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H: Lebesgue Integration Question
(Just read the bolded statements if you want to get straight to the point)
This question comes as an extension to one posed in Stein and Sakarchi's Real Analysis, and it is related to the notion that an integral of a positive function is equal to the volume bounded by its graph.
The te... |
H: Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$
Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$,
$$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$
That is, the nontrivial element in $S_2$ swaps the indetermina... |
H: Proving $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without the universal property
Let $F$ be a commutative field, and let $U$, $V$, and $W$ be finite dimensional vector spaces over $F$. How can one prove $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without using the universal property?
AI: N... |
H: Confusion on continuous linear forms
In Friedlander's book "introduction to the theory of distributions" he claimed(on page 35):
"Now the equation $$|\langle u,\phi\rangle| \le C\sum_{|a|\le N|}\sup\{|\partial^{\alpha}\phi|:x\in K\}$$ shows that $$\langle u,\phi\rangle=0$$ if the support of $\phi$ is disjoint from ... |
H: Parametric equation of a cone.
I have a cone with vertex (a, b, c) and base circumference with center $(x_0,y_0)$ and radius R. I can't understand what is the parametric representation of three dimensional space inside the cone. Any suggestions please?
AI: The parametric equation of the circle is:
$$
\gamma(u) = (... |
H: understanding equation $\|f(x)-f(a)-Df(a)(x-a)\|\leq\|x-a\|$
I failed to understand how what do author of my vector calculus textbook arrived at
below equation:
Based on equation below:
$$\lim_{x\to a}\frac{\|f(x)-[f(a)+Df(a)(x-a)]\|}{\|x-a\|}=0$$
Given $\|f(x)-f(a)-Df(a)(x-a)\|$
Thus since f is differentiable at ... |
H: how to show a pair of sequences is Cauchy while proving the Least Upper Bound Property
http://en.wikipedia.org/wiki/Least-upper-bound_property#Proof_using_Cauchy_sequences
This proof of the Least Upper Bound Property defines sequences $A_1, A_2,... $ and $B_1, B_2,...$ recursively:
Let S be a nonempty set of the re... |
H: Circular Permutation
Can someone please explain how to solve circular permutation sums. I just cannot seem to understand them.
eg.
$\text{(4)}$ The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by
$\text{(a)}$ $6!*5!$
$\text{(b)}$ $50$
$\... |
H: Implicit Function Theorem example in Baby Rudin
I am looking at example 2.29 of Baby Rudin (page 227) of my edition to illustrate the implicit function theorem. This is what the example is:
Take $n= 2$ and $m=3$ and consider $\mathbf{f} = (f_1,f_2)$ of $\Bbb{R}^5$ to $\Bbb{R}^2$ given by
$$\begin{eqnarray*} ... |
H: How to solve a differential equation containing an integral term?
I'm asking this question on behalf of another person who is not as familiar as me with computers. He has the following problem to solve and unfortunately Mathcad can't solve such type of equations.
He seeks help to find the function $u = u(x) = ?$ f... |
H: Hectic absolute values? (where $a=ix$ and $b=-ix$)
Where $a=ix$ and $b=-ix$ then what is:
$$|a+b|^2$$
$$|b-a|^2$$
And then is this equality true?
$$|a+b|^2=|a|^2+|b|^2$$
because it seems $a+b=0$!
AI: You have $a+b=0$ and $b-a=-2ix$ so $|a+b|=0$ and $|b-a|=2|x|$
Since $|a+b|^2=0$ and $|a|^2 + |b|^2 = 2|x|... |
H: Evaluating: $\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $
I'm supposed to compute the following limit:
$$\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $$
I'm looking for a resonable approach in this case, if possible. ... |
H: Curious Laplacian inequality ($\Delta f>0 \Rightarrow f$ has no maxima). Additional premises just to confuse students?
