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H: ring without zero-divisors
Suppose we have a ring (could be infinite) without zero-divisors. I have to prove that if $xy=1$ then also $yx=1$ for some $x$ and $y$ in the ring. I really need hints for this, because it seems I just cant figure it out. Thank you.
AI: $$x=(xy)x=x(yx)$$
Then $x(1-yx)=0$. |
H: Different notions of differentiability
The following is somewhat unclear to me. Let $X$, $Y$ be locally convex vector spaces, let $f: X \supseteq U \longrightarrow Y$ be a (nonlinear) continuous map. Then one can say that $f$ is $C^1$ if
a) the difference quotient
$$ Df(x)\xi := \lim_{\epsilon \longrightarrow 0} ... |
H: Computing maximum of an expression
What is the maximum of the expression?
$$x_1x_2x_3+x_2x_3x_4+\cdots+x_{2011}x_{2012}x_{2013}$$
If $x_1,x_2,..,x_{2013} \in [0,\infty)$, $x_1+x_2+\cdots + x_{2013}=2013$
AI: Hint: Compare with
$$ ( x_1 + x_4 + \ldots + x_{2011} ) ( x_2 + x_5 + \ldots + x_{2012} ) ( x_3 + x_6 + \l... |
H: Pooled Estimate of the Variance
Suppose $X_{1},\dots,X_{m}$ is iid $N(\mu_{1},\sigma^{2})$ and $Y_{1},\dots,Y_{n}$ is iid $N(\mu_{2},\sigma^{2})$. Is it true that the pooled estimate of the variance, $S_{p}^{2}$, has the property $\frac{(n-1)S_{p}^{2}}{\sigma^{2}}\sim\chi^{2}_{n-1}$, as is the case for a single sam... |
H: Volume of a wine barrel
This is a famous calculus problem and is stated like this
Given a barrel with height $h$, and a small radius of $a$ and
large radius of $b$. Calculate the volume of the barrel
given that the sides are parabolic.
Now I seem to have solved the problem incorrectly because here it see... |
H: Smooth approximation
How one can show, that if $f(x_1,\ldots,x_n)$ is a continuous function on an open subset $U\subset \mathbb{R}^n$, then for every $\varepsilon > 0$ and every open $V\subset U$, such that $\bar V \subset U$, there exists a function $g(x_1,\ldots,x_n)$, such that:
1) $g$ is smooth on $V$;
2) $g|_{... |
H: Prove that if the coefficient of $ax^3+bx+c$ are odd then it is irreducible of $\mathbb{Q}$
Let $a,b,c$ be odd integers. Prove that $p(x)=ax^3+bx+c$ has no rational root.
AI: Hint: Suppose to the contrary that there is a rational root $\dfrac{p}{q}$. We can assume that $p$ and $q$ are relatively prime. So at least ... |
H: Calculus Implicit Differentiation
I'm learning implicit differentiation and I've hit a snag with the following equation.
$$
f(x, y) = x + xy + y = 2
$$
$$
Dx(x) + Dx(xy) + Dx(y) = Dx(2)
$$
$$
1 + xy' + y + y' = 0
$$
$$
xy' + y' = -1 - y
$$
$$
y'(x + 1) = 1 + y
$$
$$
y' = \dfrac{(1 + y)}{(x + 1)}
$$
$$
y... |
H: sketch the region R bounded by the graphs of the equations and find the volume of the solid generated by revolving R about the indicated axis
Let $R$ be the region bounded by
$$y=1/x,\quad y=1, \quad y=2,\quad x=0$$
Consider the solid generated by rotating $R$ about the y-axis
Sketch the region, the solid, and a ty... |
H: Difference Between Imagespace and Columnspace of a Matrix?
I don't understand the difference between the columnspace of a matrix and the imagespace of a matrix. They are both the spanning sets of the columns of a matrix. Are they just different words for each other or is there a difference?
AI: The columnspace of a... |
H: Sketch the region, the solid, and a typical disk/washer/shell (your choice)
$y=x^2, y=1$; about the line $y=-1$
Alright so I have the sketch drawn but I cannot figure out if I'm doing this correctly because the $y=-1$ is throwing me off. The answer I got is $4/5\pi$.
I got that answer by using horizontal slices $R=... |
H: Find a basis for the range and kernel of $T$.
Find a basis for the range and kernel of $T$.
$$A =\begin{bmatrix}
2 & 0 & -1\\
4 & 0 & -2\\
0 & 0 & 0
\end{bmatrix} $$
Attempt at Solving for Basis of Range:
On finding the basis for the range, I know that the range is the same thing as the column space. So, findin... |
H: Transversal functions are smooth?
This sounds intuitively true. However, I have some counter claims:
Although transversal is defined on smooth manifolds, which implies the image of $df_x$ is smooth. But this does not say if the function $f$ itself is smooth, and the existence of $df_x$ only assumes $f$ is $\mathcal... |
H: Designing a casino cashback program
Let's say a casino is considering offering a cashback program whereby it would return 50% of player losses twice a month. The casino has a house edge of 1% on each game.
What steps could the casino take to ensure that they remain profitable?
One way would be to enforce a minimum ... |
H: Use the shell method to fine the volume of the solid generated by revolving the region bounded by:
$y=12x-11, y=\sqrt{x}$, and $x=0$.
I know I will be using $V=2\pi\int (\sqrt{x})^2-(12x-11)^2 dx$
It's the setting up of the problem I am having difficulty with. If anyone can help me without giving away the answer, ... |
H: What is the definition of "formal identity"?
In Ahlfors' Complex Analysis he remarks that harmonic $u(x,y)$ can be expressed as
$$
u(x,y) = \frac{1}{2}[f(x + i y) + \overline{f}(x - i y)]
$$
when $x$ and $y$ are real. He then writes
"It is reasonable to expect that this is a formal identity,
and then it holds ev... |
H: An image manifold that is a diffeomorphic copy of $X$ adjacent to the original.
The entire content is rather drafty, but I am especially baffled with the last comment "and thus produces an image manifold that is a diffeomorphic copy of $X$ adjacent to the original." This sentence does not make sense to me, like why... |
H: Is there an operator whose non-zero commutants are always injective?
Let $H$ be an infinite dimensional separable Hilbert space.
Is there an operator $T \in B(H)$ such that, if
$TA=AT$ with $0 \ne A \in B(H)$, then $A$ injective ?
Bonus question : what is the set of all such operators ?
AI: In finite dime... |
H: Show that the LU decomposition of matrices of the form $\left[\begin{smallmatrix}0& x\\0 & y\end{smallmatrix}\right]$ is not unique
How can I show that every matrix of the form $\begin{bmatrix}0& x\\0 & y\end{bmatrix}$ has an $LU$ factorization and that even if $L$ is unit lower triangular there is not a unique fac... |
H: Does This Condition Characterize $e^z$?
The following is a question from a Complex Analysis qualifying exam I was studying from:
Does there exist an entire function $f$, distinct from $e^z$, such that $f(0)=1$ and $f'(n)=f(n)$ for all $n\geq 1$?
My instinct is that such a function should exist, although I have trie... |
H: Help in a proof in Hungerford's book
I'm trying to understand the end of this proof:
The theorem 6.1 is:
I need help in this point.
Thanks in advance.
AI: If $f\in F[x]$ is a unit, then $f$ is non-zero because $F[x]$ is not the trivial ring, and of course by definition there is some $g\in F[x]$ such that $fg=1$. ... |
H: Reasoning about random variables and instances
Sorry if this question is so basic, it hurts. I feel like not understanding this topic well enough is holding me back. If any of this language is wrong in some fundamental way, please correct me! Moving on..
Say I make a probabilistic statement about a random variable ... |
H: Is the derivative of a function the secant line?
I am just learning derivatives and I found the derivative of $4x-x^2$ to be $4-2x$. At point $(1,3)$ the tangent line is $2x+1$. Now when I graph this, the derivative $4-2x$ cuts through the function $4x-x^2$. Does that mean the derivative is the secant line?
AI: ... |
H: Swinging Pendulum ODE
The problem is a pendulum with the ability to swing freely
I have a system of first order differential equations of the following form:
$\dfrac {d\theta} {dt}$=$\omega$
$\dfrac {d\omega} {dt}$=$-\dfrac glsin\theta-\dfrac{r(w)} {lm}$
we also know that $r(0)=0$ and $r'(0)>0$
where $\theta$ is th... |
H: numbers between two real numbers
From my intuition, I believe that between two real different numbers ($a<b$), there are infinity many:
(1) rational numbers,
(2) irrational numbers,
(3) algebraic numbers and
(4) transcendental numbers.
But the question is, if I am correct, how to prove it?
I think the proof for... |
H: Does there exist a field which has infinitely many subfields?
Does there exist a field which has infinitely many subfields? Does there exist an enormous supply of such fields?
I don't know how to begin.
AI: The complex numbers $\mathbb{C}$ is an example of such a field. It has infinitely many subfields, since you... |
H: is the Sudoku puzzle NP-complete?
