Mathematics

Publication Date: August10 , 2018                          Number of Pages: 20
In this work , we study and solve a problem , by this author , on
      geometric progressions.. Let us state the problem:
                                      PROBLEM1
        Let k be a fixed  odd positive positive integer.
            Determine/parametrically describe the set of all geometric
                progressions that satisfy two conditions:
        Condition1: Every term is a positive rational number.
        Condition2: The product of the third , fifth , and thirteenth terms
                                    is equal to the (k+1)th term.

          There are five results in this paper. The first three are well known
                        from number theory , they are stated without proof on
                              pages 3 and 4 of this paper.
                                We prove Result4 in Section2 , pages8-10.
                                    Using Results1-4 , we prove Result5 in Section3 ,
                                          pages11-20.
                                    Result5 provides a complete answer to PROBLEM1.
                        According to Result5 , the set of all geometric progressions
                    that satisfy the two conditions , can be described in terms of
                  two positive integer parameters as follows:
      1) Suppose that the odd integer k is at least 19; k>18.
          Then the first term a and the common ratio r of all such geometric
              progressions are of the form:
                        a = (t/T)^(k-18)  ,  r = (t/t)^2 ;  where t and T  can be any
                            two      relatively prime positive integers.
      2) Suppose that k is an odd integer between 1 and 17.
            Then all such  progressions  have  first term  a =(v/w)^(18-k)
                  and common ratio  r =(w/v)^2 .
                Where  v and w can be any two relatively prime
                          positive integers.

Publication Date:  August 7 ,  2018                              Number of Pages: 22
In this paper , we study and solve two Diophantine systems , by this author:
                                                          PROBLEM2
Determine all the positive solutions of the 4-variable ,  3-equation system:
                  ( x^2 )/y = w  ,  (y^2)/z=w^2 ,    (z^2)/x = w^6
                                                          PROBLEM1
            Let p , q , and r be distinct fixed prime numbers.
            And let k be a fixed positive integer.
                Determine all the positive integer solutions of the 3-variable ,
                    3-equation  system
                                                            (x^2)(y) = (pqr)^3
                                                              (y^2)(z) = (pqr)^4
                                                              (z^2)(x) = (pqr)^k
    There  are  five results in this paper.
        Result4  provides a complete answer to PROBLEM1.
        According to  Result4 ,  if k is not equal to 5; then the system
                        described in PROBLEM1  has no positive integer solutions.
                    If on the other hand k=5; the said system has a unique
                                  positive integer solution:
                        (x , y , z  )= ( pqr , pqr , (pqr)^2)
              Result4 is stated on page5 and its proof is given in Section3 ,
                            pages9-17.
                Result5  provides a comprehensive answer to PROBLEM2.
                  Result5 is stated on page6 of this proof. We present a proof
                          to Result5 in Section4 , pages 18-22.
                  According to Result5 , the  set of positive integer solutions of
                                the system described in PROBLEM2 , can be
                                      be described in terms of one positive integer
                                      parameter d:
                                  (x , y , z , w) = (d^2 , d^3 , d^4 , d)  ; where d can be
                                      any positive integer.

s: The Mathematics teachers' ability to connect Mathematics to real life problems is principal to students' interest development in Mathematics. The current study seeks to identify factors that studentsprefer that their Mathematics teacherswould improve on to help build their interest in Mathematics. The study used random sampling technique to select ten (10) high schools and 1,263 students' from the various subject areas to respond to a structured questionnaire on variables that contributes to the students' Mathematics interest building process. The study used principal component analysis together with multiple linear regression analysis to analyze the effect of teachers' ability to connect Mathematics to real life problem on the student interest in Mathematics. The study found that teachers ability to connect Mathematics to real life problem can be put into two principal component and these components significantly (p<0.001) predicts 57.4% students interest in Mathematics. The relative importance index of the factors used for the Mathematics connection to real life problem was computed and assessed. The study found that, students' will be more interested in Mathematics when Mathematics teachers dedicate quality time for practicing class exercise .The study also

The fuzzball paradigm provides a clear and concrete resolution of the information paradox. We have taken a consistent theory of quantum gravity – string theory – and found by explicit computation that black hole microstates have the structure of 'fuzzballs' where no traditional horizon exists. The fuzzball radiates from its surface like any normal body, and there is no problem with entanglement or information loss. In spite of this success, many people were reluctant at first to accept the fuzzball paradigm. Years of work with the traditional picture of the black hole had led to a strong prejudice that the horizon should be a region where the local quantum state is the vacuum. Further, many people working with standard general relativity were reluctant to accept the additional structure – strings, branes and extra dimensions-brought in by string theory. Without this additional structure one does not get fuzzballs, and the natural conclusion is that black hole evaporation leads to the information of the hole getting locked in a planck mass 'remnant'. In string theory we believe that remnants do not exist. Many string theorists, however, were also reluctant to accept that the nature of the horizon was severely altered away from the vacuum. What then was their resolution of the puzzle? We will see in this chapter that there was a pervasive belief that the puzzle could be resolved through the cumulative effect of small corrections. In this view the physics at the horizon can continue to be the physics of the local vacuum at leading order. Quantum gravity effects will, however, lead to small subleading effects to this vacuum dynamics. We know that the evaporation of the hole involves a very large number of radiated quanta. It was hoped that these small subleading effects would some how introduce 'delicate correlations' among all the quanta involved in the process, and that these small correlations would resolve the information paradox. In 2004 Hawking conceded that information was probably not lost in the process of black hole evaporation, using a version of this small corrections argument. But most other relativists did not concur. Their belief was that if we make a tiny change in the evolution at the horizon, then we will get only a tiny change in the entanglement between the emitted quanta and the remaining hole. How then could small corrections resolve the problem? In 2009 a theorem was proved showing that the paradox cannot be resolved

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