In a homework question, we've been given the following things:
Let $B=\overline{B(0,1)} \subseteq \mathbb R^n$ be the closed unit ball, $f : B\to \mathbb R^n$ a $\mathcal C^2$ function and let som... |
H: Analyzing a recurrence relation given by a Toeplitz matrix
Let $p$ be an odd prime, and $M_p$ be the $p\times p$ Toeplitz matrix over $\mathbb{F}_2$ given by $a_0=a_1=1$ and $a_{-p+1}=1$, e.g. for $p=5$ we have
$$M_5=\left[\begin{array}{ccccc}
1 & & & & 1\\
1 & 1\\
& 1 & 1\\
& & 1 & 1\\
& & & 1 & 1
\end{ar... |
H: Proving linearly independence of the functions $t^{i}e^{\lambda_{0}t}$
Following the question I asked here and is:
Let $P(\lambda)=(\lambda-\lambda_{0})^{r}$where $r$ is a positive
integer. Prove that the equation $P(\frac{d}{dt})x(t)=0$ has solutions
$t^{i}e^{\lambda_{0}t},i=0,1,\ldots,r-1$
I now wish to pro... |
H: Finding n in Fibonacci closed loop form
The nth term of the Fibonacci series is given by
$F_{n}$=$\Big\lfloor\frac{\phi^{n}}{\sqrt{5}}+\frac{1}{2}\Big\rfloor$
How do you get the following expression for n from this?
$n=\Big\lfloor\log_{\phi}\Big(F\cdot\sqrt{5}+\frac{1}{2}\Big)\Big\rfloor$
AI: Actually you can't ge... |
H: Limit of $ \sum_{n=m+1}^{\infty} \frac{m}{n\sqrt{n^2-m^2}},m\rightarrow\infty$
I would like to show that $$ \sum_{n=m+1}^{\infty} \frac{m}{n\sqrt{n^2-m^2}}\rightarrow_{m\rightarrow \infty}\frac{\pi}{2}$$
Using integrals:
$$ m\int_{m+1}^{\infty} \frac{\mathrm dx}{x \sqrt{x^2-m^2}} \leq \sum_{n=m+1}^{\infty} \frac{m}... |
H: Show the inequality $\sigma(n)\phi(n) \geq n^2(1-\frac{1}{p_1^2})(1-\frac{1}{p_2^2})\cdots(1-\frac{1}{p_r^2})$
If $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$then ,show the inequality :$$ \sigma(n) \phi(n) \geq n^2(1-\frac{1}{p_1^2})(1-\frac{1}{p_2^2})\cdots(1-\frac{1}{p_r^2})$$
I know the function $\sigma(n) \phi(n)$ ... |
H: How to prove that $A_5$ has no subgroup of order 30?
I need to show that $A_5$ has no subgroup of order 30. Any ideas?
AI: Such a subgroup would have index $2$ and so immediately be normal and non-trivial. It suffices to show that $A_5$ is simple.
Alternatively, show that the class equation is $60=1+15+20+12+12$. O... |
H: Numerator vs. denominator vs. nominator
What is appropriate usage of "numerator", "denominator", and "nominator" to refer to parts of a fraction?
I'm posting this question and answer here because I had little luck finding a clear answer through Google. I realize that it's not really mathematics, but I think it's ... |
H: Solution to a system of quadratics
I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system
$$\alpha_0^2+\alpha_1^2=1$$
$$\beta_0^2+\beta_1^2=1$$
$$\alpha_0\beta_0=1/\sqrt{2}$$
$$\alpha_1\beta_1=1/\sqrt{2}$$
has no solutions. I hav... |
H: Intuition behind Hölder space
What's the point of Hölder spaces (and parabolic Hölder spaces)? I can understand when solving some PDE say $$u_t = au_x + bu_{xx}$$ you may want the solution to lie in $C^{2,1}$ (indexed in space then time) but why do people sometimes want something like $C^{2+\alpha, 1+\alpha}$?
Also... |
H: Permutations With No Identity Elements
Possible Duplicate:
Number of permutations where n ≠ position n
There are $N!$ permutations of the set $\{1,2,\ldots,N\}$
How many of them have zero identity elements?
An identity element is an element that has a value equal to its position. ie When for some $i$, the ith e... |
H: What is the probability of two people sitting side by side each catching a home run in the same game?
Yesterday, in Edmonton, at a baseball game two of my friends caught a home run, in separate plays during the game, and I am wondering where to begin in analyzing what the probability of this happening was? They wer... |
H: Find an elegant proof of a set-theoretic equiality about relations
I am now attempting to prove the following theorem.
I am in half-underway of the proof and it seems I can do it by myself. But the proof I am now constructing is not elegant.
Could anyone provide a nice proof?
Let $n$ is an index set (it may be a f... |
H: Evaluating $\int_{0}^{\frac{\pi}{2}} e^{x+2}\sin(x) \,dx$
Could someone show me how to solve this integral?
$$\int_0^{\frac{\pi}{2}} e^{x+2}\sin(x) \,dx$$
I think that it's improper, but I'm not sure.
