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H: Do there exist some non-constant holomorphic functions such that the sum of the modulus of them is a constant
Do there exist some non-constant holomorphic functions $f_1,f_2,\ldots,f_n$such that $$\sum_{k=1}^{n}\left|\,f_k\right|$$ is a constant? Can you give an example? Thanks very much
AI: NO. Suppose $f, g$ are ... |
H: Can't that every standard positive real number is limited be proved?
I'm reading a short section about internal set theory(see here), in which
$x$ is limited in case for some standard $r$ we have $|x| ≤ r$.
while the predicate “standard” is not defined.
I'm interpreting an element $x \in A$ is standard, if it do... |
H: Calculation of ordered pair $(x,y,z)$ in $x^2 = yz\;\;,y^2=zx\;\;,z^2 = xy$
(1) Total no. of integer ordered pair $(x,y,z)$ in $x^2 = yz\;\;,y^2=zx\;\;,z^2 = xy$
(2) Total no. of integer ordered pair $(x,y,z)$ in $x+yz = 1\;\;,y+zx = 1\;\;,z+xy = 1$
My Try:: (1) Clearly $ x = 0,y = 0,z = 0$ are the solution of give... |
H: How to solve this integral $\int \frac{1+2x^2}{x^2(1+x^2)}dx$
How to solve this integral
$$\int \frac{1+2x^2}{x^2(1+x^2)}dx$$
I thought it should be $ x + 3x^2$ in the numerator so that I will take $x+x^3 = u$ then taking derivative both sides and it comes; $1+3x^2$ so this is wrong.
Please suggest how to proceed... |
H: Finding $\int_{0}^{\infty}\frac{\cos(ax)}{(x^2 + 1)^2}\,dx $
I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct.
$$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$
For this one, the function being integrated is even, so I can just take... |
H: Plus construction of a presheave factors every sheaf-valued morphism.
I'm having some trouble understanding the correctness of some proof in Sheaves in Geometry and Logic (Mac Lane, Moerdijk). It concerns the lemma III.5.3 :
If $F$ is a sheaf and $P$ a presheaf, then any map $\phi \colon P \to F$ of
presheaves f... |
H: Lengendre symbol calculation
I'm trying to calculate the lengendre symbol of (3/383) without using the Quadratic Reciprocity Law, and with not much success.
I've thought about checking if 2^191 is congroent to 1 modulo 383 but it seems too complicated.
I'd be grateful if some could point me to the solution.
Thanks ... |
H: A two-dimensional set of measure zero
I have a 2D domain $[0,1]\times[0,1]$. This domain contains some set of measure zero $A$, the last understood as the Lebesgue measure in $\mathbb{R}^{2}$.
Is the following true: for almost all $t\in[0,1]$ the set $A_{t}:=\{x\in[0,1]|(t,x)\in A\}$ has measure zero, in standard ... |
H: Solve the roots of a cubic polynomial?
I have had trouble with this question - mainly due to the fact that I do not fully understand what a 'geometric progression' is:
"Solve the equation $x^3 - 14x^2 + 56x - 64 = 0$" if the roots are in geometric progression.
Any help would be appreciated.
AI: It means the roots a... |
H: Finding $A,B,C,D \subseteq \{1,2,...n\}$ such that $A \cup B \cup C \cup D = \{1,2,...,n\}$
I have this combinatorial question:
Find the number of $(A,B,C,D)$ of sets $A,B,C,D \subseteq \{1,2,...,n\}$ such that $A \cup B \cup C \cup D = \{1,2,...,n\}$
I started by saying:
We choose the $k$ elements that are in $... |
H: Whether an infinite series can be tested by integral test
I am asked whether the following infinite series can be proved to be convergent by integral test.
$$\sum_{n=1}^\infty n e^{6 n}$$
so I integrate it
$$\int_1^{\infty}\ n e^{6n}\, dn$$
and find it diverges so I concluded that the above series also diverges by ... |
H: Is there an explicit formula for the inverse of $\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}$?
I apologize if this is trivial but I am stuck.
