-3
votes
3answers
69 views

About counting the terms of the expression $x^{m-1} + x^{m-2} +\ldots+ x^0$ given that $x=1$

I want to understand how I can count the terms of the expression $x^{m-1} + x^{m-2} +\ldots+ x^0$ when $x=1$. The result is $m$, I dont know how to count them formally, any advice would be helpful. ...
0
votes
1answer
55 views

King, Queen, 2 rooks, 2 bishops and 2 knights in 1 line -basic combinatorics [duplicate]

This is also a basic combinatorics question, but I don't understand part of its solution. We have a King, a Queen, 2 rooks, 2 bishops and 2 knights (each of the last three pieces are identical, i.e. ...
3
votes
1answer
299 views

Hodge star operator and volume form, basic properties

let $(M,g)$ be an oriented Riemannian manifold. Let $*$ be the hodge operator, I want to prove that $$*\mathrm{vol}_g =1$$ where $\mathrm{vol}_g$ is the associate volume form $\sqrt{g} e^1\wedge ...
2
votes
1answer
124 views

Convex function property

Let $ f_{1}, f_{2},..., f_{n} $ convex functions in the interval $[0,1]$ such that $ max(f_{1},f_{2},...,f_{n}) \geq 0 $ show that there exist positive real numbers $a_{1}, a_{2},...,a_{n} $ not ...
0
votes
2answers
75 views

Are there any formulas or identities that involves $\pi$ but have no obvious trigonometric interpretation?

Every formula that involves $\pi$ has an underlying trigonometric interpretation but it is not usually obvious. I wonder if there are any formula like the Gaussian integral ...
0
votes
1answer
25 views

Calculate $P(A_n)$, where $P$ uniform distribution on $[0,1]$

$P$ uniform distribution on $[0,1]$. $$A_n=\bigcup_{i=1}^{2^n-1} \left [ \frac{2i-1}{2^n}, \frac{2i}{2^n} \right ], n \in \mathbb{N}$$ To calculate $P(A_n)$ do we have to do the following?? ...
2
votes
1answer
56 views

Find the first coefficients of the inverse series of $x+x^2\sqrt{1+x}$

This is exercise 2c from chapter 2 of Wilf's generatingfunctionology. The problem is to find the inverse series $g(x)$ of the series of $f(x)=x+x^2\sqrt{1+x}$ (i.e. $f(g(x))=g(f(x))=x$). I get nowhere ...
3
votes
1answer
137 views

Finding $6$ pairwise non-isomorphic semi-direct products of order $42$

I'm stuck on an old algebra prelim question concerning semi-direct products. I am to find $6$ pairwise non-isomorphic groups of order $42$ that are semi-direct products. I think my main issue is how ...
0
votes
3answers
45 views

2nd order differential equation question

By finding a suitable particular integral, find the general solution $y$ of the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}=f(x)$ when a) $f(x)=3e^{-2x}$ b) $f(x)=1-x^2$ For part a ...
3
votes
2answers
102 views

Matrix equation $aX^{3} + bX^{2} = I$.

I want to solve the matrix equation for $X$ $$aX^{3} + bX^{2} = I,$$ where $a,b \in \mathbb{R}$ and $X \in \mathbb{R}^{n\times n}$. My thoughts: If $a = 0$ or $b = 0$, the solution is easy. If $a, ...
0
votes
1answer
124 views

Different ways to write $n$

What is a general formula $n(n)$ for this? We know that starting from below, we can see how many numbers a certain $n$ generates by counting the number of numbers contained in the column $n$ is in, ...
1
vote
2answers
101 views

Recursion with generating function.

Using generating function determine $u_n$ $$u_{n+2} -6u_{n+1} + 9u_n = 2^n + n $$ I am asking for you to give me some advices. Thanks in advance.
0
votes
2answers
99 views

(Geometric algebra) Acceleration of a particle with constant speed as a bivector-vector inner product

I've been working on (self-studying) Geometric Algebra for Physicists which, sadly, has no solutions manual. This is not a problem in general, but I feel like one of my solutions for a question asked ...
0
votes
0answers
52 views

linear programming slater condition

I am wondering if anyone could help to come up with a such example: ...
0
votes
2answers
104 views

Understanding a “matrix representation”

Consider an abstract linear transformation; $$f: V \rightarrow V$$ $V$is a polynomial vector space of degree less than or equal to 2. Thus, it has a basis $1,x,x^2$. Now, what does it mean to say ...
4
votes
1answer
187 views

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not ...
0
votes
0answers
32 views

Proving the solutions to an Equation

Consider: I am sort of confused. Without using the intermediate value theorem, directly. I suppose indirect use if fine. $1 + x^2 + \sin^2(x) > 0$ for $x \in \mathbb{R}$ Let the $LHS = I$ $$I ...
5
votes
1answer
76 views

Approximating a piecewise continuous function with a function in $\mathcal{C}^{\infty}_{0}(\mathbb{R})$

Let $\eta \in \mathcal{C}^{\infty}_{0}(\mathbb{R})$, where $\mathcal{C}^{\infty}_{0}(\mathbb{R})$ is the set of compactly supported infinitely differentiable function, be a function which is ...
1
vote
1answer
31 views

Is comaximal equivalent to simple?

