1
vote
1answer
662 views

An additive map that is not a linear transformation over $\mathbb{R}$, when $\mathbb{R}$ is considered as a $\mathbb{Q}$-vector space [duplicate]

Possible Duplicate: On sort-of-linear functions I am looking for an example of an additive map that is not a linear transformation over $\mathbb{R}$, when $\mathbb{R}$ is considered as a ...
1
vote
1answer
888 views

Cartesian product of n sets

I'm currently writing a little application in which I would like to create all permutations of n sets. For example I have the following sets (arrays): ...
5
votes
2answers
237 views

An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group

Exercise $7$, page 51 from Hungerford's book Algebra. Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and ...
0
votes
2answers
177 views

linear function

I don't get this, need some help, examples and information The linear function $f$ is given by $$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$ Enter the independent variable and the dependent ...
3
votes
6answers
2k views

Matrix-Square Root

I was wondering about matrix square roots. What is the procedure to evaluate $(X^{T}X)^{-1/2}$? Is it by a spectral decomposition of $(X^{T}X)^{-1}$ as $U\lambda U^{T}$ followed by the square root $S$ ...
2
votes
1answer
329 views

Find the convergence radius of $\sum_{n=0}^{\infty}x^n\sin (nx)$

Given this functions series : $\sum_{n=0}^{\infty}x^n\sin (nx)$, I need to find the ratio where it converges. I don't see how can I change it into a form where I'll be able to use Cauchy-Hadmard or ...
2
votes
1answer
157 views

Möbius maps and their fixed points

Why is it true that if a Möbius map, $f(z)$ fixes distinct $z_1,z_2\in \mathbb C_\infty$ that $f(z)$ either describes a rotation or has a pair of stable and unstable fixed points, such that iterations ...
1
vote
3answers
101 views

Proof of relative primality

How is it true that: If $a_1, a_2,\ldots,a_n$ are pairwise relatively prime positive integers, then $M_i = \dfrac{(a_1a_2\cdots a_n)}{a_i} $ is relatively prime to $a_i$ ? This is ...
4
votes
2answers
328 views

Confused about permutation cycles - Question on joint cycles of odd length

For some reason I'm finding permutation cycles to be strange and hard to deal with. Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2 = \beta^2$, then ...
4
votes
1answer
225 views

How to show that two vector spaces $V$ and $W$ are the same

How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help ...
0
votes
1answer
229 views

General relationship between original and rotated+translated line

I have a line $L$ in the plane expressed as the points in $L = \{(x,y) \in {\mathbb{R}}^2 : x \cos \theta + y \sin \theta = r \; \wedge \; 0 > \theta > \pi/2 \}$ (note that the line cannot be ...
0
votes
0answers
59 views

Calculate ascent from a a path with elevation data

Imagine a person takes a walk through some hilly territory, and they record their elevation at intervals as they go. How would you calculate their total ascent, given that there is some error in the ...
1
vote
2answers
107 views

Showing $\big( \frac{a}{n\bar{x}}, \frac{b}{n\bar{x}} \big)$ is an exact confidence interval for a gamma distribution

Let $X_1,\ldots,X_n$ be exponentially distributed with parameter $\lambda$ This implies that $Y=\sum_{i=1}^nX_i$ has a gamma distribution with parameters $(\lambda,n)$ Can anyone help me show that ...
1
vote
1answer
135 views

A complete Boolean algebra $B$ satisfies the $\kappa$-chain condition if and only if $B$ is $\kappa$-saturated

Let $B$ be a Boolean algebra. Then we say $B$ is $\kappa$-saturated if there is no partition $W$ of $B$ such that $|W| = \kappa$. We say that $B$ satisfies the $\kappa$-chain condition if there is no ...
20
votes
0answers
400 views

Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
3
votes
1answer
2k views

Proving a subset is closed under a binary operation on a set

Suppose that $*$ is an associative and commutative binary operation on a set $S$. Let $H=\{a\in S\mid a*a=a\}$. Show that $H$ is closed under $*$. I started this problem by listing the ...
0
votes
2answers
4k views

$T(n)=3T(n/2) + n\log n,\ T(1)=1$ [duplicate]

Possible Duplicate: $T(n) = 4T({n/2}) + \theta(n\log{n})$ using Master Theorem What is the order of this recurrence? $$T(n)=3T(n/2) + n\log n,\ \ T(1)=1$$ I found the answer where ...
1
vote
0answers
230 views

Complexity of finding solutions for a system of polynomial equations

Problem A: Given a set of polynomial equations $ f_1=0,\ldots,f_m=0 $, where the $ f_i $'s are multivariate polynomials with $ n $ variables over a field $\mathbb F$, decide whether there is a ...
8
votes
2answers
234 views

Prove $ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2 $

Prove that $$ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\frac{\log 44444444}{\log33333333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$$ where last ...
2
votes
3answers
846 views

Product of all irreducibles with degree divisible by $n$ in $\mathbb{F}_{q^n}$?