Given the bijective function $f:(0,2\pi) \to (-2,2)$ with
$$
f(x)=\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}
$$
where $\cot$ is the cotangent, how can I find an inverse $g... |
H: Is this language decidable?
Is this language decidable?
$$\{x\mid \text{$x$ is the code of a Turing machine that always halts on $y$
in less than $y^3$ steps}\}$$
I think it is, because it halts in a finite number of states. Am I right? Is there a more classic way of saying so?
AI: No, it is not computable. Fi... |
H: (Non) equivalence of regular cardinal definitions
The usual definition of a regular cardinal is "$\kappa$ is regular if $cf(\kappa) = \kappa$", which, assuming the axiom of choice, is equivalent to this definition: "$\kappa$ is regular iff it cannot be expressed as a union of a less than $\kappa$ sets, all of which... |
H: $g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.
I came across the following problem:
Let $g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.Then Which of the following maps will yield a fixed point for $g$?
The options are as foll... |
H: solving $1+\frac{1}{x} \gt 0$
In solving a larger problem, I ran into the following inequality which I must solve:
$$ 1+\frac{1}{x} \gt 0.$$
Looking at it for a while, I found that $x\gt 0$ and $x\lt -1$ are solutions. Please how do I formally show that these are indeed the solutions.
AI: There are two cases. If ... |
H: Guides/tutorials to learn abstract algebra?
I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not generally solvable".
I love set theory and stuff, but I'd like to learn... |
H: How to show that $A=B-C$
How to show that for a real symmetric matrix $A,~A$ can be written as $A=B-C$ where $B,C$ are positive definite real symmetric matrices?
Please help me ! I'm clueless.
AI: Let $C=cI$ where $I$ is the identity matrix and $c\gt0$ is chosen so that for each eigenvalue $\lambda$ of $A$, $c+\lam... |
H: Simple Diffy-Q problem
So as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem:
$$xy' = 2y$$
where I know the solution is $x^2$ but for the life of me I can't figure out how to integrate this really simple problem properly. I ke... |
H: Probability of getting split pill from bottle?
I have a bottle of 100 pills. The daily dose is 1/2 pill, so if the first pill I extract is a whole pill, I split it and put 1/2 back.
Just out of my own general curiosity, I'd like to model the probability of extracting a whole pill vs. a half pill over time, but I'm... |
H: Counting exercise
Three players a,b,c take turns in a game according to the following rules:
At the start A and B play (so C does not play). The winner of the first trial plays against C and so on until one of the players wins two trials in a row.
Possible outcomes are
aa,acc,acbb,acbaa,
bb,bcc,bcaa,bcabb et... |
H: Dimension problems
Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. Prove that there exists a subspace
$S \subseteq \mathbb{R}^n$ that verifies $\ker(f^2) = \ker(f) \bigoplus S$ and $\dim S = \dim f(S) \leq \dim(\ker(f))$.
My attempt at solution: So far I extended from the $\ker(f)$ to the $\ker(... |
H: Linear programming problem neither max nor min
Heres the actual question:
television provider broadcasts two movie channels, A and B. Channel A broadcasts 1 romantic
movie, 3 action movies and 3 comedies per month at a cost of 50 Euro. Channel B broadcasts
3 romantic movies, 4 action movies and 1 comedy per mo... |
H: On proving $\ker(TT^*+T^*T)=\ker(T^*)\cap \ker(T)$
Let $T:V\to V$ be a linear transformation of finite dimensional inner product space.
I want to show that $$\ker(TT^*+T^*T)=\ker(T^*)\cap \ker(T).$$
I showed already that $$\ker(T^*)\cap \ker(T)\subseteq \ker(TT^*+T^*T).$$
How do I showing the other inclusion?
(... |
H: Do we know if there are more primes with even leading digits or odd leading digits?
I was just wondering, out of curiosity, do we know if there are more primes with even leading digits or odd leading digits? For example, primes with even leading digits would be $23$ or $29$ and primes with odd leading digits would ... |
H: Question about bipartie graphs.
