# All Questions

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### Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$\forall f,g,h\in G:hg(f)=h(g(f))$$ Now suppose there is additional axiom, or constraint if you prefer, ...
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### prime numbers and Pythagorean triplets

What are the necessary conditions for two of the terms in the Pythagorean triplet $a^2 = b^2 + c^2$ to be prime numbers?
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### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem First we give a short introduction: A quadratic system is a polynomial vector field on ...
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### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices.
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### A kind of saturation property related to forcing notions

Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...
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### Unreasonable application of mathematics to the other areas [on hold]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science? I found ...
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+150

### Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...
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### Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$. Is there a bound of the Leopoldt defect of $M$ ...
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### Simplicial Model of Hopf Map?

The Hopf fibration is a famous map S3 --> S2 with fiber S1, which is the generator in pi_3(S2). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces ...