Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proportional Image Resizing This is quite a simple one I guess (but my mind is dead currently and is getting cluttered by other numbers).
I have an square image that is currently 256 width x 256 height.
I am variably resizing this image to a bigger size proportionally, so lets say I resize the image to 350 width to 350... | The first image has changed from $256$ to $350$, a $350/256$ change.
The second image should change by the same factor: from $416$ to $416 \cdot 350/256 = 568.75$. So choose $568$ or $569$ as the final size.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f$ is continuous, why is $f$ with the property $f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y)$ is convex?
If $f$ is continuous, why is $f$ with the property
$$f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y),$$ where $0\le x,y\le 1$
is convex?
| By induction we can prove that: if $k,m, l\in\mathbb{N} , k+m=2^l , x,y\in \mbox{domain} f $ then $$f\left( \frac{k}{2^l} \cdot x +\frac{m}{2^l} \cdot y \right)\leq \frac{k}{2^l} \cdot f(x) +\frac{m}{2^l} \cdot f(y). $$
Indeed the asertion is true when $l=1 .$ Suppose that it is true for some $l\geq 1 .$ And let $k+m=2... | {
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Different ways of proving a polynomial is irreducible in finite field. Is there a general characterization of irreducible polynomials over a finite field?
I was going through a problem in finding whether $p(x):=x^7+x^5+1$ is irreducible over $\mathbb F_2[x]$ or not.
If the polynomial is of degree less than or equal ... | The given polynomial is in fact not irreducible. There is at least one decomposition:
$$
(x^2+x+1)\cdot(x^5+x^4+x^3+x+1) = x^7+x^5+1
$$
This can be found by resolving the equality for coefficients:
$$
(x^2+ax+1)\cdot(x^5+bx^4+cx^3+dx^2+ex+1) = x^7+x^5+1
$$
which, equating term by term, and ignoring the terms of degree ... | {
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Most important Linear Algebra theorems? I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem:
A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix.
This is because it is impossible to tell when a matrix is diagonalizable, or so it seems.
I have... | The two main candidates are:
*
*The fundamental theorem of linear algebra, as popularised by Strang.
*The singular value decomposition.
From these, lots of important results follow.
| {
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"timestamp": "2023-03-29T00:00:00",
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Question regarding 3 x 3 matrices If $A$ is a $3 \times 3$ matrix with real elements and $\det(A)=1$, then are these affirmations equivalent:
$$
\det(A^2-A+I_3)=0 \leftrightarrow \det(A+I_3)=6 \text{ and } \det(A-I_3)=0?
$$
| $\Leftarrow)$ From $\det (A-I)=0$ we know that $1$ is an eigenvalue. Let $x,y$ be the two others (possibly complex, and counting multiplicities). From $\det A=1$ we know that $xy=1$. And $6=\det(A+I)=2(x+1)(y+1)$, so
$$
3=xy+x+y+1=2+x+y,
$$
so $x+y=1$. We get a system of two equations on $x,y$, namely
$$
x+y=1,\ \ xy=... | {
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Context free grammar question i have two context free grammar questions and I don't know how to do them.
*
*$$\{(a^n)b(c^n) \mid n >0 \}$$ I'm having trouble with this one because I don't know how to account for $a$ or $b$ not having empty set.
*$$\{(a^n)(b^m)(c^n) \mid n,m \ge 0\}$$
| You are close for the first:
$$S\to aAc,\qquad A\to S \mid b$$
and for the second
$$S\to aSc \mid A\qquad A\to bA\mid\epsilon $$
should work
| {
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How can I calculate $\displaystyle\lim_{x \to \infty}x^{2}\ln\left(\cos \left(\pi/x\right)\right) $? Does anybody know how to solve this?
$$\lim_{x \to \infty}x^{2}\ln\left(\cos\left(\pi \over x\right)\right)$$
| Let $x=1/t$. So we are interested in the limit as $t\to 0^+$ of $\ln(\cos(\pi t))/t^2$.
