text stringlengths 83 79.5k |
|---|
H: Doubt: "A group representation is exactly like a module over the group ring"
It is traditional to say that a representation of a group $G$ over a field $F$ is "exactly like" a module over the group ring $F[G]$.
I think it is inaccurate. I think a module over $F[G]$ encodes more than a representation of $G$. I will ... |
H: What's the most basic yet interesting Algebraic Geometry result regarding this polynomial?
Let $f(x,y,z) = x^a + y^b - z^c$, where $a,b,c \gt 0$. What is the most basic yet interesting result about this polynomial from Algebraic Geometry?
AI: You might be interested in the paper Faltings plus epsilon, Wiles plus e... |
H: Show that $X_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ and $X_{n}=\sum_{k=0}^{n} \rho^{k}$ are Cauchy sequences.
Show that $X_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ and $X_{n}=\sum_{k=0}^{n} \rho^{k}$ are Cauchy sequences.
I know that for each sequence I have to show the following:
"$\forall \epsilon>0 \text{ }\exists \text{ ... |
H: Evaluating the Average value of f(x)
Determine the average value of $f(x)$ over the interval from $x=a$ to $x=b$, where $f(x)=\frac{1}{x}$, $a=\frac{1}{10}$, and $b=10$
Can someone please explain the simplifying in this problem step-by-step?
AI: Note that $$\ln\left(\frac{1}{x}\right) = - \ln{x}$$
In particular, w... |
H: How much Differential Geometry is needed to appreciate Algebraic Geometry?
I want to start self-studying Algebraic Geometry at some point in the near future. There are plenty of posts discussing prerequisites, but one thing I couldn't find a discussion of: How much Differential Geometry is needed to appreciate Alge... |
H: Is $ \forall x(P(x) \lor Q(x)) \vdash \forall x P(x) \lor \exists xQ(x) $ provable?
I know I should be able to determine whether the following holds, but I am not able to either find a model to show that this is false nor can I prove its correctness by using natural deduction.
$ \forall x(P(x) \lor Q(x)) \vdash \... |
H: Determine all monic irreducible polynomials of degree $4$ in $\mathbb{Z_2[x]}$
Determine all monic irreducible polynomials of degree $4$ in $\mathbb{Z_2[x]}$
Well these polynomials will be of the form -
$a_0 + a_1x + a_2x^2 + a_3x^3 + x^4$
So we have four coefficients that can each have values of either $0$ or $1$.... |
H: Injective and Surjective Functions on Sets
I'm fairly new to math proofs. I've been looking for some counterexamples to the following theorems, especially the second one. I haven't been able to think of a scenario. Are the following theorems true?
If $f: S \to T$ is an injection and $A \subseteq S$, then $f^{-1}(f... |
H: $A^3 = I$. Find the possible Jordan Forms???
If $A^3 = I$, then I want to find the possible Jordan forms of the matrix. Since the minimal polynomial has degree at most three, each block is at most 3, and the eigenvalues are third roots of unity.
Is that the answer, or are there some other components I am missing... |
H: Can sign of integration by parts equation be changed to negative?
Integration by parts:
$$\int u \:dv+\int v \: du=uv$$
Are there any circumstances under which the sign can be changed so that the following is true?
$$\int u \:dv-\int v \: du=uv$$
AI: $\bullet$ For definite integrals, this is true if $\int_a^b v(t)u... |
H: What does the phrase "invariant under f" mean?
Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(\frac{x}{3}+11y^2,-2y)$.
Let $A=\{(3y^2,y)|y\in \mathbb{R}\}$.
Show that $A$ is invariant under $f$.
What is it asking?
AI: See that
$f(3y^2,y)=(\frac{3y^2}{3}+11y^2,-2y)=(12y^2,-2y)=(3(-2y)^2,(-2y))$
wh... |
H: How to apply Fubini here
Let $f$ and $g$ be integrable functions on the measure space $(X, \mathcal M, \mu)$ with the property that
$$
\mu(\{f > t\} \triangle \{g > t\}) = 0
$$
for $\lambda$-a.e. $t \in \mathbb R$. Prove that $f = g\;$ $\mu$-almost everywhere.
Here, $\lambda$ is Lebesgue measure and $\triangle$ de... |
H: Is there a single valued definition for $\sqrt[n]{z}$ on $\Bbb{C}$?
Since the $n$th root of a complex number has $n$ possibilities, which one do you choose so that its restriction to $\Bbb{R}$ is the positive square root if $r$ is positive and $\sqrt{|r|}i$ if $r$ is negative.
