2
votes
0answers
131 views

On the centralizers of $n$-cycles and conjugacy in $A_n$

I'd appreciate comments on the validity of these attempted proofs. Thanks. Let $a$ be an $n$-cycle in $S_n$. a) Show that the centralizer of $a$ in $S_n$ is $\langle a \rangle$. b) Assume that $n$ ...
1
vote
1answer
44 views

Method of moments on uniform distributions

I need help on how to find the estimates $a$ and $b$ in the uniform distribution $\mathcal U[a,b]$ using the method of moments. This is where I am at: I have found $U_1=\overline X$ and ...
3
votes
1answer
140 views

How many arrangements of the digits 1,2,3, … ,9 have this property?

How many arrangements of the digits 1,2,3, ... ,9 have the property that every digit (except the first) is no more than 3 greater than the previous digit? (For example, the arrangement 214369578 has ...
0
votes
0answers
40 views

Advanced Percentages Question

This question pertains to the game I play and enjoy, Minecraft. I am trying to calculate the chance of which a certain weapon with certain damage modifiers (some of these are not fixed and have a ...
2
votes
2answers
50 views

Is it true that $\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3} = \binom{n}{3}\binom{n-3}{k-3}$?

I was asked to find a closed formula for $$\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3}$$ To remove the $\sum$ if you will. Here's my reasoning, let's say we have a football team with $n$ players. First we ...
0
votes
1answer
67 views

Finding conditional probability of being affected by virus

I have a problem I have been asked to solve. I have tested positive for a virus affects 1 in 10000 people. The lab report says the test correctly identifies positive cases 99% of the time and ...
0
votes
1answer
88 views

Prove or disprove the following statement. $7 \ | \ (x^3 + x^2 + x + 2)$, where $x$ is an odd integer

We're learning about modulus and division (Discrete mathematics and proofs course). I'm not exactly sure how to tackle this sort of problem, is there some sort of property of cubic functions ...
2
votes
1answer
59 views

Prove that in each coloring of a $4\times7$ board in two colors there's a square that all four of it's corners are colored by the same color

Prove that in each coloring of a $4\times7$ board in two colors there's a square that all four of it's corners are colored by the same color. This is a pigeon hole principle question and I have a ...
1
vote
1answer
48 views

Expected intersection of two card draws of unequal length

here is my specific question: If two people draw independently without replacement, different numbers of cards from different (complete) decks. How do I figure out the expected number of matches or ...
0
votes
1answer
104 views

understanding into algebraic terms difference between homology and cohomology

my previous question understand quotient group was related to understanding of quotient group,i dont need to know too much detailed in group theore,just some part of algebraic topology,especially ...
1
vote
1answer
47 views

Show that,the curve $c:(0,2\pi)\rightarrow C$, with $c(t)=f(e^{\lambda t},t)$ intersects the cone at a constant angle.

Let $C:=\{(x,y,z)\in\mathbb R^3|z=\sqrt{x^2+y^2}\}\setminus\{0\}$ $f(u,v)=(u*cos(v),u*sin(v),u)$, where U is $U:=\{(u,v)\in R^2\ | 0<u,0<v<2\pi \}$ show that the trace of ...
1
vote
1answer
54 views

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
0
votes
2answers
28 views

Define variable value - General Math

I met a little problem in one of my Math - tasks. Its quite simple: I get to cases = case 1: $(476\cdot x)+220$ case 2: $(278\cdot x)+675$ The variable x have to be a value, so case 1 and case 2 ...
13
votes
3answers
409 views

Show that $\sum_{k=0}^n\binom{2n}{2k}^{\!2}-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$

How can I prove the identity: $$ \sum_{k=0}^n\binom{2n}{2k}^2-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^2=(-1)^n\binom{2n}{n}? $$ Maybe, can we expand $$ f(x)=(1+x)^{2n}? $$ Thank you.
0
votes
1answer
39 views

Proving if equation is $O(\log n)$

How do I prove if \begin{equation} 2\log(n^{2}\log n) = O(\log n) \end{equation} is true? I began by trying to find a $C$ where \begin{equation} 2\log(n^{2}\log n) < O(\log n) \end{equation} ...
1
vote
2answers
401 views

Radian and the length of a chord of a circle

Question In a circle of radius $r$, an arc of it is $2S$ long. Find the length of the chord corresponding to that arc (AB in the diagram below) . Details I got this question in a math test. And ...
2
votes
3answers
1k views

Why is integration the inverse of differentiation

Why is integration the inverse of differentiation, I mean why do I get the same function when I integrate and then differentiate the result?
2
votes
3answers
97 views

$\sigma$-Algebra: Why do we want it to contain complements as well?