I tried to solve by parts, but first I sobstitute
$$e^{x+2} = u$$
And this is what I obtained:
$$\int{u\sin(\log(u)... |
H: How many keypresses does it take to unlock a 4-digit codelock?
Possible Duplicate:
Fastest way to try all passwords
There are $10^4$ different 4-digit codes. If each code takes 4 keypresses to try, then it would take $4*10^4$ keypresses to try all possible codes.
Now the specific codelock i have in mind is of th... |
H: An attempt to prove Tutte's theorem
I'm studying Tutte's theorem. There is a proof in Graph Theory / Diestel. I took a very short glance at it before trying to prove it on my own. I am giving my proof attempt here with a specific question.
For a finite graph $G=(V,E)$ and a subset of vertices $S \subset V$, we let
... |
H: Showing a matrix is symmetric and its eigenvectors are in the columns of $(I - 2vv^T)$
Let $A \in \mathbb{R}^{N \times N}$ have the form $$A = (I - 2vv^T)D(I-2vv^T)$$ with $D = \text{diag} (\lambda_1, \lambda_2, ..., \lambda_n) \in \mathbb R^{N \times N}, v \in \mathbb{R}^n, v^Tv=1$
Show that $A$ is symmetric and ... |
H: Schwartz class estimation.
I have a function $f\in \mathcal{S}$ (i.e of Schwartz class), and I want to show there exist constants $C,k>0$ s.t
$$\|f\|_p \leq C(\sup_{x\in \mathbb{R}} |f(x)| + \sup_{\mathbb{x\in \mathbb{R}}} |x^k f(x)|)$$
for every $ p \in [1,\infty]$.
For $p=1,\infty$ it's obvious from definition, ... |
H: Positive Semi-Definite matrices and subtraction
I have been wondering about this for some time, and I haven't been able to answer the question myself. I also haven't been able to find anything about it on the internet. So I will ask the question here:
Question: Assume that $A$ and $B$ both are positive semi-definit... |
H: Question about conjugacy class of alternating group
This is problem 26 from Grove's "Algebra."
Suppose $K$ is a conjugacy class in $S_n$ of cycle type $(k_1,...,k_n)$, and that $K \subseteq A_n$. If $\sigma \in K$ write $L$ for the conjugacy class of $\sigma$ in $A_n$.
If either $k_{2m} > 0$ or $k_{2m+1} > 1$ for ... |
H: Defining a linear map via kernel and image.
Are linear maps defined in a 1-1 manner by setting their kernel and image? In other words, If I have a vector space, and I define a set to be the kernel of my would-be linear map, and another set to be it's image. Would I get a well defined, one linear map? (Given that my... |
H: What is the sum of this?
What is the sum of this
$$ \{n,n-1,...,3,2,1\}, ...... \{5,4,3,2,1\}, \{4,3,2,1\}, \{3,2,1\},
\{2,1\}, \{1\} $$
I am learning Data Structures and Algorithms now, I want to calculate the time-complexity of a nested loop. I suspect there is term and formula for this pattern.
static int c... |
H: Show a sequence is decreasing
I'm stuck trying to show that the following sequence is decreasing
$$a_{n} = \left(\frac{n+x}{n+2x}\right)^{n}$$ where $x>0$. I've tried treating $n$ as a real number and took derivatives but it didn't lead to anything promising. Any hints would be appreciated.
AI: Fix $x>0$. Our a... |
H: a set which is not measurable in relation to an outer measure
given an outer measure $\eta: 2^X \to [0, \infty]$ we call a subset $E$ of $X$ $\eta$-measurable (measureable with respect to the outer measure), if for every subset $Q$ of $X$, the following holds:
$$\eta(Q) = \eta(Q \cap E) + \eta(Q \cap E^C)$$
so here... |
H: Proving an Inequality about a function.
Assume that $f \in C^2 ([1,4])$ and for any $ \epsilon_1 , \epsilon_2 \in (0,1) $, there exsits $\lambda \in (1+ \epsilon_1 , 4 - \epsilon_2) $ such that $ | f'( \lambda ) | \leqslant | f(4 - \epsilon_2 ) | + | f ( 1 + \epsilon_1 ) | $( In fact this is by using the mean value... |
H: Does this explanation of derangements on Wikipedia make sense?
