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H: Cartesian product and union
How can we prove that
$(A\cup C)\times (B \cup D) \subset (A \times B) \cup (C \times D) \Rightarrow (C \subset A ~\land ~ D \subset B) ~~ \lor ~~ (A \subset C ~ \land ~ B \subset D) $?
I've tried to prove by contradiction but didn't manage to do it.
I'll be grateful for any help.
AI: If... |
H: How to prove $D^n/S^{n-1}\cong S^n$?
In my textbook it is said that the quotient space $D^n/S^{n-1}$ is homeomorphic to $S^n$. I can imagine it for $n=2$, but fail to make a mathematical proof for any dimension.
Can anyone provide a rigorous proof?
AI: Theorem 1: Let $X$ be compact Hausdorff, let $p \in X$. If $Y$ ... |
H: Second derivative of Brownian motion?
My question is, we give a meaning to the following expression:
$$dX(t) = \mu(t,X(t))dt + \sigma(t,X(t))dW(t), \ \ X(0)=x.$$
where $W$ is a Wiener process.
This equation can be thought as
$$\frac{dX(t)}{dt} = \mu(t,X(t)) + \sigma(t,X(t))\frac{dW(t)}{dt}, \ \ X(0)=x.$$
Now, if I ... |
H: which of the following is/are algebraic over rationals
which of the following is/are true?
$\sin 7^\circ$ is an algebraic over $\mathbb{Q}$
$\sin^{-1}(1)$ is algebraic over $\mathbb{Q}$
$\cos (\pi/7)$ is algebraic over $\mathbb{Q}$
$\sqrt{2}+\sqrt{\pi} $ is algebraic over $\mathbb{Q}(\pi)$
an algebraic number is ... |
H: Pair of compasses drawing a square (from children's fiction)
I have read a children's book where alien race of "square people" used a pair of compasses that drafted a perfect square when used. Now I wanted to explain to the child that it is not possible to have such a pair of compasses, but then I was not really su... |
H: Are there any perfect numbers which are also powerful?
Powerful numbers are discussed in this paper by R. A. Mollin and P. G. Walsh.
Wikipedia has more information.
In particular, note that OEIS A001694 does not seem to contain any (even) perfect numbers.
This is indeed the case, if we note that the Mersenne prime ... |
H: Use the Mean Value Theorem to prove $\cosh(x) \ge 1 + \frac{x^2}{2}$ in the interval $[0,x]$, given $\sinh(x) \ge x$ for all $x \ge 0$.
Use the Mean Value Theorem to prove $\cosh(x) \ge 1 + \frac{x^2}{2}$ in the interval $[0,x]$, given $\sinh(x) \ge x$ for all $x \gt 0$.
I tried using $f(x) = \cosh(x)$, but to no a... |
H: Putting the table on the ground
If I have smooth surface (which is the graph of some function $f(x,y)$) , is it true that I have 4 points of plain square lying on this surface? And is it true that the length of the edge of such square may be any prescribed (if $f$ defined on $\mathbb R^2$)?
It's motivation may be f... |
H: In any vector space, ax=bx implies a=b
The above statement is listed as false in my text, and I wanted to be sure I understood why that is. (I guess if it were written "properly" it would be $a\mathbf{x} = b\mathbf{x}$ implies $a = b$).
Given the axioms we were given, it would seem that the statement should be tru... |
H: Don't understand proof on pg 65 of Qing Liu
There is proposition in page 65 of Liu's book which is: $X$ an integral scheme with generic point $\xi$. Then if we identify $\mathcal{O}_X(U)$ with and $\mathcal{O}_{X,x}$ we have $\mathcal{O}_X(U) = \bigcap_{x \in U} \mathcal{O}_{X,x}$. His proof is like this.
"By cover... |
H: Intuitive meaning of Exact Sequence
I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all.
Can anyone explain them for me? Thanks in advance.
AI: In the linear algebra of Euclidean space (i.e. $\mathbb R^... |
H: Converting equation to $y = mx + b$
My sister have an assignment of converting below equation to slope as $y = mx + b$
$xy = 4$
Can anyone help? thanks in advance. ^_^
AI: This is not possible.
Look at the graph of xy=4. You will notice that it is hyperbola, not a line. Your sister will therefore be unable to writ... |
H: Questions about Grothendieck groups.
I have a question of the exercise 26 on page 88 of the book introduction to commutative algebra by Atiyah and Macdonald. In 26(iii), let $A$ be a field. Then finitely generated $A$-modules are finite dimensional $A$-vector spaces. Two finitely dimensional vector spaces are isomo... |
H: How to prove that the following iteration process converges?
I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$.
Q1: How to prove that this iteration process converges for every number $p_0 > p > 0 $ to the fixed point $p = g(p), p > 0$ ?
Q2: How ... |
H: Show that derivative less than 1 implies contraction.
I am told that f has a continuous derivative and that $a \leq f(x) \leq b$ and $|f'(x)| < 1 \ \forall x \in [a,b]$ and I have to show that $f$ is a contraction.
Now if I take any $x,y \in [a,b]$, the Mean-Value Theorem says that $\exists c \in (a,b)$ such that
$... |
H: Line of Symmetry for Hyperbolas
How might I find the equation for one of the lines of symmetry for the hyperbola $$y= 2 + \frac 6{x-4},\,\text{ where x cannot equal}\; 4.$$
I know that the lines of symmetry for the rational function $y=A/x$ are $y=x$ and $y=-x$...and that to find the lines of symmetry of $y=A/(x-h)... |
H: Usefulness of the concept of equivalent representations
Definition: Let $G$ be a group, $\rho : G\rightarrow GL(V)$ and $\rho' : G\rightarrow GL(V')$ be two representations of G. We say that $\rho$ and $\rho'$ are $equivalent$ (or isomorphic) if $\exists \space T:V\rightarrow V'$ linear isomorphism such that $T{\rh... |
H: numerically evaluate a continued fraction
I am looking at a continued fraction of the form
$$
F_n = \cfrac{1}{1+\cfrac{p_1}{1+\cfrac{p_2}{1+\cfrac{p_3}{1+\ldots}}}}
$$
where $p_n$ is a function I know. For simplicity I just take it to $p_n=n$ for now. I wish to evaluate this fraction numerically for a given $n$, bu... |
H: Fourier series identity
I need to prove that
$\dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \sum_{n=1}^\infty a^n \sin(nbx)$ where $|a| < 1$.
It seems that this can be proved by using Euler's formula identities for $\cos(bx)$ and $\sin(bx)$ and substituting $z = e^{ibx}$. From that, I get
$$ \dfrac{a \sin(bx)}{1 - 2a... |
H: What happen if there is several $a$ such that equation have finite number of solutions?
The motivation to this question can be found in
The equation $f(s)=a$ has a finite number of solution
My question is:
What happen (regarding the equation $f(z)=P(z)e^{g(z)}+a$) if there is several $a$ such that equation have f... |
H: Introductory books on complex analysis?
I'm a senior in my undergrad. years of college, and I haven't taken Complex Analysis yet.
I have taken Real Analysis I (covered properties of $\mathbb{R}$, set theory, limits of sequences and functions, series, (uniform) continuity, uniform convergence) and Abstract Algebra ... |
H: Proving Binomial Random Variable Identity
Good Morning All,
May I ask for a clue to the following problem?
I got stuck and I am now wondering if I understood the problem correctly.
Let X, Y be independent random variables bin(m, p), bin(n, p) respectively.
Show that X+Y is a binomial random variable with parame... |
H: Matrix inverse of $\left(A-I\right)$ given $A^{-1}$
I am wondering if the inverse of $$B = A-I$$ can be written in terms of $A^{-1}$ and/or $A$. I am able to accurately compute $A$ and $A^{-1}$, which are very large matrices. Is it possible to calculate $B^{-1}$ without directly computing any inverses?
For example... |
H: Proof for Sum of Sigma Function
How to prove:
$$\sum_{k=1}^n\sigma(k) = n^2 - \sum_{k=1}^nn\mod k$$
where $\sigma(k)$ is sum of divisors of k.
AI: Using the identity
\begin{align}
n \ \text{mod} \ k = n - k \lfloor \tfrac{n}{k} \rfloor,
\end{align}
one has
\begin{align}
\sum_{k = 1}^{n} (n \ \text{mod} \ k) =\sum_{... |
H: Limit of the root $\sqrt[k]{k}$
Haow can I calculate this limit? $$L=\lim_{k\to\infty}\sqrt[k]{k}$$
I suppose its value is one, but how can I prove it?
Thanks
AI: HINT: Note that $$\ln L=\lim_{k\to \infty}\frac{\ln k}{k}$$ Now use L'Hospital's rule. |
H: Integer pair that satisfies 42x+55y
How do I find the integer pair $(x,y)$ where $|100|\leq x,\;y\leq |200|$ that satisfies $42x+55y=1$?
AI: Hint: Use the Euclidean algorithm to find the GCD of $42$ and $55$ then work it backwards to find the linear combination of $42$ and $55$. |
H: Number of rules in my fuzzy logic
I have 6 variables with 4 membership functions such as "tiny,small,large,huge".
