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H: Homogeneous riemannian manifolds are complete. Trouble understanding proof.
I came across this proof while looking for hints on my homework, and I think it's only gotten me more confused. This is from Global Lorentzian Geometry.
Lemma 5.4 If $(H,h)$ is a homogeneous Riemannian manifold, then $(H,h)$ is complete.
P... |
H: MMSE (Minimum min square estimate) problem
I have a problem as follows. As of now, I cannot provide the definition of X and Y but can anyone provide a rough overview of what needs to be done ?
An experiment
consist of rolling a single die and the experimental outcome
determines a value for X and Y. (the random var... |
H: Probability: Normal Distribution
Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $0.95$. Approximate the
probability (by a normal distribution) that at most $10$ of the next
$150$ items produced are unacceptable.
Hits to a high-volume Web site are assumed... |
H: Hermite polynomials recurrence relation
Hermite polynomials $H_n (x)$ can be obtained using the recurrence relation
$$H_{n+1} (x)=2xH_n (x)-2nH_{n-1} (x).$$
To prove this, I started by calculating the first derivative of the Hermite's Rodrigues formula $H_n (x)=(-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2 } $. The proc... |
H: Another question of finding eigenvalues with parameters
Let
$$A=\left(\begin{matrix}a&b&c\\a-s&b+s&c\\a-t&b&c+t\end{matrix}\right)$$
Find A's eigenvalues.
So, I did some row operations without changing the value of A's determinant, so that I got
$$A'=\left(\begin{matrix}a&b&c\\-s&0&0\\-t&0&t\end{matrix}\right)$$
B... |
H: Why is this an ellipse?
On a textbook, I've arrived at the following function:
$\displaystyle \phi(z)=\log{\frac{|z-\sqrt{(z²-1})|}{2}}$
and it says that the formula has a simple interpretation: the level curves of $\phi(z)$ are the ellipses with foci $-1, 1$. I know the problem is reduced to proving $|z-\sqrt{(z²-... |
H: Probability of rolling a cuboid dice
It's easy to count the probability of events on a regular dice because we know the probabilities ($P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$) of all the basic outcomes ($P(i)=\frac{1}{6}$).
But... Is there any (simple) way how to determine the probabilities of basic outcome... |
H: Divide $x=123456$ into three factors $x=uvw$ such that $uv^iw$ is divisible by 3
I have the problem of dividing the string 123456 into three factors uvw that such $uv^iw$ as a number is divisible by three, where $\left|uv\right|\le4$ and $\left| v\right|>0$, i.e. the factors u and v should together only be 4 digits... |
H: Countability of real numbers in $(0, 1)$
I've been reading a basic book on set theory for a while now. And recently ran into an uncountability of $(0, 1)$. I guess like almost everyone who tries to understand what he or she reads I tried to come up with a rule that matches every real number in $(0, 1)$ with a corre... |
H: Prove on residue theorem
I have try to use the equation
$$
Res(f;z_0)=\lim_{z\to z_0}\frac1{(m-1)!}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^mf(z)]
$$
But very soon I stuck, is that a good way to solve it?
AI: A few hints:
1) Since you have a double pole at $z=z_0$, your $f$ has the form
$$f(z) = \frac{p(z)}{(z-z_0)^2 r(z)... |
H: Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
$a^2 + b^2 + c^2 \ge a + b + c$.
I'm supposed to prove this use AM-GM, but can't figure it out. Any hints?
AI: since
$$a^2+1\ge 2a$$
so
$... |
H: Simple module and homomorphisms
Are the following statements equivalent?
i) $M$ is an $R$-simple module.
ii) Every $R$-homomorphism (nonzero) from $M$ to an $R$-module $N$ is a monomorphism.
iii) Every $R$-homomorphism (nonzero) from an $R$-module $N$ to $M$ is an epimorphism.
AI: The answer is yes.
Hint: You can ... |
H: How to prove the compactness of the set of Hermitian positive semidefinite matrices
I am dealing with convex optimization problems. There are some useful theories for optimization problems where real-valued vector parameter, e.g., $x \in \mathbb{R}^n$, is considered. I manage to apply the theories to my semidefinit... |
H: how do we prove $p|q\cdot r \rightarrow p=q$ or $p=r$ (all primes)?
I know this is one of the most fundamental basis of arithmetic but I can't find the result by myself.
how do we prove $p|q\cdot r\rightarrow p=q$ or $p=r$? ($p, q, r$ being prime numbers)
AI: If $p\mid q$, we are done.
