1
vote
1answer
150 views

Why does Newton's method approximate $\sqrt{z}$ to be $z - \frac{z^2 - x}{2z}$?

According to a document I am reading, I can approximate the square root of some real number $x$ via "Newton's method" by repeatedly taking $z = z - \frac{z^2 - x}{2z}$ after beginning with some random ...
8
votes
3answers
513 views

What constitutes an axiom - Spivak Calculus ch. 1

In chapter 1 of Spivak's Calculus text he lays out some fundamental axioms of the integers. For instance that: $a \cdot 1 = a$, for all $a$. However he doesn't list an axiom that for instance says: $a ...
2
votes
2answers
710 views

What is the difference between $\lfloor f \rfloor (x)$ and $\lfloor f(x) \rfloor $?

I have a very basic question: What is the difference between $\lfloor f \rfloor (x)$ and $\lfloor{f(x)}\rfloor$? Are $\lfloor f \rfloor (x)$ and $f(\lfloor x \rfloor)$ equivalent?
4
votes
1answer
119 views

When do counital coalgebras have a basis of grouplike elements?

Question. Under what conditions do counital coalgebras have bases consisting entirely of grouplike elements? At least in the case of finite-dimensional coalgebras, or for bialgebras (or Hopf ...
1
vote
1answer
155 views

What happen if the argument is zero

Let us consider the following function: $\phi(s)=\phi_1(s)+i\phi_2(s)$, given uniquely by the polar form $\phi(s)=\rho(s)\exp(i\theta(s))$, where $$\rho(s)=\sqrt{\phi_1^2(s)+\phi_2^2(s)}\neq 0$$ and ...
1
vote
3answers
143 views

Graph of functions

I am trying to envision the graphs of the following equations but for some reason or the other I am not able to figure a systematic way of graphing them. I need the most basic procedure to achieve ...
3
votes
1answer
372 views

how to make a matrix a magic square?

Suppose I have a matrix $$\begin{pmatrix} & 3 & 6\\ 5 & & 5\\ 4 & 7 & \end{pmatrix}$$ How can I find the three numbers on the main diagonal such that the sum of the ...
0
votes
0answers
225 views

Maximum absolute deviation

Let be I , J a two gray scale image How I will be able to interpret the Maximum absolute deviation between the histograms of these images?
14
votes
3answers
532 views

Compute $\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$

Compute the limit $$\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$$
1
vote
1answer
1k views

Supremum of continuous functions is continuous?

If $f(t,u)$ is continuous wrt. $t$ (and $u$), then is $$\sup_{u \in H^1(\Omega)} f(t,u)$$ continuous wrt. $t$? I am unable to prove this. Help appreciated.
4
votes
2answers
226 views

Determine value of harmonic function at origin given boundary values on a regular hexagon

Let $E$ be a regular hexagon centered at the origin of $\mathbb{R}^2$. Let $f$ be the harmonic function in $E$ with boundary value 1 on one of the sides of $E$ and boundary value $0$ on each of the ...
8
votes
3answers
440 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
5
votes
1answer
426 views

Intuition behind isomorphism of algebraic varieties

Let $X \subset \mathbb A^n$, $W \subset \Bbb A^m$ be two algebraic sets. A function $\phi:X \rightarrow W$ is a morphism if there exist $m$ polynomial functions $f_1,\ldots,f_m \in K[X]$ such that for ...
2
votes
3answers
130 views

Fixed point in a continuous map [duplicate]

Possible Duplicate: Periodic orbits Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$. Does $f$ necessarily ...
0
votes
0answers
39 views

Calculating a function limit [duplicate]

Possible Duplicate: Complicated limit calculation I tried calculating this limit and what I found is the limit being 0. I'm not sure if I'm right or not and I have no answers at hand. ...
15
votes
1answer
306 views

Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
0
votes
1answer
96 views

Differential Equation

I'm stuck on this question. Originally part of a mechanics question concerning a trains's motion. I'm finding the time taken for a train to go from $75\textrm{km/hr}$ to $175\textrm{km/hr}$ The ...
1
vote
1answer
102 views

Algebra question resubmited [closed]