I tried to write the rules and came up with 200 rules but the combinations are killing me and it is still incomplete.
Can anyone tell me what is the exact number of rules that would cover every combina... |
H: Finding the smallest integer pair such that 123x+321y=1?
How do I find the smallest integer pair $(x,y)$ such that $123x+321y=1$?
AI: HINT;
Using Linear congruence theorem (Proof)
as the greatest common divisor $\textrm{gcd}(123,321)=3$ does not divide $1,$ there is no solution |
H: Finding the remainder of $11^{2013}$ divided by $61$
How am I suppose to find the remainder when $11^{2013}$ is divided by $61$?
AI: HINT:
$11^2=121\equiv-1\pmod{61}$
So, $11^{2013}=11\cdot (11^2)^{1006}\equiv11\cdot(-1)^{1006}\pmod{61}$
$\implies 11^{2013}\equiv11\pmod{61}$ |
H: How to evaluate $\lim_{n \to \infty}{\frac{a_1 \cdot a_2\cdot a_3 \ldots a_{n}}{a_{n+1}}}$, given $a_{n+1} = a_{n}^{2} -1,a_1 = \sqrt{5}$?
Let $a_{n+1} = a_{n}^{2} -1,a_1 = \sqrt{5}.$ How would one evaluate $$\lim_{n \to \infty}{\frac{a_1 \cdot a_2\cdot a_3 \ldots a_{n}}{a_{n+1}}}?$$
Added: Someone else asked me t... |
H: How to imagine a plane defined by Cartesian Plane Equation?
It isn't difficult for me to imagine a plane based on three points. Also it is quite simple to imagine a plane based on point and normal vector.
Are there some tricks to imagine a plane defined by plane equation $Ax+By+Cz+D=0?$
AI: The vector $\mathbf{a}=(... |
H: Dividing $2012!$ by $2013^n$
What's the largest power $n$ such that $2012!$ is divisible by $2013^n$?
It doesn't look like its divisible at all since $2012<2013$; am I right?
AI: $2013=3 \times 11 \times 61$. Thirty-two naturals $ \leq 2012$ are divisible by $61$ and none are divisible by $61^2$. At least thirty-tw... |
H: Inequality with Gamma function: how to prove it?
Let $\alpha \in (0,1)$ and $\Gamma(\alpha) = \int_0^{\infty}s^{\alpha - 1}e^{-s}ds$. I would like to prove that $$\int_0^{\infty}\frac{s^{-\alpha}}{1 + s}ds \le \Gamma(1 - \alpha)\Gamma(\alpha).$$
Basically I know the following two facts, but I don't know if they are... |
H: Lowerbound of maximal cardinality of the set of pairwise disjoint non-null subset of the real line
In a [3-page note] 1 on Bernstein set,
Suppose, $\operatorname{cf}(\mathfrak{c})= \mathfrak{c}$, $\operatorname{non}({\bf{L}})= \min\{|X|: X \subset \Bbb{R}, X \text{ is not a Lebesgue measure zero subset of } \Bbb{R}... |
H: Tail event, convergence of series
How to prove, that $\{ \omega : \sum_{k=1}^{\infty} X_k(\omega) \text{ is finite} \} \in \sigma(\mathcal{F}_1,\mathcal{F}_2, \ldots )$ where $\mathcal{F}_i=\sigma(X_i) = \{ X_i^{-1} (B) : B \in \mathcal{B} (\mathbb{R^n})\}$
Please write as many details as possible, I would like to ... |
H: Finding time needed for a task, given two person's distinct task/time rates
John can dig constantly at $15$ inches per minute and Linda can dig constantly at $45$ inches per minute.
A certain hole can be dug by John in $12$ hours.
If hole is dug by John for half the time and both together for the rest of the time... |
H: Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
Does the function
$$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$
where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum?
I tried asking WolframAlpha, but it appears to give an inconsistent result.
AI: $g(x)=1+\frac{1}{x... |
H: Two hyperreal numbers infinitely close to each other; $100$ and $100+\epsilon$
$100$ is a real number or we could call it a hyperreal number as every element of $\mathbb R$ is also an element of $\mathbb R^*$. If we add an infinitesimal say $\epsilon$ to $100$, the new number will be $100 + \epsilon$. We cannot giv... |
H: Fun combinatorics: How many numbers with some restrictions
I came accross this fun problem today: How many 8-digit numbers are there where:
each digit appears only one
digits 1-4 appear sequentially (though not necessarily consecutively)
5 does not appear after 4
I basically counted all possibilities but what wou... |
H: Why does $n = \sum_{k=0}^\infty \frac{1}{2^k\;e^{k-2}} \implies n = \frac{2\;e^3}{2\;e-1}$?
Consider $$n = \sum_{k=0}^\infty \frac{1}{2^k\;e^{k-2}}$$
Why does
$$n = \frac{2\;e^3}{2\;e-1}\;\;?$$
AI: HINT:
$$n = \sum_{k=0}^\infty \frac{1}{2^k\;e^{k-2}}=e^2\sum_{k=0}^\infty\frac1{(2e)^k}$$
Observe that $\sum_{k=0}^\in... |
H: Stirling numbers with $k=n-2$
Is there a more general method of calculating:
$$ \genfrac\{\}{0pt}{}{n}{n-2} $$
Like for :$$ \genfrac\{\}{0pt}{}{n}{n-1} $$ we can use $nC_2 $
AI: Dividing $n$ objects into $n-2$ sets can be done in two ways. Either you can have two sets of size 2, or one set of size 3. Hence
$$\ge... |
H: Evaluating an integral in physics question
$$U_{C} = \frac{1}{C} \int\!\frac{\cos(100\pi t + \pi/4)}{10}\,dt$$
Find $U_{C}$, the answer is $U_{C}=\left(3.2\times 10^{-4}\right)/C\times \cos(100\pi t - \pi/4)$.
Can someone show to to get this answer?
AI: $$U_{C} = \frac{1}{C} \int\!\frac{\cos(100\pi t + \pi/4)}{10}... |
H: After how many hours does a quantity becomes less than 1% initial quantity?
Life of substance reduces to half at the end of one hour i.e its quantity reduces to one half of what it was at the beginning of one hour .
In how many hours , the quantity becomes less than $1$% initial quantity..
AI: After $1$ hour, we h... |
H: Homomorphism between multiplicative group of integers modulo n
Just looking for anybody to check the following:
We have got a homomorphism $f: (\mathbb{Z}/42\mathbb{Z})^{*} \rightarrow (\mathbb{Z}/21\mathbb{Z})^{*}$, given by $f(a\text{ mod} 42)= a \text{ mod} 21$.
a.) What is the kernel of $f$ ?
b.) Is $f$ bijecti... |
H: Looking for an integer for which the $(\mathbb{Z}/n\mathbb{Z})^*$ contains elements with certain orders
I don't need a specific answer or whatever, but I'm looking for a strategy to solve this kind of problems.
The specific question I have in mind is:
Give an integer $n$ for which the multiplicative group $(\mathb... |
H: Sub-dimensional subspaces a null set
Let $m<n$ and $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ be continuously differentiable. Show $$\lambda^n(f(\mathbb{R}^m))=0$$ and conclude from this, that every linear subspace $E$ of $\mathbb{R}^n$ with $\dim E < n$ is a null set as well.
The result seems rather obvious, yet... |
H: Using either the Direct or Limit Comparison Tests, determine if $\sum_{n=2}^{\infty}\frac{1}{n\sqrt{n^2-1}}$ is convergent or divergent.
Unless I've done some calculations wrong, both tests appear to be inconclusive. I have my doubts that this is the correct outcome.
I've chosen my $\sum t_n$ to be $\sum_{n=2}^{\in... |
H: Proving that tensor distributes over biproduct in an additive monoidal category
I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus (A \otimes C)$. What I've attempted... |
H: Embedding of curves in projective spaces... typo?
I'm reading from the book "Geometry of algebraic curves", by Griffiths, Harris, Arbarello and Cornalba.
In the middle of page 5 they define the map $\phi_{\mathscr{D}}:C\to \mathbb{P}V^*$, from a curve $C$ to the projectified linear subspace $\mathbb{P}V$ of $H^0(C,... |
H: Why is $\{a + b\sqrt2 + c\sqrt3 : a\in\Bbb{Z}, b, c \in\Bbb{Q}\}$ not closed under multiplication?
The set $R = \{a + b\sqrt{2} + c\sqrt{3}: a \in \Bbb{Z}, c, b \in \Bbb{Q}\}$ is not closed on multiplication, my textbook states. Why is this?
And related to that: why then is $S = \{a + b\sqrt{2} : a, b \in \Bbb{Z}\... |
H: Necessary conditions for not having roots
Suppose $f(z)=\sum_0^\infty a_n z^n$ has a radius of convergence of $R$. What are necessary conditions, in terms of $\{a_n\}$, for $f(z)=0$ not to have any roots?
Any combinations of real/complex restrictions on coefficients/roots can be assumed.
EDIT:
Does Weierstrass Pro... |
H: Number of occurrences needed to make an event probable (>95%)
I'm trying to get a formula for the number of tries $x$ I need to make $P(A)$ occur with 95% probability $P(C)$.
$$P(C) = \sum_{i=0}^x i \bigcup A $$
$$P(C) = \sum_{i=0}^x (P(i) + P(A) - P(i)*P(A))$$
I can decompose this summation as follows:
$$P(C) = \s... |
H: Solving two greatest integer function equations
If $$x\lfloor x\rfloor =39 \quad \text{and}\quad y\lfloor y \rfloor=68.$$
What is the value of:
$$\lfloor x\rfloor+\lfloor y \rfloor $$
I don't know how to solve such problems.
I would appreciate an insight regarding the general approach to such problems.
AI: Notice ... |
H: Determine whether this integral converges: $\int_1^\infty\frac{(x+1)\arctan x}{(2x+5)\sqrt x}$
Determine whether the next integral converges: $$\int_1^\infty\frac{(x+1)\arctan x}{(2x+5)\sqrt x}$$
I has this one on a test and lost all my points on this one. Since we were given no answers to the test I still have no ... |
H: Find the expected value of an Y function
$EX = \int xf(x)dx$, where $f(x)$ is the density function of some X variable. This is pretty understandable, but what if I had, for example an $Y = -X + 1$ function and I had to calculate $EY$? How should I do this?
AI: By the linearity of expectation, we have:
$$
E(Y) = E(-... |
H: Convergence of $\sum_{n=1}^{\infty}\frac{1+\sin^{2}(n)}{3^n}$?
Using either the Direct or Limit Comparison Tests, determine if $\sum_{n=1}^{\infty}\frac{1+\sin^{2}(n)}{3^n}$ is convergent or divergent.
I seem to be completely stuck here.
I've chosen my series to be $\sum\frac{1}{3^n}$, which is clearly a converg... |
H: What does $\lim \limits_{n\rightarrow \infty }\sum \limits_{k=0}^{n} {n \choose k}^{-1}$ converge to (if it converges)?
How we can show if the sum of $$\lim_{n\rightarrow \infty }\sum_{k=0}^{n} \frac{1}{{n \choose k}}$$ converges and then find the result of the sum if it converges?
Thanks for any help.
AI: $$
\begi... |
H: Finding the exponential form of $z=1+i\sqrt{3}$ and $z=1+\cos{a}+i\sin{a}$.
Here is what I have been able to accomplish:
For the first one I found that $|z|=z\bar{z}=2$ and $\theta=\tan^{-1}{\sqrt{3}}=\frac{\pi}{3}$. So we get $2e^{\frac{\pi}{3}i}$.
For the second one I have only been able to solve the following: ... |
H: Endomorphisms of a semisimple module
Is there an easy way to see the following:
Given a $k$-algebra $A$, with $k$ a field, and a finite dimensional semisimple $A$-module $M$. Look at the natural map $A \to \mathrm{End}_k(M)$ that sends an $a \in A$ to
$$
M \to M: m \mapsto a \cdot m.
$$
Then the image of $A$ is a... |
H: Limit. $\lim_{n \to \infty}\frac{1^p+2^p+\ldots+n^{p}}{n^{p+1}}$.
Have you any idea how to find the limit of the following sum:
$$\lim_{n \to \infty}\frac{1^p+2^p+\ldots+n^{p}}{n^{p+1}}.$$
Stolz-Cesaro? any more ideas?
AI: The quickest way is using integral:
$$
\lim \sum_{k=1}^n \frac{k^p}{n^p}\frac{1}{n}=\int_0^1... |
H: experiencing methodological consternation in correctly applying Newton's method
In lecture, we were told that to find $\sqrt[3]{a}$, we use Newton's method as follows:
$$
\begin{align}
f(x) &= x^3 - a\\
f'(x) &= 3x^2\\
x_{n+1} &= x_n - \frac{f(x_n)}{f'(x_n)}\\
&= x_n - \frac{{x_n}^3-a}{3{x_n}^2}\\
&=\frac{2}{3}x_n ... |
H: Solve $ax - a^2 = bx - b^2$ for $x$
Method 1
Solve for x
$$ax - a^2 = bx - b^2$$
Collect all terms with x on one side of the equation
$$ax - bx = a^2 -b^2$$
Factor both sides of the equation
$$(a -b)x = (a+b)(a - b)$$
Divide both sides of the equation by the coefficient of $x$ (which is $a-b$)
$$x = a + b$$ ... |
H: Explain why a set is Jordan Measurable
Problem For $D\subset\mathbb{R}^3$ be region such that
$D=\{(x,y,z)\in\mathbb{R}^3:0\leq x,y,z$ and $x+y+z\leq 1\}$.
Explain why $D$ is Jordan measurable, that is, show $1_B$ is Riemann Integrable.
Thoughts Intutively, this is a hexagon in $\mathbb{R}^3$ with finite volume, bu... |
H: Infinite Expected Value of Jointly Distributed Random Variables
I am given a joint pdf function $f(x,y)$ of random variables $X$ and $Y$ such that
$f(x,y) = cxy^{-2}$ when $0 < x < 1$, $1 < y$
$f(x,y) = 0$ otherwise
where $c$ is a constant.
I have calculated the value of $c$, and computed the marginal pdf's of $X$... |
H: Probability of balls and buckets with random removal
There are 5 buckets, and I have 3 balls to place into these buckets. I cannot place more than one ball in any bucket.
After placing the balls in the buckets, 3 buckets are removed at random.
What is the probability of there being at least 1 ball in the remaining... |
H: Need help on part b of this trigonometric question.
The title of the question is: Using graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trigonometrical formula
Already done part a, I'm on part b now, I have no idea what t... |
H: how i can find mathematical way to know the number of triangles are in the photo?
how i can find mathematical way to know the number of triangles are in this photo?
I sure that the solution like sequence or series but how to find it ?
if I added new line so what is the number of triangles become?
thanks for all
AI... |
H: $\Delta u = \operatorname{div}f \ \ \mbox{in} \ \ B_1, f \in L^2 \Rightarrow \nabla u \in L^2$
I'm looking for results like, If $f \in L^p$
and
$$
\begin{array}{rclcl}
\Delta u & = & \operatorname{div}f & \mbox{in} & B_1\\
u&=&0& \mbox{on}& \partial B_1
\end{array}
$$
then
$$ \int_{B_1} \!\left| \nabla u \righ... |
H: How can we pick $f \in C(0,T;H)$ with $f(T) =0$ and $f(0) = h$, where $h$ is arbitrary?
Let $C(0,T;H)$ be the space of continuous functions $f:[0,T]\to H$ where $H$ is Hilbert.
For every $h \in H$, why is it possible to pick a function $f \in C(0,T;H)$ such that $f(0) = h$ and $f(T) = 0$?
When $H = \mathbb{R}$, OK... |
H: Repositioning and resizing when i change the size of my frame
Moderator's Note: This question is cross-posted on gamedev.SE.
I am trying to write a game.
When someone resizes the window of my game, I need all the graphics i have drawn on the screen to reposition correctly so that the ratio of the graphics' width ... |
H: How do you express the covariant cross product?
If the covariant cross product is given by $\mathbf{AxB}= \varepsilon^{ijk}A_{j}B_{k}$, then the Levi-Civita tensor must transform contravariantly for the indices to contract. But according to this physicsforums thread, it obeys the transformation law $\epsilon^{i'j'k... |
H: Question about the number of elements of order 2 in $D_n$
$$\text{Given}\;\; D_n = \{ a^ib^j \mid \text{ order}(a)=n, \text{ order}(b)=2, a^ib = ba^{-i} \}$$ $$\text{ how many elements does $D_n$ contain that have order $2$ ?}$$
My answer would be:
We can write any element as $a^ib^j$. If $j$ is odd, then we can... |
H: Can someone help me solve this problem please.
For the real numbers $x=0.9999999\dots$ and $y=1.0000000\dots$ it is the case that $x^2<y^2$. Is it true or false? Prove if you think it's true and give a counterexample if you think it's false.
AI: Since $x=y$ (that is, $y = 0.\overline{9}=\sum_{n=1}^\infty \frac9{10^... |
H: Sum of Infinite Series with the Gamma Function
I am computing the volume of an infinite family of polytopes and have run into the following sum, which I am not sure how to evaluate, as it looks similar to the Riemann zeta function, except with the gamma function being summed over instead of a regular integer $n$. T... |
H: Show that there are infinitely many triples of integers $(a,b,c)$ such that $2a^2 + 3b^2 - 5c^2 = 1997$
(Cono Sur Math Olympiad - 1997)
Show that there are infinitely many triples of integers $(a,b,c)$ such that $2a^2 + 3b^2 - 5c^2 = 1997$.
I tried to attribute a value to $a$ or $b$ to put this equation in the Pell... |
H: A question on factorial rings
Is 31 irreducible in the ring $\mathbb{Z}\left[\sqrt{5}\right]=\left\{a+b\sqrt{5}:a,b\in\mathbb{Z}\right\}$
?
And is it prime in $\mathbb{Z}\left[\sqrt{5}\right]$?
AI: Recall that for $x$ to be irreducible means that for any factorization of $x$, at least one of the factors is a unit... |
H: Question about order of elements of a subgroup
Given a subgroup $H \subset \mathbb{Z}^4$, defined as the 4-tuples $(a,b,c,d)$ that satisfy $$ 8| (a-c); a+2b+3c+4d=0$$
The question is: give all orders of the elements of $\mathbb{Z}^4 /H$.
I don't have any idea how to start with this problem. Can anybody give some hi... |
H: Concerning the tangent space of an exotic $\mathbb R^4$
My geometric intuition is very poor, so my naive approach to this question is "if $M$ is an exotic $\mathbb R^4$, then $TM$ must be something like $\mathbb R^8$, which is not exotic". Of course, my statement "like $\mathbb R^8$ " is probably rubbish.
So, my co... |
H: Formal definition of the Differential of a function
The formal definition of the differential of a differentiable function $f: x \mapsto y=f(x)$ is that it's a two-variable function, its name is $df$ and its value is $df(x,\Delta_X) = f'(x)\cdot\Delta_X$.
It's used by Courant for instance and i read in Wikipe... |
H: Does $\mathrm{Mat}_{m \times n}$ have boundary?
To me, $\mathrm{Mat}_{m \times n}$ is isomorphic to $\mathbb{R}^{mn}$, hence is boundaryless. But this disqualified the use of Sard's theorem in this question: An exercise on Regular Value Theorem. But it seems using Sard's theorem is on the right track.
Thank you~
AI... |
H: Prove the following limit $ \lim_{n\to \infty} (3^n + 4^n)^{1/n} = 4 $
How do i prove the limit below? I've tried, but i got nothing.
$ \lim\limits_{n\to \infty} (3^n + 4^n)^{1/n} = 4. $
Thanks for any help.
AI: In general, let $\alpha_1,\alpha_2,\dots,\alpha_m$ be positive numbers. Let $A=\max\limits_{1\leq i\l... |
H: Understand "i-Equivalent Valuation" that satisfies formula
From the book Logic for Mathematicians (A. G. Hamilton):
Definition 3.19
Two valuations $v$ and $v'$ are $i$-equivalent if $v(x_j)=v'(x_j)$ for all $j\neq i$.
(I know every single book out there have a different way to define this.)
Then consider $A= (\fo... |
H: Decimal pattern in division of two digit numbers by 9
Can some one explain how this is possible ?
1) 13 / 9 = 1.(1 + 3) = 1.444 ...
2) 23 / 9 = 2.(2 + 3) = 2.555 ...
3) 35 / 9 = 3.(3 + 5) = 3.888 ...
4) 47 / 9 = 4.(4 + 7) = 4.(11) → 4.(11 - 9) = 5.222 ...
5) 63 / 9 = 6.(6 + 3) = 6.(9) → 6.... |
H: $\operatorname{rank}AB\leq \operatorname{rank}A, \operatorname{rank}B$
Prove that if $A,B$ are any such matrices such that $AB$ exists, then $\operatorname{rank}AB \leq \operatorname{rank}A,\operatorname{rank}B$.
I came across this exercise while doing problems in my textbook, but am not sure where to start for t... |
H: Cauchy condition for functions
Prove that $f$ has a limit at $a$ if and only if for every $\epsilon > 0$, there exists $\delta>0$ such that if $0<|x-a|<\delta$ and $0<|y-a|<\delta$, then $|f(x)-f(y)|<\epsilon$.
Forward direction: Suppose $f$ has a limit $L$ at $a$. Fix $\epsilon$. Then for some $\delta$ we have ... |
H: maths required to complete project euler
What maths will help one complete most if not all of project Euler questions? Last I've attempted project Euler I could not understand the questions/vocabulary, etc., and could only complete a few questions. I've gone over set theory, and college algebra and looking through ... |
H: Number of distinct conjugates of a subgroup is finite
Let $H$ be a subgroup of $G$. I would like to prove that if $H$ has finite index in $G$, then there is only a finite number of distinct subgroups in $G$ of the form $aHa^{-1}$. (This is an exercise in [Herstein, Topics in Algebra], in the section on subgroups ... |
H: Finding a function
Let $\{f_j\}$ be an arbitrary sequence of positive real functions on $\mathbb{R}$. How can I find a function $f$ so that for all $n\in\mathbb{N}$: $\displaystyle\lim_{x\to\pm\infty}\dfrac{f(x)}{f_n(x)}=\infty$ ?