A finite group $G$ is called comaximal if for any non-trival irreducible representations $V$ and $W$ of $G$, it exists $n \in \mathbb{N} \ $ such that $(V^{\otimes n},W)\ge 1$. A finite group $G$ is ...
1
vote
1answer
89 views

Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$ [duplicate]

Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$ I didn't solve this problem, but I proved that f is increasing and $ x<f(x)<e^{x} $ please help
0
votes
1answer
43 views

solving systems of equations for m and b when you know they are both positive?

I am trying to make a website that runs off of this equation. I am only in algebra but I am trying to solve a systems of equation where instead of solving for x and y i am solving for m and b. Here is ...
1
vote
0answers
30 views

$ \|u(x)+u(x)-x +o(\|x\|^p)\|<r $?

Set $B_r=\{ x\in \mathbb{R}^n : \|x-0\|< r \}$ for any $r>0$. Let $C^p_0(B_r,B_r)$ the set of all smooth functions $u:B_r \to B_r$ of class $C^p$ such that $u(0)=0$. I would like to prove the ...
2
votes
2answers
133 views

Euclidean Distances Between Points on a Sphere

Let $r$, $g$, and $b$ be positive real numbers such that $r \geq g \geq b$. Consider a sphere of radius $r$ centered at the origin. If M is any point on the sphere, prove that there exist two points P ...
5
votes
1answer
81 views

What does $\alpha$ be the $1$-form and $\beta$ the $2$-form on $\mathbb{R}^3$ mean?

In an earlier post to math.stackexchange I asked a question beginning with: Let $\alpha$ be the $1$-form and $\beta$ the $2$-form on $\mathbb{R}^3$ given by $$\alpha=(x+y)\,dy+(x^2-y^2)\,dz$$ ...
4
votes
2answers
117 views

Finding maximum and minimum with 2 constraints

Let $C$ be the curve of intersection of the plane $x+y-z=0$ and the ellipsoid $$\frac{x^2}4+\frac{y^2}5+\frac{z^2}{25}=1$$ Find the points on $C$ which are farthest and nearest from the origin When ...
1
vote
1answer
33 views

Modelling an energy profile with/without catalyst

I'm trying to make a data table that can be plot to model the energy profile of a chemical reaction with/without catalyst. example image This is essentially what I want to plot, and I'm at a loss for ...
3
votes
1answer
121 views

Geometric Transformations of Images

Related to image processing, I'm familiar with different types of geometric transformations. Translation, scaling, similarity, Affine, Perspectivity, and Projective Transformation. Is there a ...
4
votes
3answers
119 views

Solve $\sin \frac{\alpha - \beta}2 + \sin \frac{\alpha - \gamma}2 + \sin \frac{3\alpha}2 =\frac 3 2$

Solve the following trigonometric eqation where $\alpha, \beta, \gamma$ are angles in a triangle ($\alpha + \beta + \gamma = 180$): $$\sin \frac{\alpha - \beta}2 + \sin \frac{\alpha - \gamma}2 + \sin ...
1
vote
1answer
65 views

Help explain this “atom” and what this $G$ is.

Following are some slides from a lecture that you can watch here (starting at 20:40). It is explaining the Hierarchical Dirichlet Process. https://www.youtube.com/watch?v=PxgW3lOrj60 In the first ...
1
vote
2answers
86 views

If $T \models \phi$ then there is a finite subtheory $T' \subset T$ such that $T' \models \phi$

Use the Compactness Theorem to show: if $T \models \varphi$ then there is a finite subtheory $T' \subset T$ such that $T' \models \varphi$. I don't see how I can use the compactness theorem here. ...
2
votes
0answers
261 views

Compute the wedge product

This is my first time computing the wedge product, I am not sure if I have done it correctly as I do not have solutions, if I have gotten the answer right or am doing the right method please say. ...
2
votes
1answer
78 views

Integrating certain functions over the unit sphere $\mathbb{S}^2$

Let $ \mathbb{S}^2$ the unit sphere, and $ \vec a$, $ \vec b$ two constant vectors. I have to prove that: $$ \iint\limits_{\mathbb{S}^2} \langle \vec x , \vec a \rangle \langle \vec x , \vec b ...
0
votes
0answers
68 views

Well-formed formulas: Difference between $\forall x(x \in A \implies x \in B)$ and $(\forall x \in A) x \in B$? [duplicate]

Let $A$ and $B$ be sets. There seem to be two ways of writing $A \subseteq B$: \begin{equation} \forall x(x \in A \implies x \in B) \end{equation} or \begin{equation} (\forall x \in A) x \in B ...
0
votes
3answers
60 views

Proving monotonicity of a sequence $x_{n} = \frac{1-n}{\sqrt{n}}$

I need to prove whether this sequence is monotone $x_{n} = \frac{1-n}{\sqrt{n}}$ I've got that $x_{n} = \frac{1-n}{\sqrt{n}} = \frac{1}{\sqrt{n}} - \sqrt{n}$ and the limit of it approaches $-\infty$ ...
4
votes
2answers
91 views