AI: Define
$$\sqrt[n]{r e^{i\theta}} ... |
H: Is it the case that every quotient ring ${F[x]}/{(p(x))}$ is a PID?
Let $F$ be a field and $F[x]$ the univariate polynomial ring over $F$. Is it the case that every quotient ring ${F[x]}/{(p(x))}$, where $p(x) \in F[x]$, is a PID (principal ideal domain)?
This is just a minor issue that's been bothering me for th... |
H: Is it theotically possible to create a machine that could randomly generate any elementary geometric problems/theorem?
Refering back to this quesion: Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry? , we know that an... |
H: $A=\{1,2,3,4,5\}$. How many functions $f : A \to A$ so that $(f\circ f)(1) = 3$
$A=\{1,2,3,4,5\}$
How many functions $f : A \to A$ so that $f$ is onto?
$5!$
is this correct?
How many functions $f : A \to A$ so that $(f\circ f)(1) = 3$?
$f(1)=1, f(1)=2, f(1)=3, f(1)=4, f(1)=5$?
I don't know what to do...
How ma... |
H: Proving that $0^\infty$ is not indeterminate?
Suppose that $f(x)>0$ for all $x$, and for some $a$ both $\lim_{x\to a} f(x) = 0$ and $\lim_{x\to a} g(x) = \infty$ are true. How would I go about proving:
$$\lim_{x\to a} f(x)^{g(x)} = 0$$
I am stuck after attempting to evaluate $\lim_{x\to a} \ln(f(x)^{g(x)})$. After ... |
H: Non-isomorphic simple graphs: order $n$, size $\displaystyle \frac{na}{2}$, degree sequence $(a,a,a,...,a) \in \mathbb{N}^n$
If a simple graph has order $n$, size $\displaystyle \frac{na}{2}$ and degree sequence $(a,a,a,...,a) \in \mathbb{N}^n$ then is it unique up to isomorphism?
I thought of this question while ... |
H: Block matrix characteristic polynomial
Let $n \in \Bbb N^*$ and $A \in \cal M_n(\Bbb R)$ a square matrix. Let the block matrix
$$B = \begin{pmatrix}A&A\\A&A \end{pmatrix} \in \cal M_{2n}(\Bbb R)$$
Calculate the characteristic polynomial $\chi_B$ using $\chi_A$.
Proof that $A$ is diagonalisable $\iff$ $B$ is d... |
H: Derivation of Symmetry Property of Metric Spaces
I am given the following modified triangle inequality property of metric spaces, where for any $x_1$, $x_2$, $x_3 \in X$, we have $d(x_1, x_2) \le d(x_1, x_3)+d(x_2, x_3)$. I am tasked to show that the symmetry property follows from the use of this and the positive ... |
H: Symbol for "is closest to"?
I am writing a paper on probabilities and we have to find a $k$ such that $P_n(k)$ is "closest to" $P_0$.
$P_0$ is getting 4-of-a-kind in a five card hand in a standard 52 card deck.
$P_n(k)$ is probability of getting $k$-of-a-kind in an $n$ card hand in some modified 88 card deck.
I ... |
H: showing a bijection is true for the assumption that it takes 4 points to have 1 intersection in a circle
Problem: 15 points are taken on the circumference of a circle, and through any two of them a chord is drawn. Suppose that no three chords intersect at the same point inside the circle. How many points of interse... |
H: How to prove the $f(x) = \sqrt{x + \sqrt{x}}$ is injective.
The function is
$$
f(x)=\sqrt{x+\sqrt{x}}
$$
I know that you need to set up the equation $$\sqrt{x_1+\sqrt{x_1}}=\sqrt{x_2+\sqrt{x_2}}$$
and you have to solve step by step until you get $x_1=x_2$. But I am having difficulty figuring out what to do. I hav... |
H: Find the Laurent series and residue of $\frac{z}{(\sin(z))^2}$ at $z_0 = 0$.
Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point.
$$\frac{z}{(\sin(z))^2}\quad \text{at}\quad z_0 = 0 \quad\text{four terms of the Laurent series}$$
I am not sur... |
H: Clarification on solution to a problem:
The problem is as follows:
How many zeroes do we write when we write all the integers from 1 to 243 in base 3?
The given solution starts as follows:
The 1-digit numbers don't have any zeroes.
The 2-digit numbers use 2 zeroes: 10 and 20.
There are $3^2 = 9$ three-digit numbe... |
H: About the symmetry of Riemann Tensor
It is a problem in my homework.
