This set is due the last day of lecture, Friday May 6.

Let be an entire function,

,

where the series converges for all complex numbers .

Basic results about power series give us that the series converges absolutely, i.e.,

for all , and that for any , if is a series such that , then converges as well.

Given a finite dimensional inner product space , and a , we want to define , in a way that it is again a linear operator on . The most common example is when . This “exponential matrix” has applications in differential equations and elsewhere.

To make sense of , we define making use of the power series of :

Of course, the problem is to make sure that this expression makes sense. (Use the results of Homework 4 to) show that this series converges, and moreover

Fixing a basis for , suppose that is diagonal. Compute in that case. In particular, in , find where

Show that, in general, the computation of reduces to the computation of for a matrix in Jordan canonical form.

For

a Jordan block, show that in order to actually find reduces to finding formulas for for Find this formula, and use it to find a formula for . It may be useful to review the basics of Taylor series for this.

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When we’re supposing that T is diagonal, do you mean the matrix associated to T is diagonal? Are we still computing f(T) or then f(M(T))? I think I’m a little confused as to when we’re using the matrices or the linear operators in the function f.

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]

I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]

I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]

The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]

If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]

I changed the last matrix to have a non-diagonalizable example.

Hello Dr. Caicedo,

When we’re supposing that T is diagonal, do you mean the matrix associated to T is diagonal? Are we still computing f(T) or then f(M(T))? I think I’m a little confused as to when we’re using the matrices or the linear operators in the function f.

thanks,

Hi Rachel: Yes; once we fix a basis , we can identified with the matrix , and by saying that is diagonal I meant that is.