0
votes
1answer
389 views

Analytical geometry - circles

How do you find the point for a circle and find the radius when $x^2$ has a co-efficient?
0
votes
1answer
303 views

Indices Again - Expressing Negative Fraction Powers as powers of given number

If someone could provide me with a website or link with information about this I'd be grateful but otherwise: I have the number two (with the ability to add a power) and the number ...
3
votes
4answers
424 views

Help with showing how $\sin\alpha\cos\beta$ $=$ $\frac{1}{2}(\sin (\alpha + \beta) + \sin(\alpha-\beta))$ using Eulers formula.

I need help with understanding how one can rewrite: $\sin\alpha\cos\beta$ to be equal to: $\frac{1}{2}(\sin (\alpha + \beta) + \sin(\alpha-\beta))$ using Eulers formula. I know that it probably ...
1
vote
3answers
422 views

Prove that a simple graph with $2n$ vertices without triangles has at most $n^2$ lines.

Prove that a simple graph with $2n$ vertices without triangles has at most $n^2$ lines. I've been struggling with this exercise for some time, but I can't come up with a decent proof.
1
vote
1answer
67 views

Taylor polynomial

I need your help to solve this question. I tried something, but i can't finish my proof. Let $f(x)$ be a differentiable function in $(0, \infty )$, so that $|f'(x)|$ is bounded there. Prove that ...
1
vote
3answers
252 views

Differential equation initial value problem - hard!!

I have been asked to solve $x' = t/(1 + t^2) - x(t/(1+t^2))$ and determine the maximal interval where the solution exists. I have tried to solve this in many different ways but must be using the ...
0
votes
2answers
392 views

Binomial random variable with number of trials being a Poisson random variable

Let $Y$ be the number of heads in a an $X$ toss sequence of flipping a coin with probability $p$ of heads. Show that $Y \sim \mathrm{Pois}(p \lambda)$ if $X \sim \mathrm{Pois}(\lambda)$.
2
votes
2answers
256 views

Kuratowski Definition of Ordered Pairs, ZFC

I came accross the elegant definition given by Kuratowski of an ordered pair which is: $$(x,y) := \{\{x\},\{x,y\}\}$$ and wondered, if the existence of this set presupposes (the axioms of) ZFC?
0
votes
1answer
226 views

A question on Right half-open Interval topology

Is Right half-open interval topology the same as Sorgenfrey line? I think it is, however I am not sure.
1
vote
0answers
81 views

Symmetrization of Powersum polynomials

Let $n\in\mathbb{N}$. Then for $i\in\mathbb{N}$ the $i-$th power sum if defined to be $p_i^{(n)}:=\sum_{j=1}^n x_j^i$. Then let $\lambda:=(\lambda_1,\ldots,\lambda_l)$ be a partition of $d$. We can ...
1
vote
1answer
47 views

Prove that $A^HB^H=(BA)^H$

I want to prove this statement where A and B are matrices and H the Hermitian. Ok, Here is a proof from MathWorld: ...
0
votes
1answer
34 views

Integral of a Stepfunction: Analytical Motivation?

For a stepfunction defined as: f = $\sum_{k=1}^{K}c_k \chi_{I_k}$ where $\chi$ is the indicator function, $c_k \in \mathbb R$, the disjoint union of the closed intervals $I_k$ is [a,b] $\in \mathbb ...
2
votes
2answers
72 views

Least value of $a$ for which at least one solution exists?

What is the least value of $a$ for which $$\frac{4}{\sin(x)}+\frac{1}{1-\sin(x)}=a$$ has atleast one solution in the interval $(0,\frac{\pi}{2})$? I first calculate $f'(x)$ and put it equal to $0$ to ...
2
votes
1answer
281 views

When is the localization of a module trivial?

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. Let $S^{-1}M$ be localization of $M$, where $S$ is a multiplicatively closed subset of $R$. How to show that $S^{−1}M =0$ if ...
4
votes
1answer
269 views

Multidimensional Hensel lifting

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ... Let $p$ be ...
0
votes
1answer
53 views

Interval for $\alpha$, so that the given line is normal to $xy=4$

Find the interval for $\alpha$ so that $(3-\alpha)x+{\alpha}y+({\alpha}^2-1)=0$ is normal to the curve $xy=4$. I don't understand why do we need an interval for $\alpha$? The curve is a hyperbola ...
2
votes
1answer
117 views

Truth functional completness for quantifiers

There is a definition for truth-functional completness for a set of propositional connectives. Is there a definition for truth-functional completness of a set of quantifiers and propositional ...
1
vote
2answers
2k views

Need help using De Moivre's theorem to write $\cos 4\theta$ & $\sin 4\theta$ as terms of $\sin\theta$ and $\cos\theta$

I need help with the following question: "Use De Moivre's theorem to write $\cos 4\theta$ & $\sin 4\theta$ as terms of $\sin\theta$ and $\cos\theta$" You could write the problem as: ...
4
votes
0answers
98 views

Has this function a name?

