Ah, this is a common misunderstanding, due to a notation issue. If f : A \rightarrow B is any function and B_0 \subset B, then by f^{-1}(B_0) people always mean the set \{ x \in A | f(x) \in B_0 \}. Notice that this has meaning even if f is not a bijection. Likewise, if A_0 \subset A, we have...
Most people I know of (theoretical physicists and mathematicians) use either Mathematica (analytical/numerical), Maple (more often for analytical things) and/or Matlab (more often numerical). Tons of things there have been programmed for you, and if you want to write a new program you can do...
I is the current, so the amount of charge passing through the surface you get when you cut the wire perpendicular to the direction of flow (area A ), per second. Try to imagine, before starting your watch, which electrons will pass through the surface during one second (say the flow is from...
( n \geq 2 , of course) I tried to find an inductive formula by setting n = 2, n = 3 and n = 4 , but don't find anything interesting. Of course we already knew that the thing is symmetric, symbolically it is also \displaystyle\sum_{i=1}^{n-1}i^{\beta}(n-i)^{\alpha} , but that's about all I...
Hello,
I would love some help on calculating the following sum for \alpha, \beta \in \mathbb{N} and n \in \mathbb{N} \backslash \{0\}:
\displaystyle\sum_{i=1}^{n-1}i^{\alpha}(n-i)^{\beta}.
Thanks in advance,
Latrace
Picture the guy walking on the axis of integers (starting in 0). p^{n_1}q^{n_2} would be the probability of the guy getting to position n_1 - n_2 after N steps, via anyone allowed path. For example, if we keep track of the sequence in which the steps were taken, q \underbrace{p p ... p}_{n_1...
Maybe you can use that because A and B are positive semi-definite, they can be written as the Gram-matrix of certain vectors a_1, a_2, ..., a_n and b_1, b_2, ..., b_n respectively, and the determinant of a Gram-matrix is the content (or volume) squared of the n-dimensional box spanned by the...
I didn't check Part I because you said the trouble was in Part II. After the part
, I would proceed as follows:
Firstly, suppose c is divisible by p. Because C \neq G, we may use the induction hypothesis to find an element of order p in C. This element also has order p in G.
Suppose then that...