Suppose that $p\not\mid q$. S... |
H: Probability question with trees and fruit using probability generating functions
Each year a tree of a particular type flowers once and the probability that it has n flowers is $(1-p)p^n$, $n=0,1,2...,$ where $0<p<1$. Each flower has probability $1/2$ of producing a ripe fruit, independently of all other flowers. ... |
H: How to solve for $g(x)$ in $f(x)=\int_a^b g(x) dx$
$$f(x)=\int_a^b g(x) dx$$
Do you just take the derivative and evaluate $f(x)$ from $a$ to $b$? Sorry if it seems obvious, but I have never done this before.
AI: $$\text{If}\;\;F(x) = \underbrace{\int g(x) \,dx}_{\text{indefinite integral}}\;\text{ then }\;\underbra... |
H: Definite Integral questions
Evaluate the definite integral of the function.
$$ \int_{-\pi/4}^{\pi/2} \; |\sin x| \; dx $$
My solution was :
$$ \cos \left(\frac{\pi}{2}\right) - \cos\left(\frac{-\pi}{4}\right)$$
$$ \frac{-\sqrt2}{2} $$
But the answer in the book is $ 2 - \frac{\sqrt2}{2} $. So what's wrong with my ... |
H: Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$
Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$
Here is my attempt at solving this using a generalized Euclid's Algorithm. Does it look alright?
Step 1
$2 + 3i = M(1-i) + N$
$$\frac{2 + 3i}{1-i} = \frac{2 + 3i}{1-i}\frac{1 + i}{1 + i}... |
H: Orthogonal Projections in Hilbert space
I am stuck with the following exercise about projections in Rudin 12.26.
Let $H$ be a Hilbert space $P,Q\in B(H)$ self-adjoint projections (A projection has the property that $P^2=P$), then the following are equivalent.
(a) $P\geq Q$
(b) $R(P)\supset R(Q)$
(c) $PQ=Q$
(d) $QP=... |
H: A basic theorem on field isomorphisms
I'm reading A Book Of Abstract Algebra by Charles C. Pinter. On page 314 is the following theorem:
Let $h:F_1\to F_2$ be an isomorphism, and let $p(x)$ be irredicible in $F_1[x]$. Suppose $a$ is a root of $p(x)$, and $b$ is a root of $h(p(x))$. Then $h$ can be extended to an i... |
H: How to show that $\operatorname{Spec}(A)=\bigcup_{i=1}^n D(g_i)$ implies that $(g_1, \ldots, g_n)=A$?
$\newcommand{\Spec}{\operatorname{Spec}}$
Let $A$ be an algebra and $\Spec(A)$ the scheme consisting of all prime ideals of $A$.
How to show that $\Spec(A)=\bigcup_{i=1}^n D(g_i)$ implies that $(g_1, \ldots, g_n)=A... |
H: Notation of a function that maps a random element
Let there be a functions $f$ and $g$ such that,
$$f:A \times B \mapsto \Re$$
$$g: B \mapsto A$$
where $\forall b \in B$, $g(b)$ is some $a$ such that, $\forall a' \in A, f(a,b) \geq f(a',b)$. (This is because there could be many elements in $A$ that suffices $\fo... |
H: Generator for the ideal $I + J$ where $I = (2 + 3i)$ and $J = (1 - i)$
On a related question I calculated the GCD of $I = (2 + 3i)$ and $J = (1 - i)$ to be $1$.
Now I know that $\mathbb{Z}[i]$ is a principal ideal domain. And I also know that the greatest common divisor of two elements $a, b$ in a P.I.D. generates ... |
H: How to show that h is constant by using **Liouville's theorem**
Let $h$ be an entire function.
$\exists$ some $R \gt 0$ and $z_0\in \Bbb C$ s.t. open ball $B_R(z_0)$ isnt in the range of $h$. $B_R(z_0)\cap h(\Bbb C) \not = \emptyset{}{}$
How to show that h is constant by using Liouville's theorem
I dont have any... |
H: Help on how to read and understand a question on vector space and subspace
V is a vector space, S is its subset. Determine if the following subsets are its subspace:
$$\begin{align} &S_1=\{(x_1,x_2,...,x_n)|x_1+x_2...+x_n=0\};V=\Bbb R^n \\
&S_2=\{(x_1,x_2,...,x_n)|x_1x_2=0\};V=\Bbb R^n \end{align}$$
I'm not sure I... |
H: When should we take direct limit and when should we take inverse limit?
