0
votes
0answers
11 views

What is a $\mathbb{Z}$-form of an algebra?

A homework problem I have is to describe the Lie algebra associated to a Kac-Moody root datum $\mathcal{K} =(I,A,\Lambda ,(c_i)_{i\in I},(h_i)_{i\in I})$ as well as to describe the universal ...
2
votes
1answer
9 views

Bound on expectation of function of standard normal, $\mathbb{E}[\exp(Z^a)]$

I'm trying to find the maximum (or sup) of the value of $a$ such that $$\mathbb{E}[\exp(Z^a)]<+\infty$$ where $Z\sim \mathcal{N}(0,1)$. Obviously for $a=1$ the expectation is finite since it is the ...
4
votes
1answer
30 views

Show that $3^n-2^n\cdot 5$ is composite for infinitely many $n$

I came across this problem: Show that $3^n-2^n\cdot 5$ is composite for infinitely many $n$ and do not know how to solve it. I only know that it is true for $n=7$, since then $1547=17\cdot 91$.
1
vote
2answers
14 views

Laplace Transform of tsin(at) using only the definition

Hello I' am stuck on how to get the final result of the laplace transform of $f(t)=tsin(at)$using (a is a constant) only the definition of $$\int_0^{\infty}f(t)e^{-st}dt$$, I know $sin(at)= {1 \over ...
2
votes
0answers
11 views

3-Coloring a graph using propositional formulas

Hello everyone I am studying for an exam on logic and computability, I am trying to tackle a specific problem so any help would be greatly appreciated: Let $G = (V,E)$ be an undirected graph ...
1
vote
1answer
15 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
0
votes
2answers
32 views

Construct a bijection from (0,1] to (0,1) and [0,1] to (0,1)

I have proved this but my teacher wants me to put more but I have no idea what to add. He says he wants a proof that they are explicitly in fact a bijection. For the first one this is what I did ...
0
votes
2answers
16 views

Is there any statistical method to compare two curves?

Is there any statistical method to visually compare two curves? What is the bast and correct way to compare two similar curves and calculate the error/difference in percentage? I have created a ...
0
votes
0answers
8 views

Question about Classifying singular points and finding corresponding residues from Laurent Series

I wanted to check if I had the right idea : Singularities have 3 classification 'essential'.'removabe' and 'pole order x' a singularity is essential if when you expand it,it is a never ending series ...
0
votes
0answers
7 views

Vectors with kronecker product [on hold]

If a is p-vector then what means a**3 where ** is the same mark as a kronecker product? Is it just that every element in the vector is **3?
0
votes
1answer
13 views

Proving that a function is onto in an interval [on hold]

I got this question: Let $f(x)=\dfrac{1}{\sin x}+\dfrac{1}{x-1}$. Prove that in the interval $(0,1)$, $f$ is onto $\mathbb{R}$ (that is, prove that $f((0,1))=\mathbb{R}$). Thanks.
0
votes
0answers
10 views

Help Finding Orthonormal Frame and Coframe Based on First Fundamental Form

Given the metric (for $x^2<1$) $g=dx^2+2x dxdy+dy^2$ (in first fundamental form) I'm trying to find the orthonormal frame and its coframe I have found (I think) the orthonormal frame to be ...
-2
votes
0answers
10 views

Help defining the statistical number that pi posses a philosophical question in the first 15 characters. [on hold]

I am looking for assistance calculating the statistical possibilities that pi posses a philosophical question of life in the first 15 characters. Additionally, it has the Fibonacci Sequence and the ...
0
votes
0answers
35 views

why $c$ in Einstein equation is Squared? [on hold]

why the speed of light is Squared in $E=mc^2$ - I want explain I am not a specialist in mathematics
0
votes
0answers
5 views

Showing that s'm is a common multiple of m and n

so in class teacher gave us this algorithm GCD(m,n)=GCD(n mod m, m). after that we used it to find s and t. for example we found GCD of 453 and 174 and their s and t by making a table like this ...
1
vote
1answer
16 views

In homology, when we operate the boundary twice we get zero, that is, $\partial^2=0$. Need help understanding proof.

