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H: Tetromino Proof
Prove that an 8 x 8 board cannot be covered by 15 L-tetrominos and one square tetromino (an L-tetromino is a plane figure shown below, constructed from four unit squares arranged in the form of L; a square tetromino is a plane figure shown below, constructed from four unit squares arranged in the fo... |
H: Radon integrals and Radon charges
I'm reading chapter 6 (measure theory) of Pedersen's book Analysis now and I'm a bit puzzled in his passage from Radon integrals to Radon charges.
The book in 6.1.2 defines a Radon integral to be a linear complex valued positive functional (i.e which maps positive functions in posi... |
H: Set where function has high values is small
Let $\mu$ be a probability measure on a set $A$, and let $f:A\rightarrow\mathbb{R}$ be a random variable. Given $\epsilon>0$, is it true that we can find $n$ such that $\mu\{|f(x)|>n\}<\epsilon$?
Intuitively it looks like it should be true (just choose $n$ large enough s... |
H: Predicate logic: "Everybody knows somebody who knows Alice"
I'm stuck on an undergraduate CS exercise: I am to translate "Everybody knows somebody who knows Alice" into predicate logic.
I'm having trouble bending my head around it (being a complete beginner), but this is what I'm thinking:
$x,y \in $ "the set of a... |
H: Integral of characteristic function is infinitely differentiable
Let $X$ be a set, $F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given a random variable $f:X\rightarrow\mathbb{R}$, define $$\chi_f(t)=\int_Xe^{itf}d\mu$$ We can show that $\chi_f$ is continuous and $|\chi_f|\leq 1$.... |
H: Proving sets A and B are countable
a) Let A and B be disjoint sets, which are both countable. Prove that $A$ U $B$ is also countable.
b) Use part (a) to show that the set of all irrational real numbers is not countable.
So for part a I understand that disjoint sets share no elements in common (other than the empty ... |
H: Convergence of $|a_{n+1} - a_n| \le Cq^n$
Be $C\gt 0$, $0\le q\lt 1$ and $(a_n)_{n\ge 1}$ a sequence in $\mathbb R$ with
$$|a_{n+1} - a_n| \le Cq^n$$
Show that $(a_n)_{n\ge 1}$ converges.
AI: Given $m,n \in \mathbb{N}$ with $n>m$ we have
$$
|a_n-a_m|\le\sum_{i=m}^{n-1}|a_i-a_{i+1}|\le C\sum_{i=m}^{n-1}q^i=C\frac{q^... |
H: If $T$ is injective then there exists $\alpha>0$ such that $||Tx||\geq \alpha||x||$
Is this proof correct? I'm proving that if $T$ is a linear operator whose is injective then exist $\alpha>0$ such that $$||Tx||\geq\alpha||x||$$ for all $x$.
By contrapositive. Assume that for all $$\alpha>0$$ there is a $x$ such t... |
H: The limit of integer valued random variables must be integer valued?
I saw something like this:
If $D_n$ are all integer valued random variables, and $D_n$ converges in distribution to $D$, then $D$ must also be integer valued.
I am a little bit suspicious about the statement. A trivial "counter example" might be... |
H: Discrete math number problem
How would I justify the following statement.
Two integers are consecutive if and only if one is more than the other. Any product of four consecutive integers is one less than a perfect square.
I think this is true.
because for example
$2<3<4<5$
$2*4*5*3=120$
Which one less than 121 a pe... |
H: How to solve this limit 0/0 indetermination?
How can I solve this 0/0 indetermination, where the limit has t going towards 0?
$$\lim_{t\to 0} \frac{t^2}{\sin^2 t}$$
AI: Hint: Use the fact that
$$\lim_{t \to 0} \frac t {\sin{t}} = 1$$
This is easily proven either by L'Hospital's rule, or Taylor series (or by geometr... |
H: What mathematical structure models arithmetic with physical units?
In physics we deal with quantities which have a magnitude and a unit type, such as 4m, 9.8 m/s², and so forth. We might represent these as elements of $\Bbb R\times \Bbb Q^n$ (where there are $n$ different fundamental units), with $(4, \langle1,0,0... |
H: How many triples of positive integers $(a,b,c)$ satisfy $a\le b\le c$ and $abc=1,000,000,000?$
How many triples of positive integers $(a,b,c)$ satisfy $a\le b\le c$ and $$abc=1,000,000,000?$$
I find combinatorial questions like this quite difficult.
I expressed $1,000,000,000$ as $2^95^9$.