I have a fairly basic inquiry but i would sleep better at night if i saw a proof of it.
Q: i know that if i take a connected subgraph with at least 2 vertices of any simple bipartite graph G that it has to be bipartite.
how would one go about proving that this is the case for any si... |
H: Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.
I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and $W_1, W_2$ are vector subspaces of V, is connected. I'm fol... |
H: Given $G$, when can we find a division ring $R$ with $R^*=G$?
This is motivated by a characterization of finite cyclic groups, in which one proves
Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic.
The proof is not actually difficult, but a unnecessarily complicated idea... |
H: length of sum of two submodule
Let $M$ be a $R$-module with finite length and $K$ and $N$ be a submodule of $M$. Prove that
$l(K+N)+l(K\cap N)=l(K)+l(N)$.
My proof: First, by assuming that $K\cap N=\{0\}$, we can conclude that $l(K+N)=l(K)+l(N)$.
The detail: Let $n=l(K)$ and $m=l(N)$.
Let $\{0\}\subseteq K_0\subse... |
H: Geometric sequence, finding the first term using only the sum, the number of terms and value of one term.
In Geometric series: S = 56, a(2) = 16 and n = 3
S - sum, a(2) - second term, n - number of terms
Is it possible to get a(2) and a(3) from here? (If yes, hints would be awesome)
Thank You!
AI: So we have $S =... |
H: The set of all functions mapping set $A$ to set $B$
Is $F$ as defined here the set of all functions from set $A$ to set $B$?
$F=\{f\in 2^{A\times B}:\forall x(x\in A\rightarrow\exists y (y\in B\wedge (x,y)\in f))\wedge \forall x,y_1,y_2 ((x,y_1)\in f\wedge (x,y_2)\in f\rightarrow y_1=y_2)\}$
If $A$ was non-empty, t... |
H: What's the intuition behind this equality involving combinatorics?
What is the intuition behind
$$
\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}
$$
? I can't grasp why picking a group of $k$ out of $n$ bijects to first picking a group of $k-1$ out of $n-1$ and then a group of $k$ out of $n-1$.
AI: We have ... |
H: Properties of basic set theory
The question is about a set:
$$B=\{a_1,a_2,a_3,...,a_n\} \subset \mathbb R$$
And would like to know how to calculate $B^n$ where $n \in\Bbb N$?
AI: Assuming $B = \{a_1,a_2,a_3,...,a_n\} \subset \mathbb R$, with $|B| = n$,
$B^n$ is the set of all ordered n-tuples of elements of $B$:
T... |
H: Second Derivative of basic fraction using quotient rule
I know this is a very basic question but I need some help.
I have to find the second derivative of:
$$\frac{1}{3x^2 + 4}$$
I start by using the Quotient Rule and get the first derivative to be:
$$\frac{-6x}{(3x^2 + 4)^2}$$
This I believe to be correct.
Follow... |
H: how to apply hint to question involving the pigeonhole principle
The following question is from cut-the-knot.org's page on the pigeonhole principle
Question
Prove that however one selects 55 integers $1 \le x_1 < x_2 < x_3 < ... < x_{55} \le 100$, there will be some two that differ by 9, some two that differ by 10... |
H: Hilbert space with all subspaces closed
Does there exist an infinite-dimensional Hilbert space with all subspaces closed?
AI: If $H$ is an infinite dimensional Hilbert space and $\{x_n:n\geq 1\}$ is a countably infinite set of linearly independent elements, then the abstract span of these elements, which is an infi... |
H: If the union of $A$ and $B$ is linearly independent then the intersection of the spans $= \{0\}$
$\newcommand{\sp}{\operatorname{sp}}$ Let $V$ be a vector space over $F$ field, and let $A,B$ be two different, disjoint, non-empty sets of vectors from $V$.