We know that we can take direct limit for a direct system and inverse limit for an inverse system. For example, when can defined the stalk of a presheaf $\mathcal{F}$ on a topological space $X$ at a point $P\in X$ by $$ \lim_{\rightarrow_{p \in... |
H: Find the rest of the division when $23^{84292}$ is divided by $7$, is the procedure and the result correct?
I want to know if the procedure I have followed in order to get the result for the next problem is correct. The problem is this:
Find the rest of the division when $23^{84292}$ is divided by $7$.
As $7$ is ... |
H: Finding the solutions of $\cos (x) +x = a$
What is the approach to finding the solutions of the following function? I was not able to analytically resolve the solutions - but rather resorted to a graphical approach.
$$\cos (x) + x = 1$$
or in general where $a$ could be any real number,
$$\cos(x) + x = a.$$
EDIT:
O... |
H: Matrix question regarding symmetric
I need help with the following problem
Express the matrix $$B=\begin{pmatrix} 2 &-2 &-4 \\
-1 &3 &4\\ 1 &-2 &-3 \end{pmatrix}$$ as the and sum of a symmetric and a skew symmetric matrices.
AI: In general,
$$\mathbf{M} = \dfrac{\mathbf{M} + \mathbf{M}^t}{2} + \dfrac{\mathbf{M} -... |
H: Rows of Change of Basis Matrix
Maybe a stupid question, but since I don't find a confirmation to my doubt on the Internet, I'll also ask here.
To change basis from $A$ to $B$ we use a matrix whose columns are the basis vectors of $A$ expressed in the new basis $B$. But we can also say that its rows are the basis ve... |
H: Is an integer a sum of two rational squares iff it is a sum of two integer squares?
Let $a\in \mathbb Z$. Is it true that $a$ is a sum of two squares of rational numbers if and only if it is a sum of two squares of integers?
I came to face this problem while dealing with quaternion algebras.
AI: Yes, if an integer ... |
H: What is the difference between Symmetric vs Skew Symmetric?
I want to know the difference between Symmetric Symmetric vs Skew Symmetric?
AI: A symmetric matrix satisfies $A^T = A$
A skew-symmetric matrix satisfies $A^T = -A$.
Additionally, it is a fact that every matrix can be written as the sum of a symmetric matr... |
H: Calculate the probability that $x \lt 5$ given Poisson distribution states the mean is $6$
Two grocers agree that the daily demand for a particular item has Poisson distribution. However, grocer $A$ claims that the mean demand is $3$ items per day, while grocer $B$ claims that the mean demand is $6$ items per day. ... |
H: $L^p $ is not uniformly convex for $p=1, \infty$
As the title says: how can we prove that $L^p(\mathbb R^n)$ is not uniformly convex for $p=1$ and $p=\infty$.
Does anyone knows a counter-example for the cases $ p=1$ and $ p =
\infty$ for the space $L^p( \mathbb R^n)$ ?
I am studying N.L. Carothers book "A short ... |
H: Extending exponentiation to reals
I've been reading through a course on exponential functions, starting from integer-valued exponents to rational ones as in: $x^r$ from $r\in \Bbb{N}$ to $\Bbb{Z}$, and combining them to rigorously construct for $r\in\ \Bbb{Q}$. Still, this book adresses high-schoolers, and therefor... |
H: Functions on finite metric spaces
Is a function f from a finite metric space M to itself always continuous? I have tried proving it, but I have gotten stuck. Any help would be appreciated.
AI: Hint: If $M$ is a finite metric space, then which subsets of $M$ are open? |
H: Trouble with l'Hôpital's rule for $\lim_{x\rightarrow 0} \frac{4x+4\sin x}{10x+10\cos x}$
This is an assignment and I am stuck:
Find the limit, whether finite or infinite, or indicate that the limit does not exist. Use l'Hôpital's Rule if appropriate.
$$\lim_{x\rightarrow 0} \frac{4x+4\sin x}{10x+10\cos x}.$$
Whe... |
H: Does this packing problem even have an optimal solution?
Under this answer, user Bruno Joyal asks:
This might be a naive question, but... how do we know there is a best possible solution?
I (but that's just me) assume that he might be thinking of a logical possibility that there is a sequence of ever better solut... |
H: Find the Second Point Of a 3D Line Segment Given Some Information
Given this information about a 3D line segment:
XYZ of the first point
Length of the line
Pitch and yaw angles
How do I determine the XYZ of the second point?