Hello I have a quick algebra question. If I have the following expression $\large \frac{1}{x^2+1}$, and I multiply the numerator and the denominator by $(x^2+1)$. Is there any way I can get ...
3
votes
1answer
416 views

calculation proof of complex form of green's theorem

Complex form of Green's theorem is $\int _{\partial S}{f(z)\,dz}=i\int \int_S{\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\,dx\,dy}$. The following is just my calculation to show both ...
1
vote
1answer
206 views

Simultaneous equations in four variables

I'm solving the following equations, $$x+y=zw$$ $$z+w=xy$$ How many solutions $(x,y,z,w)$ exist, if the variables are reals?
0
votes
2answers
378 views

Set of all finite subsets of $\mathbb{N}$ is a countable set [duplicate]

Possible Duplicate: Show that the set of all finite subsets of $\mathbb{N}$ is countable. How can I prove in a proper way that the "set of all finite subsets of $\mathbb{N}$ (the set of ...
0
votes
1answer
340 views

Finding composition factors of a group of order $48$

I have a question which is: find the composition factors of a group of order $48$. But I can't see how I can do this. If I choose my group to be $C_{48}$ then it has composition series : ...
2
votes
1answer
601 views

Plotting $(x^2 + y^2 - 1)^3 - x^2 y^3 = 0$

I have no idea how this equation: \begin{equation} (x^2 + y^2 - 1)^3 - x^2 y^3 = 0 \end{equation} Produces this picture: Can someone provide a general explanation of plotting this function?
5
votes
2answers
198 views

A property of finite field of order $2^n$

Suppose $a$ and $b$ are elements of a finite field of order $2^n$ with $n$ odd and $a^2+ab+b^2=0$. Is it necessary that both $a$ and $b$ must be zero ? I understand that the field has characteristic ...
3
votes
1answer
104 views

Equicontinuity and uniform boundedness for “distributions”

Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space $$ \mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \} $$ with the topology induced by ...
4
votes
2answers
882 views

Gamma function of negative argument

Is there any relation between the limiting behaviour of $\Gamma({\epsilon})$ and $\Gamma(-1+{\epsilon})$? I have seen the relation such as $\Gamma(-1+{\epsilon})$ $=$ ...
5
votes
1answer
136 views

What is a “set-like class”?

Just / Weese contains the following theorem (p 126): Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf ...
2
votes
1answer
51 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
0
votes
1answer
69 views

Modular arithmetic with different moduli

What is the cheapest and fastest way to find the remainder of the modular arithmetic $\pmod {n}$ when we have the reminder for $\pmod {n-1}$ or $\pmod {n+1}$ ? As an example, if: $$ 3^{60} \equiv ...
0
votes
2answers
120 views

Derive a simpler expression for $\gcd(a+b,\text{lcm}(a,b))$?

Let $a$ and $b$ be two positive integers, such that neither $a$ nor $b$ is divisible by a perfect square. Is there be a simplified formula for $$\gcd(a+b,\text{lcm}(a,b)) ?$$ and is there a way to ...
2
votes
3answers
673 views

Rotation and fixed points

I have a rotation of the form: $$(z(s),w(s))=B(s)(u(s),v(s))$$ where $z(s),w(s),u(s),v(s)$ are in $\mathbb{R}$ and $s$ is a complex number and $B(s)$ is a $2\times 2$ matrix defined by $$ B(s) = ...
8
votes
3answers
670 views

Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$

How would you approach $$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$ The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ...
1
vote
1answer
100 views

A differentiable injective function with Lipschitzian Inverse

I'm having difficulty with the following question which was given to me following studying the inverse mapping theorem. Let $U\subseteq\mathbb{R}^{n}$ be an open set and let $f:U\to\mathbb{R}^{n}$ ...
1
vote
2answers
1k views

How can I systematically find the roots of $ x^4 + 1?$ Is there some algorithm? [duplicate]

Possible Duplicate: How to find the root of $x^4 +1$ What algorithms can be used for finding all roots of the given polynomial: \begin{equation} x^4 + 1 = 0 \end{equation}
3
votes
0answers
185 views

Cluster point of $a_{n}:=n+(-1)^{n}n$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution? The subsequence diverges for increasing even $n$ since $2n$ grows infinitely. The ...
1
vote
0answers
60 views