On the Wikipedia page on derangements, the following description is given about how to count derangements:
Suppose that there are $n$ persons numbered $1,2,\ldots,n$. Let there be $n$ hats also numbered $1,2,\ldots,n$. We have to find the number of way... |
H: Total order in the power set of the real line
Is it possible to define constructively a total order in the power set of the real line ?
AI: You don't. It is consistent with ZF that there is no linear ordering of $\mathcal P(\mathbb R)$.
Andres Caicedo wrote a rather detailed answer to this on this MathOverflow thre... |
H: wedge product with the exterior derivative of the form $ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdots + x_n \, dy_n $.
Write the coordinates on $ \mathbb {R} ^{2n+1}$ as $ \displaystyle{ (x_1 , y_1, x_2, y_2, \cdots ,x_n, y_n ,z)}$. Define the 1-form $ \displaystyle{ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdot... |
H: Solution to simple recursive series
Possible Duplicate:
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
Value of $\sum\limits_n x^n$
I'm working on some math programming and ran into the following recursive series.
$\displaystyle\sum\limits_{i=1}^n a_n$.
Where $a_{n} = Ca_{n-1}$ and $0 \l... |
H: Question on a transfinite construction in algebra
I am studying out of Matsumura's Commutative Ring Theory, and in the first section on modules he proves (following Kaplansky) that every projective module over a local ring is free.
My questions have more to do with an application of transfinite induction than the a... |
H: Flipping Cards Probability
You have a deck of cards, 26 red, 26 black. These are turned over,
and at any point you may stop and exclaim "The next card is red.". If
the next card is red you win £10.
What's the optimal strategy? Prove this is the optimal strategy.
I feel like the optimal strategy is whenever yo... |
H: Maximizing the function $\binom{n+\epsilon}{k}\binom{2n-k}{n}$
I need to find an upper bound for
$$\binom{n+\epsilon}{k} \binom{2n-k}{n}$$ where $\epsilon>0$ and $k,n$ are positive integers with $0 \leq k \leq n$.
I think an upper bound should be with $k=n/2$ or something around there, not sure how to prove it.
AI... |
H: A map from unit ball to itself .
Let $\Omega=B_1(0)$ and $u\in C(\Omega, R^n) \cap C^2(\Omega, R^n)$ be a vector valued map into the unit ball ( ie. $|u(x)|\le 1$ for all $x\in \Omega $, such that
$$|\triangle u(x)| \le |\triangledown u(x)|^2$$ for all $x\in \Omega$
How can i show that $v:= |u^2| $ is a subharmoni... |
H: a ring of fractions which has finitely many maximal ideals
Let $R$ be a commutative ring and $P_1,\ldots ,P_n$ be prime ideals of $R$. If $S=\bigcap_{i=1}^n (R\setminus P_i)$ then show that the ring of fractions $S^{-1}R$ has only finitely many maximal ideals.
The above result will also follow if we can show that t... |
H: Non-Decreasing Digits
A number is said to be made up of non-decreasing digits if all the digits to the left of any digit is less than or equal to that digit.
For example, the four-digit number $1234$ is composed of digits that are non-decreasing. Some other four-digit numbers that are composed of non-decreasing di... |
H: Estimate total song ('coupon') number by number of repeats
If shuffle-playing playlist ×100 resulted in [10 13 10 3 2 2] different songs being repeated [1 2 3 4 5 6] times, what is the estimate for the total number of songs? (assuming shuffle play was completely random)
Update: (R code)
k <- 50 # k number of son... |
H: Orthogonal fitted values
I have two regression models
$$Y=X\beta+\varepsilon,\quad \beta\in\mathbb{R}^k$$
$$Y=Z\alpha+u\quad \alpha\in\mathbb{R}^m$$ it is known that using OLS estimates $\hat{\beta},\hat{\alpha}$ fitted values $\hat{Y}_x,\hat{Y}_z$ are orthogonal. I have to find estimates and fitted values of $$Y... |
H: Evaluation of a product of sines
Possible Duplicate:
Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
I am looking for a closed form for this product of sines:
\begin{equation}
\sin \left(\frac{\pi}{n}\right)\,\sin \left(\frac{2\pi}{n}\right)\dots\sin \left(\frac{(n-1)\pi}{n}\right),
\end{eq... |
H: Probabilistic regression on outliers
I have a given data set $D = \{ x_i, y_i \}_{i=1}^n$ for a regression problem. When I plot the data, it looks like there is an underlying parabola (2nd
order linear model) and some outliers.
