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H: Why is $\frac{987654321}{123456789} = 8.0000000729?!$
Many years ago,
I noticed that $987654321/123456789 = 8.0000000729\ldots$.
I sent it in to Martin Gardner at Scientific American
and he published it in his column!!!
My life has gone downhill since then:)
My questions are:
Why is this so?
What happens beyond t... |
H: Rudin: Real & Complex Analysis Thm 1.10
$\textbf{Theorem:}$ If $\mathcal{F}$ is any collection of subsets of $X$, there exists a smallest $\sigma$-algebra $\mathcal{M}^{*}$ in $X$ such that $\mathcal{F} \subset \mathcal{M}^{*}$.
$\textbf{Proof:}$ Let $\Omega$ be the family of all $\sigma$-algebras $\mathcal{M}$ in ... |
H: need to show image of $f$ contains the unit disk.
$f$ be non constant analytic on the closed unit disk,$|f|=1$ if $|z|=1$,we need to show image of $f$ contains the unit disk.
My thoughts:
whenever $|\omega|<1$ if I show that $g(z)=f(z)-\omega$ has a root in $D$ the we are done
On $|z|=1$ $|f(z)-g(z)|=|\omega|<1=|f(... |
H: The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$
I'm really confused with this one...
How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality?
Does $$\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)=\{\bar{0},\bar{1},\overline{0+\sqrt{-5}}\}?$$
AI: Recall that
$$\mat... |
H: To show that function is constant
Let $f$ be defined on $\mathbb{R}$ and suppose that |$f(x)$ - $f(y)$| $\leq$ $(x-y)^2$ $x,y \in\mathbb{R}$. Here I have to show that $f$ is a constant function.
I think I have to show that $f'(x)$ = 0 for all $x$.
But I don't know from where to start this. I tried taking it as (|$f... |
H: Is it possible to write any bounded continuous function as a uniform limit of smooth functions
Is $C^\infty(\mathbb{R})\subset C_b(\mathbb{R})$ dense? I.e. is any continuous bounded function $f:\mathbb{R}\to\mathbb{R}$ the uniform limit of smooth functions?
On any bounded interval this is true since by Stone-Weie... |
H: Neumann problem, stuck on a boundary condition.
I am stuck on a problem that I am trying for exam practice and I would very much appreciate a hint to help me out, here is the section where I am stuck:
A solution is sought to the Neumann problem for $\nabla^2 u = 0$ in the half plane $z > 0$:
$u = O(|x|^{−a}),
\f... |
H: Question about inverse with respect to convolution product.
Let $\mathcal{I}(X)$ be the collection of real valued functions $f:X\times X\to \mathbb{R}$ with the property that $f(x,y)=0$ when $x>y$. The convolution product $f*g$ for $f,g\in \mathcal{I}(X)$ is defined by $$ f*g(x,y)=\begin{cases}
\sum_{\lbrac... |
H: Irreducible polynomials have distinct roots?
I know that irreducible polynomials over fields of zero characteristic have distinct roots in its splitting field.
Theorem 7.3 page 27
seems to show that irreducible polynomials over $\Bbb F_p$ have distinct roots in its splitting field (and all the roots are powers of o... |
H: Deducing a coefficient from a cubic polynomial given a divisor and remainder?
I got this question which I don't understand:
"Suppose $x^3 - 2x^2 + a = (x + 2) Q(x) + 3$ where $Q(x)$ is a polynomial. Find the value of a."
I know the identity: $P(x)=A(x)Q(x)+R(x)$, but I'm not sure how to apply it in this question.
A... |
H: Is a maximal-square-covering unique?
Let X be a shape in 2-dimensional space.
Define a square covering of X as a set of axis-aligned squares, whose union exactly equals X.
Note that some shapes don't have a finite square covering, for example, a circle or a triangle.
Define a maximal square covering of X as a squar... |
H: Conditional series convergence guess; Prove/ Disprove
I ran into this question:
Prove or disprove:
If $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is conditionally convergent.
