Let q: X->Y and r:Y->Z be covering maps; let p=(r(q(x))). Show if r^(-1)(z) is finite for each z in Z, p is a covering map.
let x0 and x1 be points of the path-connected space X.Show that Pi_1(X,x0) is abelian iff for every pair a and b of paths from x0 to x1, we have a'=b'. where a'([f])=[a-]*[f]*[a];( a- means the reverse of a.) and [f] belongs to Pi_1(X,x0). a':Pi_1(X,x0)->Pi_1(X,x1).
Suppose that V is an inner-product space. Prove that if T: V-->V is a positive operator and trace(T)=0, then T=0.
1)Let X={1,2,...,n}and let R be the Boolean ring of all subsets of X. Define f_i:R->Z_2 by f_i(a)=[1] iff i is in a.Show each f_i is a homomorphism and thus f=(f_1,...,f_n):R->Z_2*Z_2*...*Z_2 is a ring homomorphism.Show f is an isomorphism. 2)If T is any ring,an element e of T is called an idempotent provided e^2=e.The el ...continues
1)I understand what a standard R-module (ring-module)is, but I have heard talk of modules associated with representations. Could someone please give me some idea of what these are? 2) I am trying to find all modules over Z-the Integers; so far, I have only come up with additive groups. How can I find all others?
It can be shown that R (the set of all real numbers) is an infinite-dimensional vector space over Q (field of rationals). Is it true that any basis (by basis I mean algebraic basis or Hamel basis) of R over Q has to be uncountable ?
Suppose that f:(0,1)-->R is uniformly continuous. Show that lim f(x) exists. x->0
What was the annual interest rate?
A business invests $10,000 in a savings account for two years. At the beginning of the second year, an additional $3500 is invested. At the end of the second year, the account balance is $15,569.75. What was the annual interest rate?
wide can the path be if there is enough gravel for 516 square feet?
A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet?
Find the number of characters used by the designer.
A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.