I want to design an approach using a probabilistic model with a latent binary variable $... |
H: Relationship: Rank of a matrix $\leftrightarrow$ # of eigenvalues
Can someone tell, why the number of the nonzero eigenvalues (counted
according to their algebraic multiplicities) of a matrix of type $A^{*}A$,
where $A$ is an arbitrary real or complexvalued matrix, is equal
to the rank of $A$ ? Here, in step 2, it ... |
H: Is this the correct counter example?
I encountered the following problem in Berkeley problems in Mathematics:
(Sp84): Prove or supply a counterexample: If the function $f$ from $\mathbb{R}$ has both a left limit and a right limit at each point of $\mathbb{R}$, then the set of discontinuities of $f$ is, at most, cou... |
H: Inherited Morita similar rings
Let $R$ and $S$ be Morita similar rings.
If a ring $R$ with the following property:
every right ideal is injective. How do I prove that the ring $S$ has this property?
If a ring $R$ with the following property:
$R$ is finitely generated. How do I prove that the ring $S$ has this prop... |
H: A particular isomorphism between Hom and first Ext.
Let $R$ commutative ring and $I$ an ideal of $R$.
How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ?
This question is an exercise of the course but has a chance of being false.
AI: There's really only one thing... |
H: What is the correct way to solve $\sin(2x)=\sin(x)$
I've found two different ways to solve this trigonometric equation
$\begin{align*}
\sin(2x)=\sin(x) \Leftrightarrow \\\\ 2\sin(x)\cos(x)=\sin(x)\Leftrightarrow \\\\ 2\sin(x)\cos(x)-\sin(x)=0 \Leftrightarrow\\\\ \sin(x) \left[2\cos(x)-1 \right]=0 \Leftrightarrow ... |
H: properties of a special set in the proof that there is no Lebesgue measure for all subsets of $\mathbb{R}$
i am working through a proof that there could be no measure on $\mathbb{R}$ such that
$\lambda([a,b]) = b - a$
$\lambda(A) = \lambda(A + \{c\})$
First a set $A \subset [0,1]$ is constructed with
$$
\forall ... |
H: How to guess whether a language is regular or not
I have a few languages and I am not given whether they are regular or not.
If I had to prove their irregularity, then it would not have been difficult.
How do I go about finding if the language is regular/irregular and then justifying my answer.
AI: What I do is th... |
H: Does a surjective ring homomorphism have to be surjective on the unit groups?
I know ring homomorphisms map units to units, which made me curious about the following. Suppose $f:R\to R'$ is a surjective ring homomorphism, mapping $1$ to $1'$. Is it necessarily surjective from $U(R)\to U(R')$?
I know if $f(u)$ is a ... |
H: Gauss Multiplication?
$xy = 2^nx_Ly_L + 2^{n/2}(x_Ly_R + x_Ry_L)+ x_Ry_R$
where $n$ is the size of the number (in bits) and $x_{L/R}$ and $y_{L/R}$ are the right and left bit halves of $x$ and $y$..
Initially, you have 4 multiplication problems:
$x_Ly_L$
$x_Ly_R$
$x_Ry_L$
$x_Ry_R$
Gaus's multiplication "simpl... |
H: Take set of values and change scale
I have a large array of variable-integer keypairs. The integer values range from -5 to 5. I'd like to scale that data to a range of 0 to 2. Logically, -5 would become 0, 0 would become 1 and 5 would become 2. How should I go about doing this on a large scale and with different in... |
H: Formal definition of Big-O Notation?
Big O Notation is formally defined as:
Let $f(n)$ and $g(n)$ be function from positive integers to positive reals. We say $f = \theta(g)$ (which means that "$f$ grows no faster than $g$*) if there is a constant $c>0$ such that $f(n) ≤ c ⋅ g(n)$.
Using this definition how is:
... |
H: For a finite field of characteristic $p$, $p-1$ divides $|F|-1$?
Let $F$ a finite field of characteristic $p$. Show that $p-1$ divides $|F|-1$. (We shall see later that $|F|$ is a power of $p$.)
I am able to solve this by first showing $|F|$ is a power of $p$. If $q$ divides $|F|$ for another prime $q$, then by C... |
H: Automorphisms in unit disk
Let $\mathbb{D}=\{|z|<1,\ z\in\mathbb{C}\}$. Are there any other automorphisms in $\mathbb{D}$ except the Blaschke factor $\displaystyle B_{a}(z)=\frac{z-a}{1-\overline{a}z},\ a\in\mathbb{D}$?
I denote with $\overline{a}$ the complex conjugate of $a$.
Thank you for your time,
Chris
AI: $f... |
H: Is the characteristic of simple rings necessarily prime when finite?
I'm familiar that fields and integral domains of finite characteristic have prime characteristic. What about the case when $R$ is a simple ring, so that the only ideals are $0$ and $R$?
I've been trying to use the usual tricks of assuming the char... |
H: Antiderivative simply connected region
Why do analytic functions always have an antiderivative on a simply connected region?
Thank you for your time,
Chris
AI: Cauchy's Theorem tells us that $\displaystyle\int_{\gamma_1} f(x) dz = \displaystyle\int_{\gamma_2} f(x) dz$ whenever $\gamma_1 \gamma_2$ are homotopic, sim... |
H: Coefficients in products and powers of large polynomials
Let $f\in \mathbb{Z}[x_1,\dots,x_n]$ be a polynomial. I want to show that a certain monomial $m$ shows up with non-zero coefficient in the $r^{th}$ power of $f$.
If you're lucky, you can do this as follows:
Regrade $\mathbb{Z}[x_1,\dots,x_n]$ such that $m$ a... |
H: Harmonic Function which cannot be described as real part of a holomorphic function
We define $f:\mathbb{C}\rightarrow\mathbb{C},\ f(z)=\log|z|$. $f$ is harmonic. Why can't we describe $f$ as a real part of a holomorphic (analytic) function?
Thank you very much for your time,
Chris
AI: Any harmonic function on a si... |
H: Homotopy type of 8-holed torus
I would like to determine the homotopy type of a torus with 8 punctures.
(I have come across this problem studying deformations of discontinuous groups of Heisenberg groups...)
Other than trying really hard to visualize, are there any other methods for finding homotopy types of punctu... |
H: Recalling result of tensor product of polynomial rings
Let $k$ be a field (alg closed if you want). Now let $I_{i}$ be an ideal of $k[x_{i}]$ for every $i \in \{1,2,\ldots,n\}$. Is it always true that:
$$k[x_1,x_2,\ldots,x_n]/ \langle I_1,I_2,\ldots,I_n \rangle \cong k[x_1]/I_1 \otimes_k k[x_2]/I_2 \otimes_k \cdots... |
H: Offset Alternating Series
I have the following alternating series that I would like to determine whether it is absolutely convergent, conditionally convergent, or divergent:
$$ \sum\limits^{\infty}_{n=1} \frac{1+2(-1)^n}{n} $$
I have applied some tests and I find it reasonable to conclude that it is divergent.
As ... |
H: Confusion regarding convex and affine set
I am a bit confused regarding convex and affine set. When they mention set, does it mean the set consisting of all the points belonging to the line or shape respectively?
AI: Yes. A line or a shape is typically defined as a set of points, namely the set of points that you t... |
H: Understanding the Analytic Continuation of the Gamma Function
So my book proves the convergence of $\Gamma(z) = \int_0^{\infty}t^{z-1}e^{-t}dt$ in the right half plane $Re(z) > 0$, and then goes on to prove the initial recurrence relation $\Gamma(z+1)=z\Gamma(z)$ by applying integration by parts to $\Gamma(z+1)$:
$... |
H: Analog of Compact Operators
This is kind of vague question, but I'll try to make it more precise.
$T$ is a compact operator on a Hilbert Space, $H$, if $\overline{T(D)}$ is compact in $H$, where of course, $D$ is the closed unit ball in $H$. So we use the underlying Hilbert space in order to define these operators.... |
H: Commutative Algebra without the axiom of choice
It is well known that in a commutative ring with unit, every proper ideal is contained in a maximal ideal. The proof uses the axiom of choice. This fact, and others that are proved using essentially the same argument, anchor a large part of commutative algebra.
Suppos... |
H: Measure on Baire space
Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$?