I let $a=2^p5^s$, $b=2^... |
H: Solve for z(t) from the simultaneous equation using Laplace transform
Solve for $z(t)$ from the simultaneous equation using Laplace transform
$$
y' + 2y + 6 \int\limits_0^t z \mathrm{d}t = -2 u(t) \\
y' + z' + z = 0$$
subject to $y(0) = -5$ and $z(0) = 6$.
AI: First note that if $u(t)$ is the Heaveside step funct... |
H: If a function has no critical points, then its zero set has no interior points
Let $f \in C^1$ in the open set $\Omega$ and have no critical points there. Let $E$ be the set where $f=0$. Show that $E$ has no interior points.
The back of the book says "If $f$ is constant on an open set then $Df=0$ there." I see... |
H: How to show that an automorphism of $S_n$ is inner?
Given an automorphism $\phi:S_n\rightarrow S_n$ such that it maps all the transpositions on the transpositions, how do I show that this map is given by a conjugation with an element $s\in S_n$?
Thanks in advance!
AI: Let's suppose that we are looking at $\phi:\ S_... |
H: Non-commutative indeterminates in polynomial rings.
Described below are some observations I have made while fiddling around with polynomials. In addition to the two questions below, I am looking for any sort of relevant information so I can read more.
Preliminary question:
Suppose I want to consider the polynomial ... |
H: Conditional Probability with Independent Discrete Random Variables
Let X, Y be two independent Poisson random variables with lambda of X = 1, lambda of Y = 2. Find P(X = 40 | X + Y = 100).
I know P(X|Y) = P(X, Y) / P(Y), and since X and Y are independent P(X, Y) = P(X) P(Y). But P(X) and P(X + Y) are not independen... |
H: Generating function - Partial fractions
I was Reading a book about generating functions, and it said me to do this:
$$\frac{1 - 2x + 2x^2}{(1-x)^2(1-2x)} = \frac{A}{(1-x^2)} + \frac{B}{1-x}+\frac{C}{1-2x}$$
It says:
Multiply both sides by $(1-x^2)$ so:
$$\frac{(1 - 2x + 2x^2)(1-x^2)}{(1-x)^2(1-2x)} = \frac{A(1-x... |
H: Definitions of adjoints (functional analysis vs category thy)
If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where,
$(fx,y)=(x,f^*y)$ for all $x,y$.
I am confused because this definitions is very different to the definition of an adjoint used in category theory.
Is... |
H: Show and prove if $\sum_{j=1}^\infty{ \frac{(1+(1/j))^{2j}}{e^j}}$ converges or diverges
Show and prove if the following series converges or diverges
$$\sum_{j=1}^\infty{ \frac{(1+(1/j))^{2j}}{e^j}}$$
"I tried the comparison test, the root test, and the ratio test, but got messed up..."
Thanks!
AI: Apply the $\,n... |
H: Confusion with this definition of the derivative
This function is from my text: $$p(\theta) = \sqrt{13\theta}$$
It states that the derivative of the function $p(\theta)$ with respect to the variable $\theta$ is the function $p'$ whose value at $\theta$ is given by the following formula, provided that the limit exi... |
H: Proving $n^3$ is even iff $n$ is even
I am trying to prove the following statement:
Prove $n^3$ is even iff n is even.
Translated into symbols we have:
$n^3$ is even $\iff$ $n$ is even
Since it's a double implication, I started assuming n is even, then eventually concluded:
$$n \;\text{ is even }\;\implies \; n... |
H: Show the series $ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges
Show the following series converges
$$ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} . $$
I tried to use the comparison test and tried to compare it with the series of $\dfrac{2^j}{3^j}$ because this is the geometric series. However, $\... |
H: Discrete Gaussian density function sum drawn from Gaussian distribution
Doing some analysis of my problem,
I have come up with the following equation
(I tried to search this problem but no luck)
$S_N = \sum \limits_{i=1}^{N} \exp{\left(-\frac{X_i^2}{2\sigma^2}\right)}$
where $X_i \ \ \text{~} \ \ {N(0, \sigma^2)}$... |
H: Partial Fraction Decomposition in $ \frac{s^3-1}{(s^2+6)^2(s+12)^2} $.
What special considerations do you need to take when decomposing the following fraction and why?
