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The hotspots conjecture can be expressed in simple English as: ...is only believed to hold for ''generic'' solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic e ...

...oject, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we woul ...ors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>. ...

...chniques of Maynard. This project concluded with a bound of H=246, as well as additional bounds on H_m (see below). ...ecture (or more precisely, a generalization of this conjecture, formulated as Conjecture 1 in [BFI1986]), without this assumption, and without EH or the ...

We will need effective upper bounds on various integrals that occur as error terms, with explicit constants. Here is a basic tool to do this: 4. With the same hypotheses as part 3, we also have ...

...also have the variants <math>D_{n,0}=D_n, D_{n,1}, D_{n,2}</math> defined as where <math>\Gamma_{a,b,c}</math> is defined as the strings in <math>[3]^n</math> with <math>a</math> 1s, <math>b</math> 2s ...

...i</math> at one dimension is automatically inherited by higher dimensions, as are shifted versions of this inequality (in which <math>\alpha_i</math> is The 43-point solutions have distributions (a,b,c,d,e) as follows: ...

...oring <math> \sigma \circ {\mathbf c} \circ T^{-1}</math> has the same law as <math>{\mathbf c}</math>. This gives the color permutation invariance ...a monochromatic triangle. One can use other graphs for this purpose, such as the 278-vertex graph <math>M_1</math> or <math>V \oplus V \oplus H</math>. ...

A '''unit distance graph''' is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an ed ...n numbers] <math>1,3,4,7,9,12,\dots</math>. These numbers arise naturally as the apex angle of a <math>\sqrt{t}, \sqrt{t}, 1</math> isosceles triangle, ...

for <math>s</math> as above and <math>h=O(1)</math>. .../math> independent of <math>N</math>, we can thus write the left-hand side as ...

* [http://www.ams.org/notices/200210/fea-granville.pdf It's As Easy As abc], Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2 ...chmuller Theory], Shinichi Mochizuki: "Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint (perhaps even a sort of Ros ...

that is proven as follows. as required. ...

Known colloquially as "The Erd&#337;s discrepancy problem", this question has remained unanswered ...h> contained in <math>\{1, 2, ..., N\}</math>, we can restate the question as whether <math>\scriptstyle \delta(\mathcal{HAP}(N),x) \to \infty</math> for ...

...Obviously, this is a project that can't be completed in a reasonable time, as the largest completely multiplicative sequence has length 127645 (See [http This seems to be as far as we can get with the assumption f(2) = 1. ...

The integral here can be evaluated as an Airy integral, but perhaps a Taylor expansion of the last term is a bett ...isson summation without using the Taylor expansion. We first observe that as <math>\exp( \pi i n^2 - \pi i n ) = 1</math> for all integers <math>n</math ...

Let the weight function be defined as: w(x)=3 if x=1, 2, 3, 4, 5, 6, 6, 7, 8, 0, and 2 otherwise. The target weig ..., and vice versa for those with 4 of 1, 2, 3, 4, 5 and 1 of 6, 7, 8, 9, 0, as follows: ...

...us automorphism <math>z \mapsto z^2</math>. We will use this finite field as our choice of colors. '''Proof''' As in Lemma 1, we have ...

...thcal{A}}</math> is again a lattice: the join of any two 'old' elements is as before; the join of <math>A_J</math> with some <math>K\in\mathcal{A}</math> ...vee P</math> is surjective. Write any given <math>B\in\uparrow\{P\}</math> as a join of join-irreducibles. Since <math>P</math> is prime, <math>P</math> ...

...x = \{A \in \mathcal{A} : x \in A\}</math>, contains at least as many sets as <math>\mathcal{A}_x = \{A \in \mathcal{A} : x \notin A\}</math>. A slightly ...}</math>, the set <math>\mathcal{A}_{xy}</math> has at least as many sets as <math>\mathcal{A}_\emptyset</math>. Therefore, depending on which of <math> ...

The only possible set in this case is the <math>C_5</math> set 12345 (as any smaller sets would violate the assumption), and it has weight exactly 3 ...ath>C_5</math> set 12345 has surplus 5, so the surplus is at least as much as the deficit in this case. ...

...ss of generality that the left-hand side is positive, then we may write it as ...th>. Hence we can actually just use the same value of <math>\sigma</math> as in the toy case. ...