First I was asked to show
$$
\nabla_a\nabla_bA_c-\nabla_b\nabla_aA_c=R_{a,b,c}^{\;\;\;\;\;d}A_d
$$
where $A$ is a (0,1)-tensor and $R_{a,b,c}^{\;\;\;\;\;d}$ is the Riemann curvature tensor, which is defined by
$$
\nabla_a\nabla_bV^c-\nabla_b\nabla... |
H: If $G_1$ and $G_2$ are simple groups, what can we say about normal subgroups of $G_1 \times G_2$?
If $G_1$ and $G_2$ are simple groups, what can we say about normal subgroups of $G_1 \times G_2$? I remember when I was taking Algebra I this was brought up in the class but at the time the professor left it for us to ... |
H: If $Y: G\to H$ is a group homomorphism and $G$ is abelian, prove that $Y(G)$ is also abelian.
What I got: Suppose $Y$ is a homomorphism and that $G$ is abelian. Then for all $a,b \in G$, $ab=ba$, and thus $Y(ab)=Y(ba)=Y(a)Y(b)=Y(b)Y(a)$.
However, this seems too simple and I was confused on what $Y(G)$ meant (since ... |
H: Work done by a force field $F$ via the line integrals
Let $F:\mathbb R^2\to \mathbb R^2$ be the force field with
$$F(x,y) = -\frac{(x,y)}{\sqrt{x^2 + y^2}}$$
the unit vector in the direction from $(x,y)$ to the origin. Calculate the work done against the force field in moving a particle from $(2a,0)$ to the origin... |
H: Indefinite integral...for calc 1?
a calc 1 student I know was given this integral do via u-substitution, and even though I think it's an obvious typo, I know the answer is somewhat simple thanks to Wolfram, but am not sure how to arrive at it.
$\int{x^2\sqrt{1+x}}$ $dx$.
Any help would be appreciated!
AI: let $u=1+... |
H: Consumer surplus for demand curve at the given sales level $x$
The square root is throwing me of doing the integration. Can someone please show the steps of integration.
Answer is $\$26.19$
Ok then can someone show me how we got the integration here?
AI: The height of the curve at $x = 50$ is $\sqrt{16 - 0.14\tim... |
H: Determine angle from radius of curvature
This is another grade school problem that's giving me trouble (posting on someone else's behalf).
I can see that a 36 inch semi-circumference yields a radius of 36/Pi or about 11.46 inches.
However, I can't see how to use this information to calculate the angle. Given the w... |
H: Power series method to solve Airy’s differential equation
Using power series method, solve Airy’s equation
$$y′′+ xy = 0.$$
How do I start solving this? Thanks in advance!
AI: Given:
$$\tag 1 y''+ x y = 0$$
Solve this using Power Series.
We assume:
$$y = \sum_{m=0}^\infty a_mx^m$$
Thus we have:
$$y'' = \sum_{m=2}^... |
H: Does a function that has an exponential analog to $log(xy) = log(x) + log(y)$ exist?
Similar to how $log(xy) = log(x) + log(y)$, does a nontrivial function exist that has the property $f(x^y) = f(x)f(y)$? How would one attempt to derive such a function?
AI: Then $f(x^y) = f(y^x) \Rightarrow f(1^y) = f(y^1) \Rightar... |
H: $1 + \frac{1}{1+2}+ \frac{1}{1+2+3}+ ... + \frac{1}{1+2+3+...+n} = ?$
How do I simplify the following series
$$1 + \frac{1}{1+2}+ \frac{1}{1+2+3}+ \frac{1}{1+2+3+4} + \frac{1}{1+2+3+4+5} + \cdot\cdot\cdot + \frac{1}{1+2+3+\cdot\cdot\cdot+n}$$
From what I see, each term is the inverse of the sum of $n$ natural numbe... |
H: Basis for vector space given combination of vector components
The following is the first step in a homework problem of mine:
Find a basis for the vector space $S = \{(x,y,z,w) \in \mathbb{R}^4 \mid x - y - 2z + w = 0\}$.
The actual problem involves computing the orthonormal basis using Gram-Schmidt, which I know... |
H: What is $a_5$, given the recurrence $a_{n+1}=a_n+2a_{n-1}$ and we know that $a_0 = 4, a_2 =13$
I am having a very hard time figuring this out.
So far I have been able to do the following:
Writing the recurrence as a characteristic polynomial = $x^2-x-2=0$ so there are roots, $x=2, x=-1$.