Prove or disprove the following:
If $A \cup B$ are linearly ... |
H: How to directly show that Figure 8 injective immersion is not a monomorphism
I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an embedding. Now embeddings are the monomorph... |
H: Integral involving gaussian function
I would like to calculate the following integral:
$$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\quad (x^2+y^2)^k\exp\left(-\dfrac{(x-x_0)^2+(y-y_0)^2}{a^2}\right)\,\mathrm dx\,\mathrm dy$$
Any clue on how to proceed?
Thanks
AI: You can use the fact that
$$(x^2+y^2)^k = \su... |
H: Planar Geometry question
Suppose I have three points $p$, $f_1$, and $f_2$. I want to place a third point $f_3$ such that if you extend the line segments $pf_1$ and $pf_2$ into full lines, $f_3$ is going to be on the opposite side of $pf_1$ as $f_2$ is, and $f_3$ will also be on the opposite side of $pf_2$ from whe... |
H: What about this $\lim_{x \to \infty}\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}$?
When I saw this limit, I didn't even try to solve it by an algebraic method. I thought about the assyntotic concept.
In the example,
$$\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}\sim \frac{3x}{\sqrt[5]{x^9}}(x \to \infty)$$
So,
$$\lim_{x \to \infty}\frac... |
H: How to integrate $\int \sqrt{x^2+a^2}dx$
$a$ is a parameter. I have no idea where to start
AI: I will give you a proof of how they can get the formula above. As a heads up, it is quite difficult and long, so most people use the formula usually written in the back of the text, but I was able to prove it so here goes... |
H: Group extension reference request
I'm looking for a reference for the following "well known" result
Let $C$ be an abelian group and $G$ a finite group, and let $$0 \rightarrow C \rightarrow W \rightarrow G \rightarrow 0$$ be a group extension of $G$ by $C$ corresponding to the fundamental class in $H^2(C,G)$. Now... |
H: Constructing Distribution By Coin Flipping
I am interested in any example of construction distribution by coin flipping.
Actually I want to show the process of construction a random variable $X$ with distribution $(p_1,...,p_n)$ by coin flipping and to prove expectation of the number of coin flipping is at least e... |
H: Prove $\gcd\left((a^{2m}−1)/(a+1),a+1\right)=\gcd(a+1,2m)$
Show or prove that
$$
\gcd\left(\frac{a^{2m}−1}{a+1},a+1\right)=\gcd(a+1,2m),
$$
and that
$$
\gcd\left(\frac{a^{2m+1}+1}{a+1},a+1\right)=\gcd(a+1,2m+1).
$$
AI: Hint
$$\frac{a^{2m}-1}{a+1}=\frac{a^2-1}{a+1}(a^{2m-2}+a^{2m-4}+...a^2+1)$$
$$=(a+1)(a^{2m-2}+a^{... |
H: Combinations problem help
Four couples have reserved seats in a row for a concert. In how many different ways can they be seated if the two members of each couple wish to sit together?
At first I thought that this was 8!/4!*4! but then I was told by a teacher that this was wrong. I really don't understand the logic... |
H: There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$?
There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has lim... |
H: Determining whether a quadratic congruence is solvable using Legendre symbol
I'm trying to detect whether the quadratic congruence $2x^2 + 5x - 9$ is congruent to $0$ modulo $101$. I've think I'll be able to detect whether there is or there is no solution using Legendre symbol, but I can't figure out how.
I'll be ... |
H: Normal intersecting a sphere
Let $\textbf{x}$ and $\textbf{y}$ be two points on the sphere. Show that the normal to the plane determined by the great circle through $\textbf{x}$ and $\textbf{y}$ intersects the sphere at the points $\pm \textbf{z}$, where
$$\bf z = \frac{x \times y}{||x \times y||}$$
Now obviously,... |
H: Understanding the average slope of a curve
This question mainly asks: Is my understanding of the average slope correct? This question is somewhat related to my previous question. However, its different from the previous question to a certain extent.
If $P(x_0,y_0)$ and $Q(x_0+\Delta x, y_0+\Delta y)$ are two point... |
H: Is this function from $[-1,1] \rightarrow \mathbb{R} \cup \infty$ continuous?
just a short question. So I was wondering about functions from compact sets into
$\mathbb{R} \cup \{+\infty\}$.
