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H: Intertwiner of symmetric group representations (Basic)
I am preparing for an exam and there is an excercise which I have to solve but I got stuck. The excercise states:
Let $V=\mathbb{C}^3$ be the permutation representation of the symmetric group $S_3$. Show that any map of representations (intertwiner) $I: V \righ... |
H: How many real roots for $ax^2 + 12x + c = 0$?
If $a$ and $c$ are integers and $2 < a < 8$ and $-1 < c$, how many equations of the form $$ax^2+12x+c=0$$ have real roots?
AI: As @DonAntonio give hint a quadratic eqn have real roots iff
$$b^2-4ac\ge0$$
$$144-4ac\ge0\implies ac\le36$$
since $2<a<8$ and $c>-1$ so $a$ ... |
H: True or false: if $f'(c)<0$, then $f$ is concave down at $x=c$?
How can I determine the following statement is true or false?
If $f'(c)<0$, then $f$ is concave down at $x=c$ ?
AI: Recall that the sign of $f'(c)$ determines only if a function is increasing if $f'(c) > 0$, decreasing if $f'(c)\lt 0$, or neither, e.g.... |
H: Number of solutions in a finite field
let $F$ be a field of $p^b$ elements, $p$ prime and $b \in \mathbb{N}$.
Suppose I have $(a_1, a_2, \ldots, a_s) \in F^s$ and an equation
$$
0 = a_1 x_1 + \dots + a_s x_s.
$$
I was wondering if anybody could help me figure out what the number of solution $(x_1, x_2, \dots, x_s) ... |
H: Prove that in a tree, a path is a hamilton path iff it is an euler path
Prove that in a tree graph $T$, a path is a hamilton path iff it is an euler path.
so I said this:
==>: Let $<x_1,x_2,...,x_n>$ be a hamilton path and let us suppose by contradiction that it is not an euler path. Then there is an edge that s... |
H: If I know the Conjugacy classes of a group, do I know the group?
I know that a group has Conjugacy classes of size 1, 3, 6, 6, 8 and I know that this matches with the Conjugacy classes of the group $S_4$. But could there be a different group, with the same Congucy classes?
AI: I think that in general this question ... |
H: If $\phi_i$s are linearly dependent, $\det [\phi_i(v_j)] = 0$ - is the proof legit?
Given $v_1, \ldots, v_k \in V$ and $\phi_1, \ldots, \phi_k \in V^*$. If $\phi_1, \ldots, \phi_k \in V^*$ are linearly dependent, prove that $\det[\phi_i(v_j)] = 0.$ Here $k$ is the dimension of $V$, but I need to show this also wor... |
H: questions in channel capacity
Q)
Suppose we have a set of t coins, all but two of which have uniform weight $0$. and two counterfeit coins have different weights$>0$. If one can only use a spring scale, what is the best solution to the problem of finding the counterfeit coins? Find an algorithm by using binar... |
H: Encryption using modular addition and a key
Problem i'm facing says:
The value representing each row is encrypted using modular addition
with a modulus of 32 and a key of 27.
I sort of figured out what modular addition is for myself an hour ago but the key thing confuses me.
What does this
key of 27
mean? And ... |
H: Is it true that a complex function has a global antiderivative if and only if it integrates to zero over every closed curve?
I am somehow thinking that these properties must be equivalent, unfortunately I do not know a theorem that says it:
$f$ has a global antiderivative iff the line integral $ \int_{\gamma}f$ ove... |
H: Proof that operator is an isometry
A linear operator $L$ between complex spaces with inner product $U$ and $V$ is an isometry, only if $\left < Lu_i, Lu_j \right > = \left < u_i, u_j \right >$ for all $u_j, u_i$ from a basis of $U$ (not necessary orthonormal).
I would like to see a proof of this statement.
AI: EDI... |
H: Transformation of $\mathbb R^2$
Let $T:\mathbb R^2\to \mathbb R^2$ be s.t. for all $x,y\in\mathbb R^2$, $\mid\mid T(x)-T(y)\mid\mid = \mid\mid x-y\mid\mid.$ Then, 2 questions:
If $T(0)=0$, does it follow that $T$ must be linear?
Show that $T$ is a translation, then a rotation, and then a reflection through the $x... |
H: How do I know when to use nCr button and when to use nPr button on my calculator in which situation?