So there will be the genera... |
H: distribution of objects
Find the number of ways to distribute $n$ distinct objects to $5$ distinct boxes, such that boxes $1$, $3$ and $5$ must hold an odd number of objects and boxes $2$ and $4$ must hold an even number of objects.
AI: Is the value of $n$ odd or even? |
H: calculus interval and concave up and down
f(x)=3(x)^(1/2)e^-x
1.Find the interval on which f is increasing
2.Find the interval on which f is decreasing
3.Find the local maximum value of f
4.Find the inflection point
5.Find the interval on which f is concave up
6.Find the interval on which f is concave down
Anyone c... |
H: How to derive duration of unemployment?
The average monthly flow out of unemployment pool of $7.0$ million people each month is $3.1$ million. Put another way, the proportion of unemployed leaving unemployment equals $\frac{3.1}{7.0}$ or about $44 \%$ each month. Put yet another way, the average duration of unempl... |
H: Calculating the Jacobian Matrix
I am working with the system:
$$ u' = v $$
$$ v' = -w^{2}sin(3\pi+u)-cv $$
where $c$ and $w$ are positive constants.
I'm computing the Jacobian matrix:
$$J=
\begin{pmatrix}
F_u & F_v \\
G_u & G_v \\
\end{pmatrix}
$$
$$ J=
\begin{pmatrix}
... |
H: If $G$ is solvable, is it true that for any $m,n\in\operatorname{cd}(G)$, there exists a prime $p$ such that $p\mid m,n$?
Let $G$ be a finite group and let $\operatorname{cd}(G)$ be the set of degrees of irreducible characters of $G$. It is known that if for any $m,n \in \operatorname{cd}(G) \setminus \{1\}$, there... |
H: To find Z-transform of given sequence
How to find the $z$-transform of $\left[a^{n}\sin\left(bn\right)\right]/n!$ where "!" denotes factorial of a number and b is constant??
AI: Hints:
i)
$$\sin( bn )=\frac{1}{2i}(e^{ibn}-e^{-ibn}).$$
ii) Z-Transform of $a^ne^{ibn}$ is given by
$$F(z) = \sum_{n=0}^{\infty} a^n... |
H: Question about the Monte Carlo Algortihm
I was reading the Monte Carlo algorithm for finding the area under a curve, say $y=f(x)$.
The algorithm considers, $0\le f(x)\le M$ over the closed interval $a\le x\le b$.
My question is,that why is it necessary for $f(x)\ge 0$ for the algorithm to work why can't it simply b... |
H: Constructing a bijection between intervals
So I am trying to solve questions below
Let $A = \{(\alpha_1,\alpha_2,\alpha_3,\ldots): \alpha_i \in
\{0,1\}, i \in N\}$, i.e., $A$ is the infinite cartesian product of the
set $\{0,1\}$. Show that $A$ is uncountable.
Prove that the intervals $[0,\infty)$ and $(-1,4)$ ... |
H: Closed Compact Subset of Product Space Must Have Empty Interior
Suppose that $\{X_{\alpha}\}_{\alpha\in A}$ is a nonempty family of topological spaces. Suppose also that there is an infinite index set $B\subseteq A$ such that $X_{\alpha}$ is not compact for any $\alpha\in B$. Furthermore, assume that
$K\subseteq\p... |
H: $a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$
Define a sequence $\left\lbrace a_{n}\right\rbrace$ by
$\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor a_{n}^{2}\right\rfloor=n$
The substitution $... |
H: Are monics and epics in the category of finite dimensional vector spaces actually injective and surjective linear transformations?
Consider the category of finite dimensional vector spaces with morphisms being linear transformations.
Is it still true that monics and epics are actually injective and surjective line... |
H: If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
I was checking my old set of homework problems that I found this one:
If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only
if $a$ is a cube in $G$.
One direction is easy. If $a$ is a cube in $G$ then... |
H: Four lines are drawn on a plane with no two parallel....
I am stuck on the following problem:
Four lines are drawn on a plane with no two parallel
and no three concurrent.Lines are drawn joining the
points of intersection of the previous four lines. Number of new lines obtained
this way is : $\,\,3,\,\, 5... |
H: Prove the following $(a \cap(\neg b))\cup (a \cap c)=a\cap (\neg(b\cap(\neg c))) $
I want to prove the following:
$$(a \cap(\neg b))\cup (a \cap c)=a\cap (\neg(b\cap(\neg c))) $$
What I tried to do so far is to minimize the LHS but I dont know if it enough:
$$(a \cap(\neg b))\cup (a \cap c)=a\cap (\neg b \cup c)$$
... |
H: Converges of integral, knowing the derivative has a limit
$f: [0,\infty) \to \Bbb R$ we know that f>0 and f ' exists .$
$$\lim_{x \to \infty} (\ln f)'(x)=L<0$. Prove that $\int _0^\infty f$ converges
Any hint would be helpful!
AI: Hint: if $(\ln f)'(x) \le c < 0$ for all $x \ge x_0$, then
$\ln f(x) \le \ln f(x_0)... |
H: Find the multiplicative inverse of $\,x^2+(x^3-x+2)$ in the quotient $\,F_3[x]/(x^3-x+2)$
Find the multiplicative inverse of $x^2+(x^3-x+2)$ in the quotient $F_3[x]/(x^3-x+2)$ .
I've proved that $x^3-x+2$ is irreducible polynomial in $F_3[x]$, and that $x^2$ and $x^3-x+2$ are coprime integers, therefore gcd $(x^2,... |
H: How can I prove a coordinate ring is not isomorphic to a polynomial ring
Let $Z$ be the plane curve $xy=1$. I would like to prove that $A(Z)$ is not isomorphic to a polynomial ring in one variable over $k$.
I'm already prove that the coordinate ring is $A(Z)=k[x,y]/(xy-1)$, but I couldn't finish the question.
I hav... |
H: Prove a certain limit involving integrals and series
Question:
Let f>0 be an descending function. $f:[0,\infty)\to \Bbb R$. Prove that for all a>0 that:
$$\lim_{a^+\to 0} a\sum_{n=1}^\infty f(na)= \int _0^\infty f(x)dx$$
What we know
We figured using the series comparison theorem that $\sum_{n=1}^\infty f(na)$ conv... |
H: Proving that if these quadratics are equal for some $\alpha$, then their coefficients are equal
Let
$$P_1(x) = ax^2 -bx - c \tag{1}$$
$$P_2(x) = bx^2 -cx -a \tag{2}$$
$$P_3(x) = cx^2 -ax - b \tag{3}$$
Suppose there exists a real $\alpha$ such that
$$P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$$
Prove
$$a=b=c$$
Equat... |
H: Sequence in $l^2$ that does not converge (but it should?)
Let $l^2$ be the space of all real sequences $x = (x_1,x_2, x_3,\;...)$ for which $\sum_{n=1}^\infty |x_n|^2$ converges. It can be easily verified that map
$$\langle x,y\rangle = \sum_{n=1}^\infty x_ny_n$$
defines an inner product on $l^2$ and that $x\to ||... |
H: Find a solution of $x\frac{dy}{dx} = y^2 -y$ that passes through the points (1/2, 1/2)
I do not understand how my instructor simplified the part marked with red circle.
Did he make a mistake? Could anyone help me out here.
AI: Nope, he is right
$\frac{1}{y-1}-\frac{1}{y}=\frac{y-(y-1)}{y(y-1)}=\frac{1}{y(y-1)}$ |
H: Why is $\zeta'/\zeta(s) = -\sum_p \log p/(p^{s}-1)$?
I'm going through the prime number theorem and there's a lemma which uses the fact that $\frac{\zeta'}{\zeta}(s) = -\sum_p \frac{\log p}{p^{s}-1}$.
Can somebody explain this? Am I missing something obvious?
Here's my work, $\log \zeta (s) = -\sum_p \log(1 - p^{-s... |
H: I want to show$X : ((0,1],\mathcal B,\lambda)\to \mathbb R$ is random variable
Let $F:\mathbb R \to [0,1]$ be a distribution function of a probability measure $P$ $(i.e.,F(x)=P((-\infty,x])) $. Then show that There is a random variable $X : ((0,1],\mathcal B,\lambda)\to \mathbb R$,(where $\mathcal B$ is the borel $... |
H: Rudin Example 3.35B Why the $n$th root of $a_n$ is $1/2$?
For Baby Rudin Example 3.35(b), I understand how the $\liminf$ and $\limsup$ of the ratio test were found, but I am not clear why $\ \lim \sqrt[n]{a_n } = \frac{1}{2} $.
Please help.