I'm trying to decompose the following:
$$ \frac{s^3-1}{(s^2+6)^2(s+12)^2} $$
$$ \frac{s^3-1}{(s^2+6)^2(s+12)^2} = \frac{A}{(s^2+6)^2} + \frac{B}{... |
H: Determine the number of solutions for $x^2 = c$ such that $c \in \mathbb{F}$
I am working on the problem from number system class taught by the professor who shares his knowledge about the relation of abstract algebra and number systems. Here is the problem:
Assume $c$ and $\mathbb{F}$ are arbitrary. For $c \in \... |
H: Show that the natural number $n$ with base ten representation ($r_{k}r_{k-1}$. . . $r_{1}r_{0}$)$_{10}$ is a multiple of $4$
Show that the natural number $a$ with base ten representation ($r_{k}$$r_{k-1}$. . . $r_{1}$$r_{0}$)$_{10}$ is a multiple of 4 if and only if the number ($r_{1}$$r_{0}$)$_{10}$, consisting ... |
H: Differentiation under integral sign for exponential
This question arises from this question:
Suppose $P(x)$ is a polynomial. Why is it the case that $$\dfrac{d}{dy}\int_\mathbb{R}iP(x)e^{-x^2/2}e^{-ixy}dx=\int_\mathbb{R}xP(x)e^{-x^2/2}e^{-ixy}dx?$$ I'm thinking about using dominated convergence thm, but not sure ho... |
H: Monotonic uniformly continuous function - Unique $f(t) = t$
Let $f:[0,1] \rightarrow [0,1]$ be such that $|f(x)-f(x')| <|x-x'|$ for all $x,x'\in [0,1]$ with $x \not= x'$. Show that there is a unique point $t\in [0,1]$ such that $f(t)=t$.
I noticed that $f$ is a Lipschitz function so that $f$ is a uniformly continuo... |
H: $\mathbb{E}[e^{Xt}] = \mathbb{E}[\mathbb{E}[e^{Xt}\mid Y]] = \mathbb{E}[M_{X\mid Y}(t)]$?
$$\mathbb{E}[e^{Xt}] = \mathbb{E}[\mathbb{E}[e^{Xt}\mid Y]] = \mathbb{E}[M_{X\mid Y}(t)]$$
How do I get the above statement? I don't understand how in the 1st step $e^{Xt}=\mathbb{E}[x^{Xt}\mid Y]$ then in the 2nd $\mathbb{E}[... |
H: Irreducibility in $\mathbb{Z}_7[x]$
Is the polynomial $x^2+3$ irreducible in $\mathbb{Z}_7[x]$?
I am not sure how to even start the problem. I have been looking online trying to find help to teach myself how to do irreducibility and I am not having much luck. Can anyone offer any help as to where I should begin?
AI... |
H: Two idempotent matrix
Let $A,G$ be two $n\times n$ matrix satisfying:
$$A^2=A, GAG=G, im(G)\subset im(A).$$ Prove that $G^2=G$.
I do not know how to prove it.
AI: $im(G) \subset im(A)$,so for any vector $u$,there is a vector $v$ that
$$Gu=Av$$.
So we get:
$$Gu=GAGu=GAAv=GA^2v=GAv=GGu=G^2u$$
.
QED. |
H: Where is the error? Determining why a proof is incorrect.
Why is the following proof incorrect?
I have tried to find out why for a few hours...
aproof http://forosdelecuador.com/proof.png
The answer is:
We aren't really proving the conclusion of each case is valid for the other case.
Consider the case assumptions ... |
H: $a_n = n ( \sqrt{1+ \frac{1}{n}} - 1 )$
i have to show convergence and the limit.
$a_n = n ( \sqrt{1+ \frac{1}{n}} - 1 )$
So far, i tryed to used $a^2 - b^2$, and got
$ \frac {1}{\sqrt{1+\frac{1}{n}}+1} $
how can i go on?
AI: $a_n = n \left( \sqrt{1+ \frac{1}{n}} - 1 \right)= \frac {1}{\sqrt{1+\frac{1}{n}}+1} \to... |
H: Compositum of finite field extensions
If $L_{1}$ and $L_{2}$ are field extensions of $F$ that are contained in a common field, show that $L_{1}L_{2}$ is a finite extension of $F$ if and only if both $L_{1}$
and $L_{2}$ are finite extensions of $F$. (Patrick Morandi, Field and Galois Theory, Exercise $14$ page $14$... |
H: How to find $ \sum\limits_{n=0}^{\infty}\frac{1}{3-8n-16n^2}$?
I have to show, that this series converges and determine its limit.
$$\sum\limits_{n=0}^{\infty}\frac{1}{3-8n-16n^2}$$
So far, my idea was to transform it into a telescoping series, but I don't really know how to do it.