In general Sudoku on $n^2 \times n^2$ boards of $n \times n$ blocks is NP-complete.
Is the common Sudoku on $9 \times 9$ board NP-complete?
AI: The 9x9 board cannot be NP-complete, because there are finitely many instances of the problem. |
H: Uniform Continuity of a Function
The question is as follows:
Fix any $a>0$ and any $m \in \Bbb N$. Prove that $f\colon \Bbb Q \cap [-m,m] \to \Bbb R$ given by $f(x)=a^x$ is uniformly continuous.
Thanks for any help.
AI: Given $\epsilon \gt 0$ we want to find a $\delta$ such that $\vert a^x - a^t \vert \lt \epsilon$... |
H: Prove that at a party with at least two people, there are two people who know the same number of people.
Okay, now, I really want to solve this on my own, and I believe I have the basic idea, I'm just not sure how to put it as an answer on the homework. The problem in full:
"Prove that at a party with at least tw... |
H: Making an inequality true
$n > 10$ implies $n + 3 \leq \Box\times n$
Possible answers:
$1$
$2$
$3$
$4$
I answered "$2$" and got it wrong. Why? When $n=2$, $(11) + 3 \le 2(11)$.
$n > 1$ implies $n + 3 \leq \Box\times n$
Possible answers:
$1$
$2$
$3$
$4$
I answered "$4$" and got it right.
Why did I get the ... |
H: Find $ \int \frac {\tan 2x} {\sqrt {\cos^6x +\sin^6x}} dx $
Problem: Find $\displaystyle\int \frac {\tan 2x} {\sqrt {\cos^6 x +\sin^6 x}} dx $
Solution: $\tan 2x= \dfrac{2\tan x}{1-\tan^2 x}$
Also I can take $\cos^6x$ common from $\sqrt {\cos^6x +\sin^6x}$
I don't know whether it is good approach to the question... |
H: Continuous uniform distribution over a circle with radius R
I started to do this problem with the standard integration techniques, but I cant help but think that there has got to be something I am not seeing. Since it is a uniform distribution, even though x and y are not independent, it seems like there should be ... |
H: Is $t^4+7$ reducible over $\mathbb{Z}_{17}$?
Is $f=t^4+7$ reducible over $\mathbb{Z}_{17}$?
Attempt: I checked that $f$ has not roots in $\mathbb{Z}_{17}$, so the only possible factorization is with quadratic factors.
Assuming $f=(t^2+at+b)(t^2+ct+d)$, we have $bd=7$, $a+c=0$ and $ac+b+d=0.$ But it is cumbersome to... |
H: Prove that if $x,y\in\alpha$, then $xy$ or $x=y$.
I refer to pg.4, Lemma 12 of this article on ordinal numbers.
It says "If $x,y\in \alpha$, then $x<y, x>y, \text { or }x=y$".
My question: As $\alpha$ is an ordinal, we know $x<\alpha$ and $y<\alpha$. However, how do we know that $x$ and $y$ can be compared in thi... |
H: determine position of circle inside square
i need to determine position of circle inside square,let us suppose that we have following picture
we have following informations:
1.$ABCD$ is square
2.all small figures ,$KMCE$,$PKEF$,$NPFD$ are square as well
3.diamter of small circle is equal to $6$ cm
problem:
we hav... |
H: Pushforward Filter
While reading some notes I came across the notion of a pushforward pre-filter. If $g:X\rightarrow Y$ is a continuous map of topological spaces and $F$ is a pre-filter on $X$, then then author claims that $g(F) = \{g(f) : f\in F\}$ is a pre-filter on $Y$. I am having trouble understanding why $g(F... |
H: Are irrational numbers completely random?
As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was wondering if there is a chance that somewhere down the line in the infinite list ... |
H: Extension degree of residue field.
Let $k$ be a field, and $A$ be a finitely generated $k$-algebra with $\text{dim}(A)\leq 1$. Then for any maximal ideal $\mathfrak{m}$ of $A$, does this inequality $[A/\mathfrak{m}:k]<\infty$ hold?
Actually, what I really want to know is following; for a scheme $X$ of finite type o... |
H: $\sqrt{\sum_1^\infty(x_k+y_k)^2}\le \sqrt{\sum_1^\infty x_k^2}+\sqrt{\sum_1^\infty y_k^2}$ provided the limits exist.