Let's say we have a function $f : [-1,1] \rightarrow \mathbb{R} \cup \{+\infty\}$,
defined by
$f(x) = -1/x$ if $x \in [-1,0)$... |
H: Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically
$\eqalign{
& y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr
& \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr
& {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 \over {(1 + 2x)}} - {1 \over 2} \times {{ - 2} \over {(1 - 2... |
H: Solving dependent systems
When I'm solving a system of equations and realize that I have a dependent system, I need to express the answer in terms of y = {some value} where x is any real number, OR x = {some value} where y is any real number. How do I choose whether to
For instance, I have this problem in my homew... |
H: Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1
The task is as follows:
Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial
z}{\partial y}|_x \frac{\partial y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$
Here is my work so far:
(1) ... |
H: Finding a kernel of a linear transformation of linear transformations.
Question:
Let V,W be vector spaces over field F.
We mark L(V,W) as the vector space of linear transformations from V to W.
Let $v_0 \ne 0$. We define a transformation: $\Psi: L(V,W) \to W$ that sends a linear transformation $T \in L(V,W)$ to $T... |
H: Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$
How would you compute it?
I already solved this problem this way:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( \int_\infty^\infty e^{-x^2} \right)^3 = \pi^{3/2}$... |
H: Deduce that there exists a prime $p$ where $p$ divides $x^2 +2$ and $p≡3$ (mod 4)
I am revising for a number theory exam and have a question that I am struggling with, any help would be greatly appreciated.
First I am asked to show that for an odd number $x$, $x^2+2 ≡3$(mod 4).
I can do this part of the question,... |
H: Poisson distribution question, tips needed!
A car dealership opens every day with a fresh stock of $A$ cars. Let $N$ be the r.v. corresponding to the number of purchases per day. Suppose $N$ is distributed according to the Poisson distribution with parameter $\lambda=2$. What should $A$ be for the dealership to run... |
H: What is the reason for the name *left* coset?
Let $G$ be a group and let $H \leq G$ be a subgroup. It seems that it is now standard to call the cosets
$$gH=\{gh \ | h \in H \}$$ the left cosets of $H$ in $G$. I have to admit to being slightly annoyed with this convention: these are the orbits for the right action ... |
H: Why is this derivative incorrect?
We have to find the derivative of $$f(x) = \dfrac{\tan(2x)}{\sin(x)}$$
I would like to know why my approach is incorrect:
$$f'(x) = \dfrac{\sin(x) \cdot \dfrac{2}{\cos^2(2x)} - \tan(2x) \cdot \cos(x)}{\sin^2(x)}$$
$$ = \dfrac{ 2 \sin(x) - \tan(2x) \cdot \cos(x)}{\cos^2(2x) \cdot ... |
H: Simplify summation with factorial and binomial coefficients
I would like to know how to simplify the following summation:
$$\sum_{p=0}^n\quad n!\frac{(2p)!}{(p!)^2}\frac{(2(n-p))!}{((n-p)!)^2}$$
Which rules should I use to simplify it?
Thanks!
AI: You may start with the generating function of $\displaystyle \frac{(... |
H: How to show that if $p(A) = 0 \implies p(\lambda_0)=0$?
Let $V$ be a finite dimensional vector space and $V \ne \{ 0 \}, A\in L(V), \lambda_0 \in \sigma(A)$. If $p(\lambda)$ is an arbitrary polynomial for which the following applies: $p(A) = 0 $, prove that $p(\lambda_0)=0$
$L(V)$ = The set of all linear mappi... |
H: Example for a sequence of operators converging pointwise, but not with respect to the operator norm
I am trying to understand the following example.
Define $$T_n: l^2 \rightarrow l^2$$
$$T_n(x)=(0, ..., 0, x_{n+1}, ...).$$
It's rather clear that $T_n(x)$ converges for $0$ for every $x \in l^2$. However, the script ... |
H: Conditional expectation is square-integrable
I am given the following definition:
Let $(G_i:i\in I )$ be a countable family of disjoint events, whose union is the probability space $\Omega$. Let the $\sigma$-algebra generated by these events be $\mathcal{G}$. Let $X$ be an integrable random variable, that is $E|X|... |
H: An equivalent expression of Cauchy Criterion?
