Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized Let $A$ be a non-empty set and $n$ be the number of elements in $A$, i.e. $n:=|A|$.
I know that the number of elements of the power set of $A$ is $2^n$, i.e. $|\mathcal{P}(A)|=2^n$.
I came across the fact that exactly half of the elements of $\mathcal{P}(A... | Fix an element $a\in A$ (this is the point where $A\ne\emptyset$ is needed).
Then $$S\mapsto S\operatorname{\Delta}\{a\}$$ (symmetric difference) is a bijection from the set of odd subsets to the set of even subsets.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to find the number of roots using Rouche theorem? Find the number of roots $f(z)=z^{10}+10z+9$ in $D(0,1)$. I want to find $g(z)$ s.t. $|f(z)-g(z)|<|g(z)|$, but I cannot. Any hint is appreciated.
| First, we factor by $z+1$ to get $f(z)=(z+1)(z^9-z^8+z^7+\dots-z^2+z+9)$. Let $F(z):=z^9-z^8+z^7+\dots-z^2+z+9$ and $G(z)=9$. Then for $F$ of modulus strictly smaller than $1$, $|F(z)-G(z)|\leqslant 9|z| \lt |G(z)|$. thus for each positive $\delta$, we can find the number of zeros of $f$ on $B(0,1-\delta)$.
| {
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A function whose value is either one or zero First I apologize in advance for I don't know math's English at all and I haven't done math in almost a decade.
I'm looking for a function whose "domain/ensemble of definition" would be ℝ (or maybe ℤ) and whose "ensemble/domain of variation" would be ℕ{0, 1}
that would look ... | I think that the most compact way to write this is using the Iverson Bracket:
$f: \mathbb{R} \to \{{0,1}\}$
$$ f(x) = [x \neq 0]$$
| {
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Find the Maclaurin series for the function $\tan^{-1}(2x^2)$ Find the Maclaurin series for the function $\tan^{-1}(2x^2)$
Express your answer in sigma notation, simplified as much as possible. What is the open interval of convergence of the series.
I have the correct answer, but I would like to use another method to s... | You probably know the series for $\tan^{-1} t$. Plug in $2x^2$ for $t$.
If you do not know the series for $\arctan t$, you undoubtedly know the series for $\dfrac{1}{1-u}$. Set $u=-x^2$, and integrate term by term.
For the interval of convergence of the series for $\tan^{-1} (2x^2)$, you probably know when the series f... | {
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Prove non-zero eigenvalues of skew-Hermitian operator are pure imaginary Just like the title:
Assume $T$ is a skew-Hermitian but not a Hermitian operator an a finite dimensional complex inner product space V. Prove that the non-zero eigenvalues of $T$ are pure imaginary.
| We need the following properties of the inner product
i) $\langle au,v \rangle = a \langle u,v \rangle \quad a \in \mathbb{C}$,
ii) $ \langle u, a v \rangle = \overline{\langle a v, u \rangle} = \overline{a} \overline{\langle v, u \rangle} = \overline{a} \langle u, v \rangle \quad a \in \mathbb{C}.$
Since T is skew Her... | {
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Two questions with mathematical induction First hello all, we have a lecture. It has 10 questions but I'm stuck with these two about 3 hours and I can't solve them.
Any help would be appreciated.
Question 1
Given that $T(1)=1$, and $T(n)=2T(\frac{n}{2})+1$, for $n$ a power of $2$, and
greater than $1$. Using mathem... | 1)
Prove by induction on $k$:
If $0\le k\le m$, then $T(2^m)=2^kT(2^{m-k})+2^k-1$.
The case $k=0$ is trivial.
If we already know that $T(2^m)=2^{k-1}T(2^{m-(k-1)})+2^{k-1}-1$ for all $m\ge k-1$, then for $m\ge 2^k$ we have
$$\begin{align}T(2^m)&=2^{k-1}T(2^{m-(k-1)})+2^{k-1}-1\\
&=2^{k-1}\left(2\cdot T\left(\tfrac{2^... | {
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Convergence of Bisection method I know how to prove the bound on the error after $k$ steps of the Bisection method.
I.e.
$$|\tau - x_{k}| \leq \left(\frac{1}{2}\right)^{k-1}|b-a|$$
where $a$ and $b$ are the starting points.
But does this imply something about the order of convergence of the Bisection method? I know tha... | For the bisection you simply have that $\epsilon_{i+1}/\epsilon_i = 1/2$, so, by definition the order of convergence is 1 (linearly).
| {
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Functional analysis summary Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
| Georgei E. Shilov's Elementary Functional Analysis, 2nd Ed. (Dover books, 1996) would be a great start, and cheap, as far as textbooks go!
For a very brief (17 page) "summary" pdf document, written and posted by Dileep Menon and which might be of interest: An introduction to functional analysis. It contains both defini... | {
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Partial fractions integration Re-express $\dfrac{6x^5 + x^2 + x + 2}{(x^2 + 2x + 1)(2x^2 - x + 4)(x+1)}$ in terms of partial fractions and compute the indefinite integral $\dfrac{1}5{}\int f(x)dx $ using the result from the first part of the question.
| Hint
Use $$\dfrac{6x^5 + x^2 + x + 2}{(x^2 + 2x + 1)(2x^2 - x + 4)(x+1)}=\frac{A}{(x+1)^3}+\frac{B}{(x+1)^2}+\frac{C}{x+1}+\frac{Dx+E}{2x^2-x+4}+F$$
and solve for $A,B,C,D,E$ and $F$.
Explanation
Note that this partial fraction decomposition is a case where the degree of the numerator and denominator are the same. (Jus... | {
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How many graphs with vertex degrees (1, 1, 1, 1, 2, 4, 5, 6, 6) are there? How many graphs with vertex degrees (1, 1, 1, 1, 2, 4, 5, 6, 6) are there? Assuming that all vertices and edges are labelled. I know there's a long way to do it by drawing all of them and count. Is there a quicker, combinatoric way?
| There are none. By the hand shaking lemma we know that the number of degrees of odd degree must be even.
There are 5 vertices with odd degrees in your graph, these are the ones with degrees:
1,1,1,1,5
| {
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Verify these logical equivalences by writing an equivalence proof? I have two parts to this question - I need to verify each of the following by writing an equivalence proof:
*
*$p \to (q \land r) \equiv (p \to q) \land (p \to r)$
*$(p \to q) \land (p \lor q) \equiv q$
Thank you if you can help! It's greatly appre... | We make extensive use of the identity $(a \to b) \equiv (\lnot a \lor b)$, and leave you to fill in the reasons for some of the intermediate steps in (2).
(1) $\quad p \to (q \wedge r) \equiv \lnot p \lor (q \land r) \equiv (\lnot p \lor q) \land (\lnot p \lor r) \equiv (p \to q) \wedge (p \to r)$.
(2) $\quad(p \to q)... | {
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Showing $\sqrt{2}\sqrt{3} $ is greater or less than $ \sqrt{2} + \sqrt{3} $ algebraically How can we establish algebraically if $\sqrt{2}\sqrt{3}$ is greater than or less than $\sqrt{2} + \sqrt{3}$?
I know I can plug the values into any calculator and compare the digits, but that is not very satisfying. I've tried to s... | Method 1: $\sqrt{2}+\sqrt{3}>\sqrt{2}+\sqrt{2}=2\sqrt{2}>\sqrt{3}\sqrt{2}$.
Method 2: $(\sqrt{2}\sqrt{3})^2=6<5+2<5+2\sqrt{6}=2+3+2\sqrt{2}\sqrt{3}=(\sqrt{2}+\sqrt{3})^2$, so $\sqrt{2}\sqrt{3}<\sqrt{2}+\sqrt{3}$.
Method 3: $\frac{196}{100}<2<\frac{225}{100}$ and $\frac{289}{100}<3<\frac{324}{100}$, so $\sqrt{2}\sqrt{3}... | {
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Vector Space or Not? I have a question. Suppose that $V$ is a set of all real valued functions that attain its relative maximum or relative minimum at $x=0$. Is V a vector space under the usual operations of addition and scalar multiplications? My guess is it is not a vector space, but I can't able to give a counterexa... | As I hinted in my comment above, the terms local maximum and local minimum only really make sense when talking about differentiable functions. So here I show that the set of functions with a critical point (not necessarily a local max/min) at 0 (really at any arbitrary point $a \in \mathbb{R}$) is a vector subspace.
I... | {
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Two norms on $C_b([0,\infty])$ $C_b([0,\infty])$ is the space of all bounded, continuous functions.
Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$
First I want to prove that it is a norm on $C_b([0,\infty])$. The only thing I have problems with is the triangle inequality, I do not know how to simplify
... | For the first one: Use
$$e^{-\alpha \cdot x} \cdot |f(x)+g(x)|^2 = \left|e^{-\frac{\alpha}{2} \cdot x} \cdot f(x)+ e^{-\frac{\alpha}{2} \cdot x} \cdot g(x) \right|^2$$
and apply the triangel inequality in $L^2$.
Concerning the second one: For $a>b>0$ you have
$$e^{-a \cdot x} \cdot |f(x)|^2 \leq e^{-b \cdot x} \cdot |... | {
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How can I show some rings are local. I want to prove $k[x]/(x^2)$ is local. I know it by rather a direct way: $(a+bx)(a-bx)/a^2=1$. But for general case such as $k[x]/(x^n)$, how can I prove it?
Also for 2 variables, for example $k[x,y]/(x^2,y^2)$ (or more higher orders?), how can I prove they are local rings?
| You can use the following
Claim: A commutative unitary ring is local iff the set of non-unit elements is an ideal, and in this case this is the unique maximal ideal.
Now, in $\,k[x]/(x^n):=\{f(x)+(x^n)\;\;;\;\;f(x)\in K[x]\,\,,\,\deg(f)<n\}$ , an element in a non-unit iff $\,f(0)=a_0= 0\,$ , with $\,a_0=$ the free coef... | {
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Countable unions of countable sets It seems the axiom of choice is needed to prove, over ZF set theory, that a countable union of countable sets is countable.
Suppose we don't assume any form of choice, and stick to ZF. What are the possible cardinalities of a countable union of countable sets?
Could a countable unio... | Yes. It is consistent that the real numbers are a countable union of countable sets. For example in the Feferman-Levy model this is true. In such model $\omega_1$ is also the countable union of countable sets (and there are models in which $\omega_1$ is the countable union of countable sets, but the real numbers are no... | {
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Criteria for metric on a set Let $X$ be a set and $d: X \times X \to X$ be a function such that $d(a,b)=0$ if and only if $a=b$.
Suppose further that $d(a,b) ≤ d(z,a)+d(z,b)$ for all $a,b,z \in X$.
Show that $d$ is a metric on $X$.
| Let $X$ be a set and $d: X \times X \to X$ be a function such that $$d(a,b)=0\text{ if and only if}\;\; a=b,\text{ and}\tag{1}$$ $$d(a,b) ≤ d(z,a)+d(z,b)\forall a,b,z \in X.\tag{2}$$
There's additional criterion that needs to be met for a function $d$ to be a metric on $X$:
*
*You must have that $d(a, b) = d(b,a)$... | {
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What does this mathematical notation mean? Please excuse this simple question, but I cannot seem to find an answer. I'm not very experienced with math, but I keep seeing a notation that I would like explained. The notation I am referring too generally is one variable m floating over another variable n enclosed in parae... | This is called the binomial coefficent, often read "n choose m", since it provides a way of computing the number of ways to choose $m$ items from a collection of $n$ items, provided the order or arrangement of those items doesn't matter.
To compute the binomial coefficient: $\displaystyle \binom{n}{m}$, you can use the... | {
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Proof that interpolation converges; Reference request I am interested in the mathematical justification for methods of approximating functions.
In $x \in (C[a, b], ||\cdot||_{\infty})$ we know that we can get an arbitrarily good approximation by using high enough order polynomials (Weierstrass Theorem).
Suppose that $x... | Given an arbitrary function in $x \in C[a, b]$ and defining $y_n$ to be the linear interpolant on the uniform partition of $[a, b]$ with $n + 1$ nodes we have
\begin{equation}
\lim_{n \to \infty} ||y_n - x||_{\infty} = 0.
\end{equation}
Proof. As $x$ is continuous on the compact set $[a, b]$ it is uniformly continuou... | {
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Is $ \lim_{n \to \infty} \sum_{k=1}^{n-1}\binom{n}{k}2^{-k(n-k)} = 0$? Is it true that:
$$
\lim_{n \to \infty} \sum_{k=1}^{n-1}\binom{n}{k}2^{-k(n-k)} = 0
\;?$$
It seems true numerically, but how can this limit be shown?
| Note that $(n-k)$ is at least $n/2$ for $k$ between 1 and $n/2$. Then, looking at the sum up to $n/2$ and doubling bounds what you have above by something like:
$$\sum_{k=1}^{n-1}\binom{n}{k}2^{-kn/2}=\left(1+2^{-n/2}\right)^n-2^{-n/2}-1$$
which bounds your sum above and goes to zero.
Alternatively, use the bound
$$\b... | {
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Generating Pythagorean triples for $a^2+b^2=5c^2$? Just trying to figure out a way to generate triples for $a^2+b^2=5c^2$. The wiki article shows how it is done for $a^2+b^2=c^2$ but I am not sure how to extrapolate.
| Consider the circle $$x^2+y^2=5$$ Find a rational point on it (that shouldn't be too hard). Then imagine a line with slope $t$ through that point. It hits the circle at another rational point. So you get a family of rational points, parametrized by $t$. Rational points on the circle are integer points on $a^2+b^2=5c^2$... | {
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How do I get the conditional CDF of $U_{(n-1)}$? Let $U_1$, $U_2$ .. $U_n$ be identical and independent random variables distributed Uniform(0, 1). How can I find the cumulative distribution function of the conditional distribution of $U_{(n-1)}$ given $U_{(n)} = c$? Here, $U_{(n-1)}$ refers to the second largest of th... | To condition on $A=[U_{(n)}=c]$ is to condition on the event that one value in the random sample $(U_k)_{1\leqslant k\leqslant n}$ is $c$ and the $n-1$ others are in $(0,c)$. Thus, conditionally on $A$, the rest of the sample is i.i.d. uniform on $(0,c)$. In particular $U_{(n-1)}\lt x$ means that the $n-1$ values are i... | {
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Is this Matrix Invertible? Suppose $X$ is a real $n\times n$ matrix. Suppose $m>0$ and let $\operatorname{tr}(X)$ denote the trace of $X$. If $\operatorname {tr}(X^{\top}X)=m$, can i conclude that $X$ is invertible?
Thanks
| The fact that $\operatorname{Tr}(X^TX)$ is positive just mean that the matrix is non-zero. so any non-zero matrix which is not invertible will do the job.
| {
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Analog of Beta-function What is the multi-dimensional analogue of the Beta-function called? The Beta-function being $$B(x,y) = \int_0^1 t^x (1-t)^y dt$$
I have a function
$$F(x_1, x_2,\ldots, x_n) = \int_0^1\cdots\int_0^1t_1^{x_1}t_2^{x_2}\cdots(1 - t_1 - \cdots-t_{n-1})^{x_n}dx_1\ldots dx_n$$
and I don't know what ... | What you can look at is the Selberg integral. It is a generalization of the Beta function and is defined by
\begin{eqnarray}
S_n(\alpha,\beta,\gamma) &=& \int_0^1\cdots\int_0^1\prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1}\prod_{1\leq i<j\leq n}|t_i-t_j|^{2\gamma}dt_1\cdots dt_n \\
&=& \prod_{j=0}^{n-1}\frac{\Gamma(\alp... | {
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Proof $A\sim\mathbb{N}^{\mathbb{N}}$ Let $A=\{f\in\{0,1\}^{\mathbb{N}}\,|\, f(0)=0,f(1)=0\}$, i.e. all the infinite binary vectors, that start from $0,0$.
Need to proof that $A\sim\mathbb{N}^{\mathbb{N}}$.
Any ideas or hint?
| Hint:
Show that $\mathbb{2^N\sim N^N}$ first, then show that $A\sim 2^\mathbb N$.
The first part is the more difficult part, but recall that $\mathbb{N^N}\subseteq\mathcal P(\mathbb{N\times N})$ and that $\mathbb{N\times N\sim N}$.
| {
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How do I prove the middle-$\alpha$ cantor set is perfect? Let $\alpha\in (0,1)$ and $\beta=\frac{1-\alpha}{2}$.
Define $T_0(x) = \beta x$ and $T_1(x) = (1-\beta) + \beta x$ , $\forall x\in [0,1]$.
Recursively define $I_0 =[0,1]$ and $I_{n+1}= T_0(I_n) \cup T_1(I_n)$.
The Middle-$\alpha$ Cantor Set is defined as $\bigca... | Define $I_n^* = I_0 - I_n, \forall n\in \omega$.
Note that (i);
$I_{n+1}^*\\=I_0 \setminus I_{n+1} \\=I_0 \setminus (T_0(I_n)\cup T_1(I_n)) \\=(I_0\setminus (T_0(I_0)\setminus T_0(I_n^*)))\cap (I_0\setminus (T_1(I_0)\setminus T_1(I_n^*)) \\=T_0(I_n^*)\cup I_1^* \cup T_1(I_n^*)$.
Also(ii), it can be found that, $\forall... | {
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Conceptual question about equivalence of eigenvectors Suppose for a matrix the eigenvalue is 1 and the eigenvector is (2,-3). Then does that mean (-4,6) and (4,-6) are equivalent eigenvectors as the ratios are the same?
| Let T be a transformation, and let $\lambda$ be an eigenvalue with eigenvector $v$, ie. $T(v)=\lambda v$. Then if $c$ is any scalar, $cv$ is also an eigenvector with eigenvalue $\lambda$, since $T(cv)=cT(v)=c\lambda v=\lambda(cv)$
| {
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Trying understand a move in Cohen's proof of the independence of the continuum hypothesis I've read a few different presentations of Cohen's proof. All of them (that I've seen) eventually make a move where a Cartesian product (call it CP) between the (M-form of) $\aleph_2$ and $\aleph_0$ into {1, 0} is imagined. From w... | In order to prove the continuum hypothesis is independent from the axioms of $ZFC$ what Cohen did was to start with $ZFC+V=L$ (in which the generalized continuum hypothesis holds), and create a new model in which $ZFC$ in which the continuum hypothesis fails.
First we need to understand how to add one real number to th... | {
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Coin tossing questions I had an exam today and I would like to know for sure that I got these answers correct.
A fair coin will be tossed repeatedly.
*
*What is the probability that in 5 flips we will obtain exactly 4 heads.
*Let $X =$ # flips until we have obtained the first head. Find the conditional probability ... | It seems that the general consensus is that you are right.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/250259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In regard to a retraction $r: \mathbb{R}^3 \rightarrow K$ Let $K$ be the "knotted" $x$-axis. I have been able to show that $K$ is a retract of $\mathbb{R}^3 $ using the fact that $K$ and the real line $\mathbb{R}$ are homeomorphic, $\mathbb{R}^3$ is a normal space, and then applying the Tietze Extension Theorem. But th... | Let $f : K \to \Bbb R$ and pick a point $x \in K$.
Pick a infinite sheet of paper on the left side of the knot and imagine pinching and pushing it inside the knot all the way right to $x$, and use this to define $g$ on the space spanned by the sheet of paper into $( - \infty ; f(x))$, such that if $y<f(x)$, $g^{-1}(\{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/250343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A necessary and sufficient condition for a measure to be continuous. If $(X,\mathcal{M})$ is a measurable space such that $\{x\}\in\mathcal{M}$ for all $x\in$$X$, a finite measure $\mu$ is called continuous if $\mu(\{x\})=0$ for all $x\in$$X$.
Now let $X=[0,\infty]$, $\mathcal{M}$ be the collection of the Lebesgue meas... | It seems like the contrapositive is a good way to go. Suppose that $x\mapsto\mu([0,x])$ is not continuous, say at the point $x_0$. Then there exists an $\epsilon>0$ such that for all $\delta>0$ there is a $y$ such that $\vert x_0-y\vert<\delta$ but $\vert\mu([x_0,y])\vert\geq\epsilon$. Thus we can construct a sequen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/250397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral and Area of a section bounded by a function. I'm having a really hard time grasping the concept of an integral/area of a region bounded a function.
Let's use $x^3$ as our sample function.
I understand the concept is to create an infinite number of infinitely small rectangles, calculate and sum their area. Usin... | So, $f(C_i)$ is the value of $f$ at $C_i$, but more importantly it is the height of the specific rectangle being used in the approximation. Then $i$ is just the interval which is the base of the rectangle. As $|C_{i+1}-C_i|\rightarrow 0$, this sum becomes the area under the curve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/250455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Isomorphism between 2 quotient spaces Let $M,N$be linear subspaces $L$ then how can we prove that the following map
$$(M+N)/N\to M/M\cap N$$ defined by $$m+n+N\mapsto m+M\cap N$$ is surjective?
Originally, I need to prove that this map is bijection but I have already proven that this map is injective and well defined,... | Define $T: M \to (M+N)/N$ by
$m \mapsto m+N$.
Show that it is linear and onto.
Check the $\ker T$ and is $M\cap N$
by the first isomorphism theorem $f:M/(M\cap N) \to (M+N)/N$ is an isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/250499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The notion of complex numbers How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will not in some way contradict some statement that is an outcome of the real numbers.
For exa... | A field is a generalization of the real number system. For a structure to be a field, it should fulfill the field axioms (http://en.wikipedia.org/wiki/Field_%28mathematics%29).
It is rather easy to see that the complex numbers are, indeed, a field. Proving that there isn't a paradox hiding in the complex-number theory ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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Constructing a strictly increasing function with zero derivatives I'm trying to construct a fuction described as follows:
$f:[0,1]\rightarrow R$ such that $f'(x)=0$ almost everywhere,f has to be continuous and strictly increasing.
(I'd also conlude that this functions is not absolutely continuous)
The part in bracket i... | By $\phi$ we denote Cantor-Vitali function. Let $\{(a_n,b_n):n\in\mathbb{N}\}$ be the set of all intervals in $[0,1]$ with rational endpoints. Define
$$
f_n(x)=2^{-n}\phi\left(\frac{x-a_n}{b_n-a_n}\right)\qquad\qquad
f(x)=\sum\limits_{n=1}^{\infty}f_n(x)
$$
I think you can show that it is continuous and have zero deriv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$V$ is a vector space over $\mathbb Q$ of dimension $3$ $V$ is a vector space over $\mathbb Q$ of dimension $3$, and $T: V \to V$ is linear with $Tx = y$, $Ty = z$, $Tz=(x+y)$ where $x$ is non-zero. Show that $x, y, z$ are linearly independent.
| Let $A = \mathbb{Q}[X]$ be the polynomial ring.
Let $I = \{f(X) \in A|\ f(T)x = 0\}$.
Clearly $I$ is an ideal of $A$.
Let $g(X) = X^3 - X - 1$.
Then $g(X) \in I$.
Suppose $g(X)$ is not irreducible in $A$.
Then $g(X)$ has a linear factor of the form $X - a$, where $a = 1$ or $-1$.
But this is impposible.
Hence $g(X)$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/250706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\mathbb R^2$ a field? I'm new to this very interesting world of mathematics, and I'm trying to learn some linear algebra from Khan academy.
In the world of vector spaces and fields, I keep coming across the definition of $\mathbb R^2$ as a vector space ontop of the field $\mathbb R$.
This makes me think, Why can't... | Adding to the above answer. With the usual exterior multiplication of the $\mathbb{R}-\{0\}$ as a ring with the natural addition and multiplication you can not make a field out of $\mathbb{R}^{2}-(0,0)$
\
But there may exist other products such as the one in the answers which can make a field out of ${\mathbb{R}\times\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/250768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
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Stochastic Processes...what is it? My university is offering stochastic processes next semester, here is the course description:
Review of distribution theory. Introduction to stochastic processes, Markov chains and Markov processes, counting, and Poisson and Gaussian processes. Applications to queuing theory.
I'm torn... | "Stochastic" refers to topics involving probability -- often the treatment of processes that are inherently random in nature by virtue of being some sot of random function about a random or deterministic variable, or a process parameterized by a random quantity.
For example, Brownian motion is a stochastic process; sim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Where is the mistake in the calculation of $y'$ if $ y = \Bigl( \dfrac{x^2+1}{x^2-1} \Bigr)^{1/4} $? Plase take a look here.
If $ y = \Bigl( \dfrac{x^2+1}{x^2-1} \Bigr)^{1/4} $
\begin{eqnarray}
y'&=& \dfrac{1}{4} \Bigl( \dfrac{x^2+1}{x^2-1} \Bigr)^{-3/4} \left \{ \dfrac{2x(x^2-1) - 2x(x^2+1) }{(x^2-1)^2} \right \}\\... | I believe you forgot a power 1/4 when substituting for $y$ (in the calculation using logarithms).
