2
votes
1answer
46 views

Are Sylow p-subgroups in conjugate?

Let $G$ be an infinite group and $p$ be a prime number. Let $\mathscr{C}$ be a chain of p-subgroups of $G$ ordered by inclusion. Then, for every element of the union of $\mathscr{C}$ has an order ...
3
votes
1answer
63 views

Problems in formalizing these sentences

This is the first sentence that I have to formalize: "Every student likes at least one type of cake" Let: $S(x)$ stands for 'x is a student' $C(x)$ stands for 'x is a type of cake' $L(x,y)$ ...
0
votes
2answers
49 views

Prove that $|A\cap B| \le \frac {1}{2} |A|$ where $A,B$ are two subgroups of $G$

Suppose $G$ is a finite group, $A,B$ are subgroups of $G$ and $A$ isn't a subgroup of $B$. Prove (by using Lagrange's theorem) that $|A\cap B| \le \frac {1}{2} |A|$. $ $ This is what I have so far: ...
2
votes
3answers
110 views

Infinite group with finite order elements [closed]

Can you give me an example of an infinite group in which every element has order $3$ (except identity) ?
-1
votes
3answers
30 views

Is $c$ (as in $y=mx+c$) the $x$ or $y$ intercept?

In $y=5x+6$, is the $6$ the $y$-intercept or the $x$-intercept? I can't remember and need to know for revision.
1
vote
3answers
36 views

lim sup iff conditions - please help explain

Deciphering the definition of the upper limit, we see that $\limsup x_n=L$ is and only if the following two conditions are fulfilled: (a) $\forall\epsilon>0\;\;\exists N\in\mathbb{N}$ such that ...
2
votes
0answers
31 views

Relationship between asymptotic distribution and logarithmic sums of elements of subset of the natural numbers

Consider a subset $A$ of the natural numbers analogous to the primes (but rarer). Let $a_n$ denote the $n$th element of $A$, and $a(n)$ denote the number of elements of $A$ less than or equal to $n$ ...
1
vote
0answers
36 views

Finding Modules with Given Length of Composition Series

I study a course about commutative algebra, and I saw many questions as the following: Find an example of a $\mathbb{Q}\left[\lambda_1,\lambda_2\right]$-module with $\ell\left(M\right)=3$ (i.e., ...
0
votes
1answer
32 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
1
vote
1answer
57 views

Show that $-1=\sum_{0}^{\infty} (p-1)p^i$ in $\mathbf{Q}_p$

To show that in the field $\mathbb{Q}_p$, where $p$ is a prime, it holds that: $$-1=\sum_{0}^{\infty} (p-1)p^i$$ I did the following: It suffices to show that: $\left|\sum_0^N (p-1)p^i+1 \right|_p ...
3
votes
2answers
252 views

Probability of winning a tie-break in tennis?

The winner of a tennis tie break is the first to get to 7 points and lead by 2. Let $p$ be the probability of player 1 winning when serving, and let $m$ be the probabiliity of player 1 winning when ...
1
vote
1answer
71 views

A fast method for factorizing $2^p-1$

I know that $\forall{d,n\in\mathbb{N}}:d|n\implies2^d-1|2^n-1$. Now, suppose that $n$ is prime - is there any fast algorithm for finding a divisor of $2^n-1$? By "divisor", I am referring to a ...
0
votes
1answer
112 views

Show that ${\{x_n}\}$ is convergent and monotone

Question: For $c>0$, consider the quadratic equation $$ x^2-x-c=0,\qquad x>0. $$ Define the sequence $\{x_n\}$ recersively by fixing $x_1>0$ and then, if $n$ is an index for which $x_n$ has ...
2
votes
1answer
85 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
0
votes
0answers
23 views

Coupling complex functions

After several calculations I end up with two complex functions: $$g(z)=zA(z)+\overline{z}A(\overline{z})+z^{-1}B(z)+\overline{z^{-1}}B(\bar{z})$$ and ...
0
votes
2answers
34 views

plotting of level curves using pen and paper

I want to plot the level curve for the function $f(x,y)=\frac{y^2-x^4}{y^2+x^4}$ . I tried by substituting $f(x,y)=k$. But I am Unable to draw it using paper and pen. Kindly help me.
4
votes
2answers
451 views

Why does Gaussian elimination not preserve similarity of a matrix?