AI: The sequence in question is
$$\frac{1}{2} + 1 + \frac{1}{8} + \frac{1}... |
H: Convolution of distributions is not associative
I need some help with this exercise:
It proposes to show that convolution of distributions is not associative:
If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the Heaviside function, in $\mathbb{R}$), then:
$$T\ast(S\ast R)\neq(T\ast... |
H: If $(a+b)(b+c)(c+a)=2$, then $(a^2+bc)(b^2+ca)(c^2+ab)\leq 1$
Prove that if $a,b$ and $c$ are non negative real numbers such that $(a+b)(b+c)(c+a)=2$, then we have $$(a^2+bc)(b^2+ca)(c^2+ab)\leq 1$$
AI: You can prove the inequality
$$4(a^2+bc)(b^2+ca)(c^2+ab)\le (a+b)^2(b+c)^2(c+a)^2.$$Without loss of generality, a... |
H: properties of measures with density
Let $\nu=\mu f$ be a $\sigma$-finite measure on $(\Omega,\mathcal A)$.
Then $f$ is $\mu$-a.e. unique and $\mu$-a.e. real.
Also: If $f(\omega)>0$ for all $\omega\in\Omega$, $\mu$ is also $\sigma$-finite.
I was able to show the uniqueness, but I don't quite know how to show that $f... |
H: Is this matrix block matrix Hurwitz for some $c_1, c_2, c_3 > 0$?
Given the next well partitioned real squared matrix
$$
M = \begin{bmatrix}A & \frac{c_3}{c_1}BC^T \\
c_3c_2C & E- c_3^2\frac{c_2}{c_1}CC^T\end{bmatrix},
$$
where $A$ is Hurwitz (all the eigenvalues have negative real part), $CC^T$ positive definite, ... |
H: Describe four different elements of a union of power sets of power sets
$P(P(A))\bigcup P(P(P(A)))$
The empty is set one but other than that I'm not really sure what does a power set of a power set means.
Any help would be appreciated.
Edit: Is this how you read this for example: $P(P(\emptyset))= \{\emptyset\{\emp... |
H: Caley graphs of gruops and symmetric generating sets
There are several examples (of which Wikipedia show at least one) when the Caley graph (G,U) of a group G (where U generates the group) depend on the choice of generating set. Is requiring that the generating set U is symmetric enough to guarantee that the Caley ... |
H: Prove that if $\{K_n\}_{n\in\mathbb{N}}$ is a sequence of uncountable compact sets of $X$, then $\bigcap\limits_{n\in\mathbb{N}}K_n$ is uncountable
The question I am interested in is the following:
Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied... |
H: Complex Analysis and Probability Theory
My question is a general one. I know that in complex analysis we find some very powerful theorems but given that my main area of study is Statistics and Probability, does complex analysis have applications in those areas? I know that it is helpful for example in the character... |
H: Probability to find $1$ in a random sequence of bits
Given the random sequence of $N$ bit formed by $0$ and $1$ obtained flipping a coin and associating for example $1$ to head and $0$ to tail, suppose, after having built the sequence, we randomically pick a number $k$ with a uniform probability and $1\le k\le N$,... |
H: Non-Isomorph trees of a graph
Please consider this graph
How many non-Isomorph trees with 4 vertex has this graph?
Is there any formula that show number of non-Isomorph trees with $n$ vertices?
thanks
AI: There are only two 4-node trees up to isomorphism: the straight line and the Y-shaped graph. The above graph c... |
H: Showing the polynomial $x^4 + x^3 + 4x + 1$ is irreducible in $\mathbb{Q}[x]$.
Showing the polynomial $f(x) = x^4 + x^3 + 4x + 1$ is irreducible in $\mathbb{Q}[x]$.
I have two question relating to this which I've bolded below.
Attempt to answer
(*) $f$ has degree $4 \ge 2$ so if $f$ has a root then $f$ is reducible... |
H: Set of m-vectors
In my book (matrix computations by Gill) there's a term used quite a lot which I don't understand/find what is its meaning.
The set of all $m$-vectors that are linear combinations s of the columns of the $m \times n$ matrix $A$ is called range space, column space or simply the range of $A$.
Doe... |
H: is a number contained within this number series
I hope I can ask this question with enough clarity. It is not exactly clear in my head., but here goes
I have a number series, it starts at a random number and then increments by 8(always by 8) plus the last number.
for example in the series:
36,80,132,192,260,336,42... |
H: Prove that the cardinality of the reals and all binary funcions is not equal
Let $S$ be the the set of all real functions that bring back only two values: 0 and 1 (Binary functions).
If $f\in S$ then $f:\mathbb{R}\rightarrow \left\{0,1\right\}$.
Prove that $|\mathbb{R}| \neq |S| $.
