2".This is a mistake.As a result,[Y,Theorem 4.2]should be corrected as follows.The numbering of the references follows[Y].Theorem 4.2.Let M be a smooth complex surface of general type with c2(M)=3.Then M is an arithmetic complex two ball quotient.Moreover,M is either a fake projective plane or a Cartwright-Steger surface.Hence,there are altogether 101 such surfaces.Proof.The surface M has to be a complex two ball quotient BC2/Γas explained in[31,Proposition 2.1].

The Stable Behavior of the Augmentation Quotients of the Groups of Order p^5, I

In this article we describe the stable behavior of the augmentation quotients Qn （G） for the groups G of order p^5 with even numbers of generators, where p is an odd prime.

Some Implications of the Gessel Identity

We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.

A CLASS OF INVERSE QUOTIENT CURVATURE FLOW IN THE ADS-SCHWARZSCHILD MANIFOLD

In this paper,we study the asymptotic behavior of a class of inverse quotient curvature flow in the anti-de Sitter-Schwarzschild manifold.We prove that under suitable convex conditions for the initial hypersurface,one can get the long-time existence for the inverse curvature flow.Moreover,we also get that the principal curvatures of the evolving hypersurface converge to 1 when t→+∞.

Brain metabolic profile assessed by magnetic resonance spectroscopy in children with Down syndrome:Relation to intelligence quotient

BACKGROUND Down syndrome(DS)is one of the most common causes of intellectual disability.Children with DS have varying intelligence quotient(IQ)that can predict their learning abilities.AIM To assess the brain metabolic profiles of children with DS and compare them to standard controls,using magnetic resonance spectroscopy(MRS)and correlating the results with IQ.METHODS This case-control study included 40 children with DS aged 6-15 years and 40 age and sex-matched healthy children as controls.MRS was used to evaluate ratios of choline/creatine(Cho/Cr),N-acetyl aspartic acid/creatine(NAA/Cr),and myoinositol/creatine(MI/Cr(in the frontal,temporal,and occipital lobes and basal ganglia and compared to controls and correlated with IQ.RESULTS Children with DS showed significant reductions in NAA/Cr and MI/Cr and a non-significant reduction in Cho/Cr in frontal lobes compared to controls.Additionally,we observed significant decreases in NAA/Cr,MI/Cr,and Cho/Cr in the temporal and occipital lobes and basal ganglia in children with DS compared to controls.Furthermore,there was a significant correlation between IQ and metabolic ratios in the brains of children with DS.CONCLUSION Brain metabolic profile could be a good predictor of IQ in children with DS.

Evaluation of Some Selected Metals in Rice Cultivated in Four Local Government Areas in Enugu State, Nigeria

In this study, we investigated the presence of Zn, Fe, Cu, and Ca in rice cultivated in four local government areas (Nkanu East, Aninri, Uzo Uwani, Isi Uzo) within Enugu state, Nigeria. We employed an Atomic Absorption Spectrometer with an air acetylene flame to analyze these metals after digesting the rice samples. Risk assessment studies were carried out to determine any potential health risk to consumers by evaluating the estimated daily intake (EDI), the target hazard quotient (THQ) and hazard index (HI). The average concentration (mg/kg) of trace metals in the rice samples was within the acceptable limits established by FAO/WHO. Specifically, Zn ranged from 0.265 to 0.632 mg/kg, Fe from 2.73 to 4.131 mg/kg, Cu from 0.205 to 4.131 mg/kg, and Ca from 9.718 to 12.150 mg/kg. There were no statistically significant differences in metal concentrations among the various locations. Consequently, the rice analyzed in this study can be considered safe for consumption. The calculated EDI (mg/kg-day) values were below the maximum tolerable daily intake thresholds. THQ values also fell within safe levels, and the HI values were less than 1, signifying no potential health risks associated with consuming rice from these locations. In conclusion, there is no significant non-carcinogenic health risk associated with exposure to trace metals through the consumption of rice from these areas.