Everybody Hello, I was always wondering: (Please answers apart from historical reasons) Why do we want a $\sigma$-Algebra to possess more than just its crucial disjoint $\sigma$-union property? Say, ...
0
votes
1answer
56 views

Dimension is local on irreducible components

I am having trouble to see that dimension of a scheme is local because I have the following counter-example in mind. I have a feeling this counter-example is fake because I can't find an open ...
0
votes
1answer
87 views

Sum of Cantor Sets

Let C denote the usual cantor set(which is obtained from the interval [0,1]).Then what is the sum of three such cantor sets i.e. what is C+C+C=?
0
votes
2answers
56 views

Absolute values in logarithms in a solution of differential equation

How have the moduli signs disappeared in the following step: $$\frac1{k}\left(\ln|g+kv| - \ln|g+ku|\right) = -t$$ Therefore $$ \ln\left(\frac{g+kv}{g+ku}\right) = -kt$$ $g$, $k$ and $u$ are ...
18
votes
2answers
738 views

Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}$\ $(-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$ My attempt : ...
1
vote
3answers
465 views

Dimension of sum and intersection of vector space.

I am trying to understand the proof of the following: Suppose $U,W$ are vector subspace of $V$, then $\dim (U+W)+\dim (U \cap W)= \dim (U) +\dim (W).$ The proof goes like this: Let $S: V \rightarrow ...
1
vote
2answers
92 views

understand quotient group

i am trying to understand what does mean quotient terminology in group theory by as simple way as possible,also quotient group i want to know something about it,using internet i read that " In ...
0
votes
3answers
100 views

Why $\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}$

Why $$\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}?$$ I know that $\sum_{n=0}^{\infty}x^n=\dfrac{1}{1-x}$, so by the same token, $\sum_{n=0}^{\infty}5^nx^n=\dfrac{1}{1-5x}$. Thus $$ ...
3
votes
2answers
227 views

How many strings contain every letter of the alphabet?

Given an alphabet of size $n$, how many strings of length $c$ contain every single letter of the alphabet at least once? I first attempted to use a recurrence relation to work it out: $$ T(c) = ...
6
votes
0answers
48 views

Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
2
votes
2answers
144 views

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn't use the class equation?
2
votes
1answer
61 views

Generating function for combinatorial problem

Find the number of possibilities to divide $n$ balls into $3$ cells, such that: In the first cell there must be at least one ball. No limitations for middle cell. In right cell, the number of ...
0
votes
2answers
107 views

Why isn't $\log(n!) \leq O(n\log n)$? [closed]

Why isn't $\log(n!) \leq O(n\log n)$? I know that $\log(n!)$ is of $\Theta(n\log n)$ but why can't a function that is of $\Theta$ be $\leq$ than a function that is $O$ of the same parameter? Isn't ...
0
votes
2answers
101 views

I need to solve equation, I really need some help

I really need to solve this equation, but my knowledge is not enough to figure it out: $$\cos(-55.82) = (0.6893\cos(-70) + 0.3381\sin(-70)) \cdot (-\frac{-0.4206\cos f + 0.6423 \sin f}{\sin 67.33}) - ...
2
votes
1answer
60 views

Minimal Discriminant of An Elliptic Curve

I want to determine the minimal discriminant of $$ y^2 + xy = x^3-x^2-50x+111 $$ as an elliptic curve over the rationals. I managed to reduce it to the form $y^2=x^3+Ax+B$ where $A,B$ are rational, ...
1
vote
1answer
65 views

Transcendental number over $\{k\in K\mid f(k)=k\}$

Let $K$ be a field and $f:K\rightarrow K$ be a ring endomorphism. Prove that if $\alpha\in K\setminus f(K)$, then $\alpha$ is transcendental over the subfield of $K$, $F:=\{k\in K\mid f(k)=k\}$. My ...
2
votes
1answer
47 views

number field:How can i prove that ${\Bbb Q}[\sqrt{-3}]$ is a cyclotomic field?