AI: Factorise and use partial fra... |
H: $\left<2,x\right>$ is a maximal ideal of $\Bbb Z[x]$
I want to show that $\left<2,x\right>$ is a maximal ideal of $\Bbb Z[x]$.
My game plan is to use the 3rd isomorphism theorem to somehow get that $Z[x]/\left<2,x\right>$ isomorphic to $Z_2$ (since every this would mean that $Z[x]/\left<2,x\right>$ is a field an... |
H: Definition of rank
In Hatcher P146, the rank of a finitely generated abelian group is defined to be the number of $\mathbb{Z}$ summands when the group is expressed as a direct sum of cyclic groups.
$\mathbb{Q}$ $1$: What does "$\mathbb{Z}$ summands" really mean? For example, $H_2(K \times S^1) = \mathbb{Z \oplus Z_... |
H: Continuity of the sum of functions.
I know that if $f(x)$ and $g(x)$ are continuous, then $h(x)=f(x)+g(x)$ is continuous. What about the converse? Suppose I knew that a function $j(x)$ is continuous and knew that it is composed of a sum of functions. Furthermore, suppose that I can create different expressions whos... |
H: How to prove this equality using boolean algebra?
I have approximately no idea on how to solve the following problem, so any help would very much be appreciated:
$$x' y'+ x y = (x y' + x' y)'$$
I can't figure out how to prove the equalities using boolean algebra
Thank you for your help!
AI: Here's one way. First... |
H: Number of monic polynomials of degree $2$
I know that there are $p^2$ monic polynomials of degree $2$ over the field $\mathbb{Z}_{p}
$ but I want to prove it precisely. Help me a hint to prove that. Thanks a lot.
AI: A degree 2 polynomial has the form $ax^2 + bx +c$. It is monic when $a=1$. How many choices of $... |
H: Pumping Lemma - Clarification of Usage
I'd like to make sure my understanding of the Pumping Lemma is correct.
Consider $L=\{ 0^n1^m2^{n-m}:\, n \ge m \ge 0\}$
I'm going to give 2 solutions to prove that $L$ is not regular. One using "pumping down" and the other using a "cheap trick". I'm not sure whether either so... |
H: verify, that this sequence is cauchy-sequence
Be $ 0 < q < 1, (a_n)_{n \in \mathbb{N}} $ a sequence in $\mathbb{R}$ and $ n_0 \in \mathbb{N} $
One has: $ |a_{n+1} -a_n| \leq q |a_n-a_{n-1}|$
for all $n \geq n_o$. Show, that the sequence $(a_n)_{n \in \mathbb{N}}$ convergates.
I tryed to verify, that its a Cauch... |
H: How do I determine the maximum value for a quadratic equation on an interval?
I need to determine the maximum value for y = ax^2 + bx + c, where I know the coefficients and the upper and lower x values.
Say the input values are:
a = 5
b = 1
c = 2
x lower limit = -5
x upper limit = 5
Given these input, how do I de... |
H: The Moore Plane's topology
In the definition of the Moore plane $X=L{_1}\cup L{_2}$, where $L{_1}$ is the line $y=0$ and
$L{_2}=X\setminus L{_1}$ , I have a problem. In the Engelsking's book, for each $x\in L{_1}$
neghbourhood of $x$, is the form $U(x,1/i)\cup \{ x \}$ where $U(x,1/i)$ be the set of $X$ inside the ... |
H: Complete function space proof
An exercise in my textbook is as follows.
Let $T>0$ and $L\geq 0$. Consider $C[0, T]$, the space of all continuous real valued funcitons on $[0,T]$, with the metric $\rho$ defined by $$\rho(x, y)=\sup_{0<t\leq T}e^{-Lt}|x(t)-y(t)|.$$ Verify that $(C[0, T], \rho)$ is a complete metric ... |
H: Solving $\log(x+2) - \log(x) = 3$
I have work through the whole problem, but I cannot get passed the last step.
The original equation was: $\log(x+2) - \log(x) = 3$
I worked it out to this: $\frac{x+2}{x} = 1000$.
