Acta Universitatis Danubius. Œconomica, Vol 10, No 5 (2014)
A Comparative Analysis of Some Results from Q_{p} and R
Alin Cristian Ioan^{1}
Abstract: The paper investigates whether a series of concepts and properties available in the real analysis remains valid for padic case. There are many similarities between R and Q_{p} and also so many differences. First of all, R is an ordered field, which is not true for Q_{p}. Secondly R is archimedean (that is the absolute valuation is archimedean) while Q_{p} is not archimedean for any p prime. This means that R is a connected metric space while Q_{p} is totally disconnected. This proves that there is no analogous notion of interval in Q_{p} or a notion similar to the curve. These contrasts will cause the difference between the analysis padic and the real analysis.
Keywords: padic; sequences; series; function
JEL Classification: C02
1 Introduction
Let note Q_{p} the field of padic numbers. Before we begin, we should note that there are many similarities between R and Q_{p} and also so many differences. First of all, R is an ordered field, which is not true for Q_{p}. Secondly R is archimedean (that is the absolute valuation is archimedean) while Q_{p} is not archimedean for any p prime. This means that R is a connected metric space while Q_{p} is totally disconnected. This proves that there is no analogous notion of interval in Q_{p} or a notion similar to the curve. These contrasts will cause the difference between the analysis padic and the real analysis.
2 Sequences and Series in Q_{p}
We begin by studying the basic properties of strings and series in Q_{p}. The most important thing about Q_{p} is that the field is a complete field, therefore every Cauchy sequence is convergent. Naturally all the properties of the norm _{ }_{} on R are the same of the properties of the padic valuations (the property of being nonarchimedean being an additional property).
As a result, many of the basic theorems that occur in the real analysis, taking place also in the padic analysis. One of the great benefits of the padic analysis is that it will bring generalizations to some real questions raised in the analysis (due to the property of _{ }_{p} to be nonarchimedean).
Lemma 1
A sequence (x_{n})Q_{p} is a Cauchy sequence if and only if _{ } x_{n+1} – x_{n} = 0.
Proof
If
m=n+r>n, we get  x_{m}
– x_{n}
=  x_{n+r}
– x_{n+r1}
+ x_{n+r1}
 x_{n+r2}
+... x_{n}
max {  x_{n+r}
– x_{n+r1}
,
 x_{n+r1}
 x_{n+r2}
,..., x_{n+1}
– x_{n}
} this fact being true because _{
}_{p}
is nonarchimedean. Now for
r
N^{*}
and
> 0
N_{}_{
}
N^{*}
such that x_{m}
– x_{n}
= x_{n+r}
– x_{n}
max {  x_{n+r}
– x_{n+r1}
,  x_{n+r1}
 x_{n+r2}
, ...,x_{n+1}
– x_{n}
} <
n, m
N_{}_{.}
N_{}
is that natural number which
n
N_{}
we have x_{n+1}
– x_{n}
 < .
Therefore, the sequence (x_{n})
Q_{p}
is Cauchy so convergent.
The theory of sequences and their convergence is therefore similar with that on R except lemma above.
Proposition 2
Let (a_{n} ) Q_{p} a convergent sequence. Then we have one of two statements: either lim a_{n} = 0, or there exists an integer M such that a_{n} = a_{M} n M. In other words, the absolute value of the sequence converges to zero or it becomes constant after a rank on.
Proof
Suppose that lim a_{n} 0 > 0 such that N_{1} N^{*}, n N_{1} with a_{n} . So a number c > > 0 with a_{n} c > , n N_{1}. On the other hand N_{2} integer for which n,m N_{2}  a_{n}  a_{m} < c. We want both conditions occur so fix > 0 N = max { N_{1}, N_{2}}. Now n,m N  a_{n}  a_{m} < max { a_{n}, a_{m} } from where we get a_{n} = a_{m} after nonarchimedean property (that is, in the space Q_{p} all triangles are isosceles).
Also, for series the classical theory remains valid. For example, the following statements are true:
Proposition 3
Let (a_{n}) Q_{p}. The absolute convergence of sequence implies its convergence, ie if a series of absolute values _{ } converges in R then the series _{ } converges in Q_{p}.
Proof
The series _{ } converges in Q_{p} _{ } lim a_{n} = 0. But a necessary condition for absolute series to converges is that lim a_{n} = 0.
The next result is a strong result in real analysis, but in padic context, the previous lemma becomes an important tool to determine whether a series of padic numbers converges in Q_{p} namely:
Corollary 4
An infinite series _{ } with (a_{n}) Q_{p} is convergent _{ } _{ } a_{n} = 0. In this case we also have  _{ }  _{ }.
