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.

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.

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.

Insight Into the Separation-of-Variable Methods for the Closed-Form Solutions of Free Vibration of Rectangular Thin Plates

The separation-of-variable(SOV)methods,such as the improved SOV method,the variational SOV method,and the extended SOV method,have been proposed by the present authors and coworkers to obtain the closed-form analytical solutions for free vibration and eigenbuckling of rectangular plates and circular cylindrical shells.By taking the free vibration of rectangular thin plates as an example,this work presents the theoretical framework of the SOV methods in an instructive way,and the bisection–based solution procedures for a group of nonlinear eigenvalue equations.Besides,the explicit equations of nodal lines of the SOV methods are presented,and the relations of nodal line patterns and frequency orders are investigated.It is concluded that the highly accurate SOV methods have the same accuracy for all frequencies,the mode shapes about repeated frequencies can also be precisely captured,and the SOV methods do not have the problem of missing roots as well.

KG_2 over Rings with Division Rings of Quotients

A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.

The Estimation for Lower Bounds of the Solutions of Fermat's Equation

Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.

《牛津高阶英汉双解词典》对quotient的中文释义对吗?

【问】[461]IQ表示智商,EQ表示情商,其中字母Q的全称是quotient,根据《牛津高阶英汉双解词典》第1221页的解释是:number obtained when one number is divided by another。中文解释是'商',这没有问题,但是字典在'商'后面的括号中给出了这样的补充说明:商(除数除以被除数的得数)。请问这样的解释对吗?

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.

KG2 over Rings with Division Rings of Quotients

A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.

Quotient space model based on algebraic structure

In the quotient space theory of granular computing,the universe structure is assumed to be a topology,therefore,its application is still limited.In this study,based on the quotient space model,the universe structure is assumed as an algebra instead of a topology.As to obtain the algebraic quotient operator,the granulation must be uniquely determined by a congruence relation,and all the congruence relations form a complete semi-order lattice,which is the theoretical basis of granularities ' completeness.When the given equivalence relation is not a congruence relation,it defines the concepts of upper quotient and lower quotient,and discusses some of their properties which demonstrate that falsity preserving principle and truth preserving principle are still valid.Finally,it presents the algorithms and example of upper quotient and lower quotient.The work extends the quotient space theory from structure,and provides theoretical basis for the combination of the quotient space theory and the algebra theory.

Linear Complexity of Some Binary Sequences Derived from Fermat Quotients

We determined the linear complexity of a family of p2-periodic binary threshold sequences and a family of p2-periodic binary sequences constructed using the Legendre symbol,both of which are derived from Fermat quotients modulo an odd prime p.If 2 is a primitive element modulo p2,the linear complexity equals to p2-p or p2-1,which is very close to the period and it is large enough for cryptographic purpose.

Study on Logical Relations amongUnderground Survey Objects Using Quotient Topological Space

A soitable data model and data structure make underground survey objects maintained and operated easier. This paper gives a formal definition for underground survey objects. By making use of the quotient topological space, the author studies the logical relations among underground survey objects, a partiallyordered space uuder some conditions. An example is given to show the data model’s possible applications.

Quotient统计量在尾部相关性检验中的应用

文章尝试采用基于quotient统计量的Gamma分布对资产之间的尾部相关性进行显著性检验,在确定资产之间显著尾部相关的基础上,采用t-copula函数对资产之间的相关结构进行描述,并对资产组合尾部风险进行实证分析。

ON BEST SIMULTANEOUS APPROXIMATION IN QUOTIENT SPACES

We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.

The City Intelligence Quotient （City IQ） Evaluation System＂ Conception and Evaluation

After a systematic review of 38 current intelligent city evaluation systems （ICESs） from around the world, this research analyzes the secondary and tertiary indicators of these 38 ICESs from the perspec- tives of scale structuring, approaches and indicator selection, and determines their common base. From this base, the fundamentals of the City Intelligence Quotient （City IOD Evaluation System are developed and five dimensions are selected after a clustering analysis. The basic version, City IQ Evaluation System 1.0, involves 275 experts from 14 high-end research institutions, which include the Chinese Academy of Engineering, the National Academy of Science and Engineering （Germany）, the Royal Swedish Academy of Engineering Sciences, the Planning Management Center of the Ministry of Housing and Urban-Rural Development of China, and the Development Research Center of the State Council of China. City IQ Evaluation System 2.0 is further developed, with improvements in its universality, openness, and dy- namic adjustment capability. After employing deviation evaluation methods in the IQ assessment, City IQ Evaluation System 3.0 was conceived. The research team has conducted a repeated assessment of 41 intelligent cities around the world using City IQ Evaluation System 3.0. The results have proved that the City IQ Evaluation System, developed on the basis of intelligent life, features more rational indicators selected from data sources that can offer better universality, openness, and dynamics, and is more sen- sitive and precise.

The rough representation and measurement of quotient structure in algebraic quotient space model

Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.

Rayleigh Quotient Based Interference Alignment Spectrum Sharing in MIMO Cognitive Radio Networks

An interference alignment(IA)spectrum sharing method based on Rayleigh quotient is proposed for distributed multi-user multi-antenna cognitive radio(CR) networks.The interference from cognitive users(CUs)to the primary(PR) system is constrained through the Rayleigh quotients of channel matrices to deal with the absence of PR users(PUs) in the IA process.As a result,the IA scheme can be applied in CR networks without harmful interference to PUs.Compared with existing IA based spectrum sharing methods,the proposed method is more general because of breaking the restriction that CUs can only transmit on the idle sub-channels of the PR system.Moreover,in comparison to other four spectrum sharing methods applicable in general scene,the proposed method leads to improved performance of achievable sum rate of the CR system as well as guarantees the transmission of PUs.

A ROBUST PHASE-ONLY DIRECT DATA DOMAIN ALGORITHM BASED ON GENERALIZED RAYLEIGH QUOTIENT OPTIMIZATION USING HYBRID GENETIC ALGORITHM

A robust phase-only Direct Data Domain Least Squares (D3LS) algorithm based on gen- eralized Rayleigh quotient optimization using hybrid Genetic Algorithm (GA) is presented in this letter. The optimization efficiency and computational speed are improved via the hybrid GA com- posed of standard GA and Nelder-Mead simplex algorithms. First, the objective function, with a form of generalized Rayleigh quotient, is derived via the standard D3LS algorithm. It is then taken as a fitness function and the unknown phases of all adaptive weights are taken as decision variables. Then, the nonlinear optimization is performed via the hybrid GA to obtain the optimized solution of phase-only adaptive weights. As a phase-only adaptive algorithm, the proposed algorithm is sim- pler than conventional algorithms when it comes to hardware implementation. Moreover, it proc- esses only a single snapshot data as opposed to forming sample covariance matrix and operating matrix inversion. Simulation results show that the proposed algorithm has a good signal recovery and interferences nulling performance, which are superior to that of the phase-only D3LS algorithm based on standard GA.

QUOTIENT SPACE OF LMC-COMPACTIFICATION AS A SPACE OF z-FILTERS

The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters