text stringlengths 83 79.5k |
|---|
H: Sufficient condition for a ring to be a product of two rings
In his algebraic geometry notes, Vakil suggests the exercise (remark 3.6.3) of showing that a ring $A$ is a product $A = A_1 \times A_2$ iff $\operatorname{Spec} A$ is disconnected.
His hint is to show that both conditions are equivalent to the existence ... |
H: Expanding partial derivatives
Let $$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$. I want to show that $$\Delta=\frac{\partial^2}{\partial r^2}+\frac1r\frac\partial{\partial r}+\frac1{r^2}\frac{\partial^2}{\partial \... |
H: How to get $\sum_{k=1}^{n-1}n\binom{n-1}{k-1}y^k \bigg(\frac{1-z^k}{k}- \frac{1-z^n}{n}\bigg)$"?
I asked a question here and got answer committing :
$$(1+yz^n)(1+y)^{n-1} - (1+yz)^n=\sum_{k=1}^{n-1}n\binom{n-1}{k-1}y^k \bigg(\frac{1-z^k}{k}-
\frac{1-z^n}{n}\bigg) \tag{1}$$and$$\frac{1-z^{n}}{n}-\frac{1-z^{n+1}}{n+... |
H: Simply connected does not imply contractible. Is there a nice counter example in $R^2$?
The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were interested in a counter example i... |
H: Combinatorics error correcting code
(56) * (36)^4 * (-55) + (35)(67)(-14)^2 mod 17. Find the least non-negative residue of the expression module the given n.
First I just want to make sure I understand what the question wants. To do that, I give a simple case 15 mod 7. 1 is what we looking for right?
Then get back... |
H: probability problem(3 urns)
There are 3 urns. Urn 1 has 2 black and 3 white ball. Urn 2 has one black and 2 white balls.
Urn 3 has 2 black and 1 white.
A person who is blindfolded, picks a ball from urn 1, puts it into urn 2, picks a ball from urn 2 and puts it into urn 3 and finally picks a ball from urn 3 and pu... |
H: When written in decimal notation, every square number has at most 1000 digits that are not 0 or 1. True or false?
When written in decimal notation, every square number has at most $1000$ digits that
are not $0$ or $1$. True or false?
This question is from an admissions quiz, so no calculators should be used.
A... |
H: Mutual tangent lines
Find all points where the curves $f(x) = x^3-3x+4$ and $g(x) = 3x^2-3x$ share the same tangent line.
Graphing them I see that they look like they share a tangent line at $x=2$.
I got the derivatives of both and set them equal to each other and got $x=0$ and $x=2$.
After plugging $2$ back in I g... |
H: Affine sets and affine hull
Mathematically an affine hull can be expressed as
$ Aff[C] = \{\theta_1x_1 + \theta_2x_2 .... \theta_nx_n| x_i \in C \ \ \sum_{i=1}^{n}\theta_i = 1 \}$
Intuitively can anyone explain what this means?
Also, what is a 'hull'?
AI: A little work shows that if $a,b \in \operatorname{aff} C$... |
H: Proving A and Not A are Dependent Events
How do I prove that an event and its complement are dependent on each other? Clearly both outcomes cannot happen, but I don't know how to formally prove it.
AI: Not quite always. Two events $A$ and $B$ are independent if and only if $\Pr(A\cap B)=\Pr(A)\Pr(B)$. This automati... |
H: let (X,d) be a metric space. d is discrete iff X∩X'=∅
Let (X,d) be a metric space.
prove that:
(X,d) is discrete if only if X∩X′=∅,X′ is the set of all limit points of X
AI: HINT: $X\cap A=A$ for every subset $A$ of $X$, so what is $X'$? |
H: Question from "An introduction to measure theory" by Terence Tao
If $(x_α)_{α \in A}$ is a collection of numbers $x_α ∈ [0, +\infty]$ such that $\sum_{α∈A}{x_α} < \infty$, show that $x_α = 0$ for all but at most countably many $α \in A$, even if $A$ itself is uncountable.
AI: Consider the set of $\alpha$ for which ... |
H: Finding the singular value(s) of a given matrix without SVD.
I am struggling on a problem that asks to find the singular value(s) that are unequal to 0 in the following matrix:
$M = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{bmatrix}$
I am not supposed to ... |
H: Quotient of monotone functions is monotone?
Suppose $f,g$ are monotone( say increasing) and differentiable and nonnegative. Both go from $\mathbb{R}^{\geq 1} \to \mathbb{R} $
Is $\frac{f}{g}$ also monotone ?
AI: Counterexample:
Let $f(x) = e^x+5, g(x) = x$.
$$\frac{f}{g}(x) = \frac{e^x+5}{x}$$
Can you tell me why ... |
H: Integral coordinates Proof
Nine distinct points with all coordinates integral are selected in the space. Prove that the line segment with ends at certain two of these points contains in its interior a point with all coordinates integral.
AI: HINT: Look at the coordinates modulo $2$ and apply the pigeonhole principl... |
H: Pigeonhole Principle Proof
2004 flies are inside a cube of side 1. Show that some 3 of them are within a sphere of radius 1/11.
I am not sure how to begin the proof especially since we are asked to work on a sphere rather than the given cube.
AI: I would try and see if I can cover the cube with $1001$ balls of radi... |
H: What will be the closed formula for the following recursive function?
What will be the closed formula for the following recursive function?
F(n) = F(n/2) +1 if n is even
F(n) = F(n-1) + 1 if n is odd
F(1) = 0
How do we generate closed formula for such recursive functions?
Thanks.
AI: $\newcommand{\+}{^{\dagger}}%... |
H: Algebraic Basis vs Hilbert basis
I am confused between algebraic basis and hilbert basis. How do they differ exactly?
Can you give me examples (possibly in infinite dimensions) on when they are the same and when they are not the same?
Thanks in Advance
AI: A basis $B$ of a vector space $V$ allows you to express all... |
H: A finite set always has a maximum and a minimum.
I am pretty confident that this statement is true. However, I am not sure how to prove it. Any hints/ideas/answers would be appreciated.
AI: Let $S = \{s_1, \ldots,s_n\}$ be a nonempty finite set of size $n > 0$. We will show by induction on $n \in \mathbb N$ that th... |
H: Good Pairs in Algebraic Topology
Hatcher’s book says that $ \left( \mathbb{D}^{n},\mathbb{S}^{n - 1} \right) $ is a good pair; that is, there exists an open neighborhood $ V $ of $ \mathbb{D}^{n} $ containing $ \mathbb{S}^{n - 1} $ that deformation retracts onto $ \mathbb{S}^{n - 1} $.
What is the open neighborhood... |
H: Does every tree have an upper portion that is a non-empty chain?
Given a poset $P$, call $A \subseteq P$ an upper portion iff $A^c < A$, by which I just mean that for all $a \in A^c$ and all $b \in A$ we have that $a < b$. Then every upper portion is upward closed. Proof. Suppose for a contradiction that $a \in A$ ... |
H: Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional
Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed.
To prove that $S+T$ is closed I have to show that for any limit point $x$ of $S+T$, $... |
H: Generator of a subgroup of a cyclic group
Let $G$ be a cyclic group, and let $x \in G$ be its generator such that $|x| > 1$. Suppose $H$ is a nontrivial subgroup of $G$. Prove that if $m$ is the minimum positive integer such that $x^m \in H$, then $x^m$ generates $H$.
My try: Suppose $|x| = n > 1 $. We want to s... |
H: Does exists a function that has the following o-notation properties?
Let $p>0$ be any real positive. Does there exist a function $f(x)$ which is $o(|x|^p)$ in $x=0$ yet not $O(|x|^{p+\varepsilon})$ for any $\varepsilon>0$ ?
AI: $f(0) = 0$ and on $[1/2^n, 1/2^n-1[$, $f(x)=x^{p+1/\sqrt{n}}$
The idea is to have someth... |
H: find the limit of a sequnce
I need to find the limit:
$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {{{(a + {1 \over n})}^2} + {{(a + {2 \over n})}^2} + ... + {{(a + {{n - 1} \over n})}^2}} \right]$
any ideas here? I've tried to use "squeeze theorem" but with no luck..
AI: Let $u_n={1 \over n}\left[ {... |
H: Prove that the function is differentiable at (0,0)
The function is shown below. Its not a very complicated function.
$$ f(x,y)=\sqrt{9-x^2-y^2}$$
I was wondering is it sufficient to say that since $f_x(0,0)$ and $f_y(0,0)$ are both zero and since $f_x$ and $f_y$ are continuous (by finding the limit as it appraoche... |
H: A Milnor Differential Topology Excercise
If $m<p$, show that every map $f:M^m\longrightarrow\ S^p$ is homotopic to a constant, where $M^m$ is smooth manifold of dimension $m$.
I tried to show that $M^m$ is contractible or convex, but I couldn't. Any Idea would be helpful.
AI: Expanding on John's hint.
If $f$ is no... |
H: The group of rigid motions of an icosahedron.
Prove that group of rigid motions of icosahedron is isomorphic to $A_{5}$.
Can you help me to prove this?
What I have done is shown that the order of the group of rigid motions of icosahedron is 60, which is same as $A_5$.
AI: You can do this by considering the dodeca... |
H: How to determine sides lenght of irregular 11-sided polygon given that 10-sides are equal and polygon must be described inside circle.
I have stuck on problem. I have to draw 11-sided polygon. 10 sides of polygon must be equal, while one side must be longer (at least twice longer than any other side).
To make thing... |
H: $m \in \{2,6,42,1806,...\} $ - a problem of sum-of-$m$'th powers modulo $m$
(continuing the work for an answer for a question here in MSE and also in MO)
I'm (re-)viewing the function
$$ f(m) = \sum_{k=0}^{m-1} k^m $$
considering its residue modulo $m$:
$$ r(m) \equiv f(m) \pmod m $$
It is easy to see why for odd... |
H: Linear PDE of degree 2: general form and an example
As the general form of a linear PDE of degree 2 we wrote
$$
(Lu)(x):=\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum_{i=1}^{n}b_i(x)\frac{\partial u}{\partial x_i}+c(x)u=f(x)
$$
Now I have the PDE
$$
(1+x^2)\frac{\partial^... |
H: Why is this inequality true in this proof?
I've been studying Spivak's Calculus on Manifolds and there's one proof he gaves that made me confused. Is probably a very basic fact, however, I'm not grasping why this should be true. The Lemma being proved is: "Let $A\subset\Bbb R^n$ be a closed rectangle and $f: A\to \... |
H: Elementary set questions problem
In an exam,there are 150 students. 40 passed in paper A & B.40 passed in
paper B & C. 30 passed in paper A & C and 10 passed in all three.How many
students passed in paper B only?
and also
If no student failed find the number of student who passed in exactly one paper.
Can ... |
H: Solving $\frac{dx}{dz}-\frac{2x}{z}=1$
Please can someone solve this?
$$\frac{dx}{dz}-\frac{2x}{z}=1$$
Please this is only part of my homework question. I am stuckwith here. Please teach me this solution thank you:)
AI: To elaborate on Amzoti's suggestion:
Here is a nice step-by-step solution to help you work th... |
H: A problem on second order differentiation
If $y=\sin x$, then find the value of $$\frac{d^2(\cos^7 x)}{dy^2}$$
I have no idea on how to proceed in this problem. Please help.
AI: Hint: notice that
$$ \cos x = \sqrt{1-y^2}$$ |
H: Discrete valuations of a functional field have discrete valuation rings.
Theorem: If $\nu:F\to\mathbb R\cup\{\infty\}$ is a valuation of a functional field, then the set $$\mathfrak O_{\nu}=\{x\in F: \nu(x)\geq 0\}$$ is a local ring with maximal ideal $$\mathfrak M_{\nu}=\{x\in F: \nu(x) > 0\}$$
and a quotient fiel... |
H: Show that set is null set
Let $\mu$ be a measure on $(X,\mathcal A)$. Let $(A_k)_{k\in\mathbb N}$ be a sequence of sets in $\mathcal A$ such that $\sum_{k\in\mathbb N}\mu(A_k)<\infty$. Let $A:=\{x\in X:x\in A_k $for infinitely many $A_k\}$.
Show that $\mu(A)=0$ and that this does not hold if we don't ask for $\sum_... |
H: Finding the roots of 4096x^3-10496x^2+152576x - 961=0 (1 root and 2 complex)?
I don't know how to find the roots of 4096x^3-10496x^2+152576x - 961=0
I try using wolfram and http://en.wikipedia.org/wiki/Cubic_function. I don't really understand it can someone please explain how it is done?
AI: A cubic equation in th... |
H: Average value of recurrent function.
Given a function f(x) = a*f(x-1) where a is a number between 0 and 1,
what is the average value of f(x) for x >= 0?
Clarification:
f(1) is a constant, say 1
f is only defined for integer inputs
Disclaimer: I'm still in high school, so it's possible that I'm asking/saying somethi... |
H: A problem on finding dy/dx
If $a+b+c=0$ and $$y=\frac{1}{x^b+x^{-c}+1}+\frac{1}{x^c+x^{-a}+1}+\frac{1}{x^a+x^{-b}+1}$$then $\frac{dy}{dx}$=?
The only way which I can think of solving this is by differentiating each term. However, is there a simpler way?
AI: $$y=\frac{1}{x^b+x^{a+b}+1}+\frac{1}{x^{-a-b}+x^{-a}+1}+\f... |
H: Roots to an equation using analysis
Suppose that $a_{1}<a_{2}<…<a_{n}$. Prove that the equation
$ \frac{1}{x-a_{1}}+\frac{1}{x-a_{2}} +…+\frac{1}{x-a_{n}}=c$ has exactly $n-1$ roots if $c=0$ and $n$ roots if $c\neq 0$.
I am absolutely not sure but I think I need to use Rolle's theorem on this. Multiply both sides ... |
H: Is the derivative in $C(\overline{\Omega})$?
Let $\Omega\subset\mathbb{R}^2$ a bounded domain and consider $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I would like to know if then
$$
1+\frac{\partial u}{\partial x}\in C(\overline{\Omega})?
$$
My answer is: YES, because:
1) $f(x):=1$ is of course uniformly ... |
H: Proof Concerning Sum of Binomial Coefficients
Could anybody provide a proof of the following identity identity:
$$ \sum_{n=0}^{N-1}\binom{N-1+n}{n}=\binom{2N-1}{N}$$
possibly using Symmetry property and Pascal's rule (or another easier way):
$$\binom{a}{b}=\binom{a-1}{b-1}+\binom{a-1}{b}$$
AI: $$\begin{eqnarray}
\s... |
H: A problem on limit involving various functions
Find the value of $$\lim \limits_{x\to 0} \frac{\tan\sqrt[3]x\ln(1+3x)}{(\tan^{-1}\sqrt{x})^2(e^{5\large \sqrt[3]x}-1)}$$
Applying L'Hospital's rule does not seem to simplify the expression.
AI: Note that
$$\lim \limits_{y\to 0}\frac{\tan y}{y}=\lim_{y\to 0}\frac{\tan^... |
H: Regular expression for language
Let's have a language $L=\{\omega\in\{a,b,c\}^* | \omega $ contains $ab$ and does not contain $ba\}$ make a regular expression for this language.
I've ended up with this one
$$(a^*(b^*+(cc^{*}a^*))^*)^*ab(b^*+(cc^{*}a^*))^*$$
Is this correct? Does it generate all words containing $ab... |
H: Remainder modulo 8
A number is given: $1234513151313653211415515253$
Is there any way to find out the reminder when it divided by 8? What will be happened if I use MOD rules here?
AI: When you divide by $8$, you only need to worry about the last 3 numbers. So $1234513151313653211415515253$ mod $8$ becomes $253$ ... |
H: In statistics, what is the meaning of $Z_{0.3}$
What is the meaning of $Z_{0.3}$ and how do I calculate it?
I know it was calculated this way:
$$Z_{0.3} = -Z_{0.7} = -0.52$$
I tried to follow the General Distribution table but I can't seem to find the way to get this.
It's quite hard to look for information about a... |
H: About using functor to reduce problem into the category Sets
In the answer to this question, the author said
Now as for the proof of the Lemma, just use the Yoneda Lemma to reduce it to the case of the category of sets, where you can really see this equation immediately.
Is it suggesting that to prove any propos... |
H: Is it possible to use limit to find collision between two subjects
Let's say that in a game I have a subject $A$ placed at $(0, 5)$ and a subject $B$ placed at $(5, 0)$.
Accordingly, their distance is $5\sqrt{2}$. The subject $A$ will walk that distance at a constant velocity in a infinite loop. The subject $B$ wil... |
H: Quantative comparison
Any tips for this question, that which quantity will be greater?
AI: Quantity A: There are $r \times s \times t$ possible (distinct) combinations of one entree, one side, and one dessert.
Compare that to quantity $B$. |
H: Prove that $3n +5m = 12$ for any two natural numbers.
Prove that $\exists n,m \in \mathbb{N}$ such that $$3n+5m=12 $$
This is clearly false, but I am not sure how to conduct a proof stating it is false. Should I just give examples with $n = 1,2,3$ and then pick $m$s afterwards?
AI: Since it asks for solutions in $... |
H: Catching the right bus
If buses go every $1$ minute, every $2$ minutes, every $5$ minutes and every $15$ minutes and you turn up at the bus stop at a random point, what is the probability that the next bus to arrive is one of the every $15$ minutes ones?
We can assume that the timetable is set to make this as unlik... |
H: A problem on distribution functions
I have a quick question here. From the definition of a distribution function (DF),
$\text{A real-valued, nondecreasing, right continuous function} \; F \; \text{defined on} \;\left(-\infty,\infty\right)\;\text{satisfying}$$$ F(-\infty)=0 \; \text{and} \; F(\infty)=1 $$ $\text{is... |
H: Pade Approximations convergence acceleration
Why Pade Approximatoins accelerate the convergence of series?
Generally speaking, what is there an advantage in the sence of convergence acceleration using rational interpolation?
Thanks much in advance!!!
AI: To expand in vadim123's answer. With truncations of a series... |
H: How to simplify $(5-\sqrt{3}) \cdot \sqrt{\left(7+\frac{5\sqrt{3}}{2}\right)}$
How to simplify this:
$$(5-\sqrt{3}) \sqrt{7+\frac{5\sqrt{3}}{2}}$$
Dont know how to minimize to 11.
Thanks in advance!
AI: Hint: $7+\dfrac{5\sqrt{3}}{2} = \dfrac{1}{2^2}\left(5^2+\left(\sqrt{3}\right)^2+2\cdot5\cdot\sqrt{3}\right)$. |
H: Show that any $n+1$ vectors in a $n$ dimensional vector space forms a linearly dependent set
$V$ is an $n$-dimensional vector space. Show that $n + 1$ vectors in $V$ form a linearly dependent set.
Here is how I am approaching it:
Let $\dim V = n$, which implies that $S$ is a linearly independent set of vectors such... |
H: Solve a differential equation using the power series method
Problem
By assuming a power series solution of the form $$y(x) = \sum_{m=0}^{\infty} c_mx^m , \quad c_0 \not =0 $$
Show that the equation $ 2y'+xy=x $
has general solution $y(x)=1+Ae^{-x^2/4}$ where A is a constant.
[Hint: you may use without p... |
H: Prove that $x$ and $x+1$ are coprime numbers
Given $\{x \mid x > 1\}$, how do I prove that any given $x$ and $x+1$ are coprime?
AI: If $y$ divides $x$ and $x+1$ then it divides $(x+1)-x=1$. Conclude. |
H: prove that the operator is compact.
Let $H$ be a Hilbert space over $\mathbb C$, and $\{f_j\}$ a orthonormal set in $H$. Let $t_j\in \mathbb C$ such that $\displaystyle \lim_{n\to \infty} t_j =0$ i.e $(t_j)_{j\in \mathbb N}\in c_0$. Show that the operator $T:H\to H$ defined by:
$Tx=\sum t_j (f_j \cdot x)f_j$ is com... |
H: What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?
The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed in severa... |
H: Mathematical model for magic square
As I spent some time on magic squares, it seems like the magic squares can be formed only with a odd number of rows/columns? Is it that.? If so why? is there a mathematical model that explains magic square?
AI: It's not true that magic squares can only be formed with an odd numbe... |
H: is $p \land (p \lor q)$ a tautology?
I would just like to know whether my work is correct before I continue on with the rest of the questions.
$$p \land (p \lor q)$$
$$p \land (\lnot p \rightarrow q)$$
$$(p \land \lnot p) \rightarrow q$$
$$F \rightarrow q$$
with this, I'm going to say it is not a tautology.
AI: No,... |
H: Marginal distribution of two jointly distributed Random Variables that are dependent
Find the Marginal Distribution of $X$ (note: $c=1/8$).
Clearly, $X$ is dependent on $Y$, since the value Y takes restricts the domain of $X$. Would this affect how the marginal distribution is computed? Or would you just do the us... |
H: Find the basis of a polynomial vector space where the derivative(pi)=0
I am working in the $P_3$ space.
Let W = {p($t$) $\in P_3$: p'($\pi$) = 0}
and need to find the basis for W
What I've done so far:
p($t$) = a$t^3$ + b$t^2$ + c$t$ + $d$
p'($t$) = 3a$t^2$ + 2b$t$ + $c$
p'($pi$) = $3a(\pi$)$^2$$ + 2b(\pi$) + $c$ =... |
H: Hartshorne, exercise II.2.18: a ring morphism is surjective if it induces a homeomorphism into a closed subset, and the sheaf map is surjective
Let $\phi:A\to B$ be a ring morphism, and let $f:X=Spec(B)\to Y=Spec(A)$ be the induced map of affine schemes.
I'm trying to show that if $f$ is a homeomorphism onto a clos... |
H: Counting primes by counting numbers of the form $6k \pm 1$ which are not prime
Again, pondering on twin primes, I came upon the following result. It baffles me a bit, so could someone give more intuitive reasoning why it works.
First, define a function $P_6$ as $$P_6(n)=\begin{cases}
0, \ \ 6n-1 \not\in \mathbb P... |
H: How to tell $\overline {(a,b)}=[a,b]$, $\overline{\{\frac{1}{n}:n=1,2,3,\ldots}\}=\{\frac{1}{n}:n=1,2,3,\ldots\}\cup \{0\}$
Morning reading a book that deals with metric spaces noticed this fact: Tell that $$\overline {(a,b)}=[a,b],$$ $$\overline{\{\frac{1}{n}}\}=\{\frac{1}{n}\}\cup \{0\}.$$
I do not know much abou... |
H: (New - Updated 3/1/13) Is this argument correct in showing that this field extension does not have the stated degree?
I have been asked to prove that $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}]=4$. I have some problems believing this to be true and have the following argument that assumes it to be true and seemingl... |
H: French metro metric: difficulty to prove that $d(x, y) = 0\iff x = y$.
I think that it is related to the special definition of the metric in my book:
$$d(x, y) = \begin{cases}||x - y||,\mbox{ if }\exists \alpha\in\mathbb{R}: \alpha x + (1-\alpha) y = 0;\\ ||x|| + ||y||, \mbox{ otherwise.}\end{cases}$$
This way, for... |
H: Prove that any odd number can be expressed as $4n+1$ or $4n+3$
Prove that any odd number can be expressed as $$4n+1$$ or $$4n+3$$
I can see that this is true, but I am not certain on how to make a formal proof.
AI: Hint: Every odd number can be written as $2k+1$. Now $k$ can be even ($k=2n$) or... |
H: How to show that a function is in O($h^m$)
i try to find the biggest m that
$$f(h) = \frac{e^h-e^{-h}}{2h}-1$$
is $\in O(h^m)$ ($h \to 0, h > 0$)
I thought i have to use the definition.
So i wrote this:
$$ limsup_{h \to 0} \left | \frac{\frac{e^h-e^-h}{2h}-1}{h^m} \right | = limsup_{h \to 0} \left | \frac{e... |
H: Is there an example of fields $F \subseteq K \subseteq L$ where $L/K$ and $K/F$ are normal but $L/F$ is not normal?
I'm searching(I searched this site first) for example of fields $F \subseteq K \subseteq L$ where $L/K$ and $K/F$ are normal but $L/F$ is not normal. Presenting some fields just for $F$ or $L$, instea... |
H: Mapping on Cauchy Sequences
How can I show that $f(x)=x^2$ maps Cauchy sequences to Cauchy sequences?
That is the function preserves the Cauchy property.
AI: It is a fact that a sequence in $\mathbb R$ is Cauchy iff it is convergent.
Now if $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence then it is a convergent sequen... |
H: Setting up double integrals in polar coordinates
I am currently studying double integrals in polar coordinates and I'd like clarification on some issues I'm having.
Suppose I have a disk centered at $(1,0)$ with radius 1. How do I set up the integrals by
(a) slicing up the disk where $\theta$ is constant
(b) sli... |
H: Show that $f=g$, if $f(z)=g(z)$ for $z\in dA$ with $A$ bounded region
Let $A$ be a bounded region, $f$, $g$ continuous functions of $\bar{A}$ in the complex. Suppose that these functions are holomorphic in the region and agree on the border. Prove you are the same.
I think this problem is solved by the maximum modu... |
H: Ideal of a ring of matrices
Let $F$ be a (commutative) field.
$R$, the matrices of the form
$
\left( \begin{array}{ccc}
a & b \\
0 & c \\
\end{array} \right)$ with $a,b,c \in F$, is a subring of the ring $M_2(F)$ of 2 x 2 matrices with entries in $F$.
Give a non-trivial two-sided ideal $I$ of $R$.