One round of L'Hospital's Rule brings us to
$$\lim_{t\to 0^+} -\frac{\pi}{2\cos(\pi t)} \frac{\sin(\pi t)}{t}.$$
The first part is nicely behaved near $0$. For $\lim_{t\to 0^+} \frac{\sin(\pi t)}{t}$, use L'Hospital's Rule, or simp... | {
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Inverse image of a closed subscheme Let $f:X\to Y$ be a surjective morphism of schemes, and $Z\subset Y$ a closed subscheme with short exact sequence
$$ 0\to I_Z \to \mathcal{O}_Y \to \mathcal{O}_Z \to 0. $$
What are sufficient conditions on $X$, $Y$, $Z$ and $f$ such that the scheme-theoretic inverse image $W$ of $Z$ ... | Do you see that $f^*(I_Z) \to \mathcal{O}_X \to \mathcal{O}_W \to 0$ is always exact? So the question is basically only if $f^*(I_Z) \to \mathcal{O}_X$ is injective. This holds when $f$ is flat.
| {
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Closed sets, boundary, topology. Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not.
I think it's false but have no idea how to find a counterexample. Any help will be much appr... | Given $A$, let $B$ be any countable dense subset of $A$. (Such a $B$ exists; just take one point from each nonempty set of the form $A\cap(p,q)$ where $p<q$ are rational numbers.) So the closure of $B$ is $A$. The interior of $B$ is empty, because $B$ is countable. So the boundary of $B$ is $A$.
| {
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How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square? How would I prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
| The question is not about that big numbers...
Question is about a result that two consecutive numbers can not be squares simultaneously..
Suppose $a=b^2\text { and }a+1=c^2\Rightarrow b^2+1=c^2\Rightarrow c^2-b^2=1\Rightarrow (c+b)(c-b)=1$
w.l.o.g. assume $c+b=1$ which implies
$a+1=c^2=(1-b)^2=1+b^2-2b\Rightarrow a=b... | {
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Number of involution in symmetric group Can any one show me on how to prove that the number of involutions in the symmetric group on $n$ letters, $$\sum_{k=0}^{\lfloor n/2 \rfloor} {n \choose 2k} (2k-1)!! = \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{n!}{2^kk!(n-2k)!}$$
I've tried proving it via Young tableaux, knowing that... | Let $\tau$ be an involution. Writing $\tau$ as a product of disjoint cycles, we see that no cycle can have length greater than $3$ since $\tau$ has order $2$. It follows that $\tau$ is a product of disjoint transpositions.
Suppose that $\tau$ is a product of $k$ disjoint transpositions. Then it permutes precisely $2k$... | {
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Solve PDE in 2D Problem How should I go about solving this PDE:
$$ \phi_x+\phi_y=x+y-3c $$
Where $\phi = \phi(x,y)$, $c$ is a constant, and $\phi$ is specified on the circle
$$ x^2+y^2=1 $$
My Attempt to solve it I would like to use the method of characteristics, but then I get stuck because of the given initial cond... | As you say, the characteristic equations are $\dot{z} = x + y - 3 c$, $\dot{x} = 1$, $\dot{y} = 1$. So the characteristic curves are $x = x_0 + s$, $y = y_0 + s$, i.e. $x - y = \text{constant}$. But there's a problem with specifying the initial conditions on the circle $x^2 + y^2 = 1$: the characteristic curves throu... | {
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Constrained optimisation question
Since $f$ has a local extremum at $x_1$, then surely the LHS of equation (3) always zero? If so, then isn't lambda always simply zero too? But this cannot be, otherwise the last sentence of the theorem wouldn't be phrased the way it is. What am I missing?
| It's not $f$ that has a local extremum at $x_{1}$, but rather $f|_{S}$. Consider, for example, $f(x,y)=xy$ restricted to the line $y=1-x$. This restricted function has a local maximum at $(\frac{1}{2},\frac{1}{2})$, but the full function $f$ is not at an extremum at $(\frac{1}{2},\frac{1}{2})$.
| {
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Why/How are there infinite points in a line segment? A line may have infinite points becauase it may be expanded.But in case of a line segment it has 2 distinct points which are not movable.The distance between the end points in finite and known.