Prove: for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$. [closed]

In the following $n,m$ are natural numbers. I need to prove that for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$. Any ideas? Thanks.
1
vote
0answers
90 views

How to show that $[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9$

How can I show that $$[\mathbb Q(\sqrt[3]2,\sqrt[3]5,i\sqrt 3):\mathbb Q(i\sqrt 3)]=9?$$ My idea is: $\sqrt[3]2$ has for its minimal polynpmial $X^3-2$ over $\mathbb Q(i\sqrt 3)$, which I justify by: ...
1
vote
1answer
33 views

Roots of a sequence of function in certain disk

I have a sequence of functions defined by $$ g_n(z)=\sum_{k=1}^n \frac{z^{-k}}{k!}$$ Given $r>0$ show that we can find $M(r)$ such that if $n>M(r)$, then all the zeros of $g_n(z)$ lie within ...
2
votes
3answers
86 views

Proving $ C = D $ from $ A \triangle C = A \triangle D$ [duplicate]

I have been working on one part of the proof where my aim is to show that $C = D$. I have been able to prove that $ A \triangle C = A \triangle D$. Is it reasonable to conclude $ C = D $ from $ A ...
-4
votes
2answers
76 views

Group isomorphic to $\Bbb Z/3\Bbb Z\times \Bbb Z/3\Bbb Z$ [closed]

Is a group with $9$ elements such that all elements (excepted the natural element) are of order $3$ is isomorphic to $\Bbb Z/3\Bbb Z\times \Bbb Z/3\Bbb Z$ ?
0
votes
0answers
68 views

Algebraic treatment of Feynman potential wells?

This question refers to the discussion here: Chomsky, Feynman, Thom I will try the divide-and-conquer strategy to try to make some progress in the problem dealt with in the above-mentioned link. My ...
4
votes
1answer
1k views

Dirac delta function as a limit of sinc function

I'm looking for a rigorous proof of the statement: $\delta(x) = \lim_{\epsilon->0} \frac{\sin(x/\epsilon)}{\pi x}$ (see (37)). For any non-zero value of x, LHS of the above is by definition zero. ...
3
votes
2answers
247 views

Use AM-GM to prove upper bound.

While studying for my upcoming exams I came across a problem in the AM-GM section: If $a_n = (1+\frac{1}{n})^{n}$ , $n \in \mathbb N$ then prove that: $$2 < a_n < 4$$ Proving the ...
2
votes
1answer
100 views

how to integrate $\int (1+\sin2x)^2dx$ [closed]

I really dont see a way to integrate: $$\int(1+\sin2x)^2~dx$$ any hints?
1
vote
1answer
89 views

Find the ring of homomorphisms $\mathbb{Z} \to \mathbb{Z}$ [duplicate]

Show, which well-known ring is isomorphic to ring $End(\mathbf{Z})(+,ͦ,-,0,id)$ of homomorphisms $\mathbf{Z}$ -> $\mathbf{Z}$, where $\mathbf{Z}$ is commutative group $\mathbf{Z}(+)$ and 0 constant ...
1
vote
1answer
64 views

What is the inverse function of the Log Gamma function?

What is the inverse function of the Log Gamma function? $\log\Gamma(x)$ http://mathworld.wolfram.com/LogGammaFunction.html Can it be inverted, and why not, if not?
2
votes
0answers
113 views

Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
2
votes
1answer
61 views

Conditional probability for three events

I want to calculate $P(\neg H\mid I, \neg F)$, where $p(H) = p(\neg H) = 0.5$, $p(I\mid H) = 0.8$, $p(\neg I\mid H) = 0.2$, $p(I\mid\neg H) = 0.4$, $p(\neg I\mid\neg H) = 0.6$, $p(F\mid H) = ...
4
votes
1answer
160 views

Let $G$ to be abelian group and $|G|=mn$ when $(m,n)=1$. $G_m=\{g\mid g^m=e\}$,$G_n=\{g\mid g^n=e\}$, prove isomorphism

I want to prove $ f:G_n\times G_m\rightarrow G$ when $f(g,h)=gh $ is an isomorphism. First of all I showed that $G_m,G_n$ are subgroups of $G$ (easy). Now I want to show that for every $ a,b, ...
3
votes
2answers
80 views

Plotting a function (by hand) if the second derivative is hard to find

In plotting graphics we use the first derivative to find critical points and in which intervals the function grows and becomes smaller. We can insert the critical points in the second derivative to ...
3
votes
1answer
95 views

$R$ noetherian, $I$ injective $R$-module $\Rightarrow$ $S^{-1}I$ is injective over $S^{-1}R$

I am trying to prove that if $R$ is a noetherian ring, $S$ a multiplicative part and $I$ an injective $R$-module, then $S^{-1}I$ is an injective $S^{-1}R$-module. So far I thought: I reduce to check ...

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