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Topological transitivity of the shift map $M^{\mathbb N} \to M^{\mathbb N}$ does indeed follow if you are willing to assume that $M$ has a countable basis, by a proof which is pretty similar to the pr …

Perhaps I am misunderstanding the question, but it seems to me that $\text{deg}(V)=\text{deg}(\text{det}(V))=\text{deg}(L)$, so we are just asking if for each $n$ there is a (virtual) vector bundle $[ …

The best unconditional complexity bounds I know of are those coming from Baker's theory of linear forms in logarithms. Baker wrote a paper "The Diophantine equation $Y^2 = aX^3 + bX^2 + cX + d$" where …

There is a really nice formula which is valid for |q| < 1, and it involves the dirichlet convolution of arithmetic functions. First, let's denote $\frac{1}{n}$ to mean the arithmetic function that ta …

I assume that you have given your field extension $L=K(\alpha_1,\ldots,\alpha_n)$ in the form of a list of polynomials generating the kernel $I$ of $K[X_1,\ldots,X_n]\twoheadrightarrow L$. The interse …

In 1970, Harry Kesten proposed essentially this question in the Advanced Problems section of The American Math Monthly. Let $X_1, X_2, \ldots, X_n$ be iid random variables and $S_n = \sum_{i=1}^n …

Here is some information that you might find useful. The context is that of PageRank computation, for the purpose of updating only some of ranks (essentially entries of the eigenvector). Y. Y. Chen …

The answer is not. Calabi and Eckmann ("A class of compact, complex manifolds which are not algebraic", Ann. of Math. 58 (1953) 494-500) proved that $\Bbb R^{2n}$, $n > 1$, has a complex structure whi …

Let $\varphi_a $ be the $ a $ th partial computable function in a standard way. Given a recursively enumerable set $ W $, let $ f (n) $ be the maximum of $\varphi_a (b)$ over $ b\le n $ and $ a $ amo …

How about this proof? Let $\omega_1$ be the set of all ordinal numbers which are countable. Then $\omega_1$ is itself ordinal number, but it must be uncountable, otherwise it would contain itself. We …

There is also Shelah's very enjoyable construction using descending sequences of infinite subsets of $\omega$, close in spirit to Aronszajn's tree of rational sequences, and described in Judith Roitma …

For every $\kappa$ of uncountable cofinality there is a tree $T\subseteq 2^{<\kappa}$ such that $[T]=\kappa$. The tree $T$ is the tree of all binary sequences $f\colon \alpha \to 2$, $\alpha <\kappa$ …

No, consider the characteristic function $ f $ of a union $ A$ of infinitely many disjoint closed annuli centered on the origin in $\mathbb R^2$. If the annuli have sufficiently fast-shrinking radii t …

This doesn't seem quite correct. For one thing the final condition does not make sense as written. It should probably be " It is not the case that $d_{2i}=a_{2i+1}$ (so $d_{2i-1}=0$) for all large en …

This works, as pointed out by Samuel in a comment. Here's an easy direct argument: We are claiming that no non-trivial polynomial $$ p(x) = \sum_{j=1}^N a_j x^{n_j} $$ with $N$ terms (let me call $N$ …