How do I know when to use nCr button and when to use nPr button on my calculator?
nCr= combinations I believe
NPR= permutations
Is there a general rule I can use to figure out which one to use and when according t... |
H: Why $f$ is injective? (infinitude of primes)
In this very short paper by Dustin J. Mixon, I would like understand why the author says
$f$ is injective by the fundamental theorem of arithmetic.
In my opinion, the Fundamental Theorem of Arithmetic (FTA) is necessary to define $f$, but it isn't necessary to prove ... |
H: Mystery shapes - making shapes from vague descriptions
I am suppose to mentally create something that has base $x^2 + y^2 = 1$ and the cross sections perpendicular to the x axis are triangles whose height and base are equal.
What is going on? I tried to graph it but it was too hard, is this like a turtle? Sphere on... |
H: More three-term arithmetic progression questions
This question was inspired by a recent question about whether
$\frac1{2}$, $\frac1{3}$ and $\frac1{15}$
can be (possibly non-consecutive) terms
in an arithmetic progression.
My question(s): Which of the following sets of
three values can be (possibly non-consecutive)... |
H: How should I interpolate between values in a logarithmic series?
What's the best way to interpolate between 2 values of a logrithmic series?
More specifically, I have a process where we encode values as $b = \text{floor}(\log(x, k))$. We discard the original values, and use b as a bucket for a bunch of statistics.
... |
H: Volume of liquid needed to fill sphere to height $h$
Find the volume of liquid needed to fill a sphere of radius $R$ to height $h$.
The picture shows $h$ up to maybe a quarter, I am not sure it seems pretty ambiguous.
No clue what to do here. I just know that the formula I am suppose to memorize is
$$V = \int \pi... |
H: Hilbert's Hotel, cardinality, and equality
One of the absurdities illustrated by Hilbert's Hotel, some say, is infinity + infinity = infinity. This is absurd in the sense that "After the infinitely many new guests check in, the number of guests will remain the same as before." I think what is meant by "the number o... |
H: Rearrange formula in term of r
I do not know how to arrange following equation in term of r:
$$I = P\left(1 + {r \over 100}\right)^n $$
I know that first step is dividing both parts of equation by P:
$$ {I \over P} = \left(1 + {r \over 100}\right)^n$$
But here I got stuck. I do not know how to extract r out of the ... |
H: How to solve $dy/dx = \frac{(3y^2+2x^2)}{ (xy)}$
I got this problem in today's exam and I couldn't quite figure this out. The equation is $xy \, dy = (3y^2+2x^2) \, dx$, $M_y = 6y$ and $N_x = y$, they aren't equal so this equation is nowhere near exact, it doesn't look like I can do separable either? What to do?
AI... |
H: Congruence classes: Find the inverse
I have the following problem:
If $ [3640]$ is invertible in $\mathbb {Z}_{7297}$ then determine its inverse.
Okay. The first thing I thought was:
$$3640x\equiv 1 \pmod{7297}$$
But isn't there any easier way?
Any hint, much appreciated.
AI: We use the Euclidean Algorithm. Note... |
H: Meaning of "strong" and "weak" (formulas?) in propositional logic
I was doing some review of propositional logic from Enderton's book. In one section(pg. 26 of the 2nd edition), he explains the idea that given wffs $\sigma_1, \sigma_2, \cdots, \sigma_k$ and $\tau$, one can use truth tables to check whether or not $... |
H: Targets of Fighter plane
If fighter plane travels along the path
$$r(t)=(t-t^3,12-t^2,3-t),$$
how can we show that the pilot cannot hit any target on the x-axis? Any pointers and hints would be appreciated. I do not want full answer but a start because I am stuck in terms of how to approach this problem.
AI: Yo... |
H: Can a probability density function be used directly as probability function?
This might be something basic but it confuses me greatly.
I am reading a literature, where they use the probability density function of a Gaussian distribution, that is
$$f(x)=\frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} ... |
H: Finding the point of intersection of L and the xy-plane.
$$P = (-1, 1, 1)$$
$$Q = (1,2,3)$$
$$R = (2,1,0)$$
My textbook provides a solution to this, but I'm unclear of how it came to it. The book says;
In the xy-plane we have $0 = z = 1 + 2λ$ so $λ = \frac{-1}{2}$ and so the intersection of L with the xy-plane is ... |
H: Do I understand metric tensor correctly?
So I've been studying vectors and tensors, and I'm trying to understand metric tensors.
As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors regardless of whether their basis has changed.
So if you had a set of vector... |
H: If $v_i$s are linearly dependent, $\det [\phi_i{v_j}] = 0$ - is the proof legit?
Given $v_1, \ldots, v_k \in V$ and $\phi_1, \ldots, \phi_k \in V^*$. If $v_1, \ldots, v_k \in V$ are linearly dependent, proof $\det[\phi_i(v_j)] = 0.$ Here $k$ is the dimension of $V$, but I need to show this also works for a subspac... |
H: When precisely can we replace quotient objects with subobjects in the definition of simple objects?
In a category with zero, a simple object is one that has only two quotients - itself and zero.
Firstly - a point of confusion. The definition above says that quotient object requires a congruence, what is the one we ... |
H: Trigonometry in Triangles Without Right-Angles
Could you please help by showing me how I can find the unknown sides for the triangles below?
AI: Hint: since you do not have a right triangle, and do not know whether the second triangle is a right triangle, assume it isn't:
For the first, use the law of sines: the un... |
H: Constructing a linear map
Construct a linear map α : $\mathbb{R^4} → \mathbb{R^4}$ whose kernel is spanned by $(1, 0, 0, 1)$ and $(0, 1, 1, 0)$
I'm seeking guidance, how I could construct the required map? The thing is that I am not yet introduced to matrices. The chapter is on vector spaces, and short subchapter b... |
H: Simple closed geodesic around two hyperbolic cusps.
Consider a connected hyperbolic $2$-manifold $M$ with cusps. Consider a simple closed geodesic in $M$, which dissects $M$ into two components. Assume that one of the components contains exactly two cusps. Can you prove me, that this component is conformal to the h... |
H: Is convergence in $I^I$ topology equivalent to point-wise convergence?
The $I^I$ topology is the uncountable Cartesian product (Tychonoff) of the closed unit interval $[0,1]$. We can imagine it as a space of all the functions from $[0,1]$ to itself.
I was told that a sequence $\alpha_n$ converges to $\alpha$ in thi... |
H: Which compact (orientable) surfaces are parallelizable?
Which compact (necessarily orientable) smooth $2$-manifolds are parallelizable?
I'm aware that the sphere $\mathbb{S}^2$ is not parallelizable, whereas the torus $\mathbb{T}^2 = \mathbb{S}^1 \times \mathbb{S}^1$ is. This leaves the case of connected sums of t... |
H: Divisibility by Quadratics $b^2+ba+1\mid a^2+ab+1\Rightarrow\ a=b$
The natural numbers $a$ and $b$ are such $a^2+ab+1$ is divisible by $b^2+ba+1$. Prove that $a = b$.
I tried to algebraically manipulate it as follows:
$(b^2 + ba + 1)k = a^2 + ab + 1$
$[b(a + b) + 1]k = a(a + b) + 1$
$kb(a + b) + k = a(a + b) + 1$... |
H: Questions about the local ring of a point on a variety.
Let $Y$ be a variety. Let $\mathcal{O}_{P, Y} = \mathcal{O}_{P}$ be the ring of germs of regular functions on $Y$ near $P$. That is, an element of $\mathcal{O}_P$ is pair $\langle U, f \rangle$ where $U$ is an open subset of $Y$ containing $P$ and $f$ is a reg... |
H: Nested Interval Theorem; free of set theory
I am reading this proof on Spivak for the Nested Interval Theorem. I pretty much did exactly what he did except the last step.
For each $m$ and $n$ we have $a_n \leq b_m$, because $a_n \leq a_{n+m} \leq b_{n+m} \leq b_m$. It follows from problem 12 that $\sup \{a_n : n \... |
H: What is the exact, rigorous, full statement of Divergence (Gauss') Theorem in $\mathbb{R}^3$ (without being too complicated)?
The wolfram page http://mathworld.wolfram.com/DivergenceTheorem.html states the formula
$$
\int_{V} \nabla \cdot \mathbf{F} dS = \int_{\partial V} \mathbf{F} \cdot d\mathbf{S}
$$
but it does... |
H: When will be a point $p$ not be a limit point of $A$
I am going through the proof of the Question:
Let $A$ be a subset of a (point set) topological space $(X,T)$. When will a point $p$ not be a limit point of $A$.
Proof: $p$ is not a limit point of $A$ if there exists an open set $G \in T$ such that
$p \in G$ and $... |
H: Numbers in a circle: how many sets of consecutive numbers have positive sum?
One hundred integers are written around a circle, and it is known that their sum is $1$. We will call a subset of several successive numbers a "chain". Find the number of chains whose members have a positive sum.
I had no idea how to app... |
H: How do you learn from MITOpenCourseWare 18.06 Linear Algebra Course?
I recently finished my 9th grade and I'm beginning to want to skip ahead and learn Linear Algebra throughout the summer, mainly because I want to know it to program my own games in OpenGL with C++. I already made couple of games before, but now I ... |
H: Can we call domain as inverse image of a function?
I was going through the definition of inverse image of a function http://www.northeastern.edu/suciu/U565/MATH4565-sp10-handout1.pdf, and I was wondering if inverse image of a function is the domain of the function itself. Please give me some examples on it.
AI: "... |
H: Proof Error? A line-segment of a circle is a metric.
In O'searcoid, Metric Spaces, he provides the following example of a metric space:
Suppose C is a circle and, for each $a,b ∈ C$, define $d(a,b)$ to be the distance along the line segment from $a$ to $b$. Then $d$ is a metric on $C$.
I have decided to confirm t... |
H: On changing from '<' to '$\le$' when taking limits (with norm $|\bullet|_p$)
I'm reading Gouvêa's book on $p-$adic, and there's one problem that I don't think I really get it. Here's a proposition, and the problem attached to it. It's on page 57, 58 of the book.
Proposition 3.2.12
The image of $\mathbb{Q}$ under t... |
H: Rigorous way to show image of a set under rotation
Take $R:\mathbb{C}\rightarrow\mathbb{C}$ where
$$R(z)=ze^{i\frac{\pi}{4}}.$$
Find $R(A)$ where
$$A=\{ re^{i\theta}:r\in [0,2], 0\leq\theta\leq\pi\}.$$
I've ran in to a sort of problem (it seems) with the way that this set is defined. I haven't done a lot of... |
H: Probability of getting 'k' heads with 'n' coins
This is an interview question.( http://www.geeksforgeeks.org/directi-interview-set-1/)
Given $n$ biased coins, with each coin giving heads with probability $P_i$, find the probability that on tossing the $n$ coins you will obtain exactly $k$ heads. You have to write t... |
H: When does $\sum_{i=3}^\infty n^{-1} (\log \log n)^{-r}$ converge
For $r>0$, when does $\sum_{i=3}^\infty n^{-1} (\log \log n)^{-r}$ converge? My guess is $r>1$ (treating $\log \log$ as $\log$). Please give me some hints!
AI: Never. You probably know that $\sum \frac{1}{n\log n}$ diverges. If you have not done it y... |
H: Showing path connected matrices of a group $G$ is a normal subgroup
Let $G$ be a subgroup of $GL_n(\Bbb{R})$. Define $$H = \biggl\{ A \in G \ \biggl| \ \exists \ \varphi:[0,1] \to G \ \text{continuous such that} \ \varphi(0)=A , \ \varphi(1)=I\biggr\}$$ Show that $H$ is a normal subgroup of $G$
In this post, the a... |
H: orthogonal basis for the complement
Consider $\mathbb R^3$ with the standard inner product. Let $W$ be the subspace of $\mathbb R^3$ spanned by $(1,0, -1)$.
Which of the following is a basis for the orthogonal complement of $W$?
$\{ ( 1, 0, 1), ( 0, 1, 0)\}$
$\{(1,2,1),(0,1,1)\}$
$\{(2,1,2),(4,2,4)\}$
$\{(2,-1,2),... |
H: $(X, T_1), (Y, T_2)$ be topological spaces such that every function from $X$ to $Y$ is $T_1-T_2$ continuous
Let $(X, T_1), (Y, T_2)$ be topological spaces such that every function from
$X$ to $Y$ is $T_1-T_2$ continuous. Prove that either $T_1$ is the discrete topology or $T_2$ is the indiscrete topology.
How ca... |
H: Why is square root of 2 less than cube root of 3
Square root of 2=1.41421.....
Cube root of 3=1.44224.....
4th root of 4=1.41421......
5th root of 5=1.37972......
6th root of 6=1.34800......
7th root of 7=1.32046......
.
.
nth root of n=..........
Similarly thereafter nth root of n will converge towards 1.
Q.2 why... |
H: Help to compute the following coefficient in Fourier series $\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$
$$\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$$
where $k\geq 0$, $k\in\mathbb{N} $ and $n\in\mathbb{R} $.
it is a $a_k$ coefficient in a Fourier series.
AI: \... |
H: Fourier cosine and sine transforms of 1
What is the Fourier sine and cosine transform of $f(x)=1$? I have seen some sources refer to the transform of $f=1$ involving the Dirac Delta function, but this goes against the integral definition for the Fourier sine transform, for example, since
$$\int_0^\infty f(x)\sin(x ... |
H: Qualifying problem for real analysis: limit involving definite integral
The following problem has appeared in 2013 January qualifying exam in Purdue University, which is publicly available here.
Problem 3. Let $\{a_k\}$ be sequence of positive numbers such that
$a_n\to\infty$ as $n\to\infty$. Prove that the fol... |
H: Convergence in distribution of the log-Gamma distribution
Suppose $X$ has density $f(x)=\exp(kx-e^x)/\Gamma(k)$, $x>0$, for some parameter $k>0$. Then the moment-generating function of $X$ has the form
$$
M_X(\theta)=\frac{\Gamma(\theta+k)}{\Gamma(k)}.
$$
I want to show that
$$
\lim_{k\to\infty}M_{X^*}(\theta)=\exp... |
H: accumulation point of a subset and accumulation point of an indexed family of subsets of space $X$.
Let $\xi=\{V_n:n\in\omega\}$ be a sequence of open subsets of space $X$. For every $n\in\omega$, choose $x_n\in V_n$.
If $p\in X$ is an accumulation point of $\{x_n:n\in\omega\}$, then is is true that $p$ is an accu... |
H: Determine a basis for the Lie-Algebra $\text{sp}(\text{2n},\mathbb{C})$
Consider the Lie Group $\text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}\mid\ J=g^TJg\}$ where $J=\begin{pmatrix}
0 & 1_n \\
-1_n & 0
\end{pmatrix}
$.
The corresponding Lie Algebra is $\text{sp}(\text{2n},\mathbb{C})=\{g\in\text{Mat}_{2n}\m... |
H: Markov property of a random process (a solution of piece-wise deterministic equations)
Consider a piece-wise deterministic (Markov!) process
\begin{eqnarray}
\dot{x}(t) & = & A_{\theta(t,x(t))}x(t)\\
x(0) & = & x_0 \in \mathbb{R}^n \notag
\end{eqnarray}
where $\theta(t,x(t))\in S ={1,2,\cdots,N}$ is continuous time... |
H: Convex analysis problem
I have the following problem.
Let $f:[a,b]\to \mathbb{R}$ be continuously convex. I have to prove that there exists $c\in (a,b)$ such that $$\frac{f(a)-f(b)}{b-a}\in \partial f(c)$$
Firstly, I'm being doubt with $\frac{f(a)-f(b)}{b-a}$ (don't ensure this one is correct, may be it is $\frac{... |
H: Radius of in-circle as a function of the center
I am trying to find the radius of an in-circle in a random triangle as a function of the center of the circle. Let (x,y) in\R^2 be the center of a circle, r the radius then i need an expression of the form r(x,y).
The cirle does not have to touch all three sides of t... |
H: How many triangles in picture
How many triangles in this picture:
I know that I can just count the triangles to solve this specific problem. I would be interested to know if there is a systematic approach to doing this that can be generalized to larger diagrams of the same type.
AI: They can be counted quite easil... |
H: Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra
Consider following Lie Group:
$$
\text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid J=g^TJg\}\quad\
where\quad J=\begin{pmatrix}
0 & 1_n \\
-1_n & 0
\end{pmatrix}
$$
And the corresponding Lie Algebra:
$... |
H: Find coordinates of intersection between two circles, where one circle is centered on the other
I'm writing a program where an object needs to move from point A to point B.
A and B are points on the same circle. Point B corresponds to the intersection between the circle and another circle centered on A.
The coordi... |
H: Prove a Trigonometric Series
Question:
$$\cot^{2}\frac{\pi }{2m+1}+\cot^{2}\frac{2\pi }{2m+1}+\cdots+\cot^{2}\frac{m\pi }{2m+1}=\frac{m(2m-1)}{3}$$
$m$ is a positive integer.