I'm pretty sure it's true because I couldn't fin... |
H: Propositional Logic "Riddle/Puzzle"
I have this kind of 'riddle' as a question that i need to complete, however I'm not sure what to do of it.
This is the question:
Determine who out of the following is guilty of doping. The suspects are: Sam, Michael, Bill, Richard, Matt.
1) Sam said: Michael or Bill took drugs, b... |
H: Definition of Sobolev Space
I have a definition that says that the space of functions that satisfy$$\|u\|_{H^m}^2=\sum_{k\in\mathbb{Z}}(1+|k|^2)^m|\hat{u}_k|^2<\infty$$is called Sobolev Space and when $m=1$, this is equivalent to saying,
$$\int_0^{2\pi}(|u(x)|^2+|u'(x)|^2)dx<\infty$$
I have a few questions. The Sob... |
H: Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?
I hope that my question in the title is well formulated.
I am a little bit confused with the next exercise from a book:
Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ... |
H: Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$
Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$?
I've calculated that the recurrence relation for this integral is:
$\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} _{0} - n \cdot \int_0 ^{1} x^{n-1} e^x dx$
So if... |
H: what does "in wide sense" mean?
I came across the statement "the sequence increases(in wide sense)".
So my doubt is what does author mean by wide sense?I came across this in number theory book
AI: The sequence increases in wide sense means that it is not strictly increasing ( $a_n>a_{n-1})$ but it is non-decreasing... |
H: Is there extension of function from a curve on the whole space preserving smoothness?
Assume that $\alpha: (a,b) \rightarrow \mathbb R^3$ and $f: (a,b) \rightarrow \mathbb R$ are given smooth functions. Let $t_0 \in (a,b)$.
Do there exist a $\delta>0$ and a smooth function $V: \mathbb R^3 \rightarrow \mathbb R$ suc... |
H: Why substitution method does not work for $\int (x-\frac{1}{2x} )^2\, \mathrm dx$?
Why $$\int \ \left(x-\frac{1}{2x} \right)^2 \, \mathrm dx$$ is easy to integrate once $$\left(x-\frac{1}{2x} \right)^2$$ is expanded, but impossible using substitution method? (tried 5 different subs but of course that is not the pr... |
H: About $\mathcal{L}(V,W)$
Let $V,W$ are two vector space and let $S\subseteq V$. Define: $$S^{0}=\{T\in\mathcal{L}(V,W)\mid~T(x)=0, \forall x\in S\}$$ The problem aks me to verify $S^{0}$ is a subspace of $V$ and if $V_1,V_2$ are two subspaces, then $$(V_1+V_2)^{0}=V^0_1\cap V^0_2$$ Please help me to overcome the se... |
H: Prove divisibility using linear congruences
I need to prove that:
$$10|(53^{53} - 33^{33})$$
I can and should only use linear congruences ($a \equiv b \mod n$) - how can I do this?
AI: $$53^{53}-33^{33}\equiv 3^{53}- 3^{33}\pmod {10}$$
Now, $$3^4=81\equiv 1\pmod 4\implies 3^{53}- 3^{33}\equiv 3^1-3^1\pmod {10}\equi... |
H: Solving modular equations that gives GCD = 1
I have problems with understanding modular equations that gives GCD = 1. For example:
$$3x \equiv 59 \mod 100$$
So I'm getting $GCD(3, 100) = 1$. Now:
$1 = -33*3 + 100$
That's where the first question appears - I always guess those -33 and 1 (here) numbers...is there a w... |
H: What does this dollar sign over arrow in function mapping mean?
In a certain function mapping like this,
$x \xleftarrow{\$} \{0,1\}^k$
(Lecture Notes on Cryptography by
S. Goldwasser and M. Bellare, page 18)
I fail to understand what exactly does this \$ sign mean. This has been put
here without any explanation o... |
H: What is $-i$ exactly?
We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything?
Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen that $(-i)^2 = -1$. Then in what way is $-i$ different from $i$? If it isn't, then why do we e... |
H: convergence of series with absolute value
prove or show false:
if $\sum_{n=1}^{\infty}\left |a_{n} \right |$ converges, then $\sum_{n=1}^{\infty}\frac{n+1}{n}a_{n}$ converges as well.
Thank you very much in advance,
Yaron.
AI: Just notice that,
if $\sum_{n\geq 1}|a_n|<\infty$ , then $\sum_{n\geq 1}a_n<\infty$. ... |
H: Aproximating rational with fraction with "smallest numerator and denumerator possible"
For example $0.795=\frac{159}{200}$. But is there a way to find fraction with smaller numerator and denumerator that will represent number $0.795xyz...$ i.e. it will approximate our given number?
I need algorithm or some procedur... |
H: Trigonometric Identity Problem - Cos Tan and Sin
I have been going through my lecture notes for a structures question (the solution of a 2nd order ode for a buckling problem) when I came across a very weird trigonometric simplification which I just cannot get my head around. Could anyone shed any light on this?
Fur... |
H: Cohomology $H^ {i}(\mathbb{R}P^n, \mathbb{Z}_2)$
I have to compute $H^{i}(\mathbb{R}P^2,\mathbb{Z}_2)$. I know that is $\mathbb{Z}_2$ for $i=0,1,2$ but I'm looking for a proof without universal coefficient theorem. Have you some ideas?
AI: If you look at Hatcher page 144 he writes down the cellular chain complex fo... |
H: Method to partial fractions.
For Example:
$$\frac{ax^2 + bx+c}{(dx+e)(fx^2+g)}\equiv\frac{A}{dx+e}+\frac{Bx+C}{fx^2+g}$$
and
$$\frac{ax^4 + bx^3+cx^2+dx+e}{(x+f)(x^2+g)}\equiv Ax+B+\frac{C}{x+f}+\frac{Dx+E}{x^2+g}$$
How do you know how to format the right hand side, in the equation below in partial fractions? (what... |
H: Solving Euler-equation alike 2nd order DE with disturbing RHS
For a homework problem, I have to solve
$$ t^2 \ddot{x} - 3 t \dot{x} + 3x = t^2 $$
which seems quite similar to the Euler Equation, which I would know how to solve, apart from the disturbing $ t^2 $ on the RHS instead of a constant $0$.
How does one ap... |
H: Showing that a function has a certain absolute minimum.
Suppose we have the function $$f(x) = \frac{x}{p} + \frac{b}{q} - x^{\frac{1}{p}}b^{\frac{1}{q}}$$ where $x,b \geq 0 \land p,q > 1 \land \frac{1}{p}+\frac{1}{q} = 1$
I am trying to show that $b$ is the absolute minimum of $f$.
I proceeded as follows:
$$\frac{... |
H: What is the sum of this infinite series? Which one is it, Taylors? Binomial?
I am trying to figure which formula to use for this one.
$$\displaystyle\sum\limits_{x=-1}^{-\infty} -x(1-y)p(1-p)^{-x}+\sum\limits_{x=1}^\infty xyp(1-p)^x$$
where $0<y<1$, and $0<p<1$.
My work:
$$=\displaystyle\sum\limits_{x=1}^{\infty} -... |
H: When $\int |f|=\left|\int f\right|$ holds?
I was just wondering when did the equality hold for the following inequality:
$$\left|\int_{R^d}f(x)\, d x\right|\leq\int_{R^d}|f(x)|\, d x$$
where $f:R^d\to R$ is Lebesgue integrable on $R^d$.
Obviously, if $f\geq0$ a.e. on $R^d$, the equality will hold trivially, but I d... |
H: HINT for summing digits of a large power
I recently started working through the Project Euler challenges, but I've got stuck on #16 (http://projecteuler.net/problem=16)
$2^{15} = 32768$ and the sum of its digits is $3 + 2 + 7 + 6 + 8 = 26$. What is the sum of the digits of the number $2^{1000}$?
(since I'm a big ... |
H: Sample $x$ from $g(x)$
I got confused with all this randomness and probability functions. I was trying to implement the rejection sampling method which (apparently) is really simple. I was reading from Rejection Sampling in Wikipedia and the first step says sample $x$ from $g(x)$ and $u$ from $U(0,1)$ what does thi... |
H: Finding the steady state error in the Laplace domain
I have the following block diagram:
Now I like to find the steady state error for theta_ref being a step input and for several values of n, Td, K1 and K2.
For the moment we can assume all gains are simple scalars, so no vectors and matrices involved.
I like to f... |
H: Questions on Nilpotent Operator
I've some questions from the following theorem:
Let $T:V\to V$ be a nilpotent linear operator with index of nilpotency $k.$ Then $T$ can be expressed as a block diagonal matrix representation where each block $N$ is a Jordan nilpotent matrix. Moreover there is at least one such $N$... |
H: Equivalence Relations and functions on partitions of Sets
let $f$ be a function on $A$ onto $B$. Define an equivalence relation $E$ in $A$ by: $aEb$ if and only if $f(a)=f(b)$.
Define a function $\phi$ on $A/E$ by $\phi([a]_{E})=f(a)$.
Hint: Verify that $\phi([a]_{E})=\phi([a']_{E}) $ if $[a]_{E} = [a']_{E}$
I kno... |
H: Is this $\mathbb{Z}_2^n$?
What group is formed by binary strings of a fixed length, $n$, and the XOR operation (^)?
For example, we have:
For $n=1$:
A^B = B^A = B
A^A = B^B = A
For $n=2$:
AA ^ AB = AB = AB ^ AA
BB ^ AB = BA = AB ^ BB
For $n=3$:
ABA ^ BAA = BBA
etc
I reckon it is written $\mathbb{Z}_2^n$ but I wo... |
H: Dimension of Bil(V)
Let $V$ be a vector space of finite dimension $n$, and let $\operatorname{Bil}(V)$ be the vector space of all bilinear forms on $V$.
In some notes by Keith Conrad, he says in an exercise that $\operatorname{Bil}(V)$ has dimension $n$ as well. I am confused by this, because it seems to me that th... |
H: About two equinumerous partitions of the same set.
Let $\mathcal {A,B}$ be partitions of a set $X$ into $m$ subsets. Suppose that for any $k\leq m$ and any $A_1,\ldots,A_k \in\mathcal A$ there are at most $k$ elements of $\mathcal B$ contained in $\bigcup_{i=1}^kA_i.$ Does it imply that for any $k\le m$ and any $A_... |
H: If $\lVert f(t) \rVert:[0,T] \to \mathbb{R}$ is measurable, is $f$ measurable?
Let $f:[0,T] \to X$ be a mapping to a Banach space $X$. If its norm $\lVert f(t) \rVert$ is measurable, is $f$ itself measurable? The converse is true.
AI: No. We don't even need to leave Euclidean space. Let $V$ be a nonmeasurable set i... |
H: Which real functions have their higher derivatives tending pointwise to zero?
Let $\mathrm C^\infty\!(\Bbb R)$ be the space of infinitely differentiable functions $f:\Bbb R\rightarrow\Bbb R$, and define the subspace$$A:=\{f\in\mathrm C^\infty\!(\Bbb R):(\forall x\in \Bbb R)\lim_{n\rightarrow\infty} f^{(n)}(x)=0\},$... |
H: Using Square area finding quadrilateral area
Area of square ABCD is 169 and that of square EFGD is 49. Find area of quadrilateral FBCG
I am stuck and just thinking which way can be helpful for me finding this area of quadrilateral FBCG. Instead of hint can please suggest me perfect answer , so using this idea answ... |
H: Using induction to verify a statement
I have to prove that this statement is true.
For $n = 1, 2, 3, ...,$ we have $ 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6$
Basically I thought I'd use induction to prove this. Setting $n = p+1$, I got this so far:
Left hand side: $1² + 2² + 3² +...+ p² + (p+1)² = p² + (p+1)² ... |
H: Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that
$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$
converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am interested in proving the following inequality (preferrably usin... |
H: If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?
We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm.
Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 \\ 1 & -1 & 2 & -1 \\ 1 & -... |
H: Two ramified covers $\Rightarrow$ reducible ramification divisor?
Let $X,X',X''$ be algebraic varieties and let $X''\to X'$ and $X'\to X$ be two ramified covers. Is the ramification divisor of the composition $X''\to X'\to X$ reducible?
AI: Let $X=X'=X''=\Bbb{A}^1$ be the affine line, and let the two covers be $z \... |
H: How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ satisfy $f(1) + f(2) + \dots + f(10) = 2$?
How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ have this property: $$f(1) + f(2) + \dots + f(10) = 2.$$
I understand just $2$ functions can be $1$, the rest have to be $0$, in total there are $2^{10}$... |
H: Understanding when convergence implies uniform convergence for sequences of non-continuous functions
I am working on the following problem:
Let $(f_n)$ be a sequence of functions $[a,b] \rightarrow \mathbb{R}$ such that: (i) $f_n(x)≤0$ if $n$ is even, $f_n(x)≥0$ if $n$ is odd; (ii) $|f_n(x)|≥|f_{n+1}(x)|$ for all ... |
H: What is the functional inverse (with respect to $h$ (!)) of $f^{\circ h}(x)={F(h) +x F(h-1) \over F(1+h) +x F(h) }$?
I've considered the fractional iteration of $f(x) = {1 \over 1+x} $ for which the general expression depending on the iteration-height parameter $h$ might be assumed by the formula
$$ f^{\circ h}(x)... |
H: The unity of a subdomain
Let $D$ be an integral domain and $R$ a subdomain , I'm trying to show that both $R$ and $D$ have the same unity. I did it using the fact that $1$ and $0$ are the only idempotent elements inside any integral domain. I'm trying to do it in another way which is the following:
Since $R$ is a s... |
H: how many abelian groups of order $8$ and $21$ exist?
I have to find how many abelian groups of order $8$ and order $21$ exist. There this theorem that states that every abelian group is a direct sum of cyclic groups. So what I did is this: $8=2\cdot2\cdot2$, now I look to all possible divisors of $8$, that is $2$, ... |
H: Function such that $f(x) = -1$ for $x < 0,$ and $f(x)=1$ for $x > 0$?
What is a function to returns $-1$ if number is negative, $1$ if positive, and zero if number is equal to 0?
for example:
$$
f(-8) = -1
$$
$$
f(8) = 1
$$
$$
f(0) = 0
$$
for $$x < 0$$ maybe?
$$ f(x) = (-x-(-x-1))\cdot-1 $$
AI: This function is th... |
H: If $\lim\limits_{x \to \pm\infty}f(x)=0$, does it imply that $\lim\limits_{x \to \pm\infty}f'(x)=0$?
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and
$\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$,
there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$.
Can we say anything abo... |
H: What's the asymptotic distribution of $p^n$ (powers of primes)?
We know by the prime number theorem that
$\lim_{n\to\infty}\frac{\pi(n)}{n\,/\ln n} = 1$
An even better approximation is
$\lim_{n\to\infty}\frac{\pi(n)}{\int_2^n\frac{1}{\ln t}\mathrm{d}t} = 1$.
Is there a similar formula that approximates the distribu... |
H: How many functions $f:\{1,2,3,4\}→\{1,2,3,4\}$ satisfy $f(1)=f(4)$?
I just need a hint or a way to think a about this problem: $f(1)$ can be $1, 2, 3, 4$ and $f(4)$ can be $1,2,3,4.$
AI: To give a function between these sets you need to give the data: for each number ${1,2,3,4}$ give a number in that same set to ma... |
H: Bilinear forms on C[0,1]
Let $C[0,1]$ be the vector space of real-valued continuous functions on $[0,1]$. Then
$$B(f,g) = \int_0^1{f(x)g(x)\, dx}$$
is a bilinear form on $C[0,1]$. More generally, if $k:[0,1]^2\rightarrow \mathbb{R}$ is continuous, then
$$B_k(f,g) = \int_{[0,1]^2}{f(x)g(y)k(x,y)\,dx\, dy}$$
is a bil... |
H: Topological manifolds (dimension)
I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ such that $U$ is homeomorphic to $\mathbb{R}^n$. My question is: By the way she def... |
H: Show groups of symmetries of a cube and a tetrahedron are not conjugate in isometry group.
I've shown that the symmetry group of a cube and a tetrahedron are both isomorphic to S4, but I am now trying to show that they are not conjugate when considered as subgroups of isometries of 3D space.
I cannot think of any k... |
H: Two terms that I want to understand: weakest topology and jointly continuous (in the following context).
I was reading an article online, please help me to understand the following lines (in bold letters). -
Topological structure:
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and therefore a ... |
H: Evaluating $\int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx$
Q: Suppose $\alpha>0$ and $|\beta|<\pi/2$, show that
\begin{align*}
\textbf{(1)} \; \int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx &= \frac 1 2 \sqrt{\pi/\alpha}\cos(\beta/2)\\
\textbf{(2)} \; \int_0^... |
H: How to derive the sum of an arithmetic sequence?
I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the difference between each term; however, I am unable to find one that wor... |
H: If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
Let $f\colon\mathbb R\to\mathbb R$ be twice differentiable with $f(x)\to 0$ as $x\to\infty$ and $f''$ bounded.
Show that $f'(x)\to0$ as $x\to\infty$.
(This is inspired by a comment/answer to a different question)
AI: Let $|f'... |
H: What is $\int x^re^xdx$?
Is there any simple way to get integral of $e^{x}x^{r}, r \in \mathbb{R}$?
Basically I want to solve this:
$$\displaystyle \int \frac{e^t(4t^2+1)}{2t \sqrt{t}}dt$$
so I will appreciate any help in both of the above problems.
AI: Hint: Break up the integral into two integrals
$$
\int \frac... |
H: Topology - Dunce Cap Homotopy Equivalent to $S^2$
So I'm trying to find two spaces with isomorphic homology groups but where the spaces aren't homotopy equivalent.
From my work so far, taking the Dunce Cap as a triangle with the edges identified as $aa(a^{-1})$ if that makes sense, the homology group would be $\Bbb... |
H: Rational Expression Question. (Word problem)
Joe got a mark of $\dfrac{44}{50}$ on one test and $\dfrac{32}{x}$ on another test. If the average mark on the two tests was 80%, what value was the second test out of?
My revised attempt:
Still confused, is this correct? Seems odd that the answer I am getting is a repea... |
H: Writing a 2nd order PDE as a system of equations
I want to turn this 2nd order equation into a system of first order equations but I am unsure about whether I can get rid of the $u$ or not
$$u_{xy}-u_x+u_y+10u u_{xx}$$
To write this as a system of equations so I can determine whether its semi-linear, quasilinear or... |
H: How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
but this is not completed,
2nd
oops, is not $$x^2$$ is only "x"
$$
\int \frac 1{ x \sqrt{x-1}}dx
$$... |
H: ZF Extensionality axiom
To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of Extensionality, stated as follows:
$$\forall x,y \; [\forall z(z \in x \leftrightarrow z \in ... |
H: What is $2 - 1 + 1$?
$2-1+1$; a fairly straightforward question, but I (well, not me, but Henry Reich) found something strange.
Most people would evaluate it as $2+(-1)+1 = 2$; however, this goes against the famed, and fairly standard B.E.D.M.A.S./P.E.D.M.A.S., which states that addition goes first, and then subtra... |
H: Evaluating $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$
I am trying to evaluate $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$. Factoring and using partial fraction decomposition, I have found that the indefinite integral is:
$$\ln{|x-5|}-\ln{|x-3|} + C$$
But when evaluating the improper integral, I don't know how to deal with t... |
H: $a^b = c$, is it possible to express $b$ without logarithms?
$ a^b = c $
is it possible to express b without logarithms?
AI: Not without more information, since finding $b$ in this case is what logarithms are. Maybe in some special cases something can be said. |
H: Subset of $GL(n,R)$
I'm trying to understand why the subset of $GL(n,\mathbb{R})$ formed by the block-matrices of the following type:
$$\begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where
$$A \in GL(k,\mathbb{R}),~ C \in GL(n-k,\mathbb{R}),~ B \in M(k,n-k,\mathbb{R}).$$
is closed in $GL(n,\mathbb{R})$. I first tri... |
H: Infinite Series Problem Using Residues
Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$
I know I must use summation theorem and I calculated the residue which is:
$$Res\left(\frac{1}{z^2+a^2}, \pm ai\right)=-\frac{\pi}{2a}\coth\pi a$$
Now my question is: how do I get t... |
H: an example to show separability of a Banach space does not imply separability of the dual space
$X$ is a Banach space and it is separable, is there any simple counterexample to show the dual space $X^\ast$ is not separable?
AI: The dual of $\ell^1$ (clearly separable by finite sequences of rational numbers) is $\el... |
H: Kernel of the evaluation map on a power series ring
Let $R$ be a commutative ring with unity and $r \in R$ a nilpotent element. Is it true that if $f \in R[[\epsilon]]$ satisfies $f(r) = 0$, then $(\epsilon - r) | f$ in $R[[\epsilon]]$? I tried solving for the coefficients of $f/(\epsilon - r)$ inductively and got ... |
H: The sum of the integration of g and $g^{-1}$
Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto
$[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function.
Use geometric insight to visualize the equation $\int_a^b g + \int_A^B g^{-1} = bB - aA$.
Apply this to the func... |
H: $\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal
I am trying to understand this proof of the existence of an uncountable ordinal. I don't see why $\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal as it is said.
For example, what element of $\mathcal{P}(\ome... |
H: mixture problem
From Stewart, Precalculus, $5$th ed, p.$71$, q.$55$
The radiator in a car is filled with a solution of $60\%$ antifreeze and $40\%$ water. The manufacturer of the antifreeze suggests that, for summer driving, optimal cooling of the engine is obtained with only $50\%$ antifreeze. If the capacity of t... |
H: Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ are... |
H: Linear polynomials of finite fields
I have a final tomorrow, and I was looking over some exercises in my textbook. However, I can't seem to work this problem out.
Let $F$ be a field of $p^n$ elements and let $\alpha \in F^*$, where $F^*=F-\{0\}$.
Show $(x- \alpha)(x- \alpha^p)(x- \alpha^{p^2})\cdots(x- \alpha^{p^{n... |
H: How to frame this set of linear equations?
I have the following set of equations, as an example
$2x + 1y + 2z = A$
$0x + 2y + 2z = A$
$1x + 2y + 1z = A$
I assume this can be rewritten as a matrix? How can I check if a solution exists such that x, y, and z are nonnegative? In this case I don't believe a solution exi... |
H: Rectangle is divided into 14 identical smaller rectangles of width $x$. What is its perimeter in terms of $x$?
Rectangle region ABCD show in below partitioned into 14 identical small rectangles, each of which has width x.
What is perimeter of ABCD in terms of x?
I have used rectangle perimeter formula but answer ... |
H: How to find the factorial of a fraction?
From what I know, the factorial function is defined as follows:
$$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$
And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, $\frac{1}{2}!$, which they claim is equal to $\frac{1}{... |
H: Question about notation in a theorem about Galois theory from Lang's Algebra (chapter 6 §1, corollary 1.16)
I have a question about the notation in an assertion in Lang's Algebra, chapter 6 §1, corollary 1.16:
Let $K/k$ be finite Galois with group $G$, and assume that $G$ can be written as a direct product $$G=G_1... |
H: Dominant morphism, equal dimensions: always finite?
Let $f:X\to Y$ be a dominant morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that dim $X$ = dim $Y$.
Question: must f be finite?
It seems that f must have finite fibers by looking at dimensions. So perha... |
H: Right triangles with integer sides
Most of you know these triples:
$3: 4 :5$
$5: 12 :13$
$8: 15 :17$
$7: 24 :25$
$9: 40 :41$
More generally we can construct such triangles such as
$$2x:x^2-1:x^2+1$$
My question is why one of the sides seems to be always prime? (When there is no common divisor)
AI: It's great that ... |
H: Prove that the index of the set of left coset in a cyclic group is finite
If G is cyclic , show |G/H| < ∞ for any subgroup H except the identity.