I know the answer is $\frac{2}{999}$ but I don't know how to get there. It's probably really simple, b... |
H: Finding Fourier transform of initial condition
Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let $\hat{u}(y,t)$ and $\hat{f}(y)$ denote the Fourier transform in the $x$ variable ... |
H: Show there are distinct $\xi,\eta$ s.t. $\frac{a}{f'(\xi)}+\frac{b}{f'(\eta)}=a+b$
Let $f:[0,1]\to\mathbb{R}$ be continuous. Suppose $f$ is differentiable on $(0,1)$ and $f(0)=0,f(1)=1$. Show that for any positive real numbers $a,b$, there are distinct points $\xi,\eta\in(0,1)$ s.t.
$$\frac{a}{f'(\xi)}+\frac{b}{f'(... |
H: T- invariant subspace and continuous functions
Let $C(\mathbb R)$ be the vector space of all continuous functions over $\mathbb R$. For $f,g \in C(\mathbb R)$, define the linear map $T_f: C(\mathbb R) \rightarrow C(\mathbb R)$ by $T_f(g)=g \circ f$.
Then, $(T_f(g))(x)=g(f(x))$ for all $x\in \mathbb R$. For every s... |
H: If $a_n\leq b_n$, then $\lim \sup a_n \leq \lim b_n$
Hello I want to make sure my work is correct!
Suppose that $\{a_n\}$ and $\{b_n\}$ are sequences such that $a_n \leq b_n$ for all $n $ and $b_n \to b$. Prove that $\lim \sup a_n \leq b.$
Let $\epsilon > 0$ be given. Choose an $N$ s.t. $|b_n-b|<\epsilon$ for $n\g... |
H: convergate and limit of a series, using geometric series
I have to show, that this series convergates and determine its limit.
$\sum\limits_{n=1}^{\infty}\frac{1}{5^{2n+3}}$
Well, i tried to transform it into a Geometric series, but i dont really know how to do it.
AI: Hint: $$\sum\limits_{n=1}^{\infty}\frac{1}{5^{... |
H: Problem related to continuous complex mapping.
We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$.
1) I want to show that: $\partial(g(D))\subseteq g(\partial D).$
And further, I need two examples:
a) First... |
H: Exact Solution for Logarithmic Equation?
I am faced with this equation, and I don't really know where to begin:
$$x^2e^2 - 2e^x = 0.$$
I usually start these types of problems by factoring out a common term, but I don't see any in this particular example.
AI: Here is a the crucial step
$$ e^2x^2-2e^x =0 \implies \l... |
H: On the image of $\mathbb{R}$ under an entire $f$ satisfying $f(n^{\frac{1}{n}})\in\mathbb{R}$.
Suppose $f$ is entire on $\mathbb{C}$ and $f(n^{\frac{1}{n}})\in\mathbb{R}$. Show then that $f(\mathbb{R})\subset\mathbb{R}$.
Any suggestions on how to even begin this problem? Thanks!
AI: Consider the function $g(z) = ... |
H: Showing that $\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0$ but the matching series does not converge
I want to show that: $$\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0 $$
And also $${\displaystyle \sum_{n=1}^{\infty}{n \choose \... |
H: Series, conditional or absolute? +more
This is for a presentation, so I not just want to solve it, but also be able to talk a bit about it
$$\sum \limits_{n=2}^\infty (-1)^{n+1}\frac{n-1}{n^2}$$
1)Show that the series converges, is it absolute or conditional?
First off we quickly see that the series is alternating ... |
H: How prove this $\sqrt[5]{1782+\sqrt[3]{35+15\sqrt{6}}+\cdots}$ is positive integer numbers.
Prove that
$$\sqrt[5]{1782+405\sqrt[3]{35+15\sqrt{6}}+405\sqrt[3]{35-15\sqrt{6}}}-\sqrt[3]{35+15\sqrt{6}}-\sqrt[3]{35-15\sqrt{6}}\in N$$
This problem from this
My try: let
$$x=\sqrt[3]{35+15\sqrt{6}}+\sqrt[3]{35-15\sqrt{6}}... |
H: Countably many worlds and Universal Sentence.
This is a naive, kind of informal argument.
Suppose we have a language with just one predicate $P$ and constants $a_{1}, a_{2}, a_{3}$ and so on. Suppose also that we have countably many worlds $1, 2, 3$ and so on.
Without loss of generality, we assume that $Pa_{1}$ ho... |
H: A bounded sequence has a convergent subsequence
Let ${a_n}$ be a bounded sequence of real numbers. Prove that ${a_n}$ has a subsequence that converges to lim sup $a_n$.
Thanks!
AI: $\limsup a_n$ is, by definition, the largest limit point of the sequence $a_n$. When you work with Real numbers (or, more generally, wh... |
H: Proof of an infinite sum involving cosines
I have a homework problem that requires proving the Poisson formula from the Jensen-Poisson formula. I have all steps of the problem complete except for proving the identity (for $r < 1$) $$1 + \sum_{n=1}^\infty 2r^n \cos (n(\phi - \theta)) = \frac{1-r^2}{1+r^2 - 2r \cos ... |
H: Markov chains: simple random walk $S_n$
In the case of a simple random walk $\{S_n, n \ge 0\}$ what is $S_n$. I see this for
$P\{S_n=i |\ |S_n|=i_{n-1},...,|S_1|=i_1 \} = \frac{p^i}{p^i+q^i}$
What does this mean? is it: the probability of going to state i after n steps (not sure about that) is $\frac{p^i}{p^i+q^i... |
H: How to prove that if $-1
I am trying to prove an equivalence.
I have already proved that:
$$x^2 + x < 0 \implies -1 < x < 0 $$
using a sub-proof by cases, in which I used the fact that when $xy < 0$, $x$ and $y$ have opposite signs. So I just factored the expression and went from there.
However, I can't seem to fi... |
H: Integral of exponential with second degree exponent
I want to compute the integral $$\int_\mathbb{R}e^{-t\left(y-\dfrac{(at+x)i}{2t}\right)^2}dy$$
I know that $\int_\mathbb{R}e^{-ty^2}dy=\sqrt{\pi/t}$, but here there is an extra imaginary factor. What can I do?
AI: Integrate around a contour. Start at $-R$. Go in... |
H: Integration of exponential with square
It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$?
I can't simply change the variable $y=x+ai$, because then the domain of integration would also change.
AI: Yes it is.... |
H: How do you compute the Fourier Transform of this Unit-Impulse Function?
I have been given this problem from a textbook (not homework, trying to study for an exam. The goal is to find the Fourier transform of this function.
$\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$
Can anyone give me a hint or point me in the rig... |
H: Why are projective modules cohomologically trivial?
Let $G$ be a finite group, $H\subset G$ a subgroup, $k$ a commutative ring, $M$ a $kG$-module, $n\in\mathbb{Z}$, and $\hat{H}\,^n(H,M)$ the $n$th Tate cohomology group as defined in this question, where we consider $M$ a $kH$-module via $kH\hookrightarrow kG$.
We ... |
H: Stationary points
I've a question related to stationary point of inflection. A function
$$ f(x)=ax^5+bx^3+cx $$ has stationary points at $ (-2, 64), (2,-64)~and~(0,0). $ find the values of a, b and c.
I found the first derivative and equate it to 0. I'm getting same equation when I equate $f'(x)$ to $0$ for $f(2... |
H: The residue at $\infty$
I am stuck on the following problem :
$\,\,\,\,$*Problem*$\quad$The residue of an entire function at $\infty$ is $0$.
Solution: True. This follows from the definition of the residue at $\infty$ together with the Cauchy-Goursat Theorem. Another way to see this is to take the Taylor expansio... |
H: Understanding dimension of matrix
Consider the vector space $M_{n\times n}(\mathbb F)$, the set for all $n\times n$ matrices. The basis for this vector space has dimension $n^2$. I read somewhere that the basis has $n$ matrices that have nonzero diagonal entries. I am trying to understand this intuitively, can anyo... |
H: closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$
How can I find a closed form for
$$\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx, n\in\mathbb{N}$$
AI: Substitute $u = -\log{(1-e^{-x})}$, then after a little algebra, you will find that $dx = -du/(e^u-1)$, and the integral becomes
$$\... |
H: How to find $x$ in $\frac{x}{2}-\frac{1}{4{x^2}}=1$
I have the following equation: $\frac{x}{2}-\frac{1}{4{x^2}}=1$ and trying to find $x$ value.
I wasn't able to proceed after having common denominator $\frac{2{x^3}-1}{4x^2}=1$
AI: You now have a cubic as (cross multiply by $4x^2$ and then subtract it from both ... |
H: Primitive Root question
Question:
Show that if $m$ is a positive integer and $a$ is an integer relatively prime to $m$ such that $ord_{m}a = m-1$, then $m$ is prime.
So if you could give me guidance and explanations of answers that would be great.
The textbook answer key this question is from does it by a proof by... |
H: Let $L/K$ be a Galois extension with $Gal(L/K)=A_{4}$. Prove that there is no intermediate subfield $M$ of $L/K$ such that $[M:K]=2$.
Let $L/K$ be a Galois extension with $Gal(L/K)=A_{4}$. Prove that there is no intermediate subfield $M$ of $L/K$ such that $[M:K]=2$.
Please tell me a hint. Thanks a lot.
