Let A and B be connected subsets of [0,1]2 where π(A) = [0,1] = π(B) where π is the projection map π(x,y) = x. We construct a new set C out of A and B as follows. Let C = {(x,y,z) | (x,y) ∈ A and (x,z) ∈ B}. Is C connected always? If not … More A problem
We know that set of commutators of matrices is exactly the space of matrices of trace zero and has dimension . I discuss an interesting question related to this which was brought to my notice by Anamika: Given a matrix of trace zero can we construct two matrices and of determinant , such that . … More matrix commutators
Three problems to ponder for the next 100 years: First one may seem contrived but thinking about this would hightlight some major properties of primes. So here it is : The number 82 is the largest positive number that cannot be written as sum of two square free numbers each having odd number of prime … More Three non-idiotic guesses
We wish to better the bounds for the norm of a matrix with scalar entries when it defines a linear map with from to , that is the series , is convergent for each and . The standard bounds for matrix norm given in the text for , are and , for spaces. We give … More A bound for matrix norm
Have a look at the presentation of a group defined by the generators , and the presentation below. It is symmetric, it is recursively presented, but can you guess the cardinality of this group. Or, of the group defined by four generators and relations: Some very basic group theory but some sweating and … More Presentations and cardinality
Amalgams of groups are examples of what are known as colimits in category theory(aka abstract non-sense). With abstract non-sense, you can turn a blind-eye to the structure of the groups and concentrate wholly on its social behavior, determined by its universal property. There is nothing deep about proving results using universal properties of groups, as … More associativity of amalgams or why universal properties can be powerful
You are interested in establishing the continuity of a linear map where and are normed spaces. You would like to know some sufficient conditions for proving continuity. The thought about browsing your old fat text book makes you cringe and would like a quicker way (like what a Schaum series textbook would offer). Heres a … More cheat sheet for continuity.
Sometime back I was asked this question by our professor teaching us probability : How do we characterize random variables which have uniform distribution?. I revisited my old measure notes to give a partial answer to this question: Let be a random variable defined on the probability space with Lebesgue measure. Suppose further that is … More which random variables have uniform distribution
Bedtime mathematics 