But still why do people(in my school) say that there are infinite points ... | The concept of infinity is used for many different things, and one should not confuse them. A line, be it closed, open, straight, curved, finite in length, or infinite in length always consists of infinitely many individual points (at least for any reasonable notion of 'line'). This can be proven rigorously, and it's n... | {
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Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$ Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$
I know the answer is $2^n - 1$, but how to simplify it?
| Another "proof" (this was actually the way I had "proved" it myself until I saw the GP proof):
The first summand is 1 followed by a $n-1$ in base 2. The later ones are with one less zero every time. Adding them up results in a number composed of 1 followed by n-1 zeros, which is $2^n-1$ (1 followed by n zeros minus one... | {
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Does De Moivre's Theorem hold for all real n? I have seen the proof by induction for all integers, and I have also seen in a textbook that we can use Euler's formula to prove it true for all rational n, but nowhere in the book does it say its true for irrational n.
I have also looked over the internet and there seems ... | The formula is actually true in a more general setting: if $z$ and $w$ are complex numbers, then $\left(\cos z + i\sin z\right)^w$ is a multi-valued function while $\cos (wz) + i \sin (wz)$ is not. However, it still holds that $\cos (wz) + i \sin (wz)$ is one value of $\left(\cos z + i\sin z\right)^w$.
| {
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How to show that $A_k=(-1)^k\binom nk$?
In the identity $$\frac{n!}{x(x+1)(x+2)\cdots(x+n)}=\sum_{k=0}^n\frac{A_k}{x+k},$$prove that $$A_k=(-1)^k\binom nk.$$
My try: The given identity implies $$\frac{1\cdot2\cdots n}{x(x+1)(x+2)\cdots(x+n)}=\frac{A_0}{x}+\frac{A_1}{x+1}+\dots+\frac{A_n}{x+n}.$$
Now putting $A_k=(-1)... | HINT: Multiply both side by $x+k$ and then put $x=-k.$
| {
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prove $f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$ I need help with this proof:
$f: X\rightarrow Y$
$C,D\subseteq Y$
$f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$
Thanks.
| To solve these kinds of questions, you want to show that $$f^{-1}(C\cap D) \subseteq f^{-1}(C) \cap f^{-1}(D)$$ and $$f^{-1}(C\cap D) \supseteq f^{-1}(C) \cap f^{-1}(D).$$
I will show $\subseteq$ here; you should try the other direction.
Suppose $x \in f^{-1} (C \cap D)$. This means $f(x) \in (C\cap D)$, which further ... | {
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non degenerate bilinear map for modules Let $R$ be a commutative ring with 1. Suppose $A,B$ are $R$-modules, $P:A\times B\to R$ is
a bilinear map that satisfies the following property: if $P(a,b)=0$ for all $b\in B$, then $a=0$. Then is the $R$-linear map $P':A\to B^*$, $P':a\mapsto P(a,b)$ an isomorphism? I could prov... | Hint: What you're talking about are the induced isomorphisms of a non-degenerate (not necessarily perfect) pairing. The definition of $P'$ is, however, a bit different. Here is the right one:
$$P': A \to B^*, a \mapsto P(a,-)$$
So the image of an $a \in A$ is really a dual map $B \to R$.
Are those modules finitely gene... | {
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How to find the derivative of $\operatorname{arcsinh}(3x)$? I know
The derivative of $\operatorname{arcsinh}(x) = 1/(x^2+1)^{1/2}$
But if I derivative $\operatorname{arcsinh}(3x)$
Why it doesn't equal to
$(\operatorname{arcsinh}(3x))^{-1} (1/(9x^2+1)) (3)$ ??
