# All Questions

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### Counting all possible abstract simplicial complexes on a vertex set $[m]$

Let $[m]=\{1,2,\ldots,m\}$ and consider $2^{[m]}$ the power set on $[m]$. An abstract simplicial complex on $[m]$ is a subset $K\subset 2^{[m]}$ closed under the taking of subsets, that is if ...
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### Prove that $1+\sum_{n\geq1}\sum^n_{k=1}\frac{n!}{k!}{n-1\choose k-1}x^k\frac{u^n}{n!}=\exp\frac{xu}{1-u}$

Let $k,n\in\mathbb{N}_{>0}$. How do I get started to prove that" $$1+\left(\sum_{n\geq1}\sum^n_{k=1}\frac{n!}{k!}{n-1\choose k-1}x^k \frac{u^n}{n!}\right) = \exp\frac{xu}{1-u}$$ Hints and help ...
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### $C_G(x)$ in a solvable group

Let $G=PQ$ be a solvable group with $P$ and $Q$, P- and q-sylow subgroup of $G$ respectively. Suppose both $P$ and $Q$ are not normal and $C_G(P)=Z(G)$ and $C_G(Q)>Z(G)$. Let $x\in C_G(Q)-Z(G)$. So ...
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### Every element in a ring can be written as a product of non-units elements

I'm trying to understand a little detail in this proof: I didn't understand why in a ring we can always write an element as a products of non-units elements. I need help. Thanks in advance
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### complexity of matrix multiplication

For $n\times n$ dimensional matrices, it is known that calculating $\operatorname{tr}\{AB\}$ needs $n^2$ scalar multiplications. How many scalar multiplications are needed to calculate ...
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### Is a finite group determined by the family of all its 2-generated subgroups?

At the last week I meet my old coauthor, Oleg Verbitsky who proposed me the following question. I think that here should be an easy counterexample, but I am not a pure group theorist and I am usually ...
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### smallest $k$ s.t. $(x+y)^2\leq k(x^2-xy+y^2)$

I would appreciate if somebody could help me with the following problem Q: Find $K$? $$(x+y)^2\leq k(x^2-xy+y^2)$$. where $\forall x,y\in \mathbb{R}$
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### Intersection of topologies

Is my proof that the intersection of any family of topologies on a set $X$ is a topology on $X$ correct? Proof. We are required to show that the intersection satisfies the topology axioms. Let $\tau$ ...
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### Projective Objects in a Topos

I'm working a problem in MacLane and Moerdijk, and am not sure how to proceed. The end goal is to show that if $Sh(X)$ has enough projectives and $X$ is $T_1$, then $X$ has a basis of clopen sets. I ...
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### Evaluating $\int\cos\theta~e^{−ia\cos\theta}~\mathrm{d}\theta$

Is anybody able to solve this indefinite integral : $$\int\cos\theta~e^{\large −ia\cos\theta}~\mathrm{d}\theta$$ The letter $i$ denotes the Imaginary unit; $a$ is a constant; Mathematica doesn't ...
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### Is there a $f_n$ with two local maxima converges to f only one local maxima?

Is there a {$f_n$} with two local maxima converges(pointwise/uniform or other) to $f$ only one local maximum?
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### Connections between matrix and its inverse in $\mathbb{F}^{n^2}$

Let $A$ be an $n \times n$ matrix in $\textit{GL}(n, \mathbb{F}$). $A$ and $A^{-1}$ can be also viewed as a vectors in $\mathbb{F}^{n^2}$. Are there any non-trivial connections between $A$ and ...
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### Find domain of $\sin ^ {-1} [\log_2(\frac{x}{2})]$

Problem: Find domain of $\sin ^ {-1} [\log_2(\frac{x}{2})]$ Solution: $\log_2(\frac{x}{2})$ is defined for $\frac{x}{2} > 0$ $\log_2(\frac{x}{2})$ is defined for $x > 0$ Also ...
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### If T has a basis of Eigen Vector, how do I know that T can be represented as diagonal?