I know how to do this with a 2D segment described by one angle:
point2x = point1x + sin... |
H: For fractional ideal, why $AB=R$ implies $B=A^{-1}$?
Let $A,B$ be two fractional ideals of $R$ (an integral domain). Could anyone tell me why $AB=R$ implies $B=A^{-1}$?
AI: Also for the other readers: $A^{-1}$ is the largest fractional ideal with the property that $A A^{-1} \subseteq R$. Explicitly, we have $A^{-1}... |
H: Calculating limit of function
To find limit of $\lim_{x\to 0}\frac {\cos(\sin x) - \cos x}{x^4} $.
I differentiated it using L Hospital's rule. I got
$$\frac{-\sin(\sin x)\cos x + \sin x}{4x^3}\text{.}$$ I divided and multiplied by $\sin x$.
Since $\lim_{x\to 0}\frac{\sin x}{x} = 1$, thus I got
$\frac{1-\cos x}{4x^... |
H: Is matrix multiplication really a group operator?
A group has an operation that can be performed over ANY two elements in a set. Given that an $n \times m$ matrix can only be multiplied by an $m \times o$ matrix, doesn't that mean that matrix multiplication can't be a group operator except for sets of commonly size... |
H: Why is $ \hbox{Ext}_R^* (M,M) = H^*(\hbox{Hom}_R^*(P^*,P^*))$?
Let me first fix some notation and conventions.
Let $ R$ be a ring and $ M$ a left $R$-module. Given chain complexes $P^*$ and $Q^*$ in $R$-mod, define $ \hbox{Hom}^*_R(P^*,Q^*)$ to be the graded space such that $ \hbox{Hom}_R ^i (P^*, Q^*) = \bigoplus... |
H: Proof related with mathematical induction
I tried to prove this claim using mathematical induction.
$ a^2 + 15a + 5 ≤ 21 a^2 $ $\;\; ∀a∈\mathbb Z^+$
The way is as the following:
Basis: for a = 1 is true since 21 = 21
Inductive step: If $P(t)$ then $P(t+1)$ should be proved , so it goes
$ t^2 + 15t + 5≤21t^2 $
add $... |
H: Catalan number basic question - combinatorics
I have a question regarding catalan numbers:
1) Find the number of sequences $a_1 \leq ... \leq a_n$ where $a_i \in \mathbb N$ and $0 \leq a_i \leq i-1$ for all $i \in \{1,2,...,n\}$
For example:
When $n=1$ the only available sequence is: $0$, so only 1 sequence, which ... |
H: Help with a proof involving normality.
Prove that $A_n$ is normal in $S_n$.
We know, in order to be normal, the centers of each group need to commute with everybody. We also know, the groups are normal if every left coset is equal to the right coset.
How can we do this using these groups?
AI: Hint: $|S_n:A_n|=2$ i... |
H: multiplicative character evaluated at -1 (from Ireland and Rosen's number theory book)
I'm self studying from Ireland and Rosen's "A Classical Introduction to Number Theory" Second edition. Near the beginning (page 153 in my book) of Chapter 11 the authors discuss the number of solutions of a hypersurface in a fin... |
H: Are these equivalent?
$\forall x \in D, (P(x) \Rightarrow Q(x))$ is equivalent to $(\forall x \in D \cap P,Q(x))$.
However, is this also equivalent to $(\forall x\in D)( P(x)\land Q(x))$?
If not, what's the difference? Thank you.
AI: No.
$\forall x \in D, (P(x)\rightarrow Q(x)) \not\equiv \forall x\in D(P(x) \land ... |
H: Best approximation for $\displaystyle \sum_{k=2}^n\ln\ln k$?
I need the best approximation for
$\displaystyle{\sum_{k = 2}^{n}\ln\left(\ln\left(k\right)\right)}$. Any suggestion or hint is welcomed.
I derived $n\ln\left(\ln\left(n!\right) \over n\right)$ so is there any better one ?
AI: Euler-Maclaurin series:
$$ ... |
H: Quadrature Rule Error
Question: Suppose that $S(f,h)$ is a quadrature rule for the integral $I= \int^b_a f(x) dx $ and that the error series is $c_4h^4+c_6h^6+...$ Combine $S(f,h)$ with $S(f,\frac{h}{3})$ to find a more accurate approximation of I.
So I got the easy part, that
$I-S(f,h)=c_4h^4+c_6h^6+...$
$I-S(f,\... |
H: Residue Theorem for Denominator with $e^z$
$$ f(z)=\frac{z^3}{e^z-1} $$
Is this a simple pole at $z=0$ or some other types of pole?
