Real projective n-space
Real projective space, is defined to be the space of all lines through the origin in . Each such kind is determined by a non-zero vector in , unique up to scalar multiplication, and is topologized as the quotient space of under the equivalence relation for scalars . We can restrict to vectors of length 1, so is also the quotient spaces , the sphere with antipodal points identified. This is equivalent to saying that is the quotient space of a hemisphere with antipodal points of identified. Since with antipodal points identified is just , we see that is obtained from by attaching an -cell, with the quotient projection as the attaching map. It follows by induction on that has a cell complex structure with one cell in each dimension . Since is obtained from by attaching an -cell, the infinite union becomes a Cell Complex with one cell in each dimension. We can view as the space of lines through the origin in .
Complex Projective n-space
Complex projective n-space is space of complex lines through the origin in , that is, 1-dimensional vector subspaces of . As in the case of Real projective n-space, each line is determined by a non-zero vector in, unique up to scalar multiplication, and is topologized as the quotient space of under the equivalence relation for .
Equivalently, this is the quotient of the unit Sphere with for .
It is also possible to obtain as a quotient space of disk under the identifications for , in the following way:
The vectors in with last coordinate real and nonnegative are precisely the vectors of the form with . Such vectors form the of the function . This is a disk bounded by the sphere consisting of vectors with . Each vector in is equivalent under the identifications to a vector in , and the latter vector is unique if its last coordinate is non-zero. If the last coordinate is zero, we have just the identifications for .
From the description of as the quotient of under the identifications for , it follows that is obtained from by attaching a cell via the quotient map . So by induction on , we obtain a cell structure
with cells only in even dimensions. Similarly, has a cell structure with one cell in each even dimension.
How to Draw a Cell Complex (or CW Complex)
Let we try to construct a space by following procedure:
- Start with a discrete set , whose points are regarded as -cells.
- Inductively, form the -skeleton from by attaching -cells via maps . This means that is the quotient space of the disjoint union of with a collection of n-disks under the identifications for . Thus as a set
where each is an open disk.
- One can either stop this inductive process at a finite stage, setting for some , one can continue indefinitely, setting .
A space constructed in this way is called a CELL COMPLEX or CW COMPLEX.
A series of positive terms is convergent if from and after some fixed term , where r is a fixed number. The series is divergent if from and after some fixed term.
D’ Alembert’s Test is also known as ratio test of convergency of a series.
Definitions for Generally Interested Readers
(Definition 1) An infinite series i.e. is said to be convergent if , the sum of its first terms, tends to a finite limit as n tends to infinity.
We call the sum of the series, and write .
Thus an infinite series converges to a sum S, if for any given positive number , however small, there exists a positive integer such that
for all .
If as , the series is said to be divergent.
Thus, is said to be divergent if for every given positive number , however large, there exists a positive integer such that for all .
If does not tends to a finite limit, or to plus or minus infinity, the series is called Oscillatory
Let a series be . We assume that the above inequalities are true.
- From the first part of the statement:
, ……… where r <1.
Since r<1, therefore as
therefore =k say, where k is a fixed number.
Therefore is convergent.
- Since, then, , …….
Therefore and so on.
Therefore > . By taking n sufficiently large, we see that can be made greater than any fixed quantity.
Hence the series is divergent.
- When , the test fails.
Another form of the test–
The series of positive terms is convergent if >1 and divergent if <1.
One should use this form of the test in the practical applications.
Verify whether the infinite series is convergent or divergent.
We have and
Hence, when 1/x >1 , i.e., x <1, the series is convergent and when x >1 the series is divergent.
Take Now , a non-zero finite quantity.
But is convergent.
Hence, is also Convergent.
Let denote the set of all integers (as usually it do ).
Consider a function with the following properties:
for all . Is it possible that all positive divisors of occur as values of ?
A happy note: is actually 1748 and it is written to retain symmetry in problem.