For a sequence $\{a_n\}$, if
$$
\forall \epsilon>0 \ \exists N>0, \forall k \in \mathbf{N}, \ |a_{N+k}-a_N|<\epsilon \
$$
Then $\{a_n\}$ converges and hence is a Cauchy sequence.
Now how about changing the inequality above to $|a_{N+k}-a_N|< a_{N+k}\cdot \epsilon$, or ... |
H: Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$?
Is $\def\Aut{\operatorname{Aut}}\Aut(\mathbb{I})$ isomorphic to $\Aut(\mathbb{I}^2)$ ? ($\mathbb{I},\mathbb{I}^2$ have their usual meaning as objects in $\mathsf{Top}$).
I show some of one of my attempts. I was trying to show tha... |
H: Are Taylor series and power series the same "thing"?
I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general concept? How does either/all of these ideas relate to generating fun... |
H: Number of Solutions for Congruency Equations
I'm leaning congruency equations, so for example:
$$
ax \equiv b \pmod m
$$
I have that the number of solutions will be equal to $d$, where
$$
d = \gcd(a, m).
$$
And the solutions ae:
$$
x, x+m/d, x+2m/d, x+3m/d, \ldots , x+(d-1)m/d
$$
Now, having understood ... |
H: For natural $n$, prove $\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}$
Prove that, for a natural number $n$,
$$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}$$
This follows ... |
H: Do uniformly continuous functions map complete sets to complete sets?
Let $f: (M, d) \rightarrow (N, \rho)$ be uniformly continuous.
Prove or disprove that if M is complete, then $f(M)$ is complete.
If I am asking a previously posted question, please accept my apologies and tell me to bugger off. I saw a similar p... |
H: How can I show that $x^4+6$ is reducible over $\mathbb{R}$?
How can I show that the polynomial $x^4 + 6$ is reducible over $\mathbb{R}$ without explicitly finding factors?
I was trying to find a non-prime ideal that would generate it but I'm kind of lost as to how to proceed. Is there some sort of criterion that wi... |
H: Proof of: If $x_0\in \mathbb R^n$ is a local minimum of $f$, then $\nabla f(x_0) = 0$.
Let $f \colon \mathbb R^n\to\mathbb R$ be a differentiable function.
If $x_0\in \mathbb R^n$ is a local minimum of $f$, then $\nabla f(x_0) = 0$.
Where can I find a proof for this theorem? This is a theorem for max/min in calcu... |
H: Can countability coexist with infinity?
This question concerns the countability of the real numbers.
First I will show how I count the numbers between 0 and 1 on the real line.
It is done by reversing digits behind the coma, so
that e.g. 0,761 maps to 167. Obviously this is 1 to 1 mapping,
but there are infinite nu... |
H: Need help with Fourier transform problem
I'm trying to calculate the Fourier transform of the unit step function,
$$\mathcal{F}[u(t)] \ = \int_{-\infty}^{\infty}u(t)e^{-i\omega t}dt \ = \int_{0}^{\infty}e^{-i\omega t} dt. \tag{1}$$
This simplifies to,
$$U(\omega) = (i\omega)^{-1},\ (\omega \not = 0). \tag{2}$$
Howe... |
H: $\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$
I'm having trouble understanding how to apply the $\frac{d}{dx}$when taking the anti-derivative.
$$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$$
In class it was mentioned we'll end up taking the derivative of $e^{x^{2}}$ which is $2xe^{x^{2}}$
My gue... |
H: $\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$
$$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$$
Please help me in this integral, I've tried some substitutions, but nothing work.
Thanks in advance!
AI: First complete the square:
$$
x^2 + x + 2 = \left( x^2 + x + \tfrac{1}{4} \right) + \tfrac{7}{4} = ... |
H: Find expression for $dy/dx $ + state where it is valid
hopefully you guys can shed some insight into this question I'm working on.