In Linear algebra, suppose I have: A linear transformation $T:V\to V$ A basis of eigen vectors $(v_1,...,v_n)$ How do I conclude that $T$ can be represented with a diagonal matrix $D$ ? Thank you ...
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### Showing the existence of a limit

Please show me the existence of the limit clearly $$\lim_{\large(h,k)\to (0,0)}\dfrac{\vert hk\vert ^{\alpha} log(h^2+k^2)}{\sqrt {h^2+k^2}} =0$$ for $\alpha > \frac12$
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### Finding all ordered tuples

Suppose $a+b+c+d+e=t$ and $a,b,c,d,e \geq r$ where all the given variables are positive integers. How do you calculate all ordered tuple of $a,b,c,d,e$ such that the above equation holds. The stars ...
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### Castelnuovo-Mumford regularity

Let $R=K[x_1,\dots,x_{10}]$, where $K$ is a field. Consider $$I=(x_1x_7,x_1x_{10},x_2x_8,x_3x_9,x_4x_{10},x_1x_5x_9,x_2x_6x_{10},x_1x_4x_5x_8,x_2x_5x_6x_9,x_3x_6x_7x_{10})$$ which is a squarefree ...
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### Problem on convergence of sequences

Given that $\lim f_n=1>0$, Show that there exists a positive integer $m$ such that $f_n\ge 0 \\ \forall n \ge m$
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### Three Dimensional Vectors Question

Let $\ell_1 , \ell_2$ be two lines passing through $M_0= (1,1,0)$ that lie on the hyperboloid $x^2+y^2-3z^2 =2$ . Calculate the cosine of angle the between the two lines. I have no idea about ...
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### Infinite product of an analytic function on the right half plane

I want to know whether $G_t=\prod_{i=1}^\infty G(s)^i$ is defined and analytic on the right half plane for an analytic $G(s)$ on the right half plane. If so, is $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
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### Is there any countable ordinal number which has a member undefinable?

Let $Q\colon=\{\alpha \in Ord \mid \exists \beta \in \alpha(\beta \text{ is undefinable in } \langle\alpha,<\rangle )\}$ denote the set of all ordinal numbers which has an undefinable member. ...
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### Nature of a triangle with vertices $z_1, z_2$ and $-1$ such that $|z_1|=|z_2|=1=z_1+z_2$ [closed]

If $z_1$ and $z_2$ are distinct complex number such that $|z_1|=|z_2|=1$ and $z_1+z_2=1$, then the triangle in the complex plane with $z_1,z_2$ and $-1$ as vertices must be: equilateral. right ...
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### Altitudes Ratio

If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''? ...
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### Reciprocal Shifted Log-Normal Distribution

Let $X$ be a log-normal distribution, let $k\geq0$ be a real value and let $Y=\frac{1}{X+k}$. What is the name of the $Y$ distribution other than 'reciprocal shifted log-normal'? What is the mean of ...
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### Determinant and Inverse of a Difference of two matrices

I've got an expression of the form $$\det(I-AB)$$ and I'm wondering if there is a way to write this solely in terms of functions of $A$ and $B$. For the particular case I'm ...
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Suppose we have $k$ open balls $B_1,\dots,B_k$ in $\mathbb{R}^n$ centered at $0$ (their radii may be different) and $k$ vectors $v_1,\dots,v_k\in\mathbb{R}^n$. Is it true that $\mu\left(\cup_{i=1}^k ... 0answers 86 views ### What happens in dimension 125? In Differential topology 46 years later (page 807, bottom of left column) Milnor states that for$n \neq 4, 125, 126$if the order of the stable homotopy groups$|\Pi_n|$is known then we can compute ... 2answers 94 views ### How to show that$f(x,y)$is continuous. How to show that$f(x,y)$is continuous. $$f(x,y)=\frac{4y^3(x^2+y^2)-(x^4+y^4)2x\alpha}{(x^2+y^2)^{\alpha +1}}$$ for$\alpha <3/2$. Please show me Thanks :) 4answers 285 views ### Finding the area, general case with angle$\theta$. Inspired by this question, I am curious to know the more general case. Given the radius of the large circle as$R$and the angle$\theta \le \pi$, what is the area of the colored section? My ... 1answer 61 views ### Smooth approximation How one can show, that if$f(x_1,\ldots,x_n)$is a continuous function on an open subset$U\subset \mathbb{R}^n$, then for every$\varepsilon > 0$and every open$V\subset U$, such that$\bar V ...
If $X,Y$ are topological spaces and $h:X\rightarrow Y$ is a continuous map, is there some sort of induced map \begin{align*} h':C_b(X)\rightarrow C_b(Y) \end{align*} (or in the other direction) where ...
This is an exercise from Ahlfors' Complex Analysis text. I need to show that the mean value property holds for the function $u=\log|1+z|$ in the circle with center $z_0=0$ and radius $r=1$. The ...