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see Mollin's paper and the various references given there. You will probably also find relevant material in his book "Quadratics".

If $n=\text{dim}_{K}(H_1(\mathbf{x},A))$ then $\text{dim}_{K}(H_2(\mathbf{x},A))=\frac{n(n-1)}{2}$. This was proved by Assmus in 1958. In general, $H_*(\mathbf{x},A)$ is the exterior algebra over $H_1 …

Jurkat and van Horne showed that the $L^1$ norm is asymptotic to a constant times $\sqrt{N}$ (see Theorems 4 and 5 in their paper which compute all moments). For other related work see Jurkat and van …

I am duplicating an old problem from stackexchange: Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure …

The size of $|p^n(a)|$ can be very different depending on $a$. If $a$ is periodic point of $p$, then this sequence will be bounded. If $a$ is in the domain of attraction of infinity then $\log|p^n(a) …

Let me give another proof that the long line is not paracompact. A theorem of Tamano states that a completely regular space $X$ is paracompact if and only if $X\times\beta X$ is normal where $\beta X$ …

I do not think the general inequality holds without further qualifications. Separability of the function class is one assumption which enables application of Bousquet's version of Talagrand's inequali …

For Smirnov's result I think the easiest approach (at least on a "hand waving" level) is via empirical processes: As long as $F$ is continuous then it suffices to consider uniform distributions. Let …

You might want to try Brandt matrices and lattice methods instead. If you can write your function as a theta series, then the computation of the terms can be done very quickly through multiplication o …

This is completely worked out in Walsh's paper: P. G. Walsh, Integer Solutions to the Equation $y^2=x(x^2\pm p^k)$ (2008) In particular, for your elliptic curve $y^2=x^3+px$, there are at most 2 p …

Yes, there are a couple of alternate proofs: Here is one: An Elementary Proof of Lebesgue's Differentiation Theorem Michael W. Botsko The American Mathematical Monthly Vol. 110, No. 9 (Nov., 2003), p …

To add a more general, optimistic answer to the question in the title, rather than the question in the question --- while the isomorphism class of $\pi_1(M)$ is not determined by the Laplace spectrum, …

Thanks, Joel, for mentioning our paper. By Martin-Steel, if there exist infinitely many Woodin cardinals, then every uncountable projective set contains a perfect set, so, by the result of Hausdorff …

You can get the numbers "r" from Table 1 page 612 of the following paper: http://projecteuclid.org/download/pdf_1/euclid.tmj/1178228413 If you are not used with the notation you can find useful Rema …

This inequality cannot be true. Let us rewrite it in the more common form $$P(R_n\ge x)\le e^{-x^2/2} \tag{1} $$ for $x\ge0$, where $R_n:=S_n/b_n$, $S_n:=\sum_1^n c_iB_i$, $b_n:=\sqrt{\sum_1^n c_i^2} …