Given
$xy+y^{2}-e^{x^{2}} = 6$
find an expression for $dy/dx$ and state where it is valid.
So, what I did was differentiate it, which resulted in:
$x+3y-2xe^{x^{2}} = 0 $
Although I am... |
H: the set of countable sets of Real numbers
I would like to ask some hints towards the proof that The set of countable sets in $\mathbb{R}$ is equinumerous to the set $\mathbb{R}$
AI: Hint: $\Bbb{R^N\sim(N^N)^N\sim N^{N\times N}\sim N^N}$. Show that there exists a surjection from that set onto the set of countable su... |
H: What's the difference between arccos(x) and sec(x)
My question might sound dumb, but I don't really see why the graphics of $\arccos(x)$ and $\sec(x)$ are different.
As far as I know $\arccos(x)$ is the inverse cosine function $(\cos(x)^{-1})$ and $\sec(x)$ = $\frac{1}{\cos(x)}$ (source Wolfram|Alpha).
So why aren'... |
H: Does the definition of countable ordinals require the power set axiom?
I am trying to understand the consequences of the different axioms of ZFC. In particular, I was trying to understand what you get on ZFC-power set (ZFC minus the power set axiom). If you have any references that I could read please let me know. ... |
H: Is it possible that the union of a Bernstein set and a singleton isn't a Bernstein set?
Since the construction of a partition of two Bernstein sets is almost identical to that of a partition of three in an uncountable Polish space. It's possible that the union of a Bernstein set and a singleton is a Bernstein set... |
H: Every curve is a geodesic??
I've been reading up on how isometries send geodesics to geodesics. I recently saw a proof of another theorem that used the fact:
The set of fixed points of an isometry is a geodesic.
But isnt the Identity always an isometry, which would then imply every curve, in say the Poincare half ... |
H: Sketch the function $y = {1 \over {{x^2}}}\ln x$
I don't know where to begin with this, the ${1 \over {{x^2}}}$ part of the function throws me off, how to do I go about this? How does one generally approach a question like this?
AI: It is indeed an interesting function. You can try finding a lot of points: select a... |
H: Tricky logarithmic problem?
It is given that $\log_9 p = \log_{12} q = \log_{16} (p+q) $. Find the value of $q/p$. I can see that the bases have common factors, but I don't exactly know how to exploit that. I tried many approaches to this, but I couldn't get it. The farthest I got was probably $q/p=4/\sqrt{p}$ (if ... |
H: Are the continuous functions on $G$ dense in $L^{1}(G)$?
If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
AI: Yes. A more general result is the following:
Proposition. Let $X$ be a locally compact Hausdorff space, and let $\mu$ be... |
H: Evaluating limits involving absolute values: $\lim_{x\to\pm6} \frac{2x+12}{|x+6|}$
I have two questions regarding limits involving absolute values. How do I evaluate the following:
$\displaystyle\lim_{x\to -6} \frac{2x+12}{|x+6|}$
$ \displaystyle \lim_{x\to 6} \frac{2x+12}{|x+6|}$.
To handle the first probl... |
H: Convention for locally compact groups?
$\bf{\text{Suppose I find the phrase:}}$
Let $G$ is a locally compact group, and $\mathcal{U}$ a basis of neighborhoods of $1$.
$\bf{\text{Question:}}$
Is it a convention to automatically take each $U\in\mathcal{U}$ to be compact?
Clearly this can be done if the need arises ... |
H: Specific element in a set
Given a set $A=\{a,b,c,d\}$ how can I take, for an example, the first element?
Like this:
$A(0)=a;
A(1)=b;
A(0).A(1)=a.b$.
I think there's no way to do that with sets, because it isn't an ordered set.
AI: For a set $A$, there is no way to refer to its first element, second elements, or... |
H: Finding a linear recurrence regarding strings
The question is
Let $T(n)$ be the number of length-$n$ strings of letters $a$, $b$ and $c$, that do
not contain three consecutive $a$'s. Give a recurrence relation for $T(n)$ and
justify it. (You do not have to solve it.)