Proof
A series converges only when the sequence of partial sums converges. Now take the difference between the nth partial sum and the (n1)th. By Lemma we get that this difference tends to 0 as we wanted. Conversely we have the sequence of partial sums is Cauchy therefore convergent. If _{ }=0 we have nothing to prove. Otherwise, for any partial sum, we have  _{ } _{ }a_{n}. Since _{ } a_{n} = 0 > 0 N_{} N^{*} such that a_{n} < n > N_{} = N. Let = _{ }a_{n}. Thus we have _{ }a_{n} = _{ }a_{n}. How _{ }a_{n} does not depend on N for N _{ } we get  _{ }  _{ }, that is the conclusion.
The reciprocal question related to when a series is convergent in R implies that its general term tends to zero is not necessarily true. As a counterexample we have the harmonic series which not converges in R.
Therefore, it is much easier to establish convergence of the infinite series in padic context than in R. This seems to express that the theory of series in Q_{p} is much simpler than in R.
Now we shall consider a “double string” (b_{ij}) Q_{p} asking what happens to the two series considered after a summing with i and after j or viceversa. For this, it is necessary that, as example, b_{ij}_{ }0 when one of the indices is fixed and the other goes to infinity (otherwise obvious series will not converges). We shall say that _{ } = 0 uniformly in j if > 0 we can find an integer N which does not depend on j such that i N  b_{ij}  < j. In other words, the sequence (b_{ij}) tends to 0 when i _{ } , the convergence coming from the same rank for all j. First we prove the following lemma:
Lemma 5
Let (b_{ij}) Q_{p} and assume that:
1 ) i, _{ } b_{ij} = 0
2) _{ } b_{ij} = 0 uniformly in j
Then for any real number > 0 an integer N_{} which depends only of such that if max(i, j) N b_{ij} < .
Proof
Let > 0 fixed. The second condition says that we can choose N_{0} N^{*}, which depends on but not of j such that b_{ij} < if i N_{0}. The first condition is weaker (it says basically that i we can find N_{1}(i ) N^{*}, “the notation suggesting that the whole depends on i”) such that if j N_{1}(i ) we have b_{ij} < . Now we take N = N() = max (N_{0}, N_{1}(0), N_{1}(1), …, N_{1}(N_{0} – 1)). The choice of N was done so that if max (i,j ) N then i N_{0} when b_{ij} < regardless of j or if i < N_{0} j N and i { 0,1,2,…, N_{0} – 1} therefore j N_{1}(i ), when we have b_{ij} < .
Proposition 6
Let (b_{ij}) Q_{p} and assume that:

i, _{ } b_{ij} = 0

_{ } b_{ij} = 0 uniformly in j
Then the series _{ } and _{ } converges and their sums are equal.
Proof
From
the previous lemma we know that for a given
> 0 we can choose N such that for max (i, j)
N
b_{ij}
< .
In particular for
i, when j _{
}
or viceversa then the inner sums _{
}
and _{
}
converges (the first sum for each i and the second for each j). More,
for i
N we have  _{
}
_{
}
< .
Similarly for any j
N we have  _{
}
< .
In particular, we note that _{
}
= 0 and _{
}
= 0 therefore both series converges. It remains to show that the sums
of the two double series are equal. We will continue to use N and
as above so that the condition b_{ij}
<
i or j
N holds. We will often use the ultrametric inequality: x + y
max{ x, y} applied even at the level of series as we have seen in
the last corollary. We see first  _{
}
_{
}
_{
}
=  _{
}
_{
}
_{
}.
Now for j
N+1 we shall have b_{ij}
<
i. With ultrametric inequality it remains that  _{
}
<
i and, using again the ultrametric inequality we have that  _{
}
< .
Similarly, we obtain _{
}
< .
So, again applying this inequality we have that:  _{
}
_{
}
_{
}
< .
Reversing now i with j we get a similar inequality that is  _{
}
_{
}
_{
}
< ,
then finally
 _{
}
_{
}
_{
}
< .
But how
was arbitrarily fixed the double series are equal.
What basically says this proposition is that if the double sequence {bij} converges to 0 in a uniform way, then the double sum after i and j can be taken in any order to give the same answer.
Now if a = _{ } and b = _{ }are two convergent series, then the series _{ }+ b_{n} is convergent and has the sum a + b. Indeed, the first sum is convergent _{ } _{ }^{ }a_{n} = 0 and so the second if _{ }^{ }b_{n} = 0. In conclusion, _{ } a_{n} + b_{n} = 0, which is enough to say that the series _{ }+ b_{n} converges. Now, noting with c the sum of the series we have that _{ } = _{ } + _{ } and passing to the limit with n _{ } we have that c = a + b.
A second problem is related in some way to the top as follows: if a = _{ } and b = _{ } are two convergent series, taking c_{n} = _{ }b_{nk} then the series _{ } is convergent and its sum is ab.