If it is a simple pole, what is its residue?
Is it using this formula or other else?
$$ \lim_{z\to 0}=zf(z) $$
AI: $$e^z=\sum_{k=0}^\infty\frac{x^k}{k!}\implies\frac{z^3}{e^z-1}=\fr... |
H: Fixed roots and quotient rings
I'm reading a paper by Smart and Vercauteren on homomorphic encryption (http://eprint.iacr.org/2011/133). I don't understand a specific statement around quotient rings. The authors state that for a polynomial $F(X)$ in indeterminate $X$, elements of quotient rings such as $\mathbb F_2... |
H: Is the following set connected?
Let $R \subset \mathbb{R}^2$ denote the unit square $R = [0,1] \times [0,1]$. If $F \subset R$ is finite, is $R \backslash F$ connected?
AI: You can even prove that $R\setminus F$ is connected if $F$ is countable. In fact, it’s path-connected.
HINT: Given two points $p$ and $q$ in $... |
H: Team winning probability
If a team has a 2/3rd chance of winning any game, what is the probability that it wins at most 4 out of 5 games? The answer I got was
The chances that they win no games = 1/243
The chances that they win one game = 10/243
The chances that they win two games = 40/243
The chances that they wi... |
H: The automorphism $\varphi(k)$ in the semidirect product is $h \mapsto kh k^{-1}$ What do they mean?
Let $H \rtimes K$ be the semidirect product of groups $H$ and $K$, not necessarily subgroups of anything.
In Theorem 10, part (5) of Dummit & Foote, Algebra, there's
Identifying $H$ and $K$ with their isomorphic cop... |
H: If the diagonals of an isosceles trapezoid are perpendicular to each other, prove that the area is $S=H^2$.
Where H is the altitude of the whole trapezoid (the distance between the bases).
Thanks.
AI: Here's another way to look at it.
Let $ABCD$ be the given isosceles trapezoid with $BC=a$ the top line segment and ... |
H: Cosets & Groups.
I am a newbie to group theory. I am confused by something that I would like clarified.
Say $G$ is the integers, a group under addition, and $H$ is the even integers, a subgroup of $G$.
It is usually stated that left cosets of $G$ are a partition of $G$.
Now for all $g \in G$, the left coset of $G$... |
H: Exam question, complicated chain rule
I have this old exam question that I don't understand.
Calculate $\frac{d}{dt}ln(x^2+y^2)$ where $x=e^t+e^{-t}$ and $y=e^t-e^{-t}$. Write the answer in x and y's.
The solution given does the following:
$$\frac{dx}{dt}=e^t-e^{-t}=y$$
$$\frac{dy}{dt}=e^t+e^{-t}=x$$
then use the ... |
H: Is this element-of_{ij} - looking symbol the Levi-Civita symbol?
I'm reading this formula:
from a page
Is the symbol that looks like an element-of symbol with two indices i and j the Levi-Civita symbol?
Mathematics is my weak-side so I'm not sure.
Actually I guessed on Levi-Civita not based on my expertise but by ... |
H: Proof inequality with log
How can I prove that there exist $n_0$, $c$ such that for all $n>n_0$:
$$n^{\log_2{n}}\le c2^{n}$$
(So I mean the log of n with base 2). Can anybody help me?
AI: First, let's show that
$$
\lim_{n\to\infty}\frac{\log(n)^2/\log(2)}{\log(c)+n\log(2)}=0
$$
Using L'Hospital twice gives
$$
\beg... |
H: Show that a $5 \times 4$ Matrix of row rank $3$ has nontrivial solutions for $Ax=0$
Statement: Let $A$ be a $5 \times 4$-Matrix with row rank 3. Show that $Ax=0$ has nontrivial solutions.
This is a homework problem, as a hint I am given one formula (which we didn't discuss in class yet) but intuition wise I don't... |
H: Strong induction proof with polygon
How can we show that if a simple polygon with at least four sides is triangu-lated, then at least two of the triangles in the triangulation have two sides that border the exterior of the polygon using strong induction
AI: Proposition. If $n\ge 3$ and $AB$ is an edge of a triangul... |
H: Help with 'If, then'- and 'Only if'-sentences in Predicate Logic
So, I have to ask now, because I've spent so much time on these two translations.