In the finite field of $q^n$ elements, the product of all monic irreducible polynomials with degree dividing $n$ is known to simply be $X^{q^n}-X$. Why is this? I understand that $q^n=\sum_{d\mid ...
0
votes
1answer
85 views

surfaces of spheres are made of

Sorry in advance for lacking the appropriate terminology, please help me edit it in below. Take thease basic shapes: ...
3
votes
1answer
829 views

An isomorphism between an extension $K/F$ and a subfield of the ring of $n\times n$ matrices over $F$.

I'm working on a problem from Dummit & Foote's Abstract Algebra and I'm a bit confused about one part of the problem. The problem reads: Let $K$ be an extension of $F$ of degree $n$. ...
7
votes
1answer
151 views

matrix equation $(A-B)CA=B$

let $A,B,C$ be $n\times n$ matrices with real entries such that $A$ is invertible. if $(A-B)CA=B$ show that $AC(A-B)=B$. any Ideas??
13
votes
3answers
609 views

Does $\int_{1}^{\infty}\sin(x\log x)dx $ converge?

I'm trying to find out whether $\int_{1}^{\infty}\sin(x\log x)dx $ converges, I know that $\int_{1}^{\infty}\sin(x)dx $ diverges but $\int_{1}^{\infty}\sin(x^2)dx $ converges, more than that, ...
2
votes
3answers
169 views

Is it possible for a number in form $1987^k-1$ to end with 1987 zeros? Also few questions about number theory in general.

My fragile attempt: Note that if $1987^k-1$ ends with 1987 zeros, that means $1987^k$ has last digit 1 (and 1986 "next" ones are zeros). For this to be satisfied, $k$ has to be in form $k=4n$, where ...
1
vote
1answer
97 views

How to find $(a,n)$ such that : $5^a+1 \equiv 0 \pmod {3\cdot 2^n-1}$ and $3\cdot 2^{n-1}-1 \equiv 0 \pmod a$?

Is it possible to find such integer pair $(a,n)$ that : $\begin{cases} 5^a+1 \equiv 0 \pmod {3\cdot 2^n-1} \\ 3\cdot 2^{n-1}-1 \equiv 0 \pmod a\\ \end{cases}$ where $n \equiv 3 \pmod 4$
2
votes
1answer
711 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
14
votes
3answers
5k views

Incremental averageing

Is there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance? For example you have ...
1
vote
0answers
71 views

Calculating the number of tiles shown on an isometric map

I'm currently changing an existing computer game from a top-down view to an isometric view. The map consists of an infinite number of tiles. Only the tiles which are currently visible in the players ...
0
votes
1answer
426 views

Existence of Vertex Ordering in Greedy Algorithm to get “optimal” colouring

I am trying to prove that for any Graph there is an ordering of the vertices, sucht that the Greedy Algorithm will colour the vertices in such a way that it uses the Chromatic number of colours. I am ...
1
vote
1answer
63 views

How to solve for m in this equation?

How can I solve for $m$ in this equation, where $e$ is Euler's number, and $p,k,m \gt 0$, and $p \lt 1$? $$p = \left(1 - e^{\frac{-kn}{m}}\right)^k$$
5
votes
4answers
524 views

Evaluating $\sum\limits_{n=2}^{\infty} \frac{1}{ GPF(n) GPF(n+1)}\,$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. $\operatorname{ LPF}(n)=$Least prime factor of $n$, eg. $\operatorname{ ...
4
votes
1answer
227 views

Understanding characteristic functions in probability theory.