Proof for $S=\Delta_n=(v_0 ... \hat{v_i} ...v_n)=d_i$ $\partial=\displaystyle\sum_i^n(-1)^id_i.$ Thus, $\partial^2=[\displaystyle\sum_i^n(-1)^id_i][\displaystyle\sum_j^n(-1)^jd_j] $ ...
0
votes
0answers
14 views

Jacobi method for any b

Determine if Jacobi method converges for any b for, $$\begin{bmatrix} 2 & 2\\ 3 & 4 \end{bmatrix}$$ The solution goes on like this... D-(L+U) = $$\begin{bmatrix} 2 & 0\\ 0 & 4 ...
0
votes
1answer
20 views

$\lambda \ll \mu$, $\mu X <\infty$, then $\lambda X<\infty$

$\lambda \ll\mu$ : $\lambda$ is absolutely continuous w.r.t. $\mu$. and $\mu X \lt\infty$, where $X$ is a space how to show: $\lambda X\lt\infty$
0
votes
2answers
15 views

Difference between the definition of monoid action and group action?

The question is essentially in the title. From what I read in the wikipedia article about monoids it seems to me that we can define a monoid action in the exact same way we define a group action. Is ...
0
votes
0answers
12 views

Find spikes in data

I have some datasets and I need to find spikes in them. Imagine the data looks like trading data. If the spike is big enough, I need to log it, otherwise, proceed in the analysis. I tried with a ...
0
votes
1answer
12 views

How to determine the integers $n$ for which $Z_n$, the set of integers modulo $n$, contains elements $x$, $y$ so that $x + y = 2,2x-3y = 3$.

How to determine the integers $n$ for which $Z_n$, the set of integers modulo $n$, contains elements $x$, $y$ so that $x + y = 2$ ; $2x-3y = 3$.
1
vote
0answers
10 views

Lattice Definition

I see two Lattice definitions in Mathematics. Partial order set with each pair of elements have a least least upper bound and greatest lower bound. Integer linear combinations of vectors. Is ...
0
votes
4answers
32 views

If the sequence converges, find it's limit.

$$ \frac{17n-2}{9-\sqrt{n}} $$ I think it diverges, but I'm not sure.
0
votes
1answer
11 views

What is the definition of sigma field generated by random variable $X$? [on hold]

What is the definition of $\sigma$-field generated by a random variable $X$? And what does it mean?
1
vote
1answer
21 views

Show a 2-form is exact finding a primitive.

I have to show that $\omega=-4xy\:\mathrm{d}x\wedge \mathrm{d}y-2xz\:\mathrm{d}z\wedge \mathrm{d}x +2yz\:\mathrm{d}y\wedge \mathrm{d}z$ is exact finding a primitve of $\omega$ (by Poincare lemma I ...
0
votes
2answers
30 views

Evaluate an integral using polar $\displaystyle\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx$

How do you evaluate the following integral using polar cordinates. $$\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\:\mathrm{d}y\:\mathrm{d}x$$ I converted it to polar coordinate making it ...
0
votes
1answer
28 views

Computing $\int_\gamma { |dz| \over |z-a|^2}$

Goal: Compute $$ \int_{|z|= \rho} {|\mathrm{d}z| \over |z-a|^2} $$ under the condition $|a| \ne \rho$. Ahlfors' Hint: make use of the equations $z \bar{z} = \rho^2$ and $$ |\mathrm{d}z| = -i ...
0
votes
0answers
17 views

How to prove this polynomial expression.

Let the polynomial be in $f$ be a map from $\Bbb{Z}_2^k \to \Bbb{Z}_2$, defined by $f = 1 + \sum_{i=1}^k x_i + \sum_{i\neq j; i,j = 1}^k x_i x_j + \dots + x_1 x_2 \cdots x_k$ Then I want to show ...
0
votes
1answer
28 views

The question on a ring. [on hold]

let $$G=\{5m+7n | m, n\in \Bbb N\}$$ Firstly, I want to find the complement of $G$ In $\Bbb N $ is finite. Secondly, how do I find the Frobenius number of $G$ (I guess, the larger not ...
0
votes
1answer
26 views

$\exists a\in G-H$ such that $aHa^{-1}=H$

Let $G$ be a $p$-group with proper subgroup $H$. Show that there exists an element $a\in G -H$ such that $a^{-1} Ha = H$ Can you check my proof? Since $G$ and $H$ are $p$-groups their centers ...
0
votes
1answer
27 views

A question about a continuous function that satisfies the property $\forall x\in\mathbb{R},\exists x<y\in\mathbb{R},f(x)<f(y)$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function that satisfies the property: forall $x\in\mathbb{R}$ there exists $y \in\mathbb{R}$ such that $x < y$ and ...
1
vote
0answers
6 views

Turning a rhombus billiard into an equivalent barrier billiard

I am reading a research paper and the authors map the rhombus billiard (angles $60$-$120$ degrees) to an equivalent barrier billiard. They start with a rhombus standing upright and reflect in a ...
1
vote
1answer
18 views

Notation question: Group generated by two elements?