Attempt:
I started by showing that
$$\sin(2m+1)\theta =\binom{2m+1}{1}\cos^{2m}\theta \sin\theta -\binom{2m+1}{3}\cos^{2m-2}\theta \sin^{3}\t... |
H: Manipulating Algebraic Expression
$a + b + c = 7$ and $\dfrac{1}{a+b} + \dfrac{1}{b+c} + \dfrac{1}{c+a} = \dfrac{7}{10}$. Find the value of $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b}$.
I algebraically manipulated the second equation to get:
$\dfrac{(b+c)(c+a) + (a+b)(c+a) + (a+b)(b+c)}{(a+b)(b+c)(c+a)} = \... |
H: Improper integral - show convergence/divergence
I ran into this question:
show convergence/divergence of:
$$\int_{0}^{\infty}x^3e^{-x^2}.$$
I tried for a long time and I'm kind'a lost.
Thanks in advance,
yaron.
AI: We have
\begin{equation*}
\int_{0}^{\infty
}x^{3}e^{-x^{2}}dx=\int_{0}^{1}x^{3}e^{-x^{2}}dx+\int_{1}^... |
H: What is the nature of the homomorphism of the semidirect product of two groups $H$ and $K$?
Let $H$ and $K$ be group and $H\rtimes_{\phi} K$ is the semidirect product of those group by the homomorphism $\phi:K\rightarrow Aut(H)$
Now , i have a main question about this function $\phi$
Does $\phi$ map every element $... |
H: Adding simultaneous rates
If I have 3 rates like units/second produced by 3 separate devices and I want to get the total rate, is it ok to add the 3 rates up?
This seems basic and I'd say it's correct but I thought I'd ask.
Thanks
AI: Yes, you are correct. (In normal conditions.)
That is, if you're given a problem... |
H: Let $G$ be the division graph on $\{1,2,3,4,5,6,7,8,9,10,12,14,16\}$. Does it have a hamilton/euler path?
Given a graph G whose vertices are $V = \{1,2,3,4,5,6,7,8,9,10,12,14,16\}$, and there is an edge between two vertices $w$ and $j$ iff $w\neq j$ and $w$ divides $j$ or $j$ divides $w$.
(I) Does $G$ have a Hamil... |
H: A proposed proof by induction of $1+2+\ldots+n=\frac{n(n+1)}{2}$
Prove: $\displaystyle 1+2+\ldots+n=\frac{n(n+1)}{2}$.
Proof
When $n=1,1=\displaystyle \frac{1(1+1)}{2}$,equality holds.
Suppose when $n=k$, we have $1+2+\dots+k=\frac{k(k+1)}{2}$
When $n = k + 1$:
\begin{align}
1+2+\ldots+k+(k+1) &=\frac{k(k+1)}{2}+... |
H: Vacuous Domain Mixing For all and There Exists.
May be this is a stupid question but I was thinking we know that suppose $D = \varnothing$ then $\forall\, x \in D \,P(x)$ is true vacuously and $\exists y\in D\, P(y)$ is false. What is you mix the two like $\forall x \in D \,\,\exists \,y \in D \,Q(x, y)$. Then is $... |
H: Find the volume inside both $x^2+y^2+z^2=4$ and $x^2+y^2=1$.
What is the volume inside both $x^2+y^2+z^2=4$ and $x^2+y^2=1$?
The chapter I am working on is called Change of Variables in Multiple Integrals, for my Vector Calculus class.
I understand that we will be taking the double integral of these two shapes to f... |
H: prove that $f$ is continuous on $A$ if and only if $f^{-1}(V)$is open in $\Bbb R^n$ for every open subset $V$ of $\Bbb R^m$
Suppose that $A$ is open in $\Bbb R^n$ and $f$ is a function from $A$ to $\Bbb R^m$. Prove that $f$ is continuous on $A$ if and only if $f^{-1}(V)$is open in $\Bbb R^n$ for every open subset $... |
H: Existence of a continuous selector for a continuous optimization problem
Suppose that you have a continuous function
$$ S \colon \mathbb{R}\times [0, 1]\to [0, \infty).$$
Define an auxiliary function
$$S^\star(x)=\max_{t\in[0,1]}S(x, t).$$
Does there exist a continuous function $t(x)$ such that
$$S^\star(x)=S(x, ... |
H: Difficulty in understanding integrals of complex numbers
I understand what integration of real numbers is. I know how the definition of it is made.
I have trouble in understanding how it works for complex numbers.