AI: Hint: For example, for every $n\ge 1$, let $f(x)=n(f_1(x)+\cdots +f_n(x))$ when $... |
H: one and only one double root(quartic equation)
I want to know how I can determine all positive real values of $b$ for which this equation will have one and only one double root: $x^4 +8x^3 + (288-72b)x^2 + (1088-32b)x + (4b-136)^2 = 0$.
Any help appreciated. Thanks.
AI: HINT:
If Monic Polynomial $f(x)$ has exactly ... |
H: How to calculate Mortgage calculation
I have this formula for Mortgage calculation and now i want loan amount value using with same formula.
Loan amount = Monthly Payment/ ((1 + Interest rate per annum/100) ^ Term of loan) * Term of loan * 12
For example
Interest rate per annum is : 1.09
Term of loan is : 30 years... |
H: Continuous on $[a,b]$ implies $|f(x)-f(y)|<\epsilon$ in whole interval
Prove that if $f$ is continuous on $[a,b]$ and $\epsilon>0$, there exists $\delta>0$ such that if $|x-y|<\delta$ and $x,y\in[a,b]$, then $|f(x)-f(y)|<\epsilon$.
Since $f$ is continuous on $[a,b]$, for every $p\in[a,b]$ and every $\epsilon$ the... |
H: Quick multiplication
How can we explain this with equations ?
Quick Multiplication of any two numbers whose last two digits add up to 10 and all other numbers are the same
$32\times 38 = 3\times (3+1)|(2\times 8)\implies (3\times 4)|16\implies 12|16 \implies 1216$
$81\times 89 \implies 8\times(8+1)|(1\times9)
\imp... |
H: How can $\lim_{n\to \infty} (3^n + 4^n)^{1/n} = 4$?
If $\lim_{n\to \infty} (3^n + 4^n)^{1/n} = 4$, then $\lim_{n\to \infty} 3^n + 4^n=\lim_{n\to \infty}4^n$ which implies that $\lim_{n\to \infty} 3^n=0$ which is clearly not correct. I tried to do the limit myself, but I got $3$. The way I did is that at the step $\... |
H: Is $f(x)=o(x^\alpha)$ for every $\alpha\gt0$ enough to know that $\int_c^x dt/f(t) \sim x/f(x)$?
Let $f$ be a monotone increasing function $[c,\infty)\rightarrow \mathbb{R}^{\geq 0}$ satisfying $f(x)/x^\alpha\to 0$ for all $\alpha>0$.
Is it true that $\int_c^x \frac{dt}{f(t)} \sim \frac{x}{f(x)}$ as $x\to \infty$?... |
H: why $g_n$ is measurable in the proof of Fatou's Lemma
Fatou Lemma: Suppose $\{f_n\}$ is a sequence of measurable functions with $f_n \geq 0$. If $\lim_{n\rightarrow\infty}f_n(x)=f(x)$ for a.e. $x$, then
$$\int f \leq \liminf_{n\rightarrow\infty}\int f_n$$
Proof: Suppose $0\leq g \leq f$, where $g$ is bounded and su... |
H: To prove $f$ to be a monotone function
A open set is a set that can be written as a union of open intervals. If $f$
is a real valued continuous function on $\mathbb{R}$ that maps every open set to an
open set, then prove that $f$ is a monotone function.
AI: "Increasing" ("decreasing") means nonstrictly increasing (... |
H: Class of finite groups a Fraïssé Class?
Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following:
Joint embedding property
Amalgamation property
Hereditary property: if $G \in K$ and $H \le G$, then $H \in K$)
(1) holds because if $G, H \in K$, then $G \times H \in K$ ... |
H: Cardinality and surjective functions
Let $A$ denote a set and $P(A)$ be the power set. By definition for cardinalities $|A|\le|B|$ iff there exists an injection $A \hookrightarrow B$. Note that there is an obvious surjection $P(A) \to A$.
Without the axiom of choice now can there also be a surjection $A \to P(A)$?
... |
H: If $K$ is $w$−compact and convex, $f\in X^\ast \implies f$ attains its maximum on $K$
Let $X$ be a real Banach space
If $K\subset X$ is weakly compact and convex, then for a given $f\in X^\ast$ (dual space) we can always
find
$k\in K$ such that
$$\displaystyle \sup_{x\in K}{f(x)}=f(k)$$
Any hints would be appreci... |
H: Is there a preference between proving a total order (strict vs partial)?
I know that proving a relation $\mathcal{R}$ to be a strict total order (asymmetric, transitive,and total ) implies that the relation $S$ defined as $X\mathcal{S} Y \longleftrightarrow (X\mathcal{R}Y\vee X=Y)$ is a total order (antisymmetric,... |
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