Edited to explain further:
In your calculation, you write
\begin{align} \frac{dy}{dx} &= y\frac14 \left\{ \frac{2x}{(x^2+1)} - \frac{2x}{(x^2-1)} \right\} \\
&= \frac14 \frac{x^2+1}{x^2-1} \cdot 2x\frac{(x^2-1)-(x^2+1)}{(x... | {
"language": "en",
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"source": "stackexchange",
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Why is $E(Z | Y) = 0$? Let $Z$ be a random variable distributed $\mathcal{N}(0, 1)$. Let $Y = Z^2$.
Apparently, $E(Z \mid Y) = E(Z \mid Z^2) = 0$ due to "symmetry." Why is that?
| For completion, note that for all mesurable $f$ such that $E(|f(Y)|) < \infty$, $$E(f(Y)\mid Z) = \frac{f(Z) + f(-Z)}{2}.$$
Here $f\colon x\mapsto x$ is odd, hence $E(Y\mid Z) = \frac{Z-Z}{2}=0$.
Another example of interest : if you take $f(x)=e^{i\theta x}$, you get $E(e^{i\theta Y} \mid Z) = \cos(\theta Z)$
| {
"language": "en",
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When generating set is not a basis If a generating set of a vector space being made up of linearly independent vectors constitues a basis, when such a set is not a basis does it mean that its vectors are linearly dependent?
| Yes. Let $S$ be the generating set and let $B\subset S $ be a basis. If $B,S$ are not equal, then there exists $u\in S-B$ since $B$ is a basis, it follows that $u=c_1v_1+c_2v_2+...+c_nv_n$ for some $v_1,v_2,...,v_n\in B$. From this it follows that {$u,v_1,...,v_n$} are linearly dependent. Hence, $S$ is not linearly ind... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to simplify polynomials I can't figure out how to simplify this polynominal $$5x^2+3x^4-7x^3+5x+8+2x^2-4x+9-6x^2+7x$$
I tried combining like terms
$$5x^2+3x^4-7x^3+5x+8+2x^2-4x+9-6x^2+7x$$
$$(5x^2+5x)+3x^4-(7x^3+7x)+2x^2-4x-6x^2+(8+9)$$
$$5x^3+3x^4-7x^4+2x^2-4x-6x^2+17$$
It says the answer is $$3x^4-7x^3+x^2+8x+17$... | You cannot combine terms like that, you have to split your terms by powers of $x$.
So for example $$5x^2+5x+2x^2 = (5+2)x^2+5x = 7x^2+5x$$ and not $5x^3+2x^2$. Using this, you should end up with your answer.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b $ has degree of extension 3 Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b \in \mathbb Q[x]$ has degree of extension 3 over $\mathbb Q$.
I'm trying solve do this question, it seems very difficult to me, maybe because ... | Partial answer: Let $f(x)=x^3+ax+b$, let $K$ be its splitting field, and $\alpha$, $\beta$ and $\gamma$ be the roots of $f$ in $K$.
First of all, $f$ has to be irreducible, which is the same as saying it doesn't have a rational root: if it's not, and say $\alpha$ is rational, then $f(x)$ factors as $(x-\alpha)g(x)$ wit... | {
"language": "en",
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"source": "stackexchange",
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If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? Any help would be appreciated.
If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?
| HINT: The possible degrees of a simple graph with $n$ vertices are the $n$ integers $0,1,\dots,n-1$. However, a simple graph on $n$ vertices cannot have both a vertex of degree $0$ and a vertex of degree $n-1$; why?
That means that either the degrees of the $n$ vertices are all in the set $\{0,1,\dots,n-2\}$, or they’r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving every montonic function on an interval is integrable I am trying to understand the proof of every monotonic function that is on an interval is integrable.
This is what I have $U(f, P) - L(f, P) = \sum\limits_{k=1}^n(f(t_k) - f(t_{k-1}))\cdot (t_k - t_{k-1})$
Now my book says that this is equal to:
$= (f(b) - f(... | Note that
\begin{equation*}
\sum_{k=1}^{n}(f(t_{k})-f(t_{k-1}))=(f(t_{1})-f(t_{0}))+(f(t_{2})-f(t_{1}))+(f(t_{3})-f(t_{2}))+...+(f(t_{n})-f(t_{n-1})),
\end{equation*}
so in $i$:th term of the sum, $-f(t_{i-1})$ always eliminates the $i-1$:th appearing term $+f(t_{i-1})$. Hence you are only left the endpoints, i.e. $f(b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $\|u_1\|^2+\|w_1\|^2=\|u_2\|^2+\|w_2\|^2$ If $u_1+w_1=u_2+w_2$ and $\langle u_1,w_1\rangle=0=\langle u_2,w_2\rangle$, how can we prove that $$\|u_1\|^2+\|w_1\|^2=\|u_2\|^2+\|w_2\|^2$$
I know I can open this to $$\langle u_1,u_1\rangle+\langle w_1,w_1\rangle=\langle u_2,u_2\rangle+\langle w_2,w_2\rangle$$ b... | From $u_1+w_1=u_2+w_2$ and $\langle u_1,w_1\rangle=0=\langle u_2,w_2\rangle$, we have
$\|u_1\|^2+\|w_1\|^2$
$=\langle u_1,u_1\rangle+\langle w_1,w_1\rangle+2\langle u_1,w_1\rangle$
$=\langle u_1+w_1,u_1+w_1\rangle$
$=\langle u_2+w_2,u_2+w_2\rangle$
$=\langle u_2,u_2\rangle+\langle w_2,w_2\rangle+2\langle u_2,w_2\rangl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Conditions for the mean value theorem the mean value theorem which most of us know starts with the conditions that $f$ is continuous on the closed interval $[a,b]$ and differentiable on the opened interval $(a,b)$, then there exists a $c \in (a,b)$, where
$\frac{f(b)-f(a)}{b-a} = f'(c)$.
I'm guessing we're then able ... | The answer is No. Consider $y=f(x)=x^3$ and $c=0$. $f'(c)=0$ but no secant line has a zero slope as ${{f(r)-f(s)}\over{r-s}}=r^2+s^2+rs>0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Is $Y_s=sB_{1\over s},s>0$ a brownian motion Suppose $\{B_s,s>0\}$ is a standard brownian motion process. Is $Y_s=sB_{1\over s},\ s>0$ a brownian motion or (stardard). I have found that $Y_0=0$ and $Y_s\sim N(0,1)$ as $B_s\sim N(0,s)$, so it remains to show that it is stationary increment and independent increment. But... | Have you heard of Gaussian processes ? If you have, you only have to check that $(Y_s)$ has the same covariance function as the Brownian motion.
If you haven't, don't worry, it's very simple here: you are interested in the law of the couple $(sB_{1/s},tB_{1/t}-sB_{1/s})$ when $0 < s <t$. This is a 2 dimensional centere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Representation of linear functionals I have always seen the linear functionals in $R^n$ expressed at $\ell(x) = \sum_{i=0}^n a_ix_i$ And in an countable metric space $\ell(x) = \sum_{i=0}^{\infty} a_ix_i$. I guess that this follows directly from http://en.wikipedia.org/wiki/Riesz_representation_theorem, for Hilbert spa... | I can think of a nice representation theorem that holds in a non-Hilbert space. It goes by the name Riesz-Kakutani-Markov:
Let $X$ be a compact Hausdorff space and $(C(X),\|\cdot\|_\infty)$ the space of continuous real valued functions on $X$ endowed with the maximum norm. Then, every bounded linear functional $F$ on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does Lenstra ECM work? I came across Lenstra ECM algorithm and I wonder why it works.
Please refer for simplicity to Wikipedia section Why does the algorithm work
I NOT a math expert but I understood first part well enough (I suppose), what I miss is
When this is the case, $kP$ does not exist on the original curve... | Read this paper.
this is just a piece of hole paper.
| {
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"timestamp": "2023-03-29T00:00:00",
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Simple and intuitive example for Zorns Lemma Do you know any example that demonstrated Zorn's Lemma simple and intuitive? I know some applications of it, but in these proof I find the application of Zorn Lemma not very intuitive.
| Zorn's lemma is not intuitive. It only becomes intuitive when you get comfortable with it and take it for granted. The problem is that Zorn's lemma is not counterintuitive either. It's just is.