Let there be $H$ subgroup of symmetric group $S_4$, so that $H= \langle (12)(34),(234) \rangle$. What does the notation $\langle (12)(34),(234) \rangle$ mean? I know that if there's one elements, then ...
0
votes
2answers
13 views

Sylow p-subgroups: Understanding a proof

I don't understand the last part of this proof: http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup where they say: $p \nmid \left[{N : P \cap N}\right]$, thus, $P ...
0
votes
0answers
15 views

Ellipse equation. What does it need to be in order for $b > a$?

We have the quadratic equation: $$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive. How can I, by looking at that only, determine whether $b$ (the length of the semi-minor ...
2
votes
2answers
17 views

Showing uniform convergence of series

Show that $\displaystyle \sum_{j=1}^{\infty} \frac{-2j}{(x^2 + j^2)^2}$ converges uniformly. Don't know how to do this problem since $x$ and $j$ are in the expression together. Is there a convergence ...
0
votes
1answer
15 views

My proof regarding composition of permutations came to the same conclusion as the answer sheet, but through different methods. Is it valid?

Let $S_3$ be a set of all permutations of elements in $\{1,2,3\}$. Prove that there doesn't exist f $\in S_3$ where $\{f,f^2,f^3,f^4,f^5,f^6\} = S_3$. Where $f^n = f \circ f \circ \:... \circ \:f$ ...
0
votes
0answers
21 views

Let $f:S\rightarrow\mathbb{R}$ be a continious function and $S$ be a closed subset of $\mathbb{R}$ does it imply that $f(S)$ is a closed set? [duplicate]

Let $f:S\rightarrow\mathbb{R}$ be a continious function and $S$ be a closed subset of $\mathbb{R}$ does it imply that $f(S)$ is a closed set? Can somebobly please explain me this.thanks for your kind ...
2
votes
1answer
43 views

A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
2
votes
3answers
33 views

Arc length in polar coordinates: Why isn't $dS=r×d\theta$

As a sort of exercise, I tried to derive the formula for arc length in polar coordinates, using the following logic: $$dS = r(\theta)d\theta\\ \implies S=\int r(\theta)d\theta$$ However, it turns ...
0
votes
1answer
10 views

Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
1
vote
1answer
28 views

given x>0 and n contained in N show show that there exists a unique positive real number $r$ such that $x = r^{n}$

I have proved half of this proof but I'm stuck on the other half because it is a little harder than the first due to negatives. I have considered that $(r+h)$ so that $(r+h)^n$ is less than $x$ and ...
2
votes
1answer
22 views

determine if $G = \{ f : \mathbb{R_+} \to \mathbb{R_+} \}$ is a group

I'm confused about how the identity was formed - if $e(x) = x$, then how does one get from $f(x)e(x) = f(x)\cdot 1$
1
vote
3answers
57 views

Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
0
votes
3answers
33 views

Find the limit of the sequence if it converges, otherwise state divergence.

Find the limit of the sequence given by $$\frac{10+12n+20n^4}{7n^4 + 5n^3 - 20}$$ I think the answer is $\frac{20}{7}$ after dividing, but is that right?
1
vote
2answers
27 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
0
votes
1answer
27 views

Inequality, $\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$

How do I show that $$\left(\frac{2}{x}+2\right)^{n}-\left(\frac{2}{x}-2\right)^{n}\leq \left(\frac 4 x \right)^n$$ for $x\in\left(0,1\right]$ and $n\in\mathbb N$?
0
votes
1answer
19 views

A Criterion for being Sylow p-group

Show that if $H$ is a $p$-group of finite group $G$ and $N_G(H)=H$ then $H$ is a Sylow $p$-group of $G$? Or prove the following more general property,$$[G:H]\equiv1\ (\mod\ p)$$
0
votes
0answers
3 views

Lambda calculus: encoding lists and projectors

Given a pair $[M_1,M_2]$ there is an easy encoding $\lambda x.x M_1 M_2$. For the n-tuple we have two options. First encoding: $$\lambda x.x M_1, M_2, \ldots , M_n$$ Second encoding: $$[M_0, [M_1 , ...
0
votes
0answers
8 views

kolmogorov equations for continious markov chains

I'm trying to find the for for forward equations for a birth and death processes when all $\lambda$ coefficients are zero. The forward equation for a Birth and Death Process has the form: ...

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