Thanks all
| Where did the factor $(\operatorname{arcsinh} (3x))^{-1}$ come from? Alpha agrees with $\frac 3{\sqrt{9x^2+1}}$
| {
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Why is $K(\alpha) = \left\lbrace \frac{f(\alpha)}{g(\alpha)} : f,g\in K[X],\, g(\alpha)\neq 0\right\rbrace$? I know from definition that: $K(\alpha)$ denotes the smallest subfield of $L$ that contains both $K$ and $\alpha$.
I've read here that this is equivalent with:
$$K(\alpha) = \left\lbrace \frac{f(\alpha)}{g(\alph... | Let $K(\alpha)$ be the smallest subfield containing $\alpha$ and let
$$K'(\alpha) = \left\lbrace \frac{f(\alpha)}{g(\alpha)} : f,g\in K[X],\, g(\alpha)\neq 0\right\rbrace.$$
You want $K'(\alpha) = K(\alpha)$.
First convince yourself that $K'(\alpha)$ is a subfield containing $\alpha$. Then by definition you have $K(\a... | {
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How I can calculate $\sum_{n=0}^\infty{\frac{n}{3^n}}$? For more than one series of any search function known not seem to find the sum of this series:
$$\sum_{n=0}^\infty{\frac{n}{3^n}}$$
I've found that converges with the quotient criterion.
Could you give me some suggestions for finding the sum of this series?
Thanks... | HINT:
If $|a|<1$,$$a\sum_{0\le n<\infty}r^n=\frac a{1-r}$$ (Proof)
Differentiate wrt $r$
| {
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Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$? The successive difference of powers of integers leads to factorial of that power. Here's the formula:
$$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$
Can anyone give a proof of this result?
Note: The original question was to ... | Let $A:=\{ x_1,..,x_n \}$ and $B=\{y_1,..,y_m \}$.
Lets count the number of onto functions $f:A \to B$. There are $m^n$ functions from $A$ to $B$. Lets count now the ones which are not onto:
Define
$$P_i= \{ f : A \to B |y_i \notin f(A) \}$$
Then we need to figure out the cardinality of $\cup_i P_i$.
By the inclusion ... | {
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"timestamp": "2023-03-29T00:00:00",
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Subgroup of order $n-1$ of a group of order $n$ Here is question 2.1.5 from Dummit and Foote : Prove that $G$ cannot have a subgroup $H$ with $|H| = n-1$, where $n = |G| > 2$.
How can one show this without using Lagrange's theorem (which is in chapter 3 of Dummit).
Thank you
| Let $g$ be an element of $G$ that is not in $H$. Let $h$ be any nonidentity element of $H$. Now show that $gh$ is not in $H$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How does one get better at real analysis proofs? How does one proceed through a math proof in real analysis? My instructor always says make a diagram, but I am not a visual learner. It seems that whenever I write out the definition of an assumption, then I cannot make the next logical step. Also, when I go to try to ve... | I think the best thing to do is to learn and understand the proofs of the theorem you do in class. The key thing about analysis (as opposed to algebra) is that all the proofs have a pattern to them.
For example: The proof of sequence of continuous functions converges uniformly to a continuous function uses the idea cal... | {
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Brouwer Fixed Point Theorem $f(S^1)\subset B$ I have a question about the Brouwer Fixed Point Theorem:
Theorem 1.(Brouwer Fixed Point Theorem) Let $B=\{x\in \mathbb R^2 :∥x∥≤1\}$ be the closed unit ball in $\mathbb R^2$ . Any continuous function $f:B\rightarrow B$ has a fixed point.
Theorem 2. Let $B=\{x\in \mathbb ... | Since Theorem $1$ obviously generalises to arbitrary finite radii, we can deduce Theorem $2$ from Theorem $1$ in a simple way:
Since $B$ is compact, $f(B)$ is contained in a ball $B_R$ of some finite radius $R > 0$.