The idea is that if every chain has an upper bound, then there is a maximal element. I think that the most intuitive usage is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Periodic parametric curve on cylinder Given a cylinder surface $S=\{(x,y,z):x^2+2y^2=C\}$. Let $\gamma(t)=(x(t),y(t),z(t))$ satisfy $\gamma'(t)=(2y(t)(z(t)-1),-x(t)(z(t)-1),x(t)y(t))$. Could we guarante that $\gamma$ always on $S$ and periodic if $\gamma(0)$ on $S$?
| We can reparameterize $S=\{(\sqrt{C}\cos u,\frac{\sqrt{C}}{\sqrt{2}}\sin u, v): u,v\in \mathbb{R}\}$ since $(\sqrt{C}\cos u)^2+2\left(\frac{\sqrt{C}}{\sqrt{2}}\sin u\right)^2=C$. Let $r(t)= (x(t),y(t),z(t))$ and $r(0)=(x_0,y_0,z_0)$. Define $V(x,y,z)=x^2+2y^2$. Since $V(x,y,z)=C$ then $\frac{dV}{dt}=0$. But, by chain ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Advanced integration, how to integrate 1/polynomial ? Thanks I have been trying to integrate a function with a polynomial as the denominator.
i.e, how would I go about integrating
$$\frac{1}{ax^2+bx+c}.$$
Any help at all with this would be much appreciated, thanks a lot :)
ps The polynomial has NO real roots
$${}{}$$... | If $a=0$, then
$$I = \int \dfrac{dx}{bx+c} = \dfrac1b \log (\vert bx+c \vert) + \text{constant}$$
$$I = \int \dfrac{dx}{ax^2 + bx + c} = \dfrac1a \int\dfrac{dx}{\left(x + \dfrac{b}{2a}\right)^2 + \left(\dfrac{c}a- \dfrac{b^2}{4a^2} \right)}$$
If $b^2 < 4ac$, then recall that
$$\int \dfrac{dt}{t^2 + a^2} = \dfrac1a\arct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Empirical distribution vs. the true one: How fast $KL( \hat{P}_n || Q)$ converges to $KL( P || Q)$? Let $X_1,X_2,\dots$ be i.i.d. samples drawn from a discrete space $\mathcal{X}$ according to probability distribution $P$, and denote the resulting empirical distribution based on n samples by $\hat{P}_n$. Also let $Q$ b... | In addition to the last answer, the most popular concentration inequality for the KL divergence is for finite alphabets. You can look for Theo. 11.2.1 of "Elements of Information Theory" by Thomas Cover and Joy Thomas:
$$\mathbf{P}\left(D(\hat{P}_n\|P)\geq\epsilon\right)\leq e^{-n\left(\epsilon-|\mathcal{X}|\frac{\log(... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving the uncountability of $[a,b]$ and $(a,b)$ I am trying to prove that $[a,b]$ and $(a,b)$ are uncountable for $a,b\in \mathbb{R}$. I looked up Rudin and I am not too inclined to read the chapter on topology, for his proof involves perfect sets.
Can anyone please point me to a proof of the above facts without poi... | $$\tan \left( \frac{\pi}{(b-a)} (x-\frac{a+b}{2})\right)$$
Basically $f(x)=\frac{\pi}{(b-a)} (x-\frac{a+b}{2})$ is the linear function such that $f(a)=-\frac{\pi}{2}$ and $f(b)=\frac{\pi}{2}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Maximize $x_1x_2+x_2x_3+\cdots+x_nx_1$ Let $x_1,x_2,\ldots,x_n$ be $n$ non-negative numbers ($n>2$) with a fixed sum $S$.
What is the maximum of $x_1x_2+x_2x_3+\cdots+x_nx_1$?
| I have to solve this in 3 parts. First for $n=3$, then for $n=4$ and finally for $n>4$.
For $n=3$ we can take a tranformation $x'_1=x'_2=(x_1+x_2)/2$ and $x'_3=x_3$. $\sum x_i$ remains fixed while $\sum{x'_i*x'_{i+1}}-\sum{x_i*x_{i+1}} = (x_1+x_2)^2/4-x_1*x_2 = (x_1^2+2x_1x_2+x_2^2)/4-x_1*x_2 = (x_1^2-2x_1x_2+x_2^2)/4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/252103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Proving that $x^a=x^{a\,\bmod\,{\phi(m)}} \pmod m$ i want to prove $x^a \equiv x^{a\,\bmod\,8} \pmod{15}$.....(1)
my logic:
here, since $\mathrm{gcd}(x,15)=1$, and $15$ has prime factors $3$ and $5$ (given) we can apply Euler's theorem.
we know that $a= rem + 8q$, where $8= \phi(15)$,
$x^a \equiv x^{rem}. (x^8)^q \pmod... | If $b\equiv a\pmod m, b$ will be equal to $a\iff 0\le a<m$
For example, $m-2\equiv m-2\pmod m, 13\equiv 13\pmod {15}$
but, $m+2\equiv 2\pmod m, 17\equiv 2\pmod {15}$
If $b\equiv c\pmod{\phi(m)} ,$i.e., if $b=c+d\phi(m)$
$y^b=y^{c+d\phi(m)}=y^c\cdot(y^{\phi(m)})^d\equiv y^c \pmod m$ for $(y,m)=1$
Here $\phi(15)=\phi(3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Element Argument Proofs - Set theory This is an exercise on my study guide for my discrete applications class.
Prove by element argument: A × (B ∩ C) = (A × B) ∩ (A × C)
Now I know that this is the distributive law, but I'm not sure if this proof would work in the exact same way as a union problem would, because I know... | No, you're not doing it completely right, the cartesian product produces an element that is a pair of elements from both subsets.
The definition of the cartesian product.
Def. $X\times Y = \{ (x,y) : x \in X\text{ and }y \in Y \}$.
PROOF.
$Z = A \times (B \cap C) = \{ (a,y) : a \in A\text{ and }y \in B \cap C \}$
$W =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definition of a tangent I've been involved in a discussion on the definition of a tangent and would appreciate a bit of help.
At my high school and my college, I was taught that a definition of a tangent is 'a line that intersects given curve at two infinitesimally close points.' Aside from the possibility that tangent... | Given a curve $y = f(x)$ in an $xy$-coordinate system a tangent to the curve at the point $(a,f(a))$ is a straight line ($y = mx + b$) with slope $m = f'(a)$.
I have never heard about the definition that you talk about. There are ways to "think" about what a tangent is. If you consider the definition of a derivative th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Proof: in $\mathbb{R}$, $((0,1),|\cdot|)$ is not compact. Let $(M,d)$ be a metric space, and $A\subset M$. By definition, $A$ is said to be compact if every open cover of $A$ contains a finite subcover.
What is wrong with saying that, in $\mathbb{R}$, if $I=(0,1)$, we can choose $G=\{(0,\frac{3}{4}), (\frac{1}{4}, 1)\}... | You counter example (the open cover $\cup_{n \in \mathbb N} (1/n,1)$) actually works. It has no finite subcover. Therefore, $(0,1)$ is not compact. Every cover is not necessarily infinite. Again, your $G$ is the counter example (it is a finite cover.)
Note: you can take $G$ as a finite subcover of $G$. So, $G$ does not... | {
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"source": "stackexchange",
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Optimal Coding Scheme for given Weights I'm having trouble with this homework problem. Do I create the tree by beginning with each weight being a leaf? Then combining the lowest weighted leaves, and their parent becomes the sum of their weight?
I got 85 as my answer for (b) but I'm not sure if this is the correct proc... | Yes, you first combine $10+12=22$, then $13+16=29$, then $17+17=34$, then $22+29=51$, finally $51+34=85$ (thus your answer for b).
If we always represent the first choice with 0 and the second with 1, the respective code words are
$$000,001, 010, 011, 10, 11.$$
I'm not sure if part b isn't rather referring to the weigh... | {
"language": "en",
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Showing that $W_1\subseteq W_1+W_2$ I found this question and answer on UCLA's website:
Let $W_1$ and $W_2$ be subspaces of a vector space $V$ .
Prove that $W_1 +W_2$ is a subspace of $V$ that contains both $W_1$ and $W_2$.
The answer given:
First, we want to show that $W_1 \subseteq W_1 +W_2$. Choose $x \in W_1$. Sin... | $W_1+W_2=\{w_1+w_2: w_1\in W_1,w_2\in W_2\}$. To show that an element belongs to this set, we just need to show that it can be written in the form $w_1+w_2$ for some $w_1\in W_1$ and some $w_2\in W_2$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Want to show Quantifier elimination and completeness of this set of axioms...