I already know that any subgroup of a cyclic group is also cyclic but i have no idea how to prove a the quotient of G is finite especially when Lagrange theorem also only... |
H: Find the volume of the region bounded by $z = x^2 + y^2$ and $z = 10 - x^2 - 2y^2$
So these are two paraboloids. My guess is I would want to find the intersection of these two which would be $2x^2 + 3y^2 = 10$ and construct a triple integral based on its projection. No idea how to do this but the answer comes out t... |
H: Help to understand the ring of polynomials terminology in $n$ indeterminates
In Hungerford's Algebra, page 150, the author defines a ring of polynomials in "$n$" indeterminates in the following manner:
After the author defines the operations in this ring with a theorem:
saying that the author defines another def... |
H: Proving one function is greater than another
How can I prove $f(x)$ $>$ $g(x)$ for all $x > 0$ given $f(x) = (x+1)^{2}$ and $g(x) = 4qx$ where $q$ is a constant in $(0, 1)$?
My approach was to show that $(x+1)^2 > 4qx$ for the interval endpoints, e.g. $q=0$ and $q=1$. E.g. $(x+1)^2 \geq 4x$ for all $x$ and $(x+1)^2... |
H: Self Adjoint operator $\Rightarrow$ Idempotent Operator?
If $P\in\mathcal{L}(H,H)$, with $H$ a Hilbert space, such that $P = P^*$, Is possible to show that $P^2 = P$?
If that is possible, then $P$ is a projection operator, right?
Thanks in advance.
AI: An operator $P$ satisfying $P = P^{\ast}$ is called self-adjoin... |
H: If ${ x }^{ 4 }+{ y }^{ 2 }=1$ then $x$ and $y$ can be both rational numbers?
Can you give two numbers $x,y\in\mathbb{Q}$ such that ${ x }^{ 4 }+{ y }^{ 2 }=1$?
I don't know if exists or not. I derived this equation questioning that if $\sin { \alpha } ={ x }^{ 2 }$ for $x\in \mathbb{Q}$ then for which $\alpha$,... |
H: Approximating a Poisson distribution to a Normal distribution
I have the following problem I'm trying to solve:
I know that the quantity of complains in a call center is a Poisson variable with $\lambda=18 $ costumers/hour, and that the probability of being able to solve a complain is $0.35$. They ask me about the ... |
H: Summation involving subfactorial function
Inspired by this post:
Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for the second question.
Note: !n denotes the subfactorial, also known as the num... |
H: Application of Urysohn's lemma
I am working on the following hw problem: If we have that $X$ is a compact Hausdorff space, with $\{U_\alpha\}_{\alpha\in A}$, then we can find a finite number of continous functions $f_1,...,f_k$, with $f_i:X\mapsto [0,1]$ such that $f_1(x)+...+f_k(x)=1$ for all $x$, and for each $i$... |
H: Regular $T_2$ space which is not completely regular.
Theorem 10. of
Pontryagin's Topological Groups says that:
Every Hausdorff topological group is completely regular.
But is there exists a Regular $T_2$ space which is not completely regular?
AI: This answer has a complete description of such a space, due to Joh... |
H: Bounded sequence in Hilbert space contains weak convergent subsequence
In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence.
Is there any short proof? Thanks a lot.
AI: Suppose $M$ bounds the sequence. Then, if we think of $H$ as sitting inside $H^{**}$, then for any $T \i... |
H: Is this an exact differential or not?
I have the 1-form
$$dz=2xy\, dx+(x^{2}+2y)\, dy$$
And I want to integrate it from $(x_{1},y_{1})$, to $(x_{2},y_{2})$.
If I'm not drunk, checking mixed partials, I find that $dz$ is an exact differential. BUT, when I want to calculate explicitly the integral
$$\int_{\sigma}dz$$... |
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