Simple equation misunderstanding

Im trying to use an equation on this page http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics The angle of lean, $\theta$, can easily be calculated using the laws of circular motion $$ ...
4
votes
1answer
340 views

Showing that a cyclic group of prime power order has only 1 composition series

I am trying to show that a cyclic group of prime power order has only 1 composition series. Is the following correct? Let $G=C_{p^n}$. Then as cyclic groups are abelian we have that there is a ...
-2
votes
2answers
252 views

Inverse Laplace Transform : $ F(s)=\frac{2 \omega^3}{(s^2+\omega^2)^2} $

Please help to find inverse laplace transform : $$ F(s)=\frac{2 \omega^3}{(s^2+\omega^2)^2} $$
1
vote
1answer
381 views

Integration Question on $\int \frac{1}{x^2+10x+21} \, dx$

How would i do the following indefinite integration $$\int \frac{1}{x^2+10x+21} \, dx$$ so far I've turned the bottom polynomial into $(x+7)(x+3)$ not too sure where to go from here
8
votes
4answers
741 views

How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?

How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$? I realize that any of the class of functions $f:x\to (n\cdot x)$ gives a bijection between $\mathbb{N}$ and the subset ...
1
vote
3answers
93 views

How can I find $\sum_{k=1}^{1233}f(\frac{k}{1234})$

Let $$f(x)=\frac{e^{2x-1}}{1+e^{2x-1}}$$ Then find $$\sum_{k=1}^{1233}f\left(\frac{k}{1234}\right)$$ How I proceed: ...
0
votes
1answer
197 views

Integral versus a sum for an expression that sums to one

I have an expression $f(x)$, outputting strictly real numbered values $\geq 0$ corresponding to the probability of some event, where $\sum_{i=0}^{N} f(i) = 1$. When is it true that $\int_{i=0}^{N} ...
2
votes
2answers
3k views

Convolution of two triangles

I want to convolve two triangles. The equation satisfied by one triangle is $$f(y) = \begin{cases} y + 1 & −1 < y < 0\\ \\ 1 − y & 0 \leq y < 1 \end{cases}.$$ So, the overall ...
2
votes
2answers
108 views

$f(x)=\int_{0}^{1}e^{ixz^2}dz$ as $x\rightarrow +\infty$.

Find the asymptotic behaviour as $f(x)=\int_{0}^{1}e^{ixz^2}dz$ as $x\rightarrow +\infty$. Could anyone show me how to do this with either the method of stationary phase or integration by parts? ...
2
votes
3answers
685 views

how to find a cluster point of $a_{n}:=(2+(-1)^n)\frac{n}{n+1}$

i am tryint to find a cluster point of this sequence, but i am having difficulties in definitions. the sequence is this: $(a_{n})_{n \in \Bbb{N}}$ with $a_{n}:=(2+(-1)^n)\frac{n}{n+1}$ the ...
13
votes
7answers
2k views

Is a counterexample considered a rigorous proof that a property is not true?

This is my follow-up question to my own query earlier: How can I algebraically prove that $2^n - 1$ is not always prime? Almost half of the answers said that I provided my own proof by giving ...
0
votes
2answers
151 views

A question on Taylor development of multivariate function

How can I express the Taylor development of $$f(t+\Delta t, x+h)$$ When one of the variables is fixed, I can find it but with both of them varying, I have no idea what to do. $$f(t+\Delta t, ...
2
votes
1answer
167 views

Solutions of Diophantine equations in Natural numbers

The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$ of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail ...
1
vote
1answer
130 views

Equivalence between detailed balance and time reversal

I need help in proving the following popular claim A continuous time and stationary Markov jump process obeys the detailed balance equations $$ P(x)q(x,x') = P(x')q(x',x) $$ where $q(\cdot,\cdot)$ is ...
0
votes
5answers
402 views

How can I algebraically prove that $2^n - 1$ is not always prime?

This question is from Elementary Number Theory by W. Edwin Clark. Is $2^n - 1$ always prime, or not? Prove. Is this a start? $x^n - 1 = ( x - 1)(1 + x + x^2 \cdots x^{n - 1})$. So, $2^n - 1 = ...

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