AI: Such ... |
H: Getting better at proving continuity
How can I get better at distinguishing between continuity and uniform continuity?
If I were given an exam question asking me to prove if the function is uniformly continuous or continuous then I won't be able to answer it correctly. If it were asked explicitly, then I might not... |
H: Polynomial ring with uncountable indeterminates
In Rotman's Advanced Modern Algebra second edition (2010), on page 883 (or on page 905 in its first edition (2002)), in the proof of the existence of localization of a commutative ring $R$ on its multiplicative subset $S$, he writes:
"Let $X=(x_{s})_{s\in S}$ be an ... |
H: Find the four digit number?
Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
AI: HINT:
So, we have $$1000a+100a+10b+b=11(100a+b)$$
$\implies 100a+b$ must be divisible by $11\implies 11|(a+b)$ as $100\equiv1\pmod{99}$
As $0\... |
H: Evaluating the integral $\int_0^{\frac{\pi}{2}}\log\left(\frac{1+a\cos(x)}{1-a\cos(x)}\right)\frac{1}{\cos(x)}dx$
How can I evaluate the following integral?
$$
\int_0^{\pi/2}
\log\left(\frac{1 + a\cos\left(x\right)}{1 - a\cos\left(x\right)}\right)\,
\frac{1}{\cos\left(x\right)}\,{\rm d}x\,,
\qquad\left\vert\,a\,\r... |
H: if matrix such $AA^T=A^2$ then $A$ is symmetric?
let matrix $A_{n\times n}$ is real matrix,such $AA^T=A^2$, The transpose of matrix $A$ is written $A^T$,
show that :
the matrix $A$ is Symmetric matrices
maybe this problem have more methos,because it is know that if matrix $A$ is symmetric,then we have
$AA^T=A^2$,B... |
H: How to solve gamma function integral for 4!
I have been trying to test my knowledge of the gamma function by calculating 1! and 4!. I got the right result for 1! but I cannot get 4! analytically.
To be more specific, I think I am not integrating the gamma function correctly for 4!
I have $4! = \int_0^\infty e^{-t}t... |
H: Is there a way to express matematically that B is closer to A than C is to A?
Is there a way to express mathematically that B is closer to A than C is to A?
Eg. 3 is closer to 1 than 4 is to 1
or -1 is closer to 3 than 10 is to 3
AI: If you accept the possibility that the distance could be the same:
$|A-B|\le |C-A... |
H: Positive integral solutions of $3^x+4^y=5^z$
Are there more integral solutions for $3^x+4^y=5^z$, than $x=y=z=2$ ?
If not, how do I show that? I could show that for $3^x+4^x=5^x$, but I'm stuck at the general case? Any ideas, maybe graphs, logarithms or infinite descent?
AI: I will prove that the only positive int... |
H: Prove or disapprove a propositions
Let p,q and r be three propositions. Prove or disapprove
$(p\to q) \land (q \iff r) \land (p \lor \lnot (\lnot q \lor \lnot r) \equiv p \land q \land r$
so, the way i do is
LHS = $(\lnot p\lor q) \land (\lnot q \lor r) \land (\lnot r \lor q) \land (p \lor q) \land (p \lor r)$
so w... |
H: Evaluate $\int_0^4 |\sqrt{x} - 1|~dx$
I am working on a problem set involving indefinite integrals. Currently stuck in this question:
$\int_0^4 |\sqrt{x} - 1|~dx$
I tried the following substitution:
Let $u=\sqrt{x}$ so, $du=\dfrac{dx}{2\sqrt{x}}$when $x= 4 \Rightarrow u=2$ and $x=0 \Rightarrow u=0$
This is whe... |
H: About stabilizer in group action
Let $X$ be a finite set and $x$ is an element of $X$. Let $G_x$, the stabilizer subgroup, be the subset of $S_X$ consisting of permutations that fix $x$.
The question is
Is stabilizer always a normal subgroup?
I know that $G_x=\{g\in G\mid gx=x, x \in X\}$ and I have proved that ... |
H: Convert 4-sat to 3-sat
I want to know in general how can I convert $4-SAT$ to 3-SAT.
And I have a specific case that if you can help me optimize it to 3-SAT it will be greate.
I want to do this so I be able to use sat solvers programs.
$(C \lor A \lor D) \land (C \lor B \lor D) \land (\lnot C \lor A \l... |
H: Names for certain numbers.
I am wondering if there is names for numbers with the following characteristics:
Numbers that end with 0.
Numbers divisible by 5.