I tried to start with a proof by ... |
H: Probability with bullets and walls
There are two shooters with different guns and bullets. Each shooter shoots a bullet to a different target hanging on a wall. The hit of each bullet follows a normal distribution centered on its target. Each bullet removes a piece of the wall whose volume follows a normal distribu... |
H: Function from A to A
Let $S = \{1,2,3,4,5\}$ how many functions from $S \to S$ and how many of these are $bijective$
Say we also had $A = \{1,2,3,4,5\}$ then the number of functions would be $B^A$ and the number of $bijective$ functions would be 5? But how you count the function from the same set?
AI: The number of... |
H: Inverse of Short-time Fourier transform
The Gabor transform (i.e. Short-time Fourier transform with some Gaussian window) can be defined by (see http://en.wikipedia.org/wiki/Gabor_transform) :
$G_x(t_0,\xi) = \int_{-\infty}^\infty e^{-\pi(t-t_0)^2}e^{-2i\pi \xi t}x(t)\,dt$
On this page it is mentioned that the Gab... |
H: Divisors of $x^3 + x + 1$ in $Z_3[x]$
Divisors of $f(x) = x^3 + x + 1$ in $Z_3[x]$
Do I have to manually check whether each polynomial in $Z_3[x]$ with degree less than $3$ divides $f$ or is there a better way? That's $3^3 = 27$ polynomials to check (although a few are obviously trivial).
AI: Hint: What is $f(1)$? ... |
H: Is this equivalent to continuity?
I played around a little bit with the definition of continuity and I think I got the following relations that may be equivalent to continuity. Maybe there is somebody who can check this:
$$ f(\overline{M}) \subset \overline{f(M)} \Leftrightarrow \overline{f^{-1}(A)} \subset f^{-1}... |
H: Explanation of $\mathbb{Z}/_5$
can someone explain me the following please?:
$\mathbb{Z} /_5 = \{ [0]_5, [1]_5, [2]_5, [3]_5, [4]_5\}$
Is that correct?
$[0]_5 = \{ \dotsc, -15, -10, -5, 0, 5, 10, 15, \dotsc\}$
$[1]_5 = \{ \dotsc, -14, -9, -4, 1, 6, 11, 16, \dotsc\}$
$[2]_5 = \{ \dotsc, -13, -8, -3, 2, 7, 12, 17, \d... |
H: Prove that if $f^2=g^2$ and $f(x)$ not zero then $f(x)=g(x)$ or $f(x)=-g(x)$
Suppose that $f$ and $g$ are continuous function on $R$ such that $f^2=g^2$ and $f(x)$ not zero. Show that it's either $f(x)=g(x)$ or $f(x)=-g(x)$
I tried to apply the definition of continuous function to $f$ and $g$, but I don't know how ... |
H: Comparing models through partial isomorphisms
Let $T$ be a theory of a language $\mathcal{L}$ with no function symbols. Let $\mathfrak{A}, \mathfrak{B} \models T$. For all finite sets $X \subseteq A$ and $Y \subseteq B$, there exists a function $F : X' \to Y'$ such that $$X \subseteq X' \subseteq A,$$ $$Y \subseteq... |
H: Markup reverse to get to original amount?
I am trying to figure out how to mark something up then reverse it.
I am selling an item for $100$ and I need to cover a $15\%$ advertising expense. I want to net $100$ after advertising is paid for. How do I figure out what percent I should markup the $100$ item to pay fo... |
H: Finding a limit results in division by 0.
$$
\lim_{x\to0}\dfrac{\sqrt{1+x}-1}{x^2}.
$$
Tried multiplying by $\sqrt{1+x}+1$ but got $1/(x(\sqrt{1+x}+1))$ where substitution results in $1/0$ which is illegal (or maybe $1/0$ is Infinity?). Second approach is to divide by $x^2$ which leads to the $\sqrt{1/x^2 +1)}- 1/... |
H: If $G$ is a finite semi-group and $\forall x,y,z \in G: xy=yz \implies x=z$ then $G$ is an Abelian group
If $G$ is a finite semi-group and $\forall x,y,z \in G: xy=yz \implies x=z$ then $G$ is an Abelian group.
I have no idea where to start. I'm stuck! I can't prove even the existence of the identity element :|
AI:... |
H: Finding limit of $\frac{\sin\pi3^x}{x}$ as $x\to 0$
I have to find the limit of $\frac{\sin\pi3^x}{x}$ as $x\to 0$ using ONLY notable limits
please help me.