My key is
Domain: living things
Px: x is a Pokerplayer
Cx: x is a Chessplayer
Yx: x is Professional
Rx: x is Rich
and the sentences are:
1) "Pokerplayers and Chess... |
H: Sequence with a contraction mapping of the sum
Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - f(y) \right\| \leq c \left\| x-y \right\|$, where $\left\| \cdot \ri... |
H: $\varepsilon$-$\delta$ proof of $\sqrt{x+1}$
need to prove that $ \lim_{x\rightarrow 0 } \sqrt{1+x} = 1 $
proof of that is:
need to find a delta such that $ 0 < |x-1| < \delta \Rightarrow 1-\epsilon < \sqrt{x+1} < \epsilon + 1 $ if we choose $ \delta = (\epsilon + 1)^2 -2 $ and consider $ |x-1| < \delta = (\epsilon... |
H: Conditional probability distribution question
You are given that $N$ is a Poisson random variable with mean $4$. Define a new random variable
$M=(N - 4 \mid\ N \ge 4)$. Find the mean of $M$. Lost myself trying to solve this, help would be appreciated.
AI: Hint: $E[N-4] = P(N=0) E[N-4|N=0] + \ldots + P(N=3) E[N-4|N=... |
H: What happens if we overlap two points in Pascal's theorem?
In projective geometry, we know that:
For any inscribed hexagon in a conic, the intersection of the segments formed by the union of opposite vertices, are collinear.
But now, what happens if two vertices overlap, ie, if one side of the hexagon becomes tan... |
H: Integrating $e^x$
I'm wondering if there are any rules to integrate $e^x$? For example $e^{-0.05x}$. Can I use the reversed chain rule, u-sub or something to easily integrate most examples of $e^x$?
AI: Hint: Given $\lambda \in \Bbb R$, differentiate $x\mapsto \dfrac{e^{\lambda x}}{\lambda} $.
More generally, given... |
H: Function iteration and intervals of attraction for fixed points
I am currently studying iteration sequences and I am a bit hung up on one specific bit which involves determining intervals of attraction of fixed points.
I've been given a graphical method to determine the intervals of attraction, and a more formal me... |
H: Must the intersection of connected sets be connected?
Must the intersection of two connected sets be connected?
I believe the answer is no, but I am not entirely sure. I think a counter example would be a set that intersects another set in more than one area, yet those intersections are disjoint. (Think of a cylind... |
H: Drawing the subgroup lattice of D10
I've been tasked with drawing a subgroup lattice of the dihedral group of order 10. I know from Lagrange's theorem that non-trivial subgroups must have order 2 or 5. Finding the subgroups of order 2 is straightforward, since there can be only one element besides the identity, but... |
H: How do we prove this fact about cyclic groups?
Prove that an Abelian group of order 33 is cyclic.
Can we take an element a of order 3 and an element b of order 11 and say, |ab|=33?
AI: Yes, that's definitely one way to do it. But you'll have to be explicit about how you know that you can choose $a$ and $b$ to have... |
H: Limit of $f(x)=x^2\left(1+2+\cdots+\left\lfloor \frac1{|x|} \right\rfloor\right) $
Suppose $f$ a function defined on $\left[\dfrac{-1}{2};\dfrac{1}{2}\right]$ as $f(x)=x^2\left(1+2+\cdots+\left\lfloor \dfrac{1}{|x|} \right\rfloor\right) $.
How can I prove that $f$ has a finite limite on $0$?
Calculate for $k \in \... |
H: Prove $13|19^n-6^n$ by congruences
I am trying to prove $13|19^n-6^n$. With induction its not so bad but by congruences its quite difficult to know how to get started.
Any hints?
AI: HINT: Just use the congruence $19\equiv 6\pmod{13}$. |
H: A Double Limit Question
Maybe it's easy, but: Is it true that
$$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$
If it is, could you help me prove it?
Thanks
AI: For the function $\frac{x^2}{\sqrt{x^2+y^2}}$, the limit exists. To see why, find a bound that trivially tends toward 0:
$$\frac{x^2}{\sqrt{x^2... |
H: Joukowski Conformal Mapping
I'm having trouble understanding how to map the streamlines from one plane to another using the Joukowski transform.
In the $\zeta$ plane, I'm considering flow around a cylinder, with the complex potential given by $$w(\zeta)=U(\zeta e^{-i \alpha} + \frac {a^2}{\zeta}e^{i\alpha})$$
wher... |
H: Elementary differentiation question on derivation of p.d.f. of function of random variable
Let $G(y) = \Pr(Y \le y) = 1 - F(\frac{1}{y})$.
