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For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups G of order} \le N\text{ such that } \phi \text{ is valid in } ... 0answers 3k views ### A short proof for \dim(R[T])=\dim(R)+1? If R is a commutative ring, it is easy to prove \dim(R[T]) \geq \dim(R)+1. For noetherian R, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ... 0answers 2k views ### The Ring Game on K[x,y,z] I recently read about the Ring Game on Mathoverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ... 0answers 1k views ### Identification of a curious function During computation of some Shapley values (details below), I encountered the following function:$$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$where ... 0answers 1k views ### What properties of busy beaver numbers are computable? The busy beaver function \text{BB}(n) describes the maximum number of steps that an n-state Turing machine can execute before it halts (assuming it halts at all). It is not a computable function ... 0answers 952 views ### How prove this matrix inequality \det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)? Question: let matrices A,B,C\in M_{n}(C) be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$Show that:$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)... 0answers 692 views ### Is there a categorical definition of submetry? (Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ... 0answers 2k views ### Grothendieck 's question - any update? I was reading Barry Mazur's biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ... 0answers 821 views ### Why are asymptotically one half of the integer compositions gap-free? This is a longish post about something that has been haunting me for a while about a kind of restricted composition, namely gap-free and complete compositions. First, I will define the terms that are ... 0answers 736 views ### Extending the result \int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi}  While generalizing this result, I succeeded in proving that for \alpha > 0, \beta < 1 and 1 < 2\alpha + \beta < 3, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ... 0answers 2k views ### Continuous projections in \ell_1 with norm >1 I was trying to find papers and articles about non-contractive continuous projections in \ell_1(S) where S is an arbitrary set. If it is not studied yet, I would like to know results for the case ... 0answers 601 views ### Is the sequence (a_n) defined by a_n=\tan{a_{n-1}}, a_0=1, dense in \Bbb{R}? Let a_0=1,a_n=\tan{a_{n-1}}. Then is \{a_n\}_{n=0}^\infty dense in \Bbb{R}? I've drawn a map of this dynamical system and it seems that the sequence is dense on \Bbb{R}. 0answers 923 views ### Is this similarity to the Fourier transform of the von Mangoldt function real? Mathematica knows that the logarithm of n is:\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right) The von Mangoldt function should then be: ...
Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...