In the last post I described a contravariant tensor of rank 1 and a covariant tensor of rank 1. In this post we will consider generalizations of these. We will introduce tensors of arbitrary rank 
Now before we go ahead, I would like to clarify that the above represents just 1 single component of the tensor , out of the 
In this equation, the left hand side represents the value of 1 component out of the 
Wow! We have stated the general transformation law and now we will proceed to an example. Before we do that, I would like to state our general roadmap in the future posts. Remember that our goal is to understand the equations of General Relativity. We are studying tensors just because we want to understand the weird symbols and equations in the profound equations laid down by Einstein. So in the next post, we will study general operations on tensors like - addition, subtraction, inner products etc. Along with this we will state some rules to recognize tensors. We will state a general rule by which we can ascertain that a set of number form the components of a tensor. We will use this general rule to prove that some collection of numbers are tensors in later posts. Here we will not however talk of the derivative of a tensor because that involves some more machinery - but hey .. without derivatives there is no calculus so get back we will! The next few posts will talk about the fundamental tensor which is related to measuring distance in space between two points. That will define define distance for us. In the next few posts we will then discuss derivatives of tensors. In doing so we will introduce some tensors in terms of the fundamental tensor. These will be helpful for defining derivatives. Armed with all this machinery about distances and derivatives, we will then state equations for geodesics which are shortest paths between 2 points in space. In ordinary space this is the straight line. But in spaces, where the fundamental tensor is more complicated the geodesics are not “straight” lines [Well by then we will probably wonder what straight means anyway!]. Finally we will have the notions of distance, derivatives and shortest paths in our bag so we will talk about what we really need to understand the Einstein equations - curvature. Specifically we will introduce quantities which represent how curved a space is at a point. After this we will state the Einstein equations and if I can manage I will show some simple consequences of these equations. The Einstein equations somehow relate the curvature of our playground [which is the "space" of 4 dimensional spacetime], to the distribution of mass and energy.
So much for the future posts - all that is probably going to take several posts! But for the time being lets get back to an example using familiar 3 space. In familiar 3 space, we are going to work with 2 coordinate systems - one : cartesian 
The inverse transformations equations are
See here for an illustration. Assume we are working in a small region of space where all the 
We will take a simple case here.
Example
Suppose there is a contravariant tensor that has components 
and so evaluating the partial derivatives and summing them up
Of course the above represents the value at a specific point in space. A contravariant tensor of rank 2 will have 9 such terms to sum up. We encourage the reader to evaluate the components of such a tensor.
Before the next post do think of some of the things you can do with tensors. Specifically think about
- When can two tensors be added/subtracted to produce new tensors?
- Can two tensors be multiplied to produce a new tensor?
- Consider a function
defined for each point of space. Suppose for each coordinate system
we define
. Prove that these numbers define a covariant tensor of rank 1.
Posted by cstheory
dimensional space, to study things like curves, surfaces, curvature, so common from everyday experience in two and three dimensional spaces. Many basic entities in Reimannian Geometry are best described by Tensors. Tensors are also used in many other places and in order to understand current physical theories and their mathematical formulations, one needs to know about tensors and basic differential geometry.
. A coordinate system is an assignment of an 
to each point of 
. We also assume that the 
are given by 
of the 
. The functions are assumed to be such that the 
is invertible.
can be the primed coordinate system. Or the primed coordinate system can be one with the 3 axes not necessarily at right angles to each other.
of the coordinates such that they transform according to the following law
.
. Suppose a vector 
. Consider a different coordinate system with basis vectors 
. Now in this coordinate system let 
. Now what are the 
as functions of the 
? A quick calculation shows that 
where 
is the row vector of coordinates 
is the row vector of coordinates 
is the inverse of matrix 
. Also as one can verify 
.
. Clearly the differentials in the primed coordinate system are given by
that transform according to
.
. Given a number 
, one can ask the question : Find the fraction of points in 
elements of 
of points in the new multiset 
the fraction of elements in 
. Suppose 
. Then 
of the elements were chosen from 
elements and 
of the elements were chosen from 
. Hence the probability is 
. Therefore 
. The expected value of 
and the standard deviation is 
. By the The probability that we become wrong by 
is ![Pr[|g - f| > \delta f]](http://l.wordpress.com/latex.php?latex=Pr%5B%7Cg+-+f%7C+%3E+%5Cdelta+f%5D&bg=ffffff&fg=333333&s=0 "Pr[|g - f| > \delta f]")
. By using a modified form of the 
. If 
then how large should 
be so that we are at most off by 1% of 
? A simple calculation shows that if 
then we are good. If The nice part about use of this bound is that it does not depend on the size of the underlying multiset 
then we can get away with sampling a few thousand elements. Are there any better known algorithms? In randomized algorithms that produce 
answers it is possible to increase the probability of success by repeating algorithm multiple times and then taking the majority of the output as the true answer. Is there a similar way of improving the overall accuracy of algorithms that produce one of many possible values such as the percentile algorithm?
regular bipartitie graph 
, with 
and 
is called a 
if it saisfies the properties below. For a given set 
of vertices, we denote the set of vertices to which some vertex in 
.
, 
.
with two given subsets 
with 
, such that for every 
and 
with 
, the number of disjoint paths from 
is at least 
edges are interesting for various reasons which we do not go into here.
such that for every 
, such that 
, there is a magical graph with 
.
edges for every 
. The construction is recursive. First take two copies of the magical graphs 
with 
and 
. Connect every vertex of 
to the corresponding vertex of 
and add edges between 
and 
so that the graph becomes a superconcentrator with the size of the input vertex set as 
. We claim that the resulting graph is a superconcentrator with input vertex set of size 
it is true that 
. Therefore by 
. Similarly there exists a perfect matching between vertices of 
and 
. Together with the edges of the superconcentrator between input set 
and output set 
there are at least 
. For 
, there is a subset 
of vertices with 
such that the vertices corresponding to 
in 
disjoint paths between 
disjoint paths exist as proved earlier. Hence we have a superconcentrator with input, output vertex sets of size 
be the complete bipartite graph with 
edges. The following recursion counts the edges. Let 
denote the number of edges in the superconcentrator as per this construction with input, output sets of size 
for 
and 
for 
. It can be easily seen that 
for 
.
where 
, the 
for the problem consists of a set of data points 
and 
is a set of 
where 
. The function 
defines the solution for a particular configuration 
any of the 
. The objective is to minimize the number of such updates and still be able to compute 
. An algorithm is 
updates where 
is the minimum number of updates required by any algorithm.
the areas 
can be open intervals such that the edge weight 
. The function 
.
algorithm in 
. A witness set 
update competitive when the areas are open intervals or singletons. Furthermore it is proved that, there is no 
update competitive algorithm in this case. When there are no restrictions on the areas at all, it is proved that there is no constant update algorithm for the MST problem. The paper also considers graphs with vertex position uncertainties [where vertices are points in an Euclidean space].
.