Extend $f$ to $B_R$ in the following way:
$$F(x) = \begin{cases}f(x) &, \lVert x\rVert \leqslant 1\\
f\l... | {
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How to find $\lim_{n \rightarrow +\infty } \left(\sqrt[m]{\prod_{i=1}^{m}(n+{a}_{i})}-n\right)$? I think it is zero;
$$\lim_{n \rightarrow +\infty } \left(\sqrt[m]{\prod_{i=1}^{m}(n+{a}_{i})}-n\right)$$
we can make that steps:
$$\lim_{n \rightarrow +\infty } \left(\sqrt[m]{{n}^{m}\prod_{i=1}^{m}\left(1+\frac{a_i}n\righ... | The hard part is
showing that
$ P_n
=\sqrt[m]{\prod_{i=1}^{m}\left(1+\frac{{a}_{i}}{n}\right)}
\to 1$
as $n \to \infty$.
If $n > km\max(|a_i|)$,
$1+\frac{{a}_{i}}{n}
< 1+\frac1{km}
$
so $P_n
<\sqrt[m]{\prod_{i=1}^{m}\left(1+\frac1{km}\right)}
< 1+\frac1{km}
$.
By choosing $k$ large enough,
$P_n$ can be made as close to... | {
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Reduction Transitive Relation Problem I have this problem on my homework, it's my last one left but I'm having trouble with it. Any help would be appreciated.
|
Definition:
Let $P_1$ and $P_2$ be two problems and $\mathbb{A}$ be the space of all poly-time algorithms, then
$P_1 \leq_P P_2$ if $((\exists \ A_1,\ A_2 \in \mathbb{A}):
(A_1 \text{ maps any instance } I \text{ of } P_1 \text{ to an instance } A_1(I) \text{ of } P_2) \text{ and } (A_2 \text{ maps any solution } S \t... | {
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Relations $R^2, R^3, R^i and R^*$ Consider the relation on R on the reals where $xRy$ iff $xy=1$
I need to find $R^2, R^3, R^i $ and $R^*$
Ok, so I first started off with the following:
$$xR^2z \equiv \exists y: xRy\land yRz \\ \equiv\exists y: xy=1 \land yz=1 \\ \equiv xy + yz =y(x+z)=2$$
This to me doesn't seem righ... | The definition of $R \circ S$ is: $x(R \circ S)z$ if and only if there exists a $y$ such that $xRy$ and $ySz$.
This symbol $\wedge$ is not a plus! It means "and", and like the name suggests, it means both statements must be true.
So, to the actual problem. Say $xR^2z$. Then there exists a $y$ such that $xy = 1$ and $yz... | {
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Change of sign for Brownian motion For a fixed $\epsilon > 0$, I want to show that almost surely (i.e., with probability $1$), a standard Brownian motion $W_t$ would change sign over $[0,\epsilon]$.
I thought about defining a random variable $U_t = \mbox{sign} (W_t)$ which would satisfy $(1+U_t)/2$ follows symmetric Be... | Another way to solve this problem is with the Blumenthal zero-one law. Let $\mathcal{F}_t = \sigma(B_s : 0 \le s \le t)$, and $\mathcal{F}_{0+} = \bigcap_{t > 0} \mathcal{F}_t$. The Blumenthal zero-one law asserts that $\mathcal{F}_{0+}$ is almost trivial; for each $A \in \mathcal{F}_{0+}$ we have $\mathbb{P}(A) = 0$... | {
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"timestamp": "2023-03-29T00:00:00",
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How to solve this reccurence relation? Let a,b,c be real numbers. Find the explicit formula for $f_n=af_{n-1}+b$ for $n \ge 1$ and $f_0 = c$
So I rewrote it as $f_n-af_{n-1}-b=0$ which gives the characteristic equation as $x^2-ax-b=0$. The quadratic formula gives roots $x= \frac{a+\sqrt{a^2+4b}}{-2}, \frac{a-\sqrt{a^2+... | Why not consider this?
