13
votes
2answers
504 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
2
votes
1answer
136 views

Generating function of a convolution

If I need to find generating function of such a sum: $\sum_{k=0}^{n} (n-k) a_k$, can I write $\sum_{k=0}^{n} (n-k) a_k = \sum_{n\ge 0} nx^n \cdot \sum_{n\ge 0} a_nx^n \cdot \frac{1}{1-x}$ and then ...
1
vote
1answer
101 views

Brownian motions identical distributions

Let $(B_t)_t$ be a standard Brownian motion, and $$ A = \sup\{t\leq 1\mid B_t =0 \},\qquad B = \inf\{ t\geq 1\mid B_t =0 \}. $$ I would like to show that $A$ and $B^{-1}$ are identically distributed ...
1
vote
0answers
71 views

Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space. So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a ...
3
votes
1answer
1k views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
2
votes
1answer
115 views

tangential and normal projection of a vector in the ambient vector field of a sphere

I'm having unexpected trouble to perform this computation: Let $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=3\}$ and $v_p = (1,0,0)_{(1,1,1)}$ be a vector from the ambient vector field on $M$. How do I ...
0
votes
1answer
287 views

Finite sum of products of binomial coefficients and quadratic polynomial

How can I calculate the value of such a sum? $\sum_{k=0}^{n} (2k^2-3k+1){n\choose k}$ Should I split it into three sums? But then I don't know what to do with $k^2{n\choose k}$. I know that ...
0
votes
3answers
54 views

Prove $\sum_{i=0}^{n} a^i = \frac{a^{n+1} - 1}{a - 1}$ by induction

I was assigned two induction problems that I tried to solve. One was easy to solve using the following method, but one got me stuck. Problem: Prove by induction on $n \geq 1$ that for every $a ...
1
vote
0answers
156 views

Volume with MonteCarlo

I'm asked to find the volume of a given bounded solid in $\mathbb{R}^3$ by MonteCarlo means: since it is contained in a prism, I generate random points in the prism and see what proportion of them lie ...
0
votes
1answer
124 views

A vectorial geometry problem : $a\cos{\alpha}+b\cos{\beta}+c\cos{\gamma}=0.$

Let $\triangle ABC$ be a triangle and $O$ the center of circumscribed circle of the triangle. $AB=c, BC=a,CA=b.$ We will denote with $\alpha$ the angle of the vectors : $\vec{AO}$ and $\vec{BC}$ and ...
1
vote
1answer
70 views

Complex numbers with small modulus

Let $p$ be a complex number with $|p|<1/10$. How can one prove that there exists a bounded sequence of complex number $(z_n)$ with $z_0=0$ such that $z_{n+1}=z_n^2 + p$?
4
votes
0answers
56 views

non-constant curve c with $\frac{\mathrm{d}c}{\mathrm{d}t}= \lambda\frac{\mathrm{d}^2c}{\mathrm{d}t^2}$

I don't know how to solve the following problem and would appreciate some help. Let $M$ be a submanifold of euclidean space and $c:[a,b] \to M$ a non-constant curve, such that the velocity field of ...
1
vote
2answers
97 views

Reason for existence of 'swapping' elementary matrix operation

In my Hoffman & Kunze book on Linear Algebra, we define 3 elementary row operations, the first of which is the swapping of two rows. I'm wondering why we need to even have such an elementary ...
3
votes
1answer
560 views

Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
1
vote
2answers
83 views

Show that $\frac{1}{x^2+x+1}=\frac{2}{\sqrt3}\sum_{n=0}^{\infty}\sin\bigg(\frac{2\pi(n+1)}{3}\bigg)x^n$

Need to show that: $$\frac{1}{x^2+x+1}=\frac{2}{\sqrt3}\sum_{n=0}^{\infty}\sin\bigg(\frac{2\pi(n+1)}{3}\bigg)x^n$$ There is a hint given that $x^3-1=(x-1)(x^2+x+1)$ but I don't seem to get how I ...
1
vote
1answer
56 views

Show the area of the boundary of this set is not 0

In one real analysis book, it is said that: for the points set $S=\{(x,y)|0\le x\le 1, 0\le y\le D(x)\}$, where $D(x)$ is the Dirichlet function, $D(x)= \begin{cases} 1, & \text{if $x$ is ...
2
votes
1answer
69 views

What is the mathematics of UML?

What is the mathematics of Unified Modeling Language (UML)? The concepts introduced in UML such as classes, associations, association classes, subclasses smell mathematical. Is there a mathematical ...
5
votes
2answers
1k views

How many $n$-digit palindromes are there?