Using the fact that (for $x_k,y_k\in\mathbb R~\forall~k\in\mathbb N$) $$\sqrt{\sum_1^n(x_k+y_k)^2}\le \sqrt{\sum_1^n x_k^2}+\sqrt{\sum_1^n y_k^2}$$ I need to show that $$\sqrt{\sum_1^\infty(x_k+y_k... |
H: How to rotate two vectors (2d), where their angle is larger than 180.
The rotation matrix
$$\begin{bmatrix} \cos\theta & -\sin \theta\\ \sin\theta & \cos\theta \end{bmatrix}$$ cannot process the case that the angle between two vectors is larger than $180$ degrees. (counter-clockwise rotation).
AI: The matrix $M(\th... |
H: A conservative (2D) vector field that is perpendicular to the unit square on the boundary of unit square?
I'm trying to test a PDE solver and I'm wondering if there is any 2D vector field that satisfies the following on the domain $\Omega = [0,1] \times [0,1]$:
$$\text{curl} \;\mathbf{u} = 0 \;\;\;\forall \mathbf{x... |
H: Taylor remainder of $f(x,y)=\sin x\cdot \cos y$
Given $f\colon \mathbb R^2\rightarrow\mathbb R,(x,y)\mapsto\sin x\cdot\cos y$ I want to show that there exists $M>0$ such that $$|f(x,y)-T_2(x,y)|\leq M(|x|+|y|)$$ for all $(x,y)\in\mathbb R^2$. $T_2$ is the taylor-polynomial of order $2$.
I know I have to consider th... |
H: find $\mathbb E[X|Y]$ of the following joint pmf
$$Y$$
$$
\begin{array}{c|lcr}
X & 1 & 2& 3\\
\hline
1 & \frac{2}{15}& \frac{4}{15}& \frac{3}{15}\\
2 & \frac{1}{15}& \frac{1}{15}& \frac{4}{15}
\end{array}
$$
$\mathbb E[X|Y]=\sum_xxP(X|Y)=\sum_xx\frac{P(X,Y)}{P(y)}\ldots(1)$
where $P(y)=\sum_xP(X,Y)=$
$$Y$$
$$
\beg... |
H: Convergence of $\varphi_n(x):=\frac{\varphi(nx)}{n}$ in Schwartz space
I want to find all $\varphi\in\mathcal S(\mathbb R)$ for which the sequence $\varphi_n(x):=\frac{\varphi(nx)}{n}$ converges in $\mathcal S(\mathbb R)$.
The first step, I have already managed to do by myself:
$\forall n\in\mathbb N, \varphi_n\in... |
H: Question on proof of deformation lemma
on page no 479 of partial differential equation (Evans) how the condition (iii) of deformation lemma is satisfied.
AI: After $(12)$, Evans concluded that for $u\in H$ and $t\in [0,1]$ $$\tag{1}\frac{d}{dt}I[\eta_t(u)]\leq 0,\ $$
$(1)$ is telling us that the function $v(t)=I[\... |
H: Convergence of partial sums of real sequences
For all $i\in\mathbb{N}$, let $(a_{i,n})_{n\in\mathbb{N}}$ be a real sequence that tends to $0$ for $n\rightarrow\infty$. It holds also that $|a_{i,n}|\leq1$ for all $i,n\in\mathbb{N}$. Is it possible to show that \begin{align*}
c_n:=\frac{1}{n}\sum_{i=1}^{n}a_{i,n}\xr... |
H: Let $f:[0,1]\to[0,1]$ be continuous then $f$ assumes the value $\int_0^1 f^2(t)dt$ somewhere in $[0, 1].$
True/False test: Let $f:[0,1]\to[0,1]$ be continuous then $f$ assumes the value $\int_0^1 f^2(t)dt$ somewhere in $[0, 1].$
$$f:[0,1]\to[0,1]\implies f^2:[0,1]\to[0,1]\implies 0\le\int_0^1 f^2(t)dt\le1$$
So it's... |
H: Why is $\overline{\mathbb{C}\setminus\left\{0\right\}}=\overline{\exp(\mathbb{C})}$?
Why is $\overline{\mathbb{C}\setminus\left\{0\right\}}=\overline{\exp(\mathbb{C})}$?
I do not know how one can see that...