$f_n + m = a(f_{n-1} + m) \Longrightarrow (a-1)m=b$
1) $a=1$, simple recurrence $f_n = f_{n-1} + b$, $f_n = bn+c$
2) $a\neq 1$, $m=\frac{b}{a-1}$, $f_n+m = a(f_{n-1}+m)$, geometric sequence $f_n+m=a^n(c+m)$
Hope it is helpful!
| {
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"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Bijective check with matrix My book doesn't cover the criterion for bijective transformations very well. I just want to check my understanding: is this statement true?
Let F be a linear transformation. Let A be the matrix that represents that transformation (which means that that $F(v)=Av$ for any vector $v$). We now ... | I think about this in the following manner. A bijective linear transformation should have an inverse. Hence the associated matrix should also be invertible. Therefore it's determinant is non-zero. Hope this helps.
| {
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Prove intersection of all inductive sets is inductive How to prove that the intersection of all inductive sets is inductive?
Subset A of ordered field F is inductive, when:
1) $1 \in A$
2) if $a \in A$ then $a+1 \in A$
Prove that $$\mathbb{I} = \bigcap \left\{A\in F\ {\large|} \ A \text{ is inductive }\right\}$$ is ind... | HINT: Recall that $x\in\bigcap\cal A$ if and only if for every $A\in\cal A$, $x\in A$.
| {
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Construction of Homogenous Differential Equation We have to construct a homogenous differential equation of second order which has y(t) = e^t cos(t) as solution.
I know have some knowledge of how to solve (some) differential equations, but how do I construct one (systematically)?
| You want $y=e^t cos(t)$ as a solution so lets take charecteristic equation to be the one with $i+1 $ as its root.
This will be your charecteristic equation $(\lambda-(i+1))(\lambda+(1-i)) \to \lambda ^2 - 2\lambda +2$.
And this is your ode $y''-2y'+2y=0$ .
| {
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Integrals involving Hermite Polynomials Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial (Physicist's version) and $q,\,m$ and $n$ are positive integers.
If $x^{2q}$ term were... | As noted in the comments, the solution for arbitrary $q$
$$
I_{nm}(q)=\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_n(x)H_m(x)dx
$$
are the derivatives of the integral
$$
I_{nm}(0)=\int_{-\infty}^\infty e^{-ax^2}H_n(x)H_m(x)dx
$$
with respect to $a$
$$
I_{nm}(q)=(-1)^q\frac{\partial^q}{\partial a^q}I_{nm}(0).
$$
The $q=0$ int... | {
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$R \setminus (S \cup T)$ . Where is $x$? I am sorry for the messy math symbols.
If I have the set: $R \setminus(S \cup T)$ , is it correct to assume that:
$$R \setminus(S \cup T) = \{x: x∈ \mathbb{R} \text{ and } ( x \notin S \text{ and } x \notin T) \}$$
I am confused because if I had the set $S \cup T$ I would assu... | Your statement is correct.
I think where you're getting confused is when you have to use De Morgan's Law.
This can be converted logically as follows:
$$
x \in R \setminus (S \cup T) \to x \in R \wedge \neg (x \in S \vee x \in T)\\
x \in R \wedge (\neg x \in S \wedge \neg x \in T) \\
x \in R \wedge x \notin S \wedge x \... | {
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Integrating this indefinite integral $$\int\dfrac{t-2}{t+2-3\sqrt{2t-4}}dt$$
I'm not sure whether to use substitution or what.
| Hint
With the change of variable $x=\sqrt{2t-4}$ the integral becomes
$$\int\frac{x^3}{x^2-6x+8}dx$$
then use the partial fraction decomposition (notice that $x^2-6x+8=(x-2)(x-4)$).
| {
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Stats is not maths? How mainstream is the claim that stats is not maths? And if it's right, how many people don't agree?