Some initial thoughts:
Any countably additive, translation-invariant measure has to give measure 0 to... |
H: Example of the use of differentiation rules
I am asked to differentiate the following:
$$g(t) =\sqrt t (t-1)$$
The first thing I do is apply the power rule and the constant rule to the terms inside the parentheses:
$$ g'(t) = \sqrt t(t^0-0) $$
$$g'(t)= \sqrt t(1-0) $$
$$ g'(t)= \sqrt t$$
I am wondering if this is ... |
H: What is the best approach to this problem?
I am stuck in Problem 1.3.18 in Berkeley problems in Mathematics. Without looking back in the section of solutions, I want to ask for a hint.
The problem is as follows:
Let $\{b_{i}\}$ be positive real numbers with $$\lim_{n\rightarrow \infty}b_{n}=\infty$$ and $$\lim_{n\... |
H: $H$ is a subgroup of index 5. $ a \in Z(G) , ord(a)=3$. Show that $ a \in H $
$H$ is a subgroup of $G$ so that $G:H=5$
$ a \in Z(G) , ord(a)=3$
Show that $ a \in H $
What I did:
if $ a \notin H $ then:
$ G/H = \{ H, aH, a^2H, gH, agH \} $ for some $ g \in G-H $
But I don't know how to procceed...
AI: As in Cihan's... |
H: Evaluate $\int_0^\pi xf(\sin x)dx$
Let $f(\sin x)$ be a given function of $\sin x$.
How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$?
AI: If you make the substitution $w = \pi-x$, so that $dw = -dx$, you get
\begin{align}
\int_0^\pi xf(\sin x)dx &= -\int_\pi^0 (\pi-w)f(\sin(\pi-... |
H: An inequality involving integrations.
Assume $ f \in C^2 ( \mathbb R) \cap L^2 ( \mathbb R) , \; f'' \in L^2 ( \mathbb R)$. Assume the situation, $(b-a)^2 \int_{a}^b | f''|^2 \leqslant (b-a)^{-2} \int_a^b |f|^2 $. I want to prove that there exists $ b_2 \geqslant b$ such that $$ (b_2 -a)^2 \int_a^{b_2} |f''|^2 = ... |
H: Generating function of Lah numbers
Let $L(n,k)\!\in\!\mathbb{N}_0$ be the Lah numbers. We know that they satisfy
$$L(n,k)=L(n\!-\!1,k\!-\!1)+(n\!+\!k\!-\!1)L(n\!-\!1,k)$$
for all $n,k\!\in\!\mathbb{Z}$. How can I prove
$$\sum_nL(n,k)\frac{x^n}{n!}=\frac{1}{k!}\Big(\frac{x}{1-x}\Big)^k$$
without using the expli... |
H: How to change integral bounds?
In the following integral I want to change the bounds from $(0, 2)$ to $(-1, 1)$:
$\displaystyle{\int_{0}^{2}(1+x)^3 dx}$
How do I change them? I know that a variable changing is needed, but don't know how to use it to change the bounds. Indeed my question is how do I calculate the fo... |
H: Locally closed irreducible subset of an affine scheme.
I'm self-studying some Algebraic Geometry and I have the following question.
Let us take $X=\operatorname{Spec}A$, where $A$ is a commutative ring. I am trying to show that every locally closed irreducible subset of $X$ contains an unique generic point.
This is... |
H: Can $G≅H$ and $G≇H$ in two different views?
Can $G≅H$ and $G≇H$ in two different views?
We have two isomorphic groups $G$ and $H$, so $G≅H$ as groups and suppose that they act on a same finite set, say $\Omega$. Can we see $G≇H$ as permutation groups. Honestly, I am intrested in this point in the following link. It... |
H: Identifying a map by looking at the pair of topologies that makes it continuous.
Let $\omega_X$ be the set of all topologies on $X$. Given $f:X\rightarrow X$, define $R_f \subset \omega_X \times \omega_X $ as those pairs of topologies on $X$ which make $f$ continuous. For example $\left(\text{Discrete Topology},-\r... |
H: Probability question regarding range
I am stuck with the following question. It is as follows,
| Employed | UnEmployed | Total
Male | 460 | 40 | 500
Female| 140 | 260 | 400
Total | 600 | 300 | 900
If the person is selected and the selected person is ma... |
H: Estimating maximum value of random variable
Suppose I have some random variable $X$ which only takes on values over some finite region of the real line, and I want to estimate the maximum value of this random variable. Obviously one crude method is to take many measurements, lets say $X_1$, $X_2$, $\ldots, X_n$ (wh... |
H: How to show that $\mathrm{SL}(2,\mathbb Z) = \langle A, B\rangle$?