How can one count the number of all $n$-digit palindromes? Is there any recurrence for that? Thanks. I'm not sure if my reasoning is right, but I thought that for n=1 we have 10 such numbers ...
1
vote
2answers
182 views

Proving continuity using the topological definition?

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{R}$, and suppose that $f(x) = 0$, $\forall x \in \mathbb{R}$, except when $x=c$, for a fixed $c \in \mathbb{R}$. Now, $f$ is clearly discontinuous ...
6
votes
5answers
4k views

Negating A Mathematical Statement

Regard this statement $ x \ge 0$. According to my teacher, by negating this statement, it will become $ x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?
1
vote
1answer
635 views

Reflection of a line on complex plane

Show that the reflection of $z$ on the line $ax+by=c$, where $a,b,c \in \mathbb{R}$ is given by the following: $$\frac{2ic+(b-ai)\bar z}{b+ai}$$ I know that the conjugate of $z$, which is $\bar ...
4
votes
3answers
76 views

How to prove $a_{n} < 2$ if $\displaystyle a_{n+2}=a_{n+1}+\frac{a_{n}}{3^n}$

Let $(a_{n})_{n \geq1}$ be a real sequence such that $a_{1}=a_{2}=1$ and $\displaystyle a_{n+2}=a_{n+1}+\frac{a_{n}}{3^n}, n\geq 1$. Prove that $a_{n} < 2, \forall n \geq 1.$ I write $$\sum ...
2
votes
3answers
252 views

Logical Equivalence and Corresponding English Statements

Consider the statement, "If it is Tuesday, then it is raining"; in propositional logic, the statement would read as, "$p \implies q$." Now, in accordance with the rules and definitions prescribed in ...
0
votes
2answers
40 views

Evaluate the antiderivative of a function with a different equation depending on a bracket

$$ f(x) = \begin{cases} -1 <=> x \in (-\infty,-1] \\ x <=> x \in (-1,0)\\ x^2 <=> x \in [ 0;\infty) \end{cases} $$ What have I do so far: $$ F(x) = \begin{cases} ...
0
votes
1answer
102 views

Group Isomorphism - Associativity - Change of Operator

Sorry about the title - I wasn't sure about how to be more specific. This is a homework problem, where I've only been able to write down (ii). Let (S,*) be a binary system. Define the opposite ...
1
vote
0answers
202 views

Why Does Induction Prove Multiplication is Commutative?

Andrew Boucher's General Arithmetic (GA2) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom: $$\forall x \forall y \forall ...
6
votes
1answer
296 views

Is there an irreducible polynomial vanishing on two components? (In the Zariski sense)

The polynomial $$f(x,y) = (x^2 − 1)^2 + (y^2 − 1)^2$$ is an example of an irreducible polynomial in $\mathbf{R}[x,y]$ which is irreducible but whose zero set has multiple components in the Zariski ...
8
votes
1answer
111 views

Size of the group $E(\mathbb{Q}_{p})/pE(\mathbb{Q}_{p})$

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$. Do we know anything about the order of the group $E(\mathbb{Q}_{p})/pE(\mathbb{Q}_{p})$? I know that it's finite, but do we know anything else?
2
votes
4answers
653 views

Binomial Theorem identities, evaluate the sum

This is a homework problem, please don't blurt out the answer! :) I've been given the following, and asked to evaluate the sum: $$\sum_{k = 0}^{n}(-1)^k\binom{n}{k}10^k$$ So, I started out trying ...
2
votes
3answers
373 views

Matrix notation differences

So I'm in an elementary matrix and linear algebra class and during lecture my professor uses this notation for a matrix: $$\left(\begin{matrix}a & b\\c&d\end{matrix}\right)$$ but in my ...
1
vote
1answer
134 views

linear extension of algebras homomorphism

I'm formalizing this useful algebraic method. In simple words: we have algebras $A,B$ (not necessarily commutative) over a field $F$. we define a function $\varphi:A \rightarrow B$ on a set of ...
10
votes
0answers
459 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
1
vote
2answers
317 views

Prove that if $f$ is defined and bounded in $[a,b]$ and integrable in $[c,b]$ for all $c\in(a,b)$ then $f$ is integrable in $[a,b]$

Prove that if $f$ is defined and bounded in $[a,b]$ and integrable in $[c,b]$ for all $c\in(a,b)$ then $f$ is integrable in $[a,b]$. I don't even know where to begin - I've tried to show that for ...
4
votes
1answer
96 views

solve $ y = (A+B^{-1})x $ for $x$

I wish to solve numerically for $x$, $$ y = (A+B^{-1})x $$ with $A, B$ positive definite. So, $$ x = (A+B^{-1})^{-1}y $$ I would like to avoid calculating $B^{-1}$ since that's generally bad. ...
6
votes
1answer
313 views

Inequality. $abc(a+b+c) > 3abc+ab+bc+ca.$

I want to ask you a solution for the following problem. Let $a,b,c$ be real numbers, $a,b,c > \frac{1+\sqrt{5}}{2}$. Prove that: $$abc(a+b+c) > 3abc+ab+bc+ca.$$ I don't know how "to touch" ...
2
votes
5answers
530 views

Evaluate integral with quadratic expression without root in the denominator

$$\int \frac{1}{x(x^2+1)}dx = ? $$ How to solve it? Expanding to $\frac {A}{x}+ \frac{Bx +C}{x^2+1}$ would be wearisome.
2
votes
1answer
57 views

How to find the value of $h(99)$ in the function?