AI: Hint: $\exp(\Bbb C)=\Bbb C\setminus\{0\}$. |
H: Basic First Order Linear Difference Equation(non-homogeneous)
I have this as my homework and I am not sure how to start:
Solve the first-order linear difference equation $$(k+1)x_{n+1}+x_n=k$$ for some constant $k.$ [Hint: The general solution of inhomogeneous linear difference equations also consists of a complem... |
H: Compactly supported continuous function is uniformly continuous
Let $f:\mathbb R \rightarrow \mathbb R$ be continuous and compactly supported. How can I prove that $f$ is uniformly continuous ? I was trying to prove it by contradiction but get stuck. My attempt was as follows:
Let $E$ be the compact support of $f$.... |
H: Equivalence relation and restriction
This is a HW question
Suppose $B \subseteq A$ and $R_a$ is an equivalence relation on A. Let $R_b$ the restriction of $R_a$ to B; that is, $R_b = \{(a,b) \in R_a : a,b \in B\} $ Is $R_b$ an equivalence relation on B.
I know that for am equivalence relation I need the the relat... |
H: Upper bound for the quotient of gamma functions?
I am looking for an upper bound for
$$ \frac{\Gamma(x+\beta)}{\Gamma(x)},\,\,\,\beta>0.$$
In this question it was shown that
$$ \frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta. $$
Then, I believe that there must be some sort of polynomial upper bound but I have f... |
H: equivalence relation and lexicographic order
This is a HW question
Let $A = \mathbb{Z}^+ \times \ \mathbb{Z}^ +$. Define $R$ on $A$ by $(x_1,x_2)R(y_1,y_2)$ iff $x_1+x_2=y_1+y_2$. Is $R$ an equivalence relation on A.
I dont think It is as simple as to show that the reflexive, symmetry and transitivity hold on $x_1... |
H: Combinatorics mmo problem
i've recently came across a problem that i came up with that is realted to a mmo(world of warcraft). Basicaly let's say one player deals a random number between $100$ and $200$ each strike and the opponent as $1000$ health. What would be the probability of killing my opponent in $5,6,\dots... |
H: Strange Partial Fractions Decomposition
I am trying to get from
$$\frac{z^7 + 1}{z^2(z^4+1)}$$
to
$$\frac{1}{z^2} + z - \frac{z+z^2}{1+z^4}.$$
The author did this by doing a partial fractions decomposition. I don't see how, however.. If I compute the partial fractions decomposition, I first find the roots of the de... |
H: Integral $ \dfrac { \int_0^{\pi/2} (\sin x)^{\sqrt 2 + 1} dx} { \int_0^{\pi/2} (\sin x)^{\sqrt 2 - 1} dx} $
I have this difficult integral to solve.
$$ \dfrac { \int_0^{\pi/2} (\sin x)^{\sqrt 2 + 1} dx} { \int_0^{\pi/2} (\sin x)^{\sqrt 2 - 1} dx} $$
Now my approach is this: split $(\sin x)^{\sqrt 2 + 1}$ and $(\sin... |
H: When are free modules extended
I am looking for help to understand the following:
Let $R$ be a commutative ring and $P$ a projective $R[x]$-module. If $P_{\mathfrak m}$ (localization at $R-{\mathfrak m}$, for $\mathfrak m$ a maximal ideal of $R$) is a free $R_{\mathfrak m}[x]$-module, then $P_{\mathfrak m}$ is ex... |
H: partial orders homework
This is a HW question
Let $R_1$ be a partial order on a set $A_1$ and $R_2$ be a partial order on a set $A_2$. Define a relation $R_3$ on the set $A_3 = A_1 \times A_2 = {(a_1,a_2) : a_1 \in A_1, a_2 \in A_2} $ by $(x_1,x_2) R_3 (y_1,y_2)$ iff $x_1R_1y_1$ and $x_2R_2y_2$ . Show that $R_3$ is... |
H: Is it possible to pass functions into other functions in maths?
I wanna be frank with you guys and say my mathematical education was a bit... bleh, so I'm teaching myself a lot of stuff lately, a question that has come up for me: "Is it possible to pass functions into other functions?"
Like say I have a function $g... |
H: Derivative inequality for a twice continuously differentiable function.
This is a question from a past exam. I thought that this was easy, but found no way of solving it.
Let $f: \mathbb R\rightarrow\mathbb R$ be continuously twice differentiable with $f''(x)\gt0\forall x\in\mathbb R$. Then
(a) if $a\lt b$, th... |
H: Norm map in Ideles
If $L/K$ is a finite extension, then there is a natural norm map from $\mathbf{A}^*_L$ to $\mathbf{A}^*_K$. This is a continuous homomorphism
$$\text{N}^L_K: \mathbf{A}^*_L\rightarrow \mathbf{A}^*_K$$
defined by the prescription that the $v$-component of $\text{N}^L_K$ is
$$\prod_{w\mid v}\text{N... |
H: find scalar product of vectors in rectangular
let us consider following problem and picture
we have $ABCD$ rectagular with $AB=3$ and $BC=5$,$F$ and $E$ are midpoints of rectangular sides,we should find scalar product of
my question is can i locate point $A$ arbitrary or could i take coordinates of points ... |
H: Fundamental Theorem of Calculus Q
The FTC is often written as:
If $F(x) = \int_a^x f(t)\,\mathrm{d}t$ then $F'(x) = f(x)$.