Given that it's all numbers, taught by maths departments and you get maths credits for it, I wonder whether the claim is just half-jokingly meant, like saying it's a minor part of maths, or just app... | Statistics can be thought of as an application of mathematics towards the rather specific goal of examining numerical data.
In order to understand statistics, you should probably know some mathematics. In a similar manner, in order to understand engineering, you should probably know some mathematics.
But just like in e... | {
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Necessity of a hypothesis in the fundamental theorem of calculus Baby Rudin's Fundamental Theorem of Calculus (Theorem 6.21), in my professor's words,states:
Let $f: [a,b] \to \mathbb{R}$ be a Riemann integrable function. If $F: [a,b] \to \mathbb{R}$ is an antiderivative of $f$, then $\int_a^b \! f(x) \, \mathrm{d}x = ... | The derivative of a bounded differentiable function isn't necessarily bounded or continuous. A standard example is to let $1 < \alpha < 2$ and define $f(x) = x^\alpha \sin \frac 1x$ if $x \not= 0$, and $f(0) = 0$. In this case $f'(0) = 0$ but $f'$ is unbounded in every neighborhood of $0$.
This doesn't provide a counte... | {
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Factorial, but with addition Is there a notation for addition form of factorial?
$$5! = 5\times4\times3\times2\times1$$
That's pretty obvious. But I'm wondering what I'd need to use to describe
$$5+4+3+2+1$$
like the factorial $5!$ way.
EDIT: I know about the formula. I want to know if there's a short notation.
| We should also note that the factorial function has a similar look to it as the sigma summation notation; as
$$\frac{n(n+1)}{2}=1+2+3+...+n=\sum_{k=1}^nk$$
$$n!=1 \cdot 2 \cdot 3 \cdot ... \cdot n=\prod_{k=1}^nk$$
| {
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Given $n=p_1^{k_1},...,p_r^{k_r}$. Show that $a^{lcm(\phi{(p_1^{k_1})},\phi{(p_2^{k_2})},...,\phi({p_r^{k_r})})}\equiv 1\pmod n$ I am doing revision for my final exam. This is a sample question given by TA.
I actually have some idea, but wondering it is true or not.
I guess $lcm(\phi{(p_1^{k_1})},\phi{(p_2^{k_2})},...,... | It is not the case that $\text{lcm}(\phi(p_1^{k_1}), \ldots, \phi(p_r^{k_r}))$ is necessarily equal to $\phi(n)$: Take $n=12$, for instance. In order to prove your statement, try showing that the congruence holds modulo each of the prime powers $p_i^{k_i}$ using Euler's theorem for those numbers.
| {
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"source": "stackexchange",
"question_score": "3",
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Understanding a statement: Uniform convergence of functions I'm trying to understand the following statement:
Let $f$ be a real function defined on an interval $I$ and let $\{f_n\}$ a sequence of functions that converges uniformly to $f$ on $I$. For each $k \in \Bbb N$, there is a subsequence $\{f_{n_k}\}$ such that $... | You're right, the way it is written is simply wrong. What they are saying is that we can find a subsequence $\langle f_{n_k}\rangle$ such that $|f_{n_k}-f|<2^{-k}$ over $I$.
Proof Since $f_n\to f$ uniformly on $I$, for each $\varepsilon >0$ there exists $n_\varepsilon$ such that $n\geqslant n_\varepsilon$ implies $$|f_... | {
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Confusion in proof that primes $p = 4k + 1$ are uniquely the sums of two squares I'm reading a proof in my number theory textbook that all primes of the form $p = 4k+1$ are uniquely the sum of two squares. I'm stuck right at the beginning of the proof, where they say:
To establish the assertion, suppose that
$$
p = a^2... | Have you studied Gaussian integers at all in the years since you asked this question, or prior to that? (I'm here because of a duplicate).
It might help. For example, suppose $$29 = (2 - 5i)(2 + 5i) = 2^2 + 5^2 = c^2 + d^2,$$ where $c \neq 2$ and $d \neq 5$. Then $$2^2 d^2 - 5^2 c^2 = 29(d^2 - 5^2).$$ If we set $d = 0$... | {
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"source": "stackexchange",
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Linear transformations and eigenvalues Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same.