I read a lot ... |
H: Probability task (Find probability that the chosen ball is white.)
I have this task in my book:
First box contains $10$ balls, from which $8$ are white. Second box contains $20$ from which $4$ are white. From each box one ball is chosen. Then from previously chosen two balls, one is chosen. Find probability that th... |
H: If $G, H, K$ are divisible abelian groups and $G \oplus H \cong G \oplus K$ then $H \cong K$
This is an exercise in Hungerford. But can somebody explain why is the following not a counter-example?
Let $G$ be the direct sum of $|\mathbb{R}|$ copies of $\mathbb{Q}$. Let $K$ be the direct sum of $|\mathbb{N}|$ copies ... |
H: Is the following true: $\forall x\in\mathbb R: \exists y\in\mathbb R: x^2+y^2=-1$
How would I solve the following question. And determine if its true or false.
1.$\forall x \in R , \exists y\in R, x^2+y^2=-1$
2: $\exists x\in R,\forall y \in R, x^2+y^2=-1$
For the first one I think I can justify it is false.
As for... |
H: Make the vector $[1,1]$ turn of an angle - $\pi/4$ , with complex numbers
We have $[1,1]$ and $\theta = -\pi/4$
here is my attempt:
$(\cos(-\pi/4) + i \sin(-\pi/4)) * (x+iy)$ = $(\sqrt{2}/2 - i \sqrt{2}/2) (1+i)$
= $\sqrt{2}/2 - i^2\sqrt{2}/2 $
= $[\sqrt{2}/2 + \sqrt{2}/2]$
I'm not sure if I'm adding up the parts c... |
H: Advantages to continuity at a point
A scalar field $f : \mathbb{R}^n \to \mathbb{R}$ is said to be continuous at a point $\boldsymbol{a}$ if
$$ \lim_{\boldsymbol{x} \to \boldsymbol{a}} f(\boldsymbol{x}) = f(\boldsymbol{a}) $$
So in other words, $f$ has to be defined at $\boldsymbol{a}$ and also has to have a limit ... |
H: Proving orientability of manifold
I don't know how to prove the following:
$RP^n$ is orientable manifold if n is odd?
Any help is welcome.
AI: Hint: Decide when the antipodal map on $S^n$ is orientation-preserving. |
H: Proving a few things about $ L^{p} $-spaces
I am new to $ L^{p} $-spaces and am trying to prove a few things about them. Therefore, I would like to ask you whether I have gotten the following right.
Prove that $ {L^{\infty}}(I) \subseteq {L^{p}}(I) $ for all $ p \in (0,\infty) $, where $ I = [a,b] $ is a closed bo... |
H: Why are we Multiplying here instead of Adding?
Three small towns designated by $A$, $B$ and $C$ are interconnected by by a system of two-way roads as described below
[hopefully the following answer is enough info without picture]
part a)In how many ways can Linda travel from town $A$ to town $C$?
I calculated... |
H: Is this a set in New Foundations theory of sets: A={x∈X: x∉f(x)} when X=universal set and f(x)=x?
Does the formula A={x∈X: x∉f(x)} define a set in New Foundations theory of sets (NF) when X=universal set and f(x)=x? If not: WHY?
It can be seen that many elements of universal set (X) satisfy the condition of the for... |
H: About minimal prime ideals and varieties
Let $W$ be a variety, and $I=\mathbb{I}(M)$, then we have
$$
I=\operatorname{rad}(I)=P_1\cap\cdots\cap P_n
$$
where $P_i$'s are minimal prime ideals containing $I$.
Thus we have
$$
W=\mathbb{V}(I)=\mathbb{V}(P_1\cap\cdots\cap P_n)\supset\mathbb{V}(P_1)\cup\cdots\cup\mathbb{... |
H: True or false: $\forall x \in \Bbb R,\exists y\in \Bbb R,y+x=x+y$
How would one tell if this is true or false.