Is it not also true that:
If $F(x) = \int f(x)\,\mathrm{d}x$ then $F'(x) = f(x)$?
What is the difference? Is it just the same thing written in a different way?
AI: The second case is a definiti... |
H: Fibres in a power series ring versus fibres in a polynomial ring (a simple question)
Let $A$ be a commutative ring and $p \in \operatorname{Spec} A$.
In Matsumura's Commutative Ring Theory p. 118 it is mentioned that even though $A[x] \otimes \kappa(p) = \kappa(p)[x]$, it is not true in general that $A[[ x ]] \oti... |
H: Techniques of counting
Suppose you need to answer 10 out of 13 questions at an examinatioin. How many choices do you have if you must answer the first two questions
?
I think out of 13 , 2 questions must answer, so remaining is 11.
AI: $ 11 \choose 8 $+ $ 11 \choose 9 $+ $ 11 \choose 10$+ $11 \choose 11$ as while ... |
H: find symmetric equation along line $y=-x$
generally i know that to find symmetric equation of function along line $y=x$,we should exchange $x$ and $y$ and solve back,but what about $y=-x$?should i repeat again the same procedure,but instead of $x$,should i take $-x$?let us consider following problem
consider... |
H: What log law justifies $(\lg n)^{\lg n} = n^{\lg \lg n}$?
I was reading the solution to 3.2-4 on this blog (cropped image pasted here)
notice the person says $\frac{(\lg n)^{\lg n}}{n} = \frac{n^{\lg \lg n}}{n}$
What log law justifies that?
Also, is it correct that it's an error to where they simplify $e^{\lg n}$ ... |
H: black and white balls in the box
A box contains $731$ black balls and $2000$ white balls. The following process is to be repeated as long as possible.
(1) arbitrarily select two balls from the box. If they are of the same color, throw them out and put a black ball into the box. (We have sufficient black balls for t... |
H: Is sum and product of $k$ natural numbers always different from that of some other $k$ natural numbers?
Suppose $k$ is say $3$. Let $A,B$ be a sets of $3$ natural numbers. $A$ not equal to $B$. Can sum and product of the elements in $A$ be same as that of $B$.
If the numbers are primes then the conditions should ho... |
H: Sequence of Polynomials and Weierstrass's Approximation Theorem
I've been stuck on the following problem for a some time: Let $f$ be a continuous function on $[a,b]$. Show that there exists a sequence $(p_n)$ of polynomials such that $p_n \to f$ uniformly on $[a,b]$ and such that $p_n(a) = f(a)$ for all $n$.
Since... |
H: Are there any function which's derivate is the scaled one of the original?( $f'(x) = f(cx)$ )
which function could satisfy the following, for a certain $c\ne1$
$f'(x) = f(cx)$
...beyond the trivial $f=0$
i've been thinking about it for a while.
for a simpler case:
$f'(x)= f(x+c)$
i've found $e^{xe^v}$ where $v$ is ... |
H: Conditions for multivariable differentiability
For a nonlinear system of equations: $x' = f(x)$ why is it that the following condition is equivalent to being differentiable at $x_0$.
Condition:
\begin{equation}
f(x) = A(x-x_0) + g(x)
\end{equation}
where $A$ is an $n x n$ constant matrix and $g$ satisfies:
\begin{e... |
H: Set partitions of pairs
Suppose I am given a set of $n$ pairs of items (so I have $2n$ items in the set). I wish to partition the set into 2 disjoint sets such that at least one pair of items has a member in each set. I want to know how many 2-partitions there are of the $2n$ items where the smallest set has $k$ el... |
H: Is there a result on the behaviour of power series with positive integer coefficients on their boundary?
I have a power series whose coefficients are all positive integers and whose radius of convergence $r$ is $<1$ and I wish to prove that it has a pole at $r$, or at least an infinite radial limit. Is there a gene... |
H: Simple Question on Calculating percentage
All -
It has been a while since I ever used Math. This is a very simple problem. How do I calculate the percentage of the following
Suppose I have 30 apples, and out of 30 Apples, 10 apples have become rotten, what is the percentage of apples that have become rotten?