I am trying to understand reduction of an unsymmetric real square matrix to Hessenberg form from Numerical Recipes Vol. 3. In it, the author states that one does not use Gaussian elimination for ...
1
vote
2answers
47 views

Linear transformation Df=$\frac{df}{dx}$

Let $Rx$ define vector space of all real polynomials. Let $D:Rx \to Rx$ denote map Df=$\frac{df}{dx}$, for every f. Then which of following is true. $D$ is one-to-one $D$ is onto There exist $E:Rx ...
2
votes
2answers
337 views

Trig substitution fails for evaluating $ \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral \begin{equation} \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx \end{equation} Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
1
vote
1answer
112 views

zero raised to power zero in Church encoding

In Church encoding of the natural numbers in lambda calculus raising zero to the power zero gives the answer zero. Does anybody know of an encoding where the answer is 1?
1
vote
1answer
53 views

What bound does the Hamming bound give you for the largest possible size of a $t$-error-correcting code of length $2t + 1$?

Let $\mathbb{A}$ = $\{0, 1\}$ and suppose $t$ is a positive integer. What bound does the Hamming bound give you for the largest possible size of a $t$-error correcting code of length $2t+1$? I have ...
0
votes
0answers
77 views

Blocks in a layer? $X$ layers of blocks in a triangle, which $Y$ being the total number of blocks… (w/o using triangular numbers)

I have $32$ layers of blocks in a triangle. However, for the sake of variation, let's call that $X$. I have $1000$ square blocks total. We'll call those $Y$. The first layer of the triangle has $1$ ...
0
votes
0answers
47 views

Proof the $\mathrm{rank(rows)=rank(columns)}$

Assume we have matrices $A=BC$.It is obvious that the $i$th row of $A$ is a linear combination of the rows of $C$ with coefficients from the $i$th row of $B$ or $b_{i1}C_1........b_{in}C_n$. ...
1
vote
1answer
66 views

Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
5
votes
0answers
83 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
2
votes
0answers
35 views

Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
0
votes
2answers
175 views

A digraph is a graph where every edge is directed. How many digraphs on $n$ vertices are there?

So far I have that between any two vertices (say $j$ and $k$) there are 3 options. there is no edge between $j$ and $k$ there is an edge directed from $j$ to $k$ there is an edge directed from $k$ ...
0
votes
1answer
42 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
2
votes
1answer
48 views

Possible integer roots of polynomial with real coefficents

If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower ...
1
vote
1answer
40 views

Ideal of $TV$ which trivially intersects $V$

Let $V$ be a vector space over a fied $\mathbb{K}$ and let $TV = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ be its tensor algebra. Let $b \colon V \times V \rightarrow \mathbb{K}$ be a bilinear form ...
0
votes
0answers
30 views

Showing $\{x\in \mathbb{R}:\mu(\{x\})>0\}$ is a countable set under certain conditions.

Let $\mu$ be a measure such that $(\mathbb{R}, \mathcal{B}_{\mathbb{R}}, \mu)$ is a $\sigma$-finite measure space. I have to prove that $D=\{x\in \mathbb{R}:\mu(\{x\})>0\}$ is a countable set. Let ...
4
votes
2answers
133 views

Exercise 1: Galois Theory (J. Rotman)

Definition: Let $F$ a figure in the plane, its symmetry group is defined by $\Sigma(F):=\{\sigma \in O(2,\Bbb R)\mid \sigma(F)=F\}$. Here $O(2,\Bbb R)$ denotes the real orthogonal group. Exercise 1: ...
0
votes
0answers
51 views

Filling a 5x5 array with X-s and O-s

Consider a $5x5$ array. In how many ways can we fill the array with $X$'s and $O$'s so that no two consecutive rows are identical? My tutor gave us the following answer: $2^{(25)} - [4*2^{(20)} - ...
0
votes
3answers
553 views

order of non abelian group can't be what?