If $$h(x) + h(x+1) = 2x^2$$ and $$h(33) = 99$$ What will be the value of $h(99)$?
2
votes
0answers
122 views

Group relations and generators

I have to find out what is this abelian group (in the form $\mathbb{Z}/m_1\mathbb{Z} \times ... $). Its relations are: $$6a+9b+6c=0$$ $$8a+12b+4c=0$$ with generator $a,b,c$. My solution is: ...
7
votes
3answers
200 views

Calculating value of $1000^{th}$ derivative at $0$.

I need to calculate value of $1000^{th}$ derivate of the following function at $0$: $$ f(x) = \frac{x+1}{(x-1)(x-2)} $$ I've done similar problems before (e.g. $f(x)= \dfrac{x}{e^{x}}$) but the ...
1
vote
1answer
51 views

Prove that at least two of the inequalities are true.

Let a, b, and c be positive real numbers such that $a + b + c\leq 4$ and $ab+bc+ca\geq 4$. Prove that at least two of the inequalities $|a - b|\leq2$ , $|b - c| \leq 2$, $ |c - a| \leq 2$ are ...
-1
votes
1answer
9k views

How to find the probability, mean and cdf using a pdf

Let $X$ be a random variable with pdf $f_{x}(x)= \large \frac{1}{5} e^{\frac{-x}{5}}$, $x>0$ a. Sketch the graph of $f_x$. Use the pdf to find $P(X>5)$. Find the mean of $X$. b. ...
2
votes
2answers
58 views

How does a normal vector change with respect to the points that define it?

$A$ is a nondegenerate $n \times n$ matrix with real-valued entries. If we interpret the rows of $A$ as points in $\mathbb{R}^n$, then $A$ defines a unique hyperplane that passes through each of ...
10
votes
4answers
620 views

How can I explain topology to my grandmother?

I was recently look at a post on tex.stackexchange about explaining $\LaTeX$ to the OP's grandmother. I was wondering, could the same thing be done for topology? Except in this case the "grandmother" ...
1
vote
2answers
60 views

Verification of subtraction

Here I am subtracting two cubic functions: $ax^3 +bx^2+cx+d -2(a(x+1)^3 +b(x+1)^2+c(x+1)+d) = -x^3-3x^2-3x-1$ Then I equate co-efficients to get: $-1 = -a$ $-6a-b = -3$ $-6a-4b-c = -3$ ...
1
vote
2answers
73 views

Morphisms generated by functions

Given a function $f: A \to B$, I can construct a morphism $g : A^* \to B^*$ where $X^*$ denotes some free structure generated by $X$ (Could be monoid, group, module, etc.). I'd like to study ...
5
votes
1answer
233 views

showing zero curvature implies a line

How can I show that a given (not necessarily unit-speed) parametrization $\gamma(t)$ of a curve in $\mathbb{R}^3$ which exhibits zero curvature is a line ? What I know is that zero curvature means ...
3
votes
1answer
108 views

Why was I wrong doing this problem?

"A waste disposal company averages $6.5$ spills of toxic waste per month. Assume spills occur randomly at a uniform rate, and independently of each other, with a negligible chance of $2$ or more ...
2
votes
1answer
71 views

What will be the slope of $BC$?

The vertex $A$ of triangle $\triangle ABC$ is $(3,-1)$. The equation of median $BE$ is $$6x+10y-59=0$$ and angle bisector $CF$ is $$x-4y+10=0.$$ Then what is the slope of $BC$? Let slopes of $AC, CF, ...
0
votes
2answers
205 views

Smallest value of fraction

Does the fraction $\frac{2x-3}{2x+1}$ have a minimum positive value, if so how do you find it? Thanks!
3
votes
1answer
437 views

Isomorphism from $\mathbb{R}$ to $(-1,1)$

There are many bijective functions that map $\mathbb{R}$ to $(-1,1)$, in particular: $$f\left(x\right)=\frac{e^{2x}-1}{e^{2x}+1}$$ (Of course there are others, such as ...

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