If there are names for numbers with similar characteristics, I would be happy to learn about them as well. :)
Update:
10, 20, 30, ... is called?
5, 10, 15, 2... |
H: $\iint_V |y-x^{2}| \operatorname{d}x \operatorname{d}y$ with $V = [-1,1] \times [0,2]$
it's especially difficult because i don't understand how to integrate absolute value terms.
I only know that if you function, say $x^{2}-1$, is below the $x$-axis i need to integrate $1-x^2$ between the interval $[-1,1]$.
But in ... |
H: A question about weighted forward unilateral shift operators
We define
$$
B(x_{1}, x_{2},...)=(0, \frac{x_{1}}{2}, \frac{x_{2}}{3},...,x_{n})\in l^{2}(N),
$$
How could be shown that that $B$ is a quasinilpotent?
AI: 1) By induction show that
$$
B^n(x_1,x_2,\ldots)=\left(0,0,\ldots,0,\frac{x_1}{2\cdot\ldots\cdot (... |
H: Open sets in product topology
For any two topological spaces $X$ and $Y$, consider $X \times Y$. Is it always true that open sets in $X \times Y$ are of the forms $U \times V$ where $U$ is open in $X$ and $V$ is open in $Y$?
I think is no. Consider $\mathbb{R}^2$. Note that open ball is an open set in $\mathbb{R}^2... |
H: Hard contest type trigonometry proof
Suppose that real numbers $x, y, z$ satisfy:
$$\frac{\cos x + \cos y + \cos z}{\cos(x + y + z)}
=
\frac{\sin x + \sin y + \sin z}{\sin (x + y + z )}
= p$$
Then prove that:
$$\cos (x + y) + \cos (y + z ) + \cos (x + z) = p$$
I am not even getting where to start? Please help.
AI: ... |
H: Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality?
Robin's theorem says that if
$$\sigma(n)<e^\gamma n\log\log n$$
holds for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any counterexamples, then they... |
H: How to find out if a point lie in rectangle?
I have a rectangle in $2D$ space which is determined by $2$ points (each in opposite vertice) $p_1(x,y)$ and $p_2(x,y)$ . How can I find out numerically if a other point $p(x,y)$ is lying inside plane of the rectangle?
AI: Suppose
Left-Top vertex is $P_1(x_1,y_1)$, and ... |
H: If a subgroup of a symmetric group has an odd permutation, then it has a subgroup of index 2.
I want to show that for $n\geq 2$ and $H\leq S_n$ if $H$ contains an odd permutation then it necessarily has a subgroup of index 2. I am not sure how to start, if it has an odd permutation then it might not necessarily con... |
H: Is it possible to subtract a matrix from both side?
I have this equation $AX + B = I$ and I want to find Matrix $X$.
$$(A^{-1})AX + B = (A^{-1})I$$
$$X + B = (A^{-1})I$$
My question is, is it legal to do $X + B - B = (A^{-1})I - B$?
AI: If two matrices are equal, then their differences with $B$ must be equal. How c... |
H: Homeomorphism between $D^n$ and $[0,1]^n$
I know that $D^n=\{x\in \mathbb{R}^n:\|x\|\leq 1\}$ is homeomorphic to $[0,1]^n$, but how to write down homeomorphism? How to find explicit formula?
Thanks in advance.
AI: Let $\| v \|_{p}$ denotes the $p$-norm of $v \in \Bbb{R}^{n}$. In particular,
$$ \| v \|_{2} = [ v_{1}... |
H: Finding -1 to irrational powers
I've been curious about $A=(-1)^r$ when $r$ takes different values. I know that if $r$ was an even number, $A$ will become $1$ and if it was odd, it will become $-1$. Now I want to know what will $A$ be if $r$ were an irrational number like $\sqrt{2}$.Here's what i have done: Accordi... |
H: Parseval equation for a Fourier series
Consider $f(x):=\lvert x\rvert, x\in [-\pi,\pi]$. Then the Fourier series is
$$
f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}.
$$
Now my task is to write down the related Parseval equation.
The general Parseval equation is
$$
\frac{a_0^2}{2... |
H: Argument over an induction proof
My friend gave me a problem.
Define a sequence $\langle a_n \rangle$ by the recurrence relation :$$ a_{n+2} - 6a_{n+1} + 8a_n = 0 $$ and
$a_1 = 4, a_2 = 8 $. Find the general term $a_n$ in closed form.
By inspection, I got the general term as $ a_n = 2^{n+1} $. Then I proved it ... |
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