Let $\Sigma_\infty$ be a set of axioms in the language $\{\sim\}$ (where $\sim$ is a binary relation
symbol) that states:
(i) $\sim$ is an equivalence relation;
(ii) every equivalence class is infinite;
(iii) there are infinitely many equiva... | According to the last sentence in your question, all you need is an $L$-structure that can be embedded into every model of $\Sigma_\infty$. In fact, $\Sigma_\infty$ has a "smallest" model, one that embeds into all other models of $\Sigma_\infty$. I think this should be enough of a hint to enable you to find the model... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Good software for linear/integer programming I never did any linear/integer programming so I am wondering the following two things
*
*What are some efficient free linear programming solvers?
*What are some efficient commercial linear programming solvers?
It would be nice to supply a dummy usage example with eac... | The Konrad-Zuse Institute in Berlin (ZIB), Germany provides a nice suite to solve all kinds of LP / ILP tasks. It includes:
*
*zimpl: a language to model mathematical programms
*SCIP: a mixed integer programming solver and constraint programming framework
*SoPlex: a linear programming solver
*and more
Best ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Noncontinuity and an induced equivalence relation Can someone give me an example of a map which is not continuous such that $f(\{a\}) = f(\{b \})$ induces an equivalence relation $ \{ a \} \sim \{ b \} $?
| Let $f:\{a,b,c,\}\to\{0,1\}$. Let $f(a)=f(b)=0$ and $f(c)=1$. Let $x\sim y$ precisely if $f(x)=f(y)$. Then we have
\begin{align}
a & \sim a \\
a & \sim b \\
a & \not\sim c \\ \\
b & \sim a \\
b & \sim b \\
b & \not\sim c \\ \\
c & \not\sim a \\
c & \not\sim b \\
c & \sim c
\end{align}
This is an equivalence relati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/252719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Quotient of Gamma functions I am trying to find a clever way to compute the quotient of two gamma functions whose inputs differ by some integer. In other words, for some real value $x$ and an integer $n < x$, I want to find a way to compute
$$ \frac{\Gamma(x)}{\Gamma(x-n)} $$
For $n=1$, the quotient it is simply $(x-1)... | I think you mean
$$ \frac{\Gamma(x)}{\Gamma(x-n)} = \prod_{i=1}^{n} (x - i) $$
Of course this might not be very nice if $n$ is very large, in which case you might want to first compute the $\Gamma$ values and divide; but then (unless $x$ is very close to one of the integers $i$) the result will also be enormous. If yo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is the Frobenius norm of a matrix greater than or equal to the spectral norm?
How can one prove that $ \|A\|_2 \le \|A\|_F $ without using $ \|A\|_2^2 := \lambda_{\max}(A^TA) $?
It makes sense that the $2$-norm would be less than or equal to the Frobenius norm but I don't know how to prove it. I do know:
$$\|A\... | In fact, the proof from $\left\| \mathbf{A}\right\|_2 =\max_{\left\| \mathbf{x}\right\|_2=1} \left\| \mathbf{Ax} \right\|_2$ to $\left\| \mathbf{A}\right\|_2 = \sqrt{\lambda_{\max}(\mathbf{A}^H \mathbf{A})}$ is straight forward. We can first simply prove when $\mathbf{P}$ is Hermitian
$$
\lambda_{\max} = \max_{\| \math... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can floor functions have inverses? R to R
$f(x) = \lfloor \frac{x-2}{2} \rfloor $
If $T = \{2\}$, find $f^{-1}(T)$
Is $f^{-1}(T)$ the inverse or the "image", and how do you know that we're talking about the image and not the inverse?
There shouldn't be any inverse since the function is not one-to-one, nor is it onto si... | Note to this calculating;
$[\frac{x-2}{2}]=2\Longrightarrow 2\leq\frac{x-2}{2}<3\Longrightarrow 4\leq x-2<6\Longrightarrow 6\leq x<8$
so the set $f^{-1}(\{2\})$ is equal to $[6,8)$.
Now;
$\forall y\in\mathbb{Z}\; :\; f^{-1}(\{y\})=\{x\in\mathbb{R}|[\frac{x-2}{2}]=y\}$ ... $\Longrightarrow\{x\in\mathbb{R}|x\in[2y+2,2y+4... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of ordered triplets $(x,y,z)$, $−10\leq x,y,z\leq 10$, $x^3+y^3+z^3=3xyz$ Let $x, y$ and $z$ be integers such that $−10\leq x,y,z\leq 10$. How many ordered triplets $(x,y,z)$ satisfy $x^3+y^3+z^3=3xyz$?
x,y,z are allowed to be equal.
When I tried I got any one of x,y,z to be 0. I am not sure this is correct. And... | $\textbf{Hint}$: Note that $$x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)=\frac{1}{2}(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]=0$$ if and only if either $x+y+z=0$ or $x=y,y=z$ and $z=x$. Now count the number of ordered triples for the first case using generating functions.
| {
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Subspace Preserved Under Addition of Elements? I'm trying to understand how to complete this proof about subspaces. I understand the basics about the definition of a subspace (i.e. the zero matrix must exist, and addition and multiplication must be closed within the subspace). But I'm confused as to how to show that th... | You have to show (among other things) that $Z=\{{\,a+b:a\in C,b\in D\,\}}$ is closed under addition. So, let $x$ and $y$ be in $Z$; you have to show $x+y$ is in $Z$. So, what is $x$? Well, it's in $Z$, so $x=a+b$ for some $a$ in $C$ and some $b$ in $D$. What's $y$? Well, it's also in $Z$, so $y=r+s$ for some $r$ in $C$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you find the smallest integer? $$\begin{align}
(x-1) \;\text{mod}\; 11 &= 3x\; \text{mod}\; 11\\
11&\lvert(3x-(x-1)) \\
11&\lvert2x+1\\
x &= 5?\\
\end{align}
$$
$$
\begin{align}
(a-b)\; \text{mod}\; 5 &= (a+b)\;\text{mod}\;5\\
5&\lvert a+b-a+b\\
5&\lvert2b\\
b &= 5/2\\
a &= \text{any integer}
\end{align}
$$
I do... | There are several ways; for example look at Diophantine divisors, but now I will write it ;
$ax\equiv b\;(mod\;m)\Longleftrightarrow (a,m)=d|b$ and its answers are every number in congreuent classes by modulo m like $(\frac{a}{d})^{*}(\frac{b}{d})+k\frac{m}{d}$ where $0\leq k\leq d-1$ and $(\frac{a}{d})^{*}$ in Möbius ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic definition of Inverse Limit in sheaf theory / schemes I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of limits.
My example (page 60 in the book): Let $A$ be an int... | My personal advice is to study a bit of category theory: it will let you understand all this stuff in a very clearer way. In fact you can easily realize that the first equality is not a definition, but a way to express a limit of an arbitrary presheaf, while the second is an isomorphism, not exactly an equality, given ... | {
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"source": "stackexchange",
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A Fourier transform using contour integral I try to evaluate $$\int_{-\infty}^\infty \frac{\sin^2 x}{x^2}e^{itx}\,dx$$ ($t$ real) using contour integrals, but encounter some difficulty. Perhaps someone can provide a hint. (I do not want to use convolution.)
| An idea, defining
$$f(z):=\frac{e^{itz}\sin^2z}{z^2}\,\,,\,\,C_R:=[-R-\epsilon]\cup(-\gamma_\epsilon)\cup[\epsilon,R]\cup\gamma_R$$
with
$$\gamma_k:=\{z\in\Bbb C\;;\;|z|=k\,,\,\arg z\geq 0\}=\{z\in\Bbb C\;;\;z=ke^{i\theta}\,\,,\,0\leq\theta\leq\pi\}$$
in the positive direction (check the minus sign in $\,\gamma_\epsilo... | {
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"timestamp": "2023-03-29T00:00:00",
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Applying the Thue-Siegel Theorem Let $p(n)$ be the greatest prime divisor of $n$. Chowla proved here that $p(n^2+1) > C \ln \ln n $ for some $C$ and all $n > 1$.
At the beginning of the paper, he mentions briefly that the weaker result $\lim_{n \to \infty} p(n^2+1) = \infty$ can be proved by means of the Thue-Siegel t... | Suppose that the prime factors of $n^2+1$ are all bounded by $N$ for infinitely many $n$. Then infinitely many integers $n^2 + 1$ can be written in the form $D y^3$ for one of finitely many $D$. Explicitly, the set of $D$ can be taken to be the finitely many integers whose prime divisors are all less than $N$, and whos... | {
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Why is the set of natural numbers not considered a non-Abelian group? I don't understand why the set of natural numbers constitutes a commutative monoid with addition, but is not considered an Abelian group.
| Addition on the natural numbers IS commutative ...
...but the natural numbers under addition do not form a group.