Then apply the chain rule (assuming $y \ne 0$ and $F(x)$ is differentiable at $x = 1/y$) and we have $$g(y) = \frac {d\ G(y)}{dy} = \frac{-d\ F(x)}{dx} \Large \mid_{\normalsize ... |
H: Does 1 distinct eigenvalue guarantee 1 eigenvector?
I am trying to figure out when 2x2 matrices are not diagonalizable. Right now, my conditions are:
the matrix has only 1 distinct eigenvalue
the matrix yields only 1 linearly independent eigenvector
But when I know that there are 2 eigenvalues, can I safely assum... |
H: Prove $(a + b) \bmod n = (a \bmod n + b \bmod n) \bmod n$
I find I am in trouble to prove:
$$(a + b) \bmod n = (a \bmod n + b \bmod n) \bmod n ?$$
Can anyone help?
AI: Let $a = hn + (a \bmod n)$, $b = kn + (b \bmod n)$, $h,k\in \mathbb Z$. Then the left hand side
$$\begin{align*}
(a+b)\bmod n =& [a+b-(h+k)n]\bmod n... |
H: The Tangent Disc Topology
Let $X$ be the tangent disc topology, $X=P\cup L$ where $P=\{(x,y):x,y\in \mathbb{R}, y>0\}$ and $L$ is the real line.Then,
$X$ is completely regular but not normal,
$X$ is separable,
$X$ is countably metacompact.
$X$ is not compact
I see all of them. But I wonder $X$ can be metrizable?... |
H: Application of fixed point theorem in $R^n$
Let $A=(a_{ij}) \in \mathbb R^{n \times n}$ a matrix such that $|a_{ij}|<\frac{1}{n}$ for every $i,j$. Prove that $I-A$ is invertible.
My attempt at a solution:
$I-A$ is invertible $\iff$ $(I-A)v=0$ has the trivial solution $v=0$. But $(I-A)v=0 \iff Iv-Av=0 \iff Av=v$. L... |
H: Permutation of natural numbers
Find the number of permutation of {1,2,3,4,5,6} such that the patterns 13 and 246 do not appear. Show the steps .
AI: It is easier to first find the number of permutations in which the pattern $13$ or $246$ does appear.
For the pattern $13$, tie $1$ and $3$ together to make a supersym... |
H: Why use the biconditional in the Axiom of Extensionality
I'm studying the Axiom of Extensionality in the following form:
$$
\forall a \forall b[\forall x(x\in a\leftrightarrow x\in b)\rightarrow a=b]
$$
(where quantification of a,b is restricted to sets and quantification of x can range over domain objects as well ... |
H: A question about multivariable concave functions
Consider a concave function $f: \mathbb{R}^N \rightarrow \mathbb{R}$. Is it possible for it to be convex in a single argument when I fix the remaining $N-1$ variables or does concavity of $f$ in $N$ variables imply that it should be concave in each individual argumen... |
H: Can any metric space be completed?
Completion defined in Real Analysis, Carothers, 1ed has been captured below.
Can any metric space be completed?
AI: Yes. Rudin outlines a proof in Principles of Mathematical Analysis, Chapter 3, Problem 24.
Given a metric space $(X,d)$, say two Cauchy sequences $(p_n)$ and $(q_n)... |
H: Prove that D is normal
Let G be a group, $T = G \times G$ and let $D = \{(g,g)\in G \times G | g\in G\}$. Prove that D is normal in T if and only if G is abelian.
I assume that D is normal in T, then for any $x,y \in T$ we have $(xgx^{-1}, ygy^{-1})$ for $g\in D.$ How can I show that $G$ is abelian?
AI: If $D$ is ... |
H: The vectors a and b are non-collinear. For what value(s) of m is it true that (m^2 + 2m - 3)a + (m^2 + m -6)b = 0?
Since they are both non-collinear they can never be equal to zero. So what do i do now?
Any help is appreciated. Thank you very much!
AI: Hint: Two vectors are non-collinear if and only if they are lin... |
H: Resolvent recurrence relation
Let the resolvent matrix of $\mathbf{X}$, a symmetric matrix with real entries, be defined as
\begin{align}
R_{\mathbf{X}}(\lambda):=\bigl(\mathbf{X}-\lambda\mathbf{I}\bigr)^{-1}, \qquad \lambda \in \mathbb{C}\backslash\mathbb{R}.