Show, that if $\mathbf{A}= \left( \begin{array}{cc} 1&1\\ 0&1 \end{array} \right)$, $\mathbf{B}= \left( \begin{array}{cc} 0&1\\ -1&0 \end{array} \right)$ and $\mathrm{SL}(2, \mathbb{Z}) := \{ \mathbf{C}\in\mathrm{M}(2\times 2;\mathbb{Z})\, |\, \det... |
H: Construction of $[a,b]$-fold Cartesian product over space of all real-valued functions
I am currently reading "Applied Analysis", which could be found here,
and on page 85 I don't understand example 4.16. There it is said:
Suppose that $X$ is the space of all real-valued functions on the interval $[a,b]$. We may i... |
H: Analytic functions with poles
If $f$ is meromorphic on $\mathbb{C}$ and $a \in \mathbb{C}$, why is $g=\displaystyle \frac{1}{f-a}$ holomorphic (analytic), even if there are poles?
Thank you very much for your time,
Chris
AI: The function $g(z)$ doesn't have to be holomorphic in $\mathbb{C}$, because if there are $z... |
H: differential form is exact on $ U \cup V$
Let $ U ,V \subset \mathbb R ^n$ two simply connected open sets such that $ \displaystyle{ U \cap V}$ is a connected set.
If $\omega$ is a closed 1-form wich is exact in $U$ and $V$ prove that: $\omega$ is exact in $ \displaystyle{ U \cup V}$
AI: We use the fact that a dif... |
H: Find the period of the function $y=[2x]-3*[4x]$
suppose that we have function
$y=[2x]-3*[4x]$
here $[*]$ denotes as a minimum distance till integer.
we are required to find period of this function,first of all i am confused in terms of what does mean minimum distance till integer?could you explain me it?... |
H: a question on meromorphic function
Given that $f:\mathbb{C}\rightarrow \mathbb{C}$ is meromorphic, analytic at $0$ and satisfies $f(1/n)=\frac{n}{2+n}$ I want to know whether I can say $f(z)=\frac{1}{2+z}$, I have considered $g(z)=f(z)-\frac{n}{2+n}$ and zero set of $g$ is $\{\frac{1}{n}\}$ has limit point $0\in\ma... |
H: show if function is even or odd
Suppose that we have equation:
$$f(x)=\frac{2^x+1}{2^x-1}$$
There is question if this function even or odd? I know definitions of even and odd functions, namely
even is if $f(-x)=f(x)$ and odd is if $f(-x)=-f(x)$ and when I put $-$ sign in function, found that this is neither even n... |
H: Free boolean algebra
Consider the following definition:
Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra
$A.$ We say that $A$ is free over $X$ (with respect to $e$) if for
every mapping $f:X \mapsto B$ for a boolean algebra $B$ there is
precisely one homomorphism $\overline{f}:A \mapst... |
H: If $\sum\limits_{k=1}^{\infty}a_k=S $, then $ a_4+a_3+a_2+a_1+a_8+a_7+a_6+a_5+\dots=?$
if we know that $\sum\limits_{k=1}^{\infty}a_k=S$, what can we say about the convergence of $$a_4+a_3+a_2+a_1+a_8+a_7+a_6+a_5+a_{12}+a_{11}+a_{10}+a_{9}+\dots$$ ?
If it does converges, what is the sum (in terms of $S$)?
As per t... |
H: A question Riemann's mapping theorem
Like in Riemann's mapping theorem, we have a conformal mapping $f:\Omega\rightarrow\mathbb{D}$ (so $f$ is bijective and holomorphic), where $\mathbb{D}=\{|z|<1,\ z \in\mathbb{C}\}$ is the set of the open unit disks.
Why does it follow from Liouville's theorem, that $\Omega$ is n... |
H: Commutative law in conditionally convergent series.
Whilst reading the answers to this question, one of the answers states:
"The problem of series that are not absolutely convergent is that you can't make arbitrary rearrangement of the terms."
However, my understanding was that due to the commutative law, we coul... |
H: Jordan's lemma and estimation lemma
I need a clarification on the utility of Jordan's lemma. I think I have understood the theorem and its implications. It basically implies that if you have a function like $g(z) =f(z)e^{iz}$ it suffices for $f(z)$ to tend to zero at infinity in order to have a negligible integral ... |
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