I am studying characteristic functions in probability theory and I am struggling to understand the following equality. $$\int_{-\infty}^{\infty}e^{itX}dF_X(x)=\int_{-\infty}^{\infty}e^{itX}f_X(x)dx$$ ...
5
votes
2answers
2k views

Find irreducible but not prime element in $\mathbb{Z}[\sqrt{5}]$

I have tried various numbers of the form $a+b\sqrt{5},\ a,b \in \mathbb{Z}$, but cannot find the one needed. I would appreciate any help. Update: I have found that $q=1+\sqrt{5}$ is irreducible. Now ...
0
votes
1answer
201 views

Eventually always notation

Let $X$ be a set and $A\subset X$. For a sequence $F = (f_n)_{n\geq 0}$ of elements of $X$ we say that $F$ is eventually always in $A$ if for some $N\geq0$ it holds that $f_n\in A$ for all $n\geq N$. ...
0
votes
2answers
95 views

Laplace of $x^2\frac{d^2y}{dx^2}$

How does one evaluate the Laplace of functions like $t^2\frac{d^2y}{dt^2}$ ? I wanted to solve a differential equation using Laplace Transform resembling: $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y ...
6
votes
3answers
1k views

Odd-Odd-Even-Even Sequence

I want a sequence that alternates between being an even integer and being an odd integer and I've come up with this sequence $ s_n=\lfloor \frac{n}{2} \rfloor $. So, it goes $0,1,1,2,2,3,3,\ldots$ and ...
0
votes
1answer
182 views

The inequality $e^{-t} -\left(1-\dfrac{t}{x}\right)^x \geq 0$

If $x > 0$ and $t \leq x $ Prove: $$ e^{-t}-\left(1-\dfrac{t}{x}\right)^x \geq 0 \>. $$
2
votes
1answer
135 views

Is $\{x \in \mathbb{R}^n: f(x) < 0\}$ open?

For a function $f: \mathbb{R}^n \to \mathbb{R}$, I was wondering Is $\{x \in \mathbb{R}^n: f(x) < 0\}$ open? If not, what are some sufficient and/or necessary conditions for it to be open? Is ...
2
votes
3answers
860 views

Cauchy-Schwarz Inequality

In Luenberger book Cauchy-Schwarz Inequality is defined like this: For all $x,y$ in an inner product space $|(x|y)| \le \|x\|\|y\|$. Equality holds if and only if $x = \lambda y$ or $y = \theta$. ...
1
vote
1answer
148 views

Graph, planar or not?

A graph $L_n$ has vertices $V=\{l_1,l_2,\dotsc,l_n\}\cup\{r_1,r_2,\dotsc,r_n\}$ and edges $E=\{(l_i,r_j): i \ge j\}$ . Which of these graphs $L_1$, $L_2$, etc. are planar and which are not? For ...
2
votes
2answers
70 views

Conditional convergence of $\int_{0}^{\infty}(-1)^{[ x^2]}dx$

I'm trying to prove that the following integral $\int_{0}^{\infty}(-1)^{[ x^2]}dx$ is conditional converges, ( The brackets stand for the floor function). I can't use the integral test since this is ...
0
votes
2answers
429 views

Position of 3 circles intersecting at the centre of bounding box

Here's what I feel is a neat challenge: I'm building a data visualization comprised of 3 circles of dynamic sizes. I want to have them all intersect at the centre of a bounding box that will also be ...
2
votes
3answers
558 views

Two ants on a triangle puzzle

Last Saturday's Guardian newspaper contained the following puzzle: Two soldier ants start on different vertices of an equilateral triangle. With each move, each ant moves independently and ...
0
votes
1answer
368 views

Finding probability of recording head in a sequence of coin toss

Suppose there is a biased coin that gives heads with probability $d$ such that $0<d<1$. Every time, I toss the coin two times. If both outcomes of the toss are the same, I ignore it. Else, ...
0
votes
1answer
707 views

Uniformly convergent subsequence: Arzela-Ascoli

Let $f_n$ be a sequence of differentiable functions on $[0,1]$ such that $|f_n'(x)| \le M$ (the absolute value of the derivative of $f_n$ at $x$) for all $n$ and for all $x$ in $[0,1]$. Show that ...
2
votes
1answer
70 views

What is the base for the following number system?

So there was a spaceship sent back with the following picture that assumed to constitute an addition. What is the base of the number system? ...
4
votes
2answers
4k views

Multiplying Taylor series and composition

I have two questions: A. I know the taylor series for $\arctan(x)$ and for $e^x$. How do I find the series for $\arctan(x)\cdot e^x$ ? B. Say I want to find the series for $\arctan(g(x))$, do I just ...
6
votes
2answers
637 views

How many ways are there for people to queue?

I'm stuck at the following combinatorics problem: Fifteen people queue up for cinema tickets at five (different) sales points. In how many ways can they stand in queue behind one another, if the ...
8
votes
0answers
335 views

Multiplicative Identity analog for absolute value

Is there a standard name for the function: $$ f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases} $$ And is there a potential application of this ...

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