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Further On Natural Analogarithms

My fellow students and I have of late been thinking upon an equivalence between the roots of rational numbers and an infinite dimensional rational vector space, which we have named -space, that we discovered whilst defining analogues of logarithms that were expressed purely in terms of rationals.
We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report.

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On Natural Analogarithms

Last year my fellow students and I spent a goodly portion of our free time considering the similarities of the relationships between sequences and series and those between derivatives and integrals. During the course of our investigations we deduced a sequence form of the exponential function ex, which stands alone in satisfying the equations

    D f = f
  f(0) = 1

where D is the differential operator, producing the derivative of the function to which it is applied.
This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm ln x, albeit in the sense of being expressed in terms of integers rather than being defined by equations involving sequences and series.

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Finally On A Calculus Of Differences

My fellow students and I have spent much of our spare time this past year investigating the similarities between the calculus of functions and that of sequences, which we have defined for a sequence sn with the differential operator

  Δ sn = sn - sn-1

and the integral operator
  n
  Δ-1 sn = Σ si
  i = 1
where Σ is the summation sign, adopting the convention that terms with non-positive indices equate to zero.

We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.

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Further Still On A Calculus Of Differences

For some time now my fellow students and I have been whiling away our spare time considering the similarities of the relationships between sequences and series and those between the derivatives and integrals of functions. Having defined differential and integral operators for a sequence sn with

  Δ sn = sn - sn-1

and
  n
  Δ-1 sn = Σ si
  i = 1
where Σ is the summation sign, we found analogues for the product rule, the quotient rule and the rule of integration by parts, as well as formulae for the derivatives and integrals of monomial sequences, being those whose terms are non-negative integer powers of their indices, and higher order, or repeated, derivatives and integrals in general.

We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.

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Further On A Calculus Of Differences

As I have previously reported, my fellow students and I have found our curiosity drawn to the calculus of sequences, in which we define analogues of the derivatives and integrals of functions for a sequence sn with the operators

  Δ sn = sn - sn-1

and
  n
  Δ-1 sn = Σ si
  i = 1
respectively, where Σ is the summation sign, for which we interpret all non-positively indexed elements as zero.

I have already spoken of the many and several fascinating similarities that we have found between the derivatives of sequences and those of functions and shall now describe those of their integrals, upon which we have spent quite some mental effort these last few months.

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On A Calculus Of Differences

The interest of my fellow students and I has been somewhat piqued of late by a curious similarity of the relationship between sequences and series to that between the derivatives and integrals of functions. Specifically, for a function f taking a non-negative argument x, we have
  x
  F(x) = f(x) dx
  0
  f(x) = d F(x)
  dx

and for a sequence s having terms

  s1, s2, s3, ...

we can define a series S with terms
  n
  Sn = s1 + s2 + s3 + ... + sn = Σ si
  i = 1
where Σ is the summation sign, from which we can recover the terms of the sequence with

  sn = Sn - Sn-1

using the convention that S0 equals zero.
This similarity rather set us to wondering whether we could employ the language of calculus to reason about sequences and series.

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Finally On The Wealth Of Stations

In our recent investigations we have found that games comprising of random returns upon funds, of random trades between players and of random outcomes of labour, trade and sustenance, with the latter subject to some bare minimum of expenditure, invariably rewarded a fortunate few at the expense of an unfortunate many, despite having rules that applied perfectly equitably to all.
For our final analysis, my fellow students and I have sought to develop a rule by which we might cuff the hands of providence!

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Further Still On The Wealth Of Stations

My fellow students and I have spent some time investigating my suspicion that serendipity, beyond worth, might account for the relative fortune of the few over the many. To this end we have set to creating perfectly fair games mimicking the manner in which wealth accumulates amongst the populace, that we might discover whether their outcomes should elevate some small lucky band of players well above their fellows.
Thus far we have seen that games of both random returns and losses of players' funds and random trade between them most certainly do so, but their rules failed to take into account either the value of labour or the cost of sustenance, somewhat weakening any conclusions that we might have drawn from their study.
We have consequently spent some time creating further rules to rectify these deficiencies.

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Further On The Wealth Of Stations

Recall that my fellow students and I have resolved to investigate the role of chance in the spread of wealth amongst the populace by creating a series of games with which we might approximate its ebb and flow. Our first such game was of so simplistic a construction that it offered no meaningful insights into the matter, but at least shed some light upon the manner in which we might answer such questions as what are the chances that, in a perfectly fair game, a few players might fare significantly better than the many or what are they that a player having had a run of poor luck might conclude the game ahead of a fellow who had had better?
We have since spent some mental effort improving the rules of our game and it is upon these changes that I shall now report.

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On The Wealth Of Stations

As much as I enjoy the Baron's tall tales, I am no more convinced of their veracity than I am of his fervent belief that the nobility enjoy the benefits of their station because they are of better mettle than the likes of you and I. No, I am rather of the opinion that serendipity, above worthiness, maketh man!
To investigate the possibility that luck might well account for the elevation of the few above the many, my fellow students and I naturally turned to the mathematical arts and we resolved to create a series of games with which we might model the ebb and flow of prosperity.

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