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Is a non-trivial finite perfect group of order 4n?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial. Question: Is a non-trivial finite perfect group of ...
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profit and loss related aptitude qustion? [on hold]

A sold a watch to B at a gain of 5% and B sold it to C at a gain of 4%. If C paid Rs. 1902 for it, the price paid by A is?
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Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
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Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...
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What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?

Over the years, advances in machine learning has allowed us to communicate and interact, using the same natural language, more and more semantically with computers, e.g. Google, Siri, Watson, etc. On ...
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Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$. This question was proposed (problem ...
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Orientation form on the blow up of a Kaehler manifold

Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always ...
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Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
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How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?

Grojnowski constructs for a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$ The functor $E^*_{S^1}(-)$ takes in a space $X$ ...
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Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
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Is this differential identity known?

Recently I discovered the differential identity $$\frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$ valid for any odd natural number $k$; for ...
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Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
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Do discrete groups with property $(T)$ have “modest” subgroup growth?

I saw it conjectured at http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above ...
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Swan conductor, representation Weil group

Let $F$ be a non-archimedean local field and $\mathcal W_F$ its Weil group. We consider a linear representation $\sigma$ of $\mathcal W_F$. Could someone explain to me the definition of the Swan ...
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Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...
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Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$? If ...
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Projective family of probability spaces

This is a crosspost of this question from MSE. I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions ...
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Why the Dold-Thom theorem?

Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$ It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...
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Question:Let $G$ be a finite abelian group and $X$ be a G-set. $K$ be a subgroup of $G$. let $i$ be a group homomorphism from $K$ to $G$ . I am looking for the map $$i^{*} : H^{\alpha}_{G}(X,M) ... 4answers 696 views Ore's Conjecture for perfect groups We know that Ore's Conjecture is a theorem now. Does anyone know a counterexample to Ore's conjecture for perfect groups? I believe there should be one, but I dont have a counterexample. Thanks. 16answers 5k views How helpful is non-standard analysis? So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ... 9answers 3k views What are some examples of interesting uses of the theory of combinatorial species? This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by this thread, and sigfpe's comment to Pete Clark's answer. I've often heard it claimed ... 1answer 185 views Algebras admitting quantifier elimination I apologize if this question is meaningless or trivial: What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination? I need to say ... 0answers 87 views Modular forms related to G(q) and H(q) If G(q),H(q) are the functions appearing in Rogers-Ramanujan identities$$G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})}$$and ... 3answers 648 views Non-commutator in simple group? Hi, For a group G, we say that x\in G is a commutator if there exists a,b \in G such that x=a^{1}b^{-1}ab, and we say that x is a non-commutator if there is no a,b \in G such that ... 2answers 742 views Does every finite nontrivial group have two distinct irreducible representations over the complex numbers of equal degree? Is it true that for any finite nontrivial group G, there exist two inequivalent irreducible representations of G over the complex numbers that have the same degree. If so, is there an easy proof? If ... 1answer 287 views centralizer of the order 2^k cyclic permutation matrix over F_2 Let C be the 2^k\times 2^k-permutation matrix over \mathbb{F}_2 of the 2^k-cycle. We needed to know the structure of its centralizer in \mathrm{GL}_{2^k}(\mathbb{F}_2), and we computed it - ... 2answers 958 views Groups with all normal subgroups characteristic Today in my research, I had to use fairly explicitly the rather tautological property of finite cyclic groups that every normal subgroup is characteristic, i.e. fixed by all automorphisms. This got me ... 6answers 3k views Why does one think to Steenrod squares and powers? I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ... 1answer 753 views Universal covering of compact surfaces Is there any elementary (i.e. without using analytical methods like the theory of Riemann surfaces or more elaborate results from differential geometry) way to show that the universal covering of the ... 1answer 427 views Character of parity-twisted supersymmetric VOA module — question inspired by the Stolz-Teichner program I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader: Topological modular forms (TMF) is a generalized cohomology theory whose ... 2answers 916 views varieties with points in number fields Let V be a projective variety, defined over \mathbb{Q}. Suppose that for every number field K \neq \mathbb{Q}, there is a K-point of V. Does it follow that V has a \mathbb{Q}-point? ... 3answers 3k views What was Galois theory like before Emil Artin? I read that the primitive element theorem for fields was fundamental in expositions of Galois theory before Emil Artin reformulated the subject. What are the differences between pre and post-Artin ... 0answers 8 views Stochastic integration with respect to Fractional Brownian Motion I would like to know what can be said about integral process X_t = \int_0^t e^{-sr} dB_s^H,t\in[0,\infty), where B^H_t is Fractional Brownian Motion with Hurst parameter H>\frac{1}{2}, ... 2answers 570 views About Abhyankar's conjecture I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book Rigid Geometry and Its Applications. It claims that all quasi p-group G ... 1answer 48 views A quasicompact space with a net that contains no convergent strict subnet If x:\Lambda \rightarrow X is a net in a topological space X and \Lambda '\subseteq \Lambda is a cofinal subset of the directed set \Lambda, then x|_{\Lambda '} is a subnet of x. We call ... 1answer 130 views On Cantor sets every map is C^{\infty} For a fixed Cantor set K\subset [0,1] and a continuous function g:[0,1]\to \mathbb R. Is it always possible to find a C^{\infty} map f:[0,1]\to \mathbb R such that g and f coincide in K? ... 0answers 64 views Higher algebra and terminology about 2-objects It is well known that one way to build higher category theory is to use some induction process, where an n-category has as 0-cells some n-1 categories, such that for two 0-cells \mathcal{A} ... 0answers 40 views How to obtain a permutation of a tensor product? [on hold] I am looking for a way to efficiently compute a re-ordered kronecker product from the result of another kronecker product. For example, consider$$F=A\otimes B\otimes C\otimes D\otimes E from the ...

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