How would I go about solving this problem. I fou... |
H: Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$
From numerical test, I know $x=1$ is an attractive fixed point of the function
$$
f(x)=\frac12 \left(x+\frac{1}{x}\right),
$$
on $(0,\infty)$.
Is there a way to prove it?
Since
$$
f'(x)=\frac12\left(1-\frac{1}{x^2}\right),
$$
then
$$
... |
H: How to improve mathematical creativity?
To introduce myself: I'm an undergraduate mathematics student in Germany. Currently I'm studying in the second semester and until now I'm doing well, but I still got the feeling that my ability to develop proofs (or just solving complex issues in general) needs to be improved... |
H: Find $M$, where $M^7=I$ and $M\neq I$, $M$ has only 0's and 1's.
Find a $3 \times 3 $ matrix $M$ with entries 0 and 1 only such that $M^7=I$ and $M\neq I$.
This was a short question in a recent exam. I tried with permutation matrices but couldn't find $M^{odd}=I$ except for 3.
AI: There is no such matrix over $\mat... |
H: Divide by a number without dividing.
Can anyone come up with a way to divide any given x by any given y without actually dividing?
For example to add any given x to any given y without adding you would just do:
$x-(-y)$
And to subtract any given x from any given y (that is, y-x) you could do:
$y+xe^{iπ}$
*edit: w... |
H: What model do you get from PA without addition and multiplication?
I have the feeling that this question is trivial, but I cannot figure the answer by myself nor from the stuff I have read. So the question is if addition (and multiplication) can be shown as a theorem of PA (without the arithmetical axioms but with ... |
H: Oscillation and Hölder continuity
I am studying a proof of a theorem. And I have the following situation in the proof:
Consider $\Omega$ is a bounded open set of $\mathbb R^n$ and $u: \Omega \to \mathbb R$ is a function satisfying:
$$\operatorname{osc}_{B(x_0,R)} u \leq (1-\delta) \operatorname{osc}_{B(x_0,4R)},$$ ... |
H: A simple proof about $e^x$?
Do you guys think this is correct? I am trying to prove that there is no single-term polynomial function (oxymoron, I know) $f(x)$ which is always (or at least as x approaches infinity) greater than $e^x$ (I will try to expand this to any polynomial function later). Let $f(x)$ be such a ... |
H: Finding partial derivatives for equations expressed in terms of $z$ where $z=f(x,y)$ to find tangent plane
I am having troubles finding partial derivatives.
If $f(x,y)=2x^2+y^2$ then,
$$f_x=4x$$
$$f_y=2y$$
That's simple enough. But when I see a $z$ in the equation, I get stumped. I know $z=f(x,y)$. I don't really s... |
H: What is the (parametric) intersection of a plane and a sphere?
Can someone please show me how to prove that the intersection of the plane
$$x+y+z=0$$ and the sphere
$$x^2+y^2+z^2=1$$
can be expressed as
$$x(t)=\frac{\cos t-\sqrt3 \cdot\sin t}{\sqrt6}$$
$$y(t)=\frac{\cos t+\sqrt3 \cdot\sin t}{\sqrt6}$$
$$z(... |
H: How to show C_e is closed and not dense in C.
Let $C_{e}([-1,1],\mathbb{R})$ denote the set of even functions in $C([-1,1],\mathbb{R})$
(a) Show $C_e$ is closed and not dense in $C$.
(b) show the even polynomials are dense in $C_e$, but not in $C$.
I can't start on it... I can't catch any clue..
AI: Assume there ... |
H: Proof regarding unitary self-adjoin linear operators
I'm suck on how to do the following Linear Algebra proof:
Let $T$ be a self-adjoint operator on a finite-dimensional inner product space $V$.
Prove that for all $x \in V$, $||T(x)\pm ix||^2=||T(x)||^2+||x||^2.$
Deduce that $T-iI$ is invertible and that $[(T-iI)^... |
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