Let the partial sum of order n of a and the partial sum of order n of b that is s_{n} = _{ } and t_{n} = _{ }. Now s_{n}t_{n} = _{ }. As above, we have: _{ }^{ }a_{n} = 0 and _{ }^{ }b_{n} = 0. Computing _{ } _{ }  _{ } for n N. In short, this expression is written s_{n}t_{n}  c_{n}  _{ }b_{l} = 0 where l and k go through the set of numbers 0,...,n. Finally, we have: s_{n}t_{n} – c_{n}  c_{n1}  c_{n2} +... c_{0 }– c_{n+1} ... – c_{2n} = 0 ie passing to the limit with n _{ } we get ab = _{ } that is c = ab.
3 Functions, Continuity, Differentiability in Q_{p}
The basic idea on the functions and continuity remains unchanged by the passage of real numbers to padic numbers because ultimately they depend on the metric structure. Not be able to work with intervals (nay nor related with nontrivial connected sets), so that our functions will be defined on disks (closedopen). We shall write B(a,r) for open sets of center a and radius r> 0 and _{ }(a,r) for the closed sets of center and radius r.
Definition 7
Let U Q_{p} be an open set. A function f: U _{ } Q_{p} is called continous in a U if > 0 > 0 such that x U with the property  x – a <  f(x) – f(a) < .
The base results on continuity are true in all metric spaces and therefore also in the padic fields. For example, if U is a compact set (and remember that Q_{p} is both open and compact so a subset included in it can have these properties) and f is continuous at any point in U then f is uniformly continuous. Automaticaly, the Darboux property to carry an interval within an interval is true since the intervals in Q_{p} are identified with points. In the general context, the Darboux property says that a continuous function defined on a metric space carry a connected set into another connected set.
Now, if U = Z_{p} then for any a Z_{p}, > 0, n N with x Z_{p} such that  x – a  < _{ }, we have  f(x) – f(a) < . However = _{ } for m Z. For m = 0 we have that f(x) – f(a) Z_{p} that is f(x) is in one of the neighbourhoods (closedopen) of f(a) ie f carry a local connected set into a local connected set.
Derivatives are perhaps more interesting from the fact that there is a lower analogy with the classical real case. It will make sense to define derivatives of functions f: Q_{p} _{ } Q_{p} in the usual way, namely:
Definition 8
Let U Q_{p} be an open set and let f: U _{ } Q_{p} a function. We say that f is differentiable in xU if the limit f^{ ‘}(x) = _{ }. If f^{ ‘} (x) exists for any x U we shall say that f is differentiable on U and we write: f^{ ‘}: U _{ } Q_{p} for the function x _{ } f^{ ‘} (x).
Remark
Up to a certain point, the derivative of a function with values in Q_{p} behaves as if real, that is it can be shown that a differentiable function is continuous as shown in R or C.
It is natural to ask what is the role of the derivative of a function in the padic case. But if we consider that the mean value theorem states for a and b real data in the domain of definition of a differentiable function (while continuing) between a and b such that f (b)  f (a ) = f '() (b  a), is not working in the padic case, because in fact we have not the relation of “being between” because Q_{p} is not an ordered field. But this slight inconvenience can be simply remedied if we think that in R we can define the relation “being between” saying that is between a and b if we have =at+b(1t) for 0 t 1. Nearly the same happens in the complex case. What we can now express through the mean value theorem in the padic case? We ask if there the statement holds: if we have a function f defined on Q_{p}, differentiable and continuous on Q_{p} then for any two numbers a and b in Q_{p} Q_{p} of the form: =at+b(1t) for t such that t 1, for which f(b) – f(a)=f^{ ‘} () (b – a). We shall show that the mean value theorem for padic case is false.
Proof
Let f(x) = x^{p }– x, a = 0, b = 1. We have f^{ ‘} (x) = px^{p1} – 1 and f(a) = f(b) = 0. If the statement is true, it exists Q_{p} of the form = at + b(1  t) = 1 – t with t Z_{p} such that p^{p1} – 1 = 0. But from here and Z_{p} and from p^{p1} – 1 = 0 0 1 + p Z_{p}  contradiction.
4 References
Gouvea, Fernando Q. (1997). Padic numbers. An introduction. 2^{nd} Edition. New York, Heidelberg, Berlin: SpringerVerlag.
Ioan, A.C. (2013). Through the maze of algebraic theory of numbers. Galati: Zigotto Publishing.,
Ireland, Kenneth & Rosen Michael (1990). A classical introduction to modern number theory, Springer.
Marcus, Daniel (1977). Number fields. New York: Springer Verlag.
Neukirch, Jurgen (1999). Algebraic number theory. New York, Heidelberg, Berlin: Springer Verlag.
Roquette, Peter (2003). History of valuation theory. Part I, Heidelberg
1 University of Bucharest, Faculty of Mathematics and Computer Science, Romania, Address: 412 Regina Elisabeta Blvd, Bucharest 030018, Romania, Corresponding author: alincristianioan@yahoo.com.
AUDŒ, Vol. 10, no. 5, pp. 5258
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