Thanks... |
H: Sum of geometric series $\frac{7}{8} - \frac{49}{64} + \frac{343}{512}$
$$\frac{7}{8} - \frac{49}{64} + \frac{343}{512}...$$
I make the guess that this is suppose to be represented as
$$-1^{n+1} * (\frac{7}{8})^n$$
Now I use the formula my book gives
$$\Sigma_{n=M}^\inf -1^{n+1} = \frac{cr^M}{1 - r}$$
$$ \Sigma_{n=... |
H: What does $\frac{\partial}{\partial x}(\frac{\partial f}{\partial u})$ mean when $f(x,t), u=x+ct, v=x-ct $?
I'm trying to transform the wave equation $\frac{\partial^2f}{\partial t^2}-c^2\frac{\partial^2f}{\partial x^2}$ using the substitution:
$
\\u=x+ct
\\v=x-ct
$
and using the chain rule for the partial derivati... |
H: What does the following symbol mean? (direct sum? o-plus? -- subject: matrix theory)
In this paper equation 11, the author uses a symbol that is a cross in a circle. I believe I have seen that referred to as a direct sum, but I am not completely sure what that is.
$$\bigoplus$$
This one
AI: This is an example of, ... |
H: Prove a limit using the epsilon-delta definition
$\lim_{x\to-2}\frac{4x-1}{x+1} = 9$
Given $\epsilon>0$, $$(\exists \delta(\epsilon)>0) \left( |x+2|<\delta \implies \left|{\frac{4x-1}{x+1} - 9}\right| < \epsilon \right)$$
So, if $|x+2|<\delta$, then:
$$\left|{\frac{4x-1}{x+1} - 9}\right| = \left|{\frac{4x-1-9(x+1)}... |
H: If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals?
Let $R$ be a commutative ring with $1$, if $R$ contains only finitely many ideals, then every nonzero element of $R$ is either a unit or a zero divisor. I know it's true.
How about the converse? i.e. If e... |
H: Avoiding primefactors in reducible polynomials
Take distinct pairs $(c_i,d_i) \in \mathbb Z^2$, the entries being coprime. Put $f(x) = \prod_{i=1}^k (c_i x + d_i)$. Let $\mathbb P$ denote the rational prime numbers. Which conditions (if any) need to be imposed on $f$ in order to satisfy
$$\forall \textrm{ finite se... |
H: Surjectivity of a restriction map on distributions
I'm reading Kudla's exposition of Tate's thesis in the book "An Introduction to the Langlands Program" and have gotten stuck on some analytic details. Here's the setup: let $F$ be $\mathbb{R}$ or $\mathbb{C}$, so that there is an inclusion $C^{\infty}_c(F^{\times})... |
H: algebraic manipulation question
$M_{z_n}(t)$ is a particular moment generating function, and it is given that $\lambda_n$ approaches $\infty$ as $n$ approaches $\infty$:
Could someone help me see how the above was derived?
AI: The big idea here is that
$$
e^x=1+x+\frac{x^2}{2}+O(x^3)\text{ as }x\rightarrow0.
$$
S... |
H: Question about implication
I have question about something i don't understand.
$\alpha$, $\beta$, and $\gamma$ are statements.
if $\alpha\implies\beta\lor\gamma$ then it's necessary that $\alpha\implies\beta$ or $\alpha\implies\gamma$. that answer is "no" but i can't understand why.
if it's given that $\alpha\impli... |
H: Proving an Identity involving $4^N$
I am trying to prove the following identity:
$$\sum_{k=0}^N\left({2 \, N - 2 \, k \choose N - k}{2 \, k \choose k}\right)=4^N$$
I have tried writing $4^N=2^{2N}=(1+1)^{2N}=(1+1)^N(1+1)^N$, and expanding each of these as a binomial expansion, but I have found nothing but dead ends... |
H: A 'complicated' integral: $ \int \limits_{-\infty}^{\infty}\frac{\sin(x)}{x}$
I am calculating an integral $\displaystyle \int \limits_{-\infty}^{\infty}\dfrac{\sin(x)}{x}$ and I dont seem to be getting an answer.
When I integrate by parts twice, I get:
$$\displaystyle \int \limits _{-\infty}^{\infty}\frac{\sin(x... |
H: Generate Correlated Normal Random Variables
I know that for the $2$-dimensional case: given a correlation $\rho$ you can generate the first and second values, $ X_1 $ and $X_2$, from the standard normal distribution. Then from there make $X_3$ a linear combination of the two $X_3 = \rho X_1 + \sqrt{1-\rho^2}\,X_2$ ... |
H: Do I have enough iMac boxes to make a full circle?