I am referring to the notes here: http://people.math.gatech.edu/~cain/winter99/ch4.pdf.
I understa... |
H: Matrix to power $2012$
How to calculate $A^{2012}$?
$A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$
How can one calculate this? It must be tricky or something, cause there was only 1 point for solving this.
AI: Observe that $A^2$ is $A$.
So $A^{\large2012}$ is $A$ too. |
H: What are discrete and fast Fourier transform intuitively?
I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them.
AI: So first things first: the FFT simply refers to the algorithm... |
H: Range changes for functions of stochastic variables.
I have the stochastic variables $X=U~(-1,1)$, and $Y=2X^2+1$. I need to find the cdf of $Y$ ($F_Y(y)$). I have reasoned like this:
$$
F_y(y) = P(Y<y) = P(2X^2+1<y) = P(-\sqrt{\frac{y-1}2} < X < \sqrt{\frac{y-1}2})
$$
$$
= \int_{-\sqrt{\frac{y-1}2}}^{\sqrt{\frac{y... |
H: Evaluating the $\int \frac{x+2}{x^2-4x+8}$ - a doubt
I have to find the antiderivative of $f(x) = \dfrac{x+2}{x^2-4x+8}$
I rewrote it to the form $$ \dfrac{x-2}{x^2 -4x + 8} + \dfrac{1}{\frac{1}{4} (x-2)^2 +1}$$
The next step supposedly is $$F(x) = \dfrac{1}{2}\ln|x^2-4x+8| + 2 \arctan\left(\dfrac{1}{2}(x-2)\right... |
H: Question about linear dependence and independence by using Wronskian
Here is the theorem I use:
Two solutions $\phi_1$, $\phi_2$ of $L(y)=y''+a_1y'+a_2y=0$, where $a_1$ and $a_2$ are constants, are linearly independent on an interval $I$ if, and only if, the Wronskain $W(\phi_1,\phi_2)\ne0$ for all $x\in I.$
The... |
H: Let G(V,E) undirected Graph with n vertices, where every vertex has degree less than $\sqrt{n-1}$. Prove that the diameter of G is at least 3.
Let G(V,E) undirected Graph with n vertices, where every vertex has degree less than $\sqrt{n-1}$. Prove that the diameter of G is at least 3.
Well I've thought about provin... |
H: linear operator $A$ and $B$ commute
This Is exercise problem from my lecturer
Prove that if the linier operators $A$ and $B$ commute (i.e., if $AB=BA$), then every eigen space of the operator $A$ is invariant subspace of the operator B.
AI: Hint: Let $A,B\colon X \to X$. A subspace $U \subseteq X$ is $B$-invariant ... |
H: Evaluation/Estimation of a Gaussian integral
Is there a closed form expression for the following definite integral:
$$ F(u) = \frac{1}{2}\int_{-u}^u e^{-\frac{\alpha^2}{x^2}-\beta^2 x^2}\,dx
= e^{-2\alpha\beta} \int_0^u e^{-\left(\frac{\alpha}{x}-\beta x\right)^2}\,dx? $$
Graphing the function makes it clear th... |
H: How do prove that there is a vertex with degree less than 6 in a disconnected planar graph
As we all know, in every planar graph, connected or disconnected, there is at least a vertex $v$ with $deg(v) \leq 5$. (Even we can prove that there are at least two of such vertices.)
But, as I looked for a proof to this sim... |
H: Differential forms: need to understand the 1-form $dx^i$
In [1] on page 118 the authors introduce differential k-forms $\omega$ on $U \subset \mathbb{R}^n$ by \begin{equation} \omega : U \subset \mathbb{R}^n \to\Lambda^k\mathbb{R}^n \end{equation} where $\Lambda^k$ is the space of k-vectors and $(\mathbb{R}^n)^\ast... |
H: Application of the Chinese Remainder Theorem
Three brothers A, B and C live together and they all love eating pizza. A has the habit of eating a pizza every 5 days, B every 7 days and C every 11 days. A and C both eat pizzas together on 3 January 2012 and B has a pizza the next day. When will they all three eat pi... |
H: About the Spectrum of operators
I'm studying operator theory, and a doubt come at me, we know the diference between the pontual spectrum, the continuous spectrum and the residual spectrum. And we have that $\lambda \in \sigma(T)$ iff $T-\lambda I$ is not invertible. But the fact that $T-\lambda I$ is not invertible... |
H: How to teach a High school student that complex numbers cannot be totally ordered?