Let $G$ be a non abelian group; then its order can be: $25$ $55$ $35$ $125$ I think the order cannot be $25$ and $35$. But from option $55$ and $125$ which one is not possible? Why not $25$ ...
1
vote
1answer
22 views

Reference request for a special product

I have the product $$\prod_{k=0}^n (1+a_k)$$ Does this product have a special name under which I can find some of its properties? I appreciate any reference for this product. Note: Because of the ...
1
vote
1answer
92 views

Solving Fulton and Harris exercise 2.4

Let $\rho$ be a representation of $G$ on $V$ of dimension $4$. let $g\in G$ be an element of order 4.Let $\chi$ the character of representation $V$ is known then find the eigenvalues of $\rho_g$. ...
2
votes
4answers
114 views

If $a,b,c\in\mathbb{R^+}$ such that $ abc = 1 $ and $ ab + bc + ca = 5 $. Prove that $ 17/4 \leq (a+b+c)\leq 1+ \sqrt{32}. $

If $a,b,c\in\mathbb{R^+}$ such that $ abc = 1 $ and $ ab + bc + ca = 5 $. Prove that $$ \frac{17}{4} \leq (a+b+c)\leq 1+ \sqrt{32}. $$ My attempt Tried using Vieta but it didn't work. Also I used ...
4
votes
0answers
130 views

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, ...
0
votes
1answer
35 views

What are equivalent parametric equations?

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?
0
votes
0answers
36 views

Involution on divisor

Suppose that $\pi:C^{'} \rightarrow C$ is an unramified double cover of a complex riemann surface. Let $\tau$ the involution sheet exchange. If $E$ is a divisor on $C^{'}$ What is the definition of ...
0
votes
0answers
177 views

Determining the weights of known parameters in a formula

I have a formula of the following form: $a_1*w + a_2*x + a_3*y + a_4*z$ In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the ...
1
vote
1answer
178 views

Dehn twist as isometries on hyperbolic surface

[I am editing the question to a most correct and precise one thanks to comments of Lor and studiosus] Let (S,g) be a compact hyperbolic surface. On a simple closed geodesic $\gamma $ I can realized a ...
2
votes
1answer
38 views

If the probability of 3 events with non-zero probability equals the product of the individual probabilities, are they also pairwise independent?

Consider three events $A$, $B$, and $C$, none of which has a zero probability. If $A$, $B$, and $C$ satisfy $\Pr(A \cap B \cap C) = \Pr(A) \cdot \Pr(B) \cdot \Pr(C)$, does this imply that the three ...
2
votes
1answer
622 views

Probability of winning a game in tennis?

Suppose there is a tennis singles match, where Player A plays a single game against Player B. The probability that player A will win a single point is $x$, and thus $1-x$ is the probability that ...
1
vote
1answer
63 views

Problem with Laurent series

I am trying to find a Laurent series for $\cos(\frac{1}{z})z$. I know that $$ \cos(1/z) = \frac{1}{2} e^{(-i/z)}z + \frac{1}{2} e^{(i/z)}z = z \sum_{n=0}^\infty \frac{(-1)^n ...
0
votes
0answers
50 views

Reference Request: James reduced product

I would like to quickly learn the basics of James reduced product (also called James construction). Anyone know some suitable material for beginners?
1
vote
1answer
581 views

Proof of Laurent series co-efficients in Complex Residue

Am trying to see if there is any proof available for coefficients in Laurent series with regards to Residue in Complex Integration. The laurent series for a complex function is given by $$ f(z) = ...
1
vote
2answers
99 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
2
votes
1answer
128 views

What is the process of nondimensionalizing an equation?

Question: I need to scale time by $\frac{1}{I}$ and species by $P$ for the following equation $\frac{dS}{dt}=I(1-\frac{S}{P})-\frac{ES}{P}$ where P - Size of the source pool of species on the ...
0
votes
3answers
255 views

how to show that this function is continuous for all real numbers?

I'm having hard time playing with trigonometric functions. I want to show that this piecewise function is continuous for all real numbers (from $-\infty$ to $\infty$) a) $g(x)=$ \begin{cases} ...

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