Recent content by mahler1

Homework Statement Let ##G## be a group such that its center ##Z(G)## has finite index. Prove that every conjugacy class has finite elements. Homework EquationsThe Attempt at a Solution I know that ##[G:Z(G)]<\infty##. If I consider the action on ##G## on itself by conjugation, each...

Homework Statement Determine all semisimple rings with a unique maximal ideal. The Attempt at a Solution If I call ##I## to the unique maximal ideal of ##R##, then ##I## can be seen as a simple ##R##-submodule, by hypothesis, there exists ##I' \subset R##, ##R-##submodule such that ##R=I...

Homework Statement Find the zero divisors and the units of the quotient ring ##\mathbb Z[X]/<X^3>## The attempt at a solution If ##a \in \mathbb Z[X]/<X^3>## is a zero divisor, then there is ##b \neq 0_I## such that ##ab=0_I##. I think that the elements ##a=X+<X^3>## and...

Homework Statement Find all the ideals of ##\mathbb Q[X]## The attempt at a solution Suppose ##I \subset \mathbb Q[X]## is an ideal with an element ##p(x) \neq 0##. Since ##\mathbb Q[X]## is an euclidean domain (the function ##degree(f)## is an euclidean function), then ##\mathbb...

Homework Statement Let ##X## be a discrete random variable, we say that ##x_0 \in R_X## is a most probable value for ##X## if ##p_X(x_0)=sup_{x \in R_X} p_X(x)##. 1)Show that every discrete random variable admits at least one most probable value. 2) Check that ##[(n+1)p]## is a most...

Edited, thanks for the correction.

Homework Statement Let ##E \subset \mathbb R^n## be a measurable set such that ##E=A \cup B## with ##|B|=0## (##B## is a null set). Show that ##A## is measurable. The Attempt at a Solution I know that given ##\epsilon##, there exists a ##\sigma##-elementary set ##H## such that ##E \subset...

I am given the parametrized curve ##\alpha:(-1,\infty) \to \mathbb R^2## as ##\alpha(t)=(\dfrac{3at}{1+t^3},\dfrac{3at^2}{1+t^3})##. I am asked to show that the line ##x+y+a=0## is an asymptote. So, I have to prove that when ##t \to \infty##, the curve tends to that line. My doubt is: The...

It is exactly as you've said, sorry for not making myself clear.

Homework Statement Suppose one extracts a ball from a box containing ##n## numbered balls from ##1## to ##n##. For each ##1 \leq k \leq n##, we define ##A_k=\{\text{the number of the chosen ball is divisible by k}\}.## Find ##P(A_k)## for each natural number which divides ##n##. The Attempt...

Homework Statement Let ##(\Omega,\mathcal F, P)## be a probability space and let ##A_1,A_2,...,A_n## be events in ##\mathcal F##. Prove the following inclusion-exclusion formula ##P(\bigcup_{i=1}^nA_i)=\sum_{k=1}^n## ##\sum_{\mathcal J \subset \{1,...,n\}; |\mathcal J|=k}...

Following your suggestions (haruspex,ehild), the distance between any point of the curve ##\gamma(t)## and the origin is ##||\gamma(t)-(0,0)||##. At the point ##\gamma(t_0)=P##, this function has a minimum and as the norm is a monotone increasing function, I can look at the function...

Homework Statement . Let ##C## be a curve that doesn't pass through the origin and let ##P## be the closest point on the curve to the origin. Prove that the tangent to ##C## at ##P## is orthogonal to the vector ##P##. The attempt at a solution. Suppose ##P=\gamma(t_0)##, I want to...

Homework Statement . If ##(x_{\alpha})_{\alpha \in \Lambda}## is a net, we say that ##x \in X## is an accumulation point of the net if and only if for evey ##A \in \mathcal F_x##, the set ##\{\alpha \in \Lambda : x_{\alpha} \in A\}## is cofinal in ##Lambda##. Prove that ##x## is an accumulation...

Oh, ##\bigcap_{i \in I, \mathcal A \subset \tau_i} T_i##, the intersection of all topologies which contain ##\mathcal A## is a topology (I've verified that intersection of an arbitrary number different from ##0## of topologies is a topology, and if ##\tau## is a topology such that every element...