1: $\forall x \in \Bbb R,\exists y\in \Bbb R,y+x=x+y$
2: $\exists x \in \Bbb R,\forall y \in \Bbb R,y+x=x+y$
For the first I think it would be true because if $x=8$ then
$y+8=8+y$
$y=2$
So to justify it ... |
H: How do I take the limit with invoking L'Hospital's rule? $\lim_{x \to 1}\left(\frac{x}{x-1}-\frac{1}{\ln(x)}\right)$
Need to take the limit:
$$\lim_{x \to 1}\left(\frac{x}{x-1}-\frac{1}{\ln(x)}\right) = \lim_{x \to 1}\left(\frac{x\cdot \ln(x)-x+1}{(x-1)\cdot \ln(x)}\right)=(0/0)$$
Now I can use L'Hospital's rule:
$... |
H: Inclusion $[0,1]\rightarrow\mathbb{C}$ generates $C^{1}[0,1]$ as a Banach algebra
I am trying to show that the inclusion map $x:[0,1]\rightarrow\mathbb{C}$ generates $A=C^{1}[0,1]$ as a Banach algebra. The first thing that occurred to me was to try using Stone-Weierstrass but somehow I'm not getting it.
Also, I wan... |
H: Right invertible element
My question is as follows:
$a,b \in L$ where $L$ is a ring.
We are given that $ab=1$.
I need to prove or disprove that $a$ is invertible (meaning, there is an element $x \in L : ax=xa=1$).
But how can we say that for sure? just because $ab=1$ doesn't mean that $ba=1$, ba could be something ... |
H: prove of disprove :'For every $x\in G$ there exists some $y\in G$ such that $x=y^2$, where $G$ is a group."
I am working on a question in the book: A Book of Abstract Algebra by Pinter. The question asks to prove or disprove the following statement:
For every $x\in G$ there exists some $y\in G$ such that $x=y^2$, w... |
H: Quick logic question about $P\leftrightarrow Q$, terminology
I know that if we have $P\rightarrow Q$, $P$ can be called the antecedent and $Q$ the consequent or conclusion. If we have $P \leftrightarrow Q$, are there names for what we would call $P$ and $Q$ here?
I am writing a proof where I am proving $P \leftrigh... |
H: Must the intersection of a certain decreasing sequence of open sets be non-empty
I am trying to show the open mapping theorem. As I was trying to prove it, I made the following conjecture:
Let $X$ be a complete metric space. Let $V_1,V_2,...$ be a sequence of
open sets in $X$ such that: $$V_1\supseteq \overline{... |
H: Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$.
Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$. Also suppose there exist $... |
H: Showing almost sure divergence
Doing some exercises as preparation for an upcoming exam, but im sort of stuck at this exercise: \
Assume that $X_1,X_2,...$ is an i.i.d sequence, such that $X_1 \sim \mathcal{N} (\xi , \sigma^2)$, with $\xi>0$ . Define
$$
S_n=\sum_{k=1}^n \frac{X_k}{k}
$$
Show that $X_n \to \infty $... |
H: Finding solutions for $x^3\equiv 1 \bmod n$
How can I find all the numbers mod n such that $x^3\equiv 1 \bmod n$?
Does it help if n is prime?
AI: If $n$ is a power of a prime, $n=p^r$, $r\gt1$, then, if you can solve $x^3\equiv1\pmod p$, you can lift $x$ to a solution modulo $n$ by Hensel's Lemma, q.v.
If you can ... |
H: Can any subset of $x$ be moved out of $x$?
Let $x$ be a set and let $y\subset x$. Does there exist a set $z$ such that:
(1) $z\cap x=\emptyset$ and
(2) there exists a bijection $y \to z$ ?
It is quite intuitive that the answer should be yes. My first attempt was to take a set $x'\notin x$ and to consider $z=y\ti... |
H: Set of all functions with a Lipschitz Condition
I could not find a way to start with, let alone solution. Any help would be greatly appreciated.
Let $M_K$ be the set of all functions f in $C_{[a,b]}$ satisfying a Lipschitz condition i.e., $|f(t_1)-f(t_2)|\leq K|t_1-t_2|$ for all $t_1,t_2\in[a,b]$, where $K$ is a f... |
H: Possible Class equation for a group
Determine the possible class equation for a group of order 21?
Until now I have found the following:
$1+3+3+7+7$
$1+1+1+3+3+3+9$
$1+1+1+1+1+1+1+7+7$
$1+1+1+1+1+1+1+1+1+3+3+3+3$
$1+1+1+\cdots +1 \ (21 \ \text{times})$
Is there any way to eliminate the choices from this equation? M... |
H: Is the condition "sample paths are continuous" an appropriate part of the "characterization" of the Wiener process?
Wikipedia has separate articles on "Brownian motion" and "Wiener process" (http://en.wikipedia.org/wiki/Brownian_motion and http://en.wikipedia.org/wiki/Wiener_process ). I am not an expert, but that... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.