\end{align}
with $\mathbf{I}$ the identity of matri... |
H: Calculating $\int_{0}^{\infty} x^{a-1} \cos(x) \ \mathrm dx = \Gamma(a) \cos (\pi a/2)$
My goal is to calculate the integral
$\int_{0}^{\infty} x^{a-1} \cos(x) dx = \Gamma(a) \cos (\pi a/2)$,
where $0<a<1$,
and my textbook provides the hint: integrate $z^{a-1} e^{iz}$ around the boundary of a quarter disk.
However... |
H: Calderón-Zygmund operators with positive kernel
Let $T$ be a Calderón-Zygmund operator. That is, $T$ maps $L^2(\mathbb{R}^d)$ to itself and satisfies the representation formula
$$
Tf(x) = \int_{\mathbb{R}^d}K(x,y)f(y)\, dy
$$
for all $f \in L^2$ with compact support and $x \notin \operatorname{supp} f$, for some ke... |
H: Example of a continuous bijective function on and to the closure of the complex numbers, with an inverse that is not continuous?
Note that by the closure of the complex number, I mean the union of the complex numbers and infinity.
I have been stumbling over this questions for a wile now, and I understand many examp... |
H: Local solutions over $\mathbb{Q}_p$ but no solutions over $\mathbb{Q}$
I was looking at a set of notes that states the equation $x^4-17=2y^2$ is solvable locally over $\mathbb{Q_p}$ for every $p$ , but is not solvable over $\mathbb{Q}$. Now, this is not a homework problem and is just for my own reading. Is there a ... |
H: Help debunk a proof that zero equals one (no division)?
Unlike the more common variant of proof that 0=1, this does not use division.
So, the reasoning goes like this:
\begin{align}
0 &= 0 + 0 + 0 + \ldots && \text{not too controversial} \\
&= (1-1) + (1-1) + (1-1) + \ldots && \text{by algebra}\\
&= 1 + (-1 + 1) ... |
H: Why does $( \operatorname e^x)' = \operatorname e^x?$
It's known the the derivative of exponential function $a^x$ is $xa^{x-1}$.
If I play $e$ as $a$, we'll get $(a^x = \operatorname e^x)' = x \operatorname e^{x-1}$.
Why does $(\operatorname e^x)' = \operatorname e^x$?
AI: As T. Bongers has pointed out, you don't ... |
H: Let $G$ be a group of order 6. Suppose that $a,b\in G$ with $a$ of order 3 and $b$ of order 2. Show that either $G$ is cyclic or $ab\not=ba$.
Let $G$ be a group of order 6. Suppose that $a,b\in G$ with $a$ of order 3 and $b$ of order 2. Show that either $G$ is cyclic or $ab\not=ba$.
Attempt at Answer:
Suppose $ab=b... |
H: Iran Math Olympiad 2012 (perfect power)
Prove that if $t$ is a natural number then there exists a natural number $n > 1$ such that $(n, t) = 1$ and none of the numbers $n + t, n^2 + t, n^3 + t…$ are perfect powers.
There is a solution posted at AOPS, by considering $n = t(t+1)^2 + 1$. Two cases are discussed: $t + ... |
H: If dim(V) + dim(W) > dim(R^n), show that some nonzero vector is in V & W. [GStrang P183, 3.5.45]
Inside $\mathbb{R^n}$, suppose dimension$(\mathbf{V})$ + dimension$(\mathbf{W}) > n$. Show that some nonzero vector is in both $\mathbf{V}$ and $\mathbf{W}$.
Answer : Since $\dim\mathbf{V} + \dim\mathbf{W} = \dim\m... |
H: Real Analysis: Example related to irrational number and rational limit and Cauchy
A) a sequence $(x_n)$ of irrational numbers that converges to a rational number
Are the following answer correct?
$x_n = e^1$
or
$\sqrt{2}/n$
Give a little explanation if you could
B) a sequence $(x_n)$ that is not Cauchy, but for wh... |
H: Confused about the definition of a group as a groupoid with one object.
A groupoid is defined to be a category where every morphism is an isomorphism. So sometimes a group is said to just be a groupoid with one object.
When I try to make sense of this, I denote the single object as $G$. I view the morphisms as the... |
H: Show that the multiplicative group $\mathbb{Z}_{10}^{\times}$ is isomorphic to the additive group $\mathbb{Z}_4$.
Show that the multiplicative group $\mathbb{Z}_{10}^{\times}$ is isomorphic to the additive group $\mathbb{Z}_4$.
I'm completely lost with this one.
AI: First off, we have $\mathbb{Z}_4^+=\{0,1,2,3\}$, ... |
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