Has this function a name? $$f(x) = \prod_{i=2}^{\lceil \frac{x}{2} \rceil} \sin\left( \frac {\pi x}{i}\right)$$ (the product of $\sin( \frac {\pi x}{i})$ for $i=2,3,...,\lceil x/2 \rceil$)
2
votes
1answer
78 views

Smallest $n$ for which, we can have $k$ distinct subsets of $[n]$ such that no two of them have empty intersection

Assume that, we know the number $k$, the number of distinct subsets of $[n]$ ( whose value we want to come up with ). The condition is that, no two subsets from the $k$ above should result in an empty ...
4
votes
1answer
493 views

How to show the tensor product of reduced $k$-algebras must be reduced?

Let $A$, $B$ be two reduced $k$-algebras. Then if any element of the form $$\sum a_{i}\otimes b_{j}$$ is nilpotent, we can compose it with any $k$-homomorphism $f$ from $A$ to $k$ to get a ...
1
vote
2answers
37 views

Define a $2 \times 2$ matrix that is the lower $2 \times 2$ block in $A$ (Matlab)

First of all, on this Matlab exercise sheet that I am currently working through what does the term 'the lower $2 \times 2$ block' mean in the question below? $A = \left[\begin{array}\ 1 & 2 ...
2
votes
3answers
98 views

Does this isomorphism hold?

Proposition: If $B\cong C$ then $\dfrac{A\oplus B}{C} \cong A.$ This is clearly true for vector spaces by counting the dimensions, but I am most interested to see if it holds for groups. What about ...
5
votes
0answers
116 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
1
vote
2answers
161 views

Classical solutions of Neumann Laplacian

I have a question concerning the Neumann Laplacian. Say, we consider a boundary value problem formulated on an interval $\Omega = [0,1]$: \begin{equation} -dv'' + cv = r, \quad v'(0) = v'(1) = 0, ...
6
votes
5answers
215 views

Find $\int e^{-x}\cos x\,dx$ without using complex numbers

$\int e^{-x} \cos{x} dx $ - i know how to solve with Euler complex representation, but can't figure out how to solve with integration by parts or something.
2
votes
1answer
53 views

Kolmogorov complexity of $n$

I was reading a paper on Kolmogorov complexity, but got stuck on the convergence part. It states that a common form to assign probability to integers $n$ would be $P(n)=A2^{-\log_2^*n}$, where ...
2
votes
0answers
233 views

Algebraic curves and riemann surfaces

I am a physics undergrad with no formal background in complex analysis. I have done complex analysis at the level of the first 4 chapters (till Complex integration) from Churchill and Brown. I am very ...
2
votes
1answer
232 views

Find min & max of $a(b-c)^n+b(c-a)^n+c(a-b)^n$ where $a + b+ c =1$

Find min & max of $a(b-c)^n+b(c-a)^n+c(a-b)^n$ where $a + b+ c =1;\ a,b,c\ge0; \ n \in N$ I am really stuck, I don't remember where I read this problem.
0
votes
1answer
78 views

Cleaning a signal and computing period

I am working with a signal which is a periodic square signal with some kind of noise and some outliers. I would like to know which is the best solution in order to get the period and clean the ...
0
votes
2answers
44 views

What does $\sum_{n=0}^{\infty} (-1)^n .\frac{x^{2n+1}}{2n+1}$ converge to at x= 1 and x = -1

This is what I did. $\sum_{n=0}^{\infty} (-1)^n .\frac{1^{2n+1}}{2n+1}=\sum_{n=0}^{\infty}(-1)^n .\frac{1}{2n+1}$ Now I broke it up to positive and negative. $\sum_{n=0}^{\infty}(-1)^n ...
1
vote
1answer
186 views

Does the Sorgenfrey Line have point-countable base?

Just as the title explains, does the Sorgenfrey Line have point-countable base? Thanks ahead.
1
vote
1answer
173 views

Stuck with Fourier transform

I'm trying to solve a simple mass-dashpot-spring system $$m\ddot{u}(t) + c\dot{u}(t) + ku(t) = F(t)$$ by making use of the Fourier transform defined as $$\tilde{f}(s) = \int_{-\infty}^{\infty} f(t) \, ...
1
vote
2answers
110 views

Linear function inverse

I have two monotonically increasing piecewise-linear functions: \begin{align} F_{L}(s) &= s_L\\ F_{H}(s) &= s_H \end{align} and their inverses: \begin{align} F_{L}^{-1}(s_L) &= s\\ ...
2
votes
3answers
900 views

How to find the period of a periodic function?