AI: $\sin x=\sin(\pi-x)\quad,\quad\lim_{x\to0}\frac{\sin x}x=1\quad,\quad\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=f'(a)\iff$ $$\lim_{x\to0}\frac{\sin(\pi\cdot3^x)}x=... |
H: Trouble understanding Tensor product in context of Torsion Tensor
I know that there are quite a few threads dealing with this question already. I have pored through them for quite some time and they have been informative. However there are still some clarifications that I seek. If this is too much of a duplicate, t... |
H: Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$
Is it trivial to say
$$\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k},$$
considering the fact that we know
$$\mathop {\lim }\limits_{n \to \infty } {(1 + {1 \over n})^n} = e?$$
AI: Since $\frac kn = \f... |
H: Banach algebra problem: $\left\|e^{ta}\right\|\leq Me^{-\omega t}$
Let $A$ be a unital algebra, and $a\in A$. Assume that $\sigma(a)\subset \{\lambda\in \mathbb{C}: Re\lambda < 0\}$.
Show there exists $M,\omega >0$ such that $$\left\|e^{ta}\right\|\leq Me^{-\omega t} $$
for all $t>0$.
Even if $\sigma(a)$ contains a... |
H: Can you factor out vectors?
My prof introduced eigenvalues to us today:
Let $A$ be an $n \times n$ matrix. If there a scalar $\lambda$ and an $n-1$ non-zero column vector $u$, then
$$Au = \lambda u$$
then $\lambda$ is called an eigenvalue and $u$ is called an eigenvector.
$$Au - \lambda u = 0$$
$$\implies (A - \la... |
H: subsets of a generating set of a free group
Is it true that, given any free group $F$ and any set $S\subseteq F$ which generates $F$, there is a subset $T\subseteq S$ which is a free generating set for $F$?
It would be great if you could give me some literature recommendations.
AI: Notice that $\{2,3\}$ is a genera... |
H: Transitive relations and Subsets
I have a question to prove:
If relations R is transitive, than R^2 is transitive. In the answer the professor says that if R is transitive than:
R^2 is a subset of R (I understand why, this is the definision)
Therefore, R^2*R^2 is a subset of R^2. This is the part I don't understand... |
H: Partial derivative in higher dimension .
Let us consider $g: \mathbb R^n \to \mathbb R^m$ , $f:\mathbb R^m \to \mathbb R^k$ be $C^1$ functions .
Define $F= f\circ g$. I am having the problem in finding the partial derivative of $F$,
$$\frac {\partial F}{\partial x_i} = \frac{\partial \{f_1(g(x)), .......f_k(g(x))... |
H: Series representation of $m=1^k + 2^k + \ldots n^k$.
Is there any simple form for following question?
$m=1^k+2^k+\cdots+n^k$
AI: See Faulhaber's formula, for instance. |
H: Solving roots of a sum of sinusoids
Suppose I have a sinusoid with fundamental frequency $f_0$ and $N$ harmonics (all with distinct amplitudes $a_k$. Each harmonic also has it's own corresponding phase $\phi_k$ and offset $c_k$.
$y(t) = \sum_{k=0}^{N} [a_k sin(2 \pi k f_0 t + \phi_k)+c_k]$
How could one find the z... |
H: Fix some $\delta\in \mathbb R$ and let $f:[0,\infty)\rightarrow \mathbb R$ be given by the equation
$$f(x)=\frac{\sin(x^2)}{x}+\frac{\delta x}{1+x}$$
Show that,
$\lim_{n\rightarrow\infty}\int_{0}^{a}f(nx)\ dx=a\delta$
for each $\ a>0$.
My attempt:
$\lim_{n\rightarrow\infty}\ f(nx)=\delta$ and $|f(nx)|\le (\frac{... |
H: Example of Group which is not a direct product of its Sylow-Subgroups
Can you please give me an example of a group which could not be written as the direct product of its Sylow-subgroups?
AI: The minimal example is the symmetric group $S_3$. |
H: Prove that the greatest integer function: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$
Statement: the greatest integer function int: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$
Proof: let $x \in \mathbb{R}$, then $int(x) \leq x$ and is an element of $\mathbb{Z}$. Since $\mathbb{Z}$ is an ... |
H: Stuck trying to solve this differential equation
The equation is:
$$\dfrac{dy}{dx} = \dfrac{3y}{(3y^{2/3}-x)}$$
So I wrote this as:
$$\dfrac{dx}{dy}= \dfrac{(3y^{2/3}-x)}{3y}$$
$$\therefore \dfrac{dx}{dy} + \dfrac{x}{3y} = y^{-1/3}$$
If I let $v=y^{\frac{2}{3}}$ then:
$$\dfrac{dx}{dy} = \frac{2}{3}y^{-1/3}\dfrac{... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.