My work has a bunch of iMac boxes and because of their slightly wedged shape we are curious how many it would take to make a complete circle.
We already did some calculations and also laid enough out to make 1/4 of a circle so we know how many it would take, but I'... |
H: Singular Value Decomposition & Compression
What happen when we cancel some singular values in order to compress something?
How we compress it in this way ?
AI: Let $A = U \Sigma V^*$ be an SVD of $A$ and let $\widetilde{A} = U \widetilde{\Sigma} V^*$. Take a look at Frobenius norm of $A - \widetilde{A}$:
$$\|A - \w... |
H: What is $x$ in $-x^2+2x+3 > 0$
I'm busy with a homework assignment and I do not understand how I can factorize $-x^2+2x+3$.
I can't find two numbers that when multiplied make $3$, and when added make $2$. How do I solve this problem? Also one thing that confuses me is the minus sign in front of $x^2$. All the assig... |
H: How to show if $A \subseteq B$, then $A \cap B = A$?
Hi I'm new to set theory. I need to prove that if $A \subseteq B$, then $A \cap B = A$. I would like to do this the formal way, without a Venn diagram. How should I proceed?
AI: Like this:
I hope this helps ;-) |
H: Quickest way to determine a polynomial with positive integer coefficients
Suppose that you are given a polynomial $p(x)$ as a black box (i.e. some oracle, to which you feed $x$ and it returns $p(x)$). It is known that the coefficients of $p(x)$ are all positive integers. How do you determine what $p(x)$ is in the q... |
H: Proving that a function has a removable singularity at infinity
I'm having trouble with the following exercise from Ahlfors' text (not homework)
"If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1} \Re f(z) \to0$ as $z \to \infty$, show that $\lim_{z \to \infty} f(z)$ exists. (In other words, the iso... |
H: Mental estimate for tangent of an angle (from $0$ to $90$ degrees)
Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable.
I can estimate sines and cosines quite well, but I consider div... |
H: Find the matrix for $T$ with respect to the standard bases $B = \{1,x,x^2\}$ for $P_2$.
Let $T:\ P_2 \to P_2$ be a linear operator defined by
$$T(a_0 + a_1 x + a_2 x^2) = a_0 + a_1 (x-1) + a_2 (x-1)^2.$$
Find the matrix $T$ with respect to the standard basis $B = \{1, x, x^2\}$ for $P_2$.
I know that the solution t... |
H: Calculate the projection of $g(x)=\exp(−2x^2)$ onto the subspace $S$
I have problem to getting started on this one:
"Let $f_1(x) = \exp(−x^2)$, $f_2(x) = xf_1(x)$, S the subspace of $L^2(\mathbb{R})$ spanned by $\{f_1,f_2\}$, and $P$ the projector onto $S$. Find $Pg$, where $g(x) = \exp(−2x^2)$."
AI: We have
\begin... |
H: Showing that a point lies in the intersection of the closure of some subsets of $\mathbb R^d$
Let $I$ be an index set and $D_\iota\subseteq \mathbb R^d$ for $\iota\in I$ and $x\in\mathbb R^d.$ Assume that for every $\iota\in I$ there exists a sequence $(x^\iota_n)_{n\in\mathbb N}\subseteq D_\iota$ such that $x^\iot... |
H: Math logic - What does $X\vdash a, a \in X$ mean?
Lets take, for example, the deduction theorem:
For any Well Formed Formulas group $\Sigma$ and for any 2 formula $\alpha, \beta$ ,
$$\Sigma \cup \{\alpha\} \vdash \beta\iff\Sigma \vdash (\alpha\to\beta)$$
Question:
What does $\vdash$ mean? and what does it mean in... |
H: Uniform Convergence and Injectivity
Let $f$ be a continuous function and let $\phi: [0,1] \to [a,b]$ where $\phi(x) = (b-a)x + a$. Clearly $\phi$ is injective and $f \circ \phi$ is continuous on $[0,1]$.
A Theorem in my book says that there exists a sequence of polynomials $(p_n)$ such that $p_n \to f \circ \phi$ ... |
H: Is there any arc-connected set $X\subset\mathbb{R}^n$ such that $\overline{X}$ is not arc-connected?
Could someone give me an example of an arc-connected set $X\subset\mathbb{R}^n$ such that $\overline{X}$ is not arc-connected?
Thanks.
AI: The topologist's sine curve $S:=\left\{\left(x,\sin\left(\frac1x\right)\righ... |
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