I once again need your precious knowledge! I am not sure which is the best pedagogic way to teach a High school student about why complex numbers cannot be totally ordered. When I was in High school we were simply told that we cannot... |
H: Standard models being non-standard?
If there is a ''set'' W in V which is a standard model
of ZF, and the ordinal κ is the set of ordinals which occur in W, then
Lκ is the L of W. If there is a set which is a standard
model of ZF, then the smallest such set is such a Lκ. This
set is called the minimal mode... |
H: If a group $G$ has the trivial center then $|Aut(G)|\geq |G|$
If a group $G$ has the trivial center then $|Aut(G)|\geq |G|$. Any suggestion?
AI: Hint: For $g\in G$ consider $x\mapsto g^{-1}\cdot x\cdot g$. |
H: Find the integral of $\overline{z}$
Question:
Find $\int\overline{z}$, when the contour is a parabola. Interval is from 0 to 1.
My Attempt:
$z = x + iy \Rightarrow \overline{z} = x - iy$
$f(z) = x - iy$
Since the contour is a parabola, $\gamma(t) = t + it^2$ and $y = x^2$.
$Re(\gamma) = t$, $Im(\gamma) = t^2$ ... |
H: Length of spanning and independent lists
There is this theorem in my notes that says that in a finite-dimensional vector space any linearly independent list of vectors is shorter than or equal in length to every spanning list. I understand the proof but it doesn't appear to use the assumption that the space is fini... |
H: Help with a step in rearranging this problem
i'm working through the proof of this theorem;
If $x$ is any real number other than $1$, then $$\sum_{j = 0}^{n -1} x^j = 1 + x + x^2 + \cdots + x^{n-1} = \frac{x^n-1}{x-1}$$
But i'm struggling with an intermediate step, so I hope you can help. Please don't take any larg... |
H: Finding Eigenvalues and Eigenvectors weird equations
I have matrix:
$A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$
And I have to find eigenvalues and coresponding eigenvectors.
I get $det(A-xI_3) = -x(x-1)^2$, so eigenvalues are $x_1 = 0$ and $x_2 = 1$
Now I want to find coresponding eigen... |
H: Notation in a Dirichlet Character
What is the meaning of the sign in the notation for these Dirichlet characters?
In the context of specific cases building up to a general proof of the Theorem Primes in Progressions there are several depictions of characters:
In the case of $\pmod 3$, there is the symbol $\chi_{- 3... |
H: Use maximum modulus theorem to control the number of zeros of analytic functions.
Let $f$ be a analytic function on $\bar B(0,R)$,with $\| f(z)\|\leq M$ in $\| z \| \leq R$,suppose $\|f(0)\|=a>0$,the the number of zeros of $f$ in $B(0,\frac{1}{3}R)$ is equal or less than $\log 2 \log(\frac{M}{a})$.
This result se... |
H: How to combine these 2 notations to be as simple as possible?
Consider the following question:
Find the antiderivative of $f(x) = 6x \, (x^2+1)^5$.
I have been using 2 notations and I would like to combine them.
Notice that $[x^2 + 1]' = 2x$, so $6x \, dx = 3 d(x^2+1)$
Take $(x^2 + 1) = u$, then $6x$ is $3\,u'$.... |
H: What's the ratio of triangles made by diagonals of a trapezoid/trapezium?
In the above image, what will be the ratio of areas of triangle $A$ and $B$?
From Googling, I've found that:
$\operatorname{Ar}(A) = \dfrac{a^2h}{2(a+b)}$
and
$\operatorname{Ar}(B) = \dfrac{b^2h}{2(a+b)}$
but how do I get these formulas f... |
H: Proof involving Continuous Map and Disjoint Sets
I'm a bit confused about the following proof my analysis book gives about the following Theorem: Consider metric spaces $(S,d), (S^*, d^*)$ and let $f: S \to S^*$ be continuous. If $E$ is a connected subset of $S$, then $f(E)$ is a connected subset of $S^*$.
Proof: A... |
H: Vector Space isomorphisms of $\mathbb{Q}(z)$ preserving the Galois group (where $z$ is a primitive third root of unity)
Take the field extension $\mathbb{Q}(z)$ where $z$ is a primitive third root of unity and consider the set $A$ of vector-space automorphisms of $\mathbb{Q}(z)$ so that for $T \in A$ the map $\phi ... |
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