I'm working on some numerical analysis problem, and I'm studying functions that "seem" to be periodic. Now what I would like to do, is to determine their period. Only, the methods I actually use are ...
2
votes
1answer
218 views

Ramsey Number proof

I am trying to prove: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$ I am confused as to how one can go from a $4$ color problem to a $3$ color problem by multiplying and adding. edit: $R$ is the Ramsey ...
0
votes
0answers
249 views

Application of maximum principle to a function of a complex variable.

The question is: $f:U \to \mathbb{C}$ analytic and non constant on $U$ (open and connected). Show that $|\Re(f)|$, $|\Im(f)|$ and $\Re(f)^4 + \Im(f)^4$ do not attain local maxima. I've attempted to ...
30
votes
10answers
5k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
1
vote
1answer
122 views

Null sets of $\sigma$-algebras generated by functionals

These are some questions that I couldn't answer after a class I'm teaching. I would very much appreciate help. Let $\Omega=[0,1]^N$ and suppose that $f,g\colon \Omega \to [0,1]^N$ are two functions ...
1
vote
1answer
88 views

Is $e^{\alpha\log(M)}$ equal to $M^{\alpha}$?

Supposing the matrix logarithm exists, is $e^{\alpha\log(M)}$ equal to $M^{\alpha}?$ This equality obviously holds for positive reals, but does it also hold for matrices?
4
votes
3answers
164 views

Values of $a$ for which $(a+4)x^2-2ax+2a-6 <0$ for all $x \in R$

How can we find all values of $a$ for which the inequality $(a+4)x^2-2ax+2a-6 <0$ is satisfied for all $x \in R$? For the given condition, $D >0$, therefore $ (-2a)^2-4(2a-6)(a+4) >0$. ...
1
vote
2answers
66 views

Struggling with a question on quotients in elementary Galois theory

I have started teaching myself Galois Theory. I have a problem understanding a part of the proof of the following proposition : Let $K\subseteq L$ be a field extension and $l\in L$ an element which is ...
0
votes
2answers
116 views

inverse of a mod

i can't get my head around this I got this and ned to find an inverse for Chinese Remainder Theorem $$2x=1 \mod 5$$ $$S_1 \Rightarrow 2^{-1}\cdot 2\cdot x=2^{-1} \cdot 1 \mod 5$$ $$S_2 \Rightarrow ...
1
vote
1answer
57 views

Respective complexities of a string and its substring

If $s$ is a substring so that $s \subset S$, can Kolmogorov complexity of the whole string $S$ be lower than that of the given substring, $K(S) < K(s)$?
3
votes
2answers
253 views

This sequence $\sqrt {a+\sqrt{a+\sqrt{a+\sqrt{a…}}}}$ is bounded? [duplicate]

I'm trying to prove that this sequence $(x_n)$, where $x_1 =\sqrt a$ and $x_{n+1}=\sqrt{a +x_n}$ has a limit, then I would like to find the limit of $L=\sqrt {a+\sqrt{a+\sqrt{a+\sqrt{a...}}}}$ It's ...
2
votes
1answer
103 views

Why if $\alpha$ is a logical axiom, then $\alpha^{c}_{y}$ is also a logic axiom?

On page 123, A Mathematical Introduction to Logic, Herbert B. Enderton(2ed)(A part of proof 24F,"GENERALIZATION ON CONSTANTS") $\alpha$ is a logical axiom, then $\alpha^{c}_{y}$ is also a logic ...
2
votes
1answer
98 views

Computable function example

Suppose $p(x)\in\mathbb{Z}[x]$. How can we show that the function $b\to$ the least non-negative integer root of $p(x) - b$ is computable (if there is no such root, then the function is undefined)? ...
3
votes
3answers
839 views

Finding the coefficient of $x^{25}$ in $(1 + x^3 + x^8)^{10}$?

Find the coefficient of $x^{25}$ in $(1 + x^3 + x^8)^{10}$. I've tried thinking of this combinatorially, but I couldn't get it to make sense. I've also tried applying some identities, only to ...
3
votes
1answer
172 views

Finding a minimal polynomial for a square root of an algebraic number?

I have a given monic polynomial with an algebraic root $\alpha$. How can I find the minimal polynomial with a root of $\sqrt{1-\alpha^2}$ ?
4
votes
1answer
62 views

Kleene algebra without right distributivity?

I'm facing a mathematical structure that has everything of a Kleene algebra (S, +, ., 0, 1, *), except that the multiplication '.' is not right-distributive over the addition '+'. ...

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