3
votes
0answers
52 views

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 ...
4
votes
2answers
138 views

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 ...
2
votes
1answer
80 views

$C_G(x)$ 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 ...
0
votes
1answer
132 views

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
7
votes
1answer
125 views

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 ...
13
votes
1answer
164 views

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 ...
3
votes
4answers
69 views

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 $\forall x,y\in \mathbb{R}$
9
votes
2answers
734 views

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 $\tau$ ...
2
votes
1answer
109 views

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 ...
2
votes
2answers
133 views

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 ...
0
votes
2answers
49 views

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?
0
votes
0answers
34 views

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 ...
2
votes
3answers
68 views

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 ...
2
votes
3answers
74 views

Write $100$ as sum of $n$ numbers, such that each number is twice as big as its predecessor.

I don't quite know where to start on this one. lets say we have a value 100. and we want to split it in two parts where one is twice as big as the other. That would be $v_1 = 66.666$ and $v_2= ...
2
votes
3answers
223 views

$F(x,y)$ is continuous.

Prove that $$ f(x,y)=\begin{cases}\frac{x^3-xy^2}{x^2+y^2}&\text{if }(x,y)\ne(0,0)\\0&\text{if }(x,y)=(0,0)\end{cases}$$ is continuous on $\mathbb R^2$ and has first partial derivatives ...
2
votes
5answers
599 views

Looking for a clear definition of the geometric product

In brief: I'm looking for a clearly-worded definition1 of the geometric product of two arbitrary multivectors in $\mathbb{G}^n$. I'm having a hard time getting my bearings in the world of ...
2
votes
1answer
68 views

What does $f \in L^p(I,V) + L^q(I,H)$ mean?

What does $f \in L^p(I,V) + L^q(I,H)$ mean? What does the addition mean?? I see this in PDE theory.
0
votes
1answer
107 views

Which way of writing functions is the most correct?

In functional programming it's not uncommon to bind a closure/lambda/anonymous function to a value name, i.e. $$f = x \mapsto x^2 + 3$$ so I've been wondering which is more right to do in ...
1
vote
2answers
88 views

Solving the inequality: $1\leq \cot^2(x)\leq 3$

I want to solve the following inequality: $$1\leq \cot^2(x)\leq 3.$$ but I'm unsure of how to handle the positive and negative square roots. If I take the square of the inequality, just focusing on ...
1
vote
1answer
1k views

solving heat equation under Neumann boundary conditions

I am having trouble solving following equation \begin{align*} u_t - u_{xx} = 0 & \; ;0<x<L, t>0 \\ u(0, x)=x &\; ; 0<x<L\\ u_x(t, 0) = u_x(t, L) = 0&\; ; t >0 ...
5
votes
2answers
173 views

Limit involving the Riemann zeta function, why is this identity trivial?

Mathematica knows that: $$n^k=\lim_{s\to 1} \, \frac{\zeta (s) \left(1-\frac{1}{\exp ^{s^{n^k}-1}(n)}\right)}{n}$$ Why is the above a trivial identity? What is it about the Zeta function that makes ...
4
votes
0answers
92 views

An element of $\ell^2$ wanted

I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is ...
1
vote
0answers
41 views

Computing group of exotic spheres

In Levine on page 90 it is stated that the following sequence is exact $$ 0 \to bP^{n+1} \to \Theta^n \to Coker(J_n) $$ where $\Theta^n$ is the group of exotic spheres, $bP^{n+1}$ is the subgroup of ...
3
votes
1answer
93 views

Splitting field of resolvent equals that of $f$

Lemma: Let $\Psi \in k[X_1,...,X_n]=:B$ be s.t. $stab_{S_n}(\Psi)=H \subset S_n$, $S_n/H=\{ \Psi, t_2 \Psi,..., t_e\Psi \}$, $\Delta_\Psi$ the discriminant of $L_\Psi:=\prod_{i=1}^e (X-t_i \Psi)$, and ...
1
vote
1answer
183 views

Splitting field of $x^m - 1$ over $\mathbb F_p$

I need to find the splitting field of a polynomial $x^m-1 \in\mathbb{F}_p[x]$. I know that if $(m,p)=1$ then the splitting field is $\mathbb{F}_p(z)$ where $z$ is primitive root of unity of order $m$. ...
0
votes
3answers
2k views

project a point in 3D on a given plane

A point in a 3D space is given as $ P(x,y,z) $. I want to find the position of this point projected parallel to the normal on a plane Q defined by $3$ non-collinear points $ Q1(x1,y1,z1), ...
0
votes
2answers
49 views

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 ...
0
votes
3answers
63 views

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 $\alpha > \frac12$
0
votes
1answer
55 views

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 ...
3
votes
0answers
103 views

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 ...
0
votes
2answers
88 views

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 $
0
votes
2answers
53 views

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 ...
1
vote
0answers
76 views

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 $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
0
votes
2answers
74 views

Surgery on $S^m$

On page 4 of the book "ALGEBRAIC AND GEOMETRIC SURGERY" by Andrew Ranicki, after the definition of surgery has written: Example View the $m$-sphere $S^m$ as $$S^m=\partial (D^{n+1} \times ...
2
votes
0answers
186 views

Notation for sampling a random variable

my question is pretty much the same as the one asked here: Notation for sampling random variate (I did not find a satisfying answer there...): I have a random variable $X \sim U(1, 10)$. I want to ...
3
votes
3answers
732 views

Proving existence of $T$-invariant subspace

Let $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ be a linear transformation. I'm trying to prove that there exists a T-invariant subspace $W\subset \mathbb{R}^3$ so that $\dim W=2$. How can I prove ...
0
votes
1answer
45 views

How to define a function that gives us the number of pentagons formed in between two or more hexagons?

I have been trying to make a general formula/function that helps in calculating the number of pentagons that may be formed using 2 or more hexagons. Like it is shown in the picture below: In Fig. 1 ...
6
votes
1answer
186 views

A consequence of the law of large numbers

Let $(X_k)_{k=1}$ be Poisson random variables with expectation $\mu$, let $Y_n = \sum_{k=1}^{n} X_k$. The weak law of large numbers states that, $$ \forall \delta>0, \forall \epsilon>0 \, \, ...
7
votes
3answers
200 views

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. ...
-1
votes
3answers
156 views

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 ...
0
votes
1answer
181 views

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''? ...
1
vote
0answers
161 views

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 ...
1
vote
1answer
690 views

Determinant and Inverse of a Difference of two matrices

I've got an expression of the form \begin{equation}\det(I-AB)\end{equation} 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 ...
3
votes
1answer
136 views

Balls going away from the origin

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 ...
2
votes
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 ...
2
votes
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 :)
3
votes
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 ...
0
votes
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 ...
3
votes
2answers
179 views

Do maps between topological spaces somehow induce maps between Banach spaces?

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 ...
2
votes
1answer
391 views

Mean value property for a harmonic function

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 ...

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