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Let $ X$ be a smooth surface of general type with $ q=p_g=0$ , define a set $ A=\{C\subset X|$ C is smooth rational and $ C^2<-1\}$ , let $ k=|A|$ which is cardinality of $ A$ . What is the minimal value of $ k$ can be? Can you give me some example ofRead more

This was [asked but never answered at MSE] and This question has second an open bounty,until Now no one can solve it,I think there should be a more professional here severe people may be able to solve this problem, I am looking forward to (https://math.stackexchange.com/questions/2598247/how-to-determine-the-minimal-constant-lambda-lambdan-k). Given fixed positive integers $ n,k$ , determine the minimalRead more

Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I wanted for the problem I was thinking about, but it seemed natural enough that it might have some useful interpretation or application.Read more

Let $ P(n)$ be the set of all monic polynomials of degree $ n$ with integer coefficients, such that all coefficients have absolute value at most $ 2^n$ . Given a positive integer $ n$ let us define a function $ A(n)$ as the minimal number $ m$ such that for any $ f \inRead more

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Is the following statement true: Any surjective homomorphism $ f:G\to H$ of groups with equal rank maps a minimal generating system $ x$ of $ G$ to a minimal generating system $ y$ of $ H$ ? Is it only valid for finite rank? How about infinite rank?Read more

$ A_{fin}$ algebra it is $ A_\infty$ algebra with $ m_n = 0$ for $ n >> 0$ and $ A^i = 0$ for $ |i| >> 0$ . Suppose that we have (compact) dg-algebra $ A$ , we can build $ A_\infty$ minimal model on $ H^*(A)$ what we can say about $ A$Read more

This question already has an answer here: Can you help me with my software licensing issue? 5 answers Sadly 25 Clients are required until KMS approves activations requests. Do you know if this is counted per Product (i.e. Windows and Office seperately) or in total? I need to activate Office 2016 on some machines withRead more

Let $ G$ be a (connected) graph with $ n$ vertices. Is it true that the maximum cardinality of a minimal vertex cover of $ G$ is $ \geq \lfloor \frac{n}{2} \rfloor$ ? If so, can you point out any reference? If not, what is an easy counterexample? Thanks!Read more

Let $ G = (V,E)$ be an infinite graph. $ C\subseteq V$ is called an edge cover if $ C\cap e \neq \emptyset$ for all $ e\in E$ . Question. If $ G = (V,E)$ is infinite (i.e. $ |V|\geq \aleph_0$ ), is there an edge cover $ C$ such that for all $ v\inRead more