Why not a group?
*
*if you define $\mathbb{N} = \{ n\in \mathbb{Z} \mid n\ge 1\}$, as we typically do, then it fails to be a group because it does not contain $0$, the additive identity. Indeed, $\math... | {
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If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
$P$ and $Q$ are subgroups of a group $G$. How can we prove that $P\cap Q$ is a subgroup of $G$? Is $P \cup Q$ a subgroup of $G$?
Reference: Fraleigh p. 59 Question 5.54 in A First Course in Abstract Algebra.
|
$P$ and $Q$ are subgroups of a group $G$. Prove that $P \cap Q$ is a subgroup.
Hint 1: You know that $P$ and $Q$ are subgroups of $G$. That means they each contain the identity element, say $e$ of $G$. So what can you conclude about $P\cap Q$? If $e \in P$ and $e \in Q$? (Just unpack that means for their intersectio... | {
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"timestamp": "2023-03-29T00:00:00",
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What's a proof that a set of disjoint cycles is a bijection? Consider a function $f : D \to D$ (where $D$ is a finite set) so that for every $d \in D$, there is an integer $n$ so that $f(f(...(n\text{ times})...f(d)...) = d$.
*
*Prove that $f$ is a bijection.
*Prove that if $f : D \to D$ (where $D$ is a finite set)... | For (1), it's easier to prove that if $f$ is not injective then neither is $f^{n}$ and if $f$ is not surjective then neither is $f^n$, for any $n \ge 1$. You know that $f^n$ is bijective, and hence so must $f$ be.
For (2), note that the bijections on a set form a group under composition, and that if $D$ is finite then ... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $\alpha_1u+\alpha_2v+\alpha_3w=0\Rightarrow\alpha_1=\alpha_2=\alpha_3=0$ Let $u, v, w$ be three points in $\mathbb R^3$ not lying in any plane containing the origin. Would you help me to prove or disprove: $\alpha_1u+\alpha_2v+\alpha_3w=0\Rightarrow\alpha_1=\alpha_2=\alpha_3=0.$ I think this is wrong since ot... | With $u_1=(1,1,0)$, $u_2=(1,2,0)$, $u_3=(1,3,0)$
Let $A = \begin{pmatrix}
u_1 \\
u_2 \\
u_3
\end{pmatrix} = \begin{pmatrix}1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 3 & 0 \end{pmatrix}$
we have $\det A = 0$ $\implies$ $u_1, u_2, u_3$ is linearly dependent $\implies$ you're wrong !
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Convolution Laplace transform
Find the inverse Laplace transform of the giveb function by using the
convolution theorem.
$$F(x) = \frac{s}{(s+1)(s^2+4)}$$
If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - \frac{1}{5(x+1)}$$
which gives me Laplace inverses:
$$\frac{1}{5}(\cos2t + \sin2t) -\frac{1}{5} e^{-t}... | Related techniques (I), (II). Using the fact about the Laplace transform $L$ that
$$ L(f*g)=L(f)L(g)=F(s)G(s)\implies (f*g)(t)=L^{-1}(F(s)G(s)) .$$
In our case, given $ H(s)=\frac{1}{(s+1)}\frac{s}{(s^2+4)}$
$$F(s)=\frac{1}{s+1}\implies f(t)=e^{-t},\quad G(s)=\frac{s}{s^2+4}\implies g(t)=\cos(2t).$$
Now, you use the c... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to verify this limit? Kindly asking for any hints about showing:
$$\lim_{n\to\infty}\int_0^1\frac{dx}{(1+x/n)^n}=1-\exp(-1)$$
Thank you very much, indeed!
| HINT: Just evaluate the integral. For $n>1$ you have
$$\int_0^1\frac{dx}{(1+x/n)^n}=\int_0^1\left(1+\frac{x}n\right)^{-n}dx=\left[\frac{n}{n+1}\left(1+\frac{x}n\right)^{-n+1}\right]_0^1\;;$$
evaluating that leaves you with a limit that involves pieces that ought to be pretty familiar.
| {
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"timestamp": "2023-03-29T00:00:00",
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Is one structure elementary equivalent to its elementary extension? Let $\mathfrak A,\mathfrak A^*$ be $\mathcal L$-structures and $\mathfrak A \preceq \mathfrak A^*$. That implies forall n-ary formula $\varphi(\bar{v})$ in $\mathcal L$ and $\bar{a} \in \mathfrak A^n$
$$\models_{\mathfrak A}\varphi[\bar{a}] \iff \model... | (I realise that it was answered in the comments, but I'm posting the answer so as to keep the question from staying in the unanswered pool.)
This is, of course, true, an $\mathcal L$-sentence without parameters is an $\mathcal L$-sentence with parameters, that happens not to use any parameters, so elementary extension ... | {
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"timestamp": "2023-03-29T00:00:00",
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The pebble sequence Let we have $n\cdot(n+1)/2$ stones grouped by piles. We can pick up 1 stone from each pile and put them as a new pile. Show that after doing it some times we will get the following piles: $1, 2, \ldots n$ stones.
Example: $n = 3$
Let we have 2 piles with 3 stones of each.
$$3 3 \to 2 2 2 \to 1 1 1 3... | This was originally proved by Jørgen Brandt in Cycles of Partitions, Proceedings of the American Mathematical Society, Vol. 85, No. 3 (Jul., 1982), pp. 483-486, which is freely available here. The proof of this result covers the first page and a half and is pretty terse.
First note that there are only finitely many pos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/253860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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} |
Counterexample to Fubini? I am trying to come up with a measurable function on $[0,1]^2$ which is not integrable, but such that the iterated integrals are defined and unequal.
Any help would be appreciated.
| $$
\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dy\,dx \ne \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dx\,dy
$$
Obviously either of these is $-1$ times the other and if this function were absolutely integrable, then they would be equal, so their value would be $0$. But one is $\pi/2$ and the other is $-\pi/2$, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/253921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find integral from 1 to infinity of $1/(1+x^2)$ I am practicing for an exam and am having trouble with this problem. Find the integral from 1 to infinity of $\frac{1}{1+x^2}$. I believe the integral's anti-derivative is $\arctan(x)$ which would make this answer $\arctan(\infty)-\arctan(1)$ but from here I'm lost. I did... | Essentially, your question appears to be "Why does $\lim_{x\to\infty}\arctan{x} = \frac{\pi}{2}$?"
Remember that
$$\theta = \arctan\left(\frac{y}{x}\right)$$
When will $\frac{y}{x}\to\infty$? That's when $x \to 0$.
Think of a right triangle with height $y$ and base $x$. $\theta$ is the angle between $x$ and the hyp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/253979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Find the Matrix of a Linear Transformation. It's been a few weeks since the subject was covered in my Linear Algebra class, and unfortunately linear transformations are my weak spot, so could anyone explain the steps to solve this problem?
Find the matrix $A$ of the linear transformation $T(f(t)) = 3f'(t)+7f(t)$ from ... | NOTE Given a finite dimensional vector space $\Bbb V$ and a basis $B=\{v_1,\dots,v_n\}$ of $\Bbb V$, the coordinates of $v$ in base $B$ are the unique $n$ scalars $\{a_1,\dots,a_n\}$ such that $v=\sum_{k=1}^n a_kv_k$, and we note this by writing $(v)_B=(a_1,\dots,a_n)$.
All you need is to find what $T$ maps the basis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $(a,b,c)$ is a primitive Pythagorean triplet, explain why... If $(a,b,c)$ is a primitive Pythagorean triplet, explain why only one of $a$,$b$ and $c$ can be even-and that $c$ cannot be the one that is even.
What I Know:
A Primitive Pythagorean Triple is a Pythagorean triple $a$,$b$,$c$ with the constraint that $\gcd... | Clearly they cannot all be even as a smaller similar triple could be obtained by dividing all the sides by $2$ (your final point).
Nor can two of them be even since $a^2+b^2=c^2$ and either you would have an even number plus an odd number (or the other way round) adding to make an even number or you would have an even... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uniformly convergent sequence proof. Prove that if $(f_k)$ is a uniformly convergent sequence of continuous real-valued functions on a compact domain $D\subseteq \mathbb{R}$, then there is some $M\geq 0$ such that $\left|f_k(x)\right|\leq M$ for every $x\in D$ and for every $k\in \mathbb{N}$.
