Topology

A conjecture about direct product of funcoids ★★

Author(s): Porton

Conjecture   Let $ f_1 $ and $ f_2 $ are monovalued, entirely defined funcoids with $ \operatorname{Src}f_1=\operatorname{Src}f_2=A $. Then there exists a pointfree funcoid $ f_1 \times^{\left( D \right)} f_2 $ such that (for every filter $ x $ on $ A $) $$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$ (The join operation is taken on the lattice of filters with reversed order.)

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

Keywords: category theory; general topology

Posted by porton
updated July 26th, 2012
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Number Theory

Special M ★★

Author(s): Kimberling

Let $ r $ denote the golden ratio, $ \frac{(1+\sqrt{5})}{2} $ and let $ \lfloor \rfloor $ denote the floor function. For fixed $ n $, let $ u(k) = \lfloor kr^n \rfloor $, let $ v(k) = \lfloor kr \rfloor^n $, and let $ w(k) = \left \lfloor \frac{v(k)}{k^{(n-1)}} \right \rfloor $. We can expect $ w $ to have about the same growth rate as $ u $.

Conjecture   Prove or disprove that for every fixed $ n > 0 $, as $ k $ ranges through all the positive integers, there is a number $ M $ such that $ u(k) - w(k) $ takes each of the values $ 1,2,\dots,M $ infinitely many times, and $ u(k) - w(k) \leq M $. (Can you formulate $ M $ as a function of $ n $? Generalize for other numbers $ r $?)

Keywords:

Posted by vprusso
updated June 12th, 2012
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