Solve the following problems from Chapter 3 of the textbook: 2, 4, 5, 7, 9, 10, 26.

In addition, solve the following problem:

Let be a linear map between finite dimensional vector spaces over For we denote by the preimage of under This is the set of all vectors such that

Recall that a subset of a vector space over is convex iff whenever and then also Show that is convex iff is convex. Is it necessary that is linear for this to hold, or does it suffice that is continuous?

Assume that is a subspace of and show that is a subspace of

43.614000-116.202000

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When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

First of all, $f(z)+e^z\ne 0$ by the first inequality. It follows that $e^z/(f(z)+e^z)$ is entire, and bounded above. You should be able to conclude from that.

Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$. Of course there are many possible choices […]

I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is no […]

No, the rank of a set $x$ is the least $\alpha$ such that $x\in V_{\alpha+1}$. Note that if $\alpha$ is limit, any $x\in V_\alpha$ belongs to some $V_\beta$ with $\beta

When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.

Yes, I am thinking of as and I would like a proof or a counterexample in this context.