My response: Basically, I ... | I think this is quite obvious. Noted that $D$ is compact and $f$ is continuous so $f_k(D)$ is also compact for every $k$ and hence it is bounded for every $k\in\mathbb{N}$ and we can just take $M=\sup\bigcup f_k(D)$ where $M \ne +\infty$ as every $f_k(D)$ are bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/254225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
About Central limit theorem We prove Central limit theorem with characteristic function. If we know the $X_i$ are independent but not identically distributed, is there any weaker condition which still yields the convergence to normal distribution?
| For example, suppose $X_i$ are independent with $\mathbb E(X_i) = 0$, $\text{Var}(X_i) = \sigma_i^2$, and $$\lim_{n \to \infty} \frac{1}{\sigma(n)^3} \sum_{i=1}^n \mathbb E[|X_i|^3] = 0$$
where $$\sigma(n)^2 = \text{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \sigma_i^2$$ Then $\displaystyle \frac{1}{\sigma(n)} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$ How can I calculate the following sum involving binomial terms:
$$\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$
Where the value of n can get very big (thus calculating the binomial coefficients is not feasible).
Is there a closed form for this su... | Apparently I'm a little late to the party, but my answer has a punchline!
We have
$$
\frac{1}{z} \int_0^z \sum_{k=0}^{n} \binom{n}{k} s^k\,ds = \sum_{k=0}^{n} \binom{n}{k} \frac{z^k}{k+1},
$$
so that
$$
- \int_0^z \frac{1}{t} \int_0^t \sum_{k=0}^{n} \binom{n}{k} s^k\,ds\,dt = - \sum_{k=0}^{n} \binom{n}{k} \frac{z^{k+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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"answer_id": 2
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A simple limit of a sequence I feel almost ashamed for putting this up here, but oh well..
I'm attempting to prove: $$\lim_{n\to \infty}\sqrt[n]{2^n+n^5}=2$$
My approach was to use the following inequality (which is quite easy to prove) and the squeeze theorem:
$$\lim_{n\to \infty}\sqrt[n]{1}\leq \lim_{n\to \infty}\sqr... | How about using $\root n\of{1+(1/n)}\le\sqrt{1+(1/n)}$ for $n\ge2$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/254475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
An easy way to remember PEMDAS I'm having trouble remembering PEMDAS (in regards to precedence in mathematical equations), ie:
*
*Parentheses
*Exponentiation
*Multiplication & Division
*Addition & Subtraction
I understand what all of the above mean, but I am having trouble keeping this in my head. Can you recom... | I am a step by step person.
(remembering always left to right)
1. Parenthesis
2. exponents
3. multiply/divide
4. add/subtract
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/254513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 13,
"answer_id": 9
} |
Pattern continued The following pattern:
$$\frac{3^{2/401}}{3^{2/401} +3}+\frac{3^{4/401 }}{3^{4/401} +3}+\frac{3^{6/401}}{3^{6/401} +3}+\frac{3^{8/401}}{3^{8/401} +3}$$
what will the result be if the pattern is continued $\;300\;$ times?
| If you need the sum to the nth term, you're looking at computing the sum of the first 300 terms:
$$\sum_{k=1}^{300}\left(\large\frac{3^{\frac{2k}{401}}}{3^{\frac{2k}{401}}+3}\right)$$
To sum to the nth term, you need to compute:
$$\sum_{k=1}^{n}\left(\large\frac{3^{\frac{2k}{401}}}{3^{\frac{2k}{401}}+3}\right)=
\sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Probability question with combinations of different types of an item Suppose a bakery has 18 varieties of bread, one of which is blueberry bread. If a half dozen loafs of bread are selected at random (with repetitions allowed), then what is the probability that at least one of the loafs of blueberry bread will be inclu... | I don't think that's a very good direction. It could require much work. I always fond it useful to think of a simplified version:
Suppose we pick 1 loaf. What are the chances of it being blueberry?
Suppose we pick 2 loaves. What are the chances of not having a blueberry?
3 loaves?
4 loaves?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/254641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to find its closed-form? here is a sequence defined by the below recursion formula:
$$a_n=2a_{n-1}+a_{n-2}$$
where $n \in \mathbb{N}$ and $a_0=1,a_1=2$.how to find its closed-form.
| If we write $E^ra_n=a_{n+r},$
the characteristic/auxiliary equation becomes $E^2-2E-1=0,E=1\pm\sqrt2$
So, the complementary function $a_n=A(1+\sqrt2)^n+B(1-\sqrt2)^n$ where $A,B$ are indeterminate constants to be determined from the initial condition.
$a_0=A+B,$ But $a_0=1$ So, $A+B=1$
and $a_1=A(1+\sqrt2)+B(1-\sqrt2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
On some inequalities in $L_p$ spaces Let $f$ be a function such that $\|fg\|_1<\infty$ whenever $\|g\|_2<\infty$. I would like to show that $\|f\|_2<\infty$. It seems that I should use some kind of Hölder inequalities, since we have $\|fg\|_1\leq \|f\|_2\|g\|_2$, but I don't know how. Any help would be appreciated. Tha... | You have to assume that
$$M := \sup \{ \|f \cdot g\|_1; \|g\|_2 \leq 1\}<\infty$$
... otherwise it won't work. (Assume $M=\infty$. Then for all $n \in \mathbb{N}$ there exists $g_n \in L^2$, $\|g_n\|_2 \leq 1$, such that $\|f \cdot g_n\|_1 \geq n$. And this means that there cannot exist a constant $c$ such that $\|f \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$ I am trying to prove this binomial statement:
For $a \in \mathbb{C}$ and $k \in \mathbb{N_0}$, $\sum_{j=0}^{k} {a+j \choose j} = {a+k+1 \choose k}.$
I am stuck where and how to start. My steps are these:
${a+j \choose j} = \frac{(a+j)!}{j... | One way to prove that
$$\sum_{j=0}^k\binom{a+j}j=\binom{a+k+1}k\tag{1}$$
is by induction on $k$. You can easily check the $k=0$ case. Now assume $(1)$, and try to show that
$$\sum_{j=0}^{k+1}\binom{a+j}j=\binom{a+(k+1)+1}{k+1}=\binom{a+k+2}{k+1}\;.$$
To get you started, clearly
$$\begin{align*}
\sum_{j=0}^{k+1}\binom{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why does $\ln(x) = \epsilon x$ have 2 solutions? I was working on a problem involving perturbation methods and it asked me to sketch the graph of $\ln(x) = \epsilon x$ and explain why it must have 2 solutions. Clearly there is a solution near $x=1$ which depends on the value of $\epsilon$, but I fail to see why there ... | For all $\varepsilon>0$ using L'Hospital's rule
$$\lim\limits_{x \to +\infty} {\dfrac{\varepsilon x}{\ln{x}}}=\varepsilon \lim\limits_{x \to +\infty} {\dfrac{x}{\ln{x}}}=\varepsilon \lim\limits_{x \to +\infty} {\dfrac{1}{\frac{1}{x}}}=+\infty.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/254926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
On an identity about integrals Suppose you have two finite Borel measures $\mu$ and $\nu$ on $(0,\infty)$. I would like to show that there exists a finite Borel measure $\omega$ such that
$$\int_0^{\infty} f(z) d\omega(z) = \int_0^{\infty}\int_0^{\infty} f(st) d\mu(s)d\nu(t).$$
I could try to use a change of variable... | When we have no idea about the problem, the question we have to ask ourselves is: "if a measure $\omega$ works, what should it have to satisfy?".
We know that for a Borel measure that it's important to know them on intervals of the form $(0,a]$, $a>0$ (because we can deduce their value on $(a,b]$ for $a<b$, and on fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Making a $1,0,-1$ linear commbination of primes a multiple of $1000$ Prove that with every given 10 primes $p_1,p_ 2,\ldots,p_{10}$,there always exist 10 number which are not simultaneously equal to $0$, get one of three values: $-1$, $0$, $1$ satisfied that:
$\sum\limits_{ i=1}^{10}a_ip_i$ is a multiple of 1000
| The pigeonhole principle takes care of this. There are $2^{10}-1=1023$ non-empty sums, so two are congruent modulo $1000$, etc., etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/255110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is $S = X^2 + Y^2$ distributed $Exponential(\frac{1}{2})$? Let $X$ and $Y$ be independent, standard normally distributed random variables ($\sim Normal(0, 1)$). Why is $S = X^2 + Y^2$ distributed $Exponential(\frac{1}{2})$?
I understand that an exponential random variable describes the time until a next event given... | Sum of 2 standard normals is chi squared with 2 degrees of freedom.
A chi squared with 2 degrees of freedom is equivalent to a exponential with parameter =$1/2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/255163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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