Can you help me with this ''simple'' exercise: Prove that ${\Bbb Q}[\sqrt{-3}]$ is a cyclotomic field.
2
votes
1answer
94 views

from Ireland and Rosen: when a prime remains inertial

I'm reading Ireland and Rosen's number theory book, and i'm having trouble with proposition 13.1.3 ii): Let F be $\mathbb{Q(\sqrt d)}$ where $d$ is a square free integer,and $p$ and odd prime, and ...
0
votes
1answer
127 views

Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0<\alpha<1$ an interval $I$ such that $m(A\cap I)/m(I)>\alpha$.

Let $A\subset\mathbb{R}$ a measurable where $0<m(A)<\infty$. Show that exists for each $0<\alpha<1$ an interval $I$ such that $$ \frac{m(A\cap I)}{m(I)}>\alpha. $$ MY ATTEMPT: ...
0
votes
1answer
321 views

Counting measure $\sigma$-finite / not $\sigma$-finite for different sets

Let $\zeta\colon\mathcal{P}(\Omega)\to [0,\infty]$ be the counting measure. Show that for $\Omega:=\mathbb{R}$ it is not $\sigma$-finite but for $\Omega:=\mathbb{N}$. Hello and good ...
6
votes
1answer
145 views

Existence theorem for antiderivatives by Weierstrass approximation theorem

Is there a way of proving the existence of antiderivatives (of continuous functions on a compact subset of the real line) without using tools of integration? This is an exercise in: ...
1
vote
1answer
283 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
-1
votes
2answers
58 views

A Number Theoretic Argument. Proof or counter-example! [closed]

Can we find two consecutive integers $n$ and $n+1$ that are divisible by the same prime?
1
vote
1answer
191 views

Finding the limit (using L'hopitals rule and log)

I am taking an algorithms course this semester and we have to find limits. I know the general basics of the product rule, L'hopitals rule, etc. When it comes to working with logs, I get confused. ...
2
votes
0answers
39 views

Dimension with respect to a module

This question has confused me. I appreciate anybody who can help or give a reference. Let $R$ be a ring and $A$, $X$, and $M$ be $R$-modules. We say that a map $f:M\longrightarrow A$ is an ...
0
votes
1answer
89 views

Ramanujan resummation

according to the paper from delabaere http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf $$ \sum_{n=1}^{\infty}n^{k}= \zeta (-k)+ \frac{1}{k+1} $$ and $ \sum_{n=1}^{\infty}n^{-1}= \gamma $ but ...
1
vote
5answers
190 views

$(A_1\rightarrow\wedge A_2)$ is not a well-formed formula

Let $A_1,A_2$ be sentence symbols. Could anyone advise me how to prove $(A_1\rightarrow\wedge A_2)$ is not a well-formed formula? Thank you.
0
votes
1answer
35 views

Show that if $\{X_n\}$ is bounded above and no cluster points then $\lim X_n=-\infty$

$\lim X_n=-\infty$ if for any $M$ there exists $n_0$ s.t. $X_n<M$ for every $n>n_0$. I am stuck on how to show that this limit is bounded above and that there are no cluster points so that the ...
1
vote
0answers
33 views

$\omega$ covers and Lindelöf property [duplicate]

As similarly described in a question represented here before: Let $\langle \mathcal{U}_n: n \in \mathbb{N} \rangle$ be a sequence of $\omega$-covers of $X$, and suppose that $X$ is Lindelöf. Can we ...
0
votes
1answer
98 views

Sinc series convergence?

Does $\sum_{n=-\infty}^{\infty}\textrm{sinc}(an)$ converge? And if it does to which value? Does $\sum_{n=-\infty}^{\infty}\textrm{sinc}^2(an)$ converge and to what value? $a\in\mathbb{R}$. I have read ...
2
votes
0answers
60 views

finite normal subgroup

$G$ is a subgroup of finite index in $SL(n,Z)$, $n\ge 3$, $N$ is finite normal subgroup of $G$, then I want to know why $N$ is a normal subgroup of $SL(n,Z)$. More generally, $A$ is an arithmetic ...
3
votes
2answers
100 views

Prove that a number all of whose digits are either $6$ or $0$ cannot be a perfect square [duplicate]

Let $m$ be a number all of whose digits are either $6$ or $0$. Prove that '$m$' cannot be a perfect square.
3
votes
1answer
92 views

The intersection numbers in Fermat curve

I'm a beginner in this subject and I think this "easy" exercise could help me to have more practice in basic algebraic curves. Let $F=X^{p+1}+Y^{p+1}+Z^{p+1}$ be a Fermat curve in the field $k$, with ...

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