Can anyone provide tips/hints in the right direction? I'm struggling as I try to understand this int... | Hint:
$[T]_\beta$ and $[T]_\gamma$ are similar matrices, i.e. there is a matrix $Q$ (change of coordinate matrix) so that $[T]_\beta = Q^{-1}[T]_\gamma Q$.
| {
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Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?
| Let $|F| = p^n$, where $p$ is a prime; and consider $\varphi : F\to F$ given by $x\mapsto x^2$.
*
*If $p=2$, $\varphi$ is an isomorphism, so we're done.
*If $p > 2$, check that $\varphi(x) = \varphi(y)$ iff $x = \pm y$, and hence (why?)
$$
|Im(\varphi)| \geq \frac{p^n+1}{2} := k
$$
For $z \in F$, consider $S:= \{z ... | {
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How to prove that if $\forall x \in (a,b)$ Lebesgue integral $\int_{(a,x)}fd\lambda=0$, then $f(x)=0$ $\lambda$-almost everywhere? Let $f:(a,b)\rightarrow\mathbb{R}$.
The statement to prove is that if $\forall x \in (a,b)$ Lebesgue integral $\int_{(a,x)}fd\lambda=0$, then $f(x)=0$ $\lambda$-almost everywhere.
So if it ... | Consider arbitrary $x \in (a,b)$. Then for every suitably small $\epsilon > 0$,
Let, $$ I_\epsilon : = \int_{((x-\epsilon),(x + \epsilon))} f d\lambda = \int_{(a,x+\epsilon)} f d\lambda - \int_{(a,x - \epsilon)} f d\lambda = 0 - 0 = 0. $$
Also, $$ \lim_{\epsilon \rightarrow 0} \frac{I_{\epsilon}}{2\epsilon} = \lim_{\e... | {
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Injective but not divisible module
I want to find an injective but not divisible $R$-module.
If $R$ is integral domain, every injective is divisible so it should be $R$ is not an integral domain. Is there any example?
| Consider the ring $R=F_2[Z]/(Z^2)=M$ where $F_2$ is the field of two elements. This is a self-injective ring, so $M$ is an injective $R$-module.
But now consider $x=1$ and $r=Z$, where I abuse notation for the images of $1$ and $Z$ in this ring. Saying that there exists $y\in R$ such that $yZ=1$ implies that $Z$ is a u... | {
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How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$? I'd like to find out why
\begin{align}
\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6
\end{align}
I tried to rewrite it into a geometric series
\begin{align}
\sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=0}^{\infty} \Big(\frac{1}{2}\Big)^nn^2
\end{align}
But I don't know... | Let me show you a slightly different approach; this approach is very powerful, and can be used to compute values for a large number of series.
Let's think of this in terms of power series. You noticed that you can write
$$
\sum_{n=0}^{\infty}\frac{n^2}{2^n}=\sum_{n=0}^{\infty}n^2\left(\frac{1}{2}\right)^n;
$$
so, let'... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Matrix and eigenvectors
$\quad$The matrix $\mathbf A=\frac19\begin{bmatrix}
7 & -2 & 0 \\
-2 & 6 & 3 \\
0 & 2 & 5 \\
\end{bmatrix}$ has eigenvalues $1$, $\frac23$ and $\frac13$n with the corresponding eigenvectors $ \mathbf v_1=\begin{bmatrix}
-2 \\
2 \\
1 ... | Here is a HINT: $$x_k = A^k x_0$$
| {
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Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$ I have no idea how to do this. Any help would be appreciated. The chapter I'm on is about differentiation and the mean value theorem.
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
| Hint: Notice that if we divide by $b$ the inequality is equivalent to $$(a/b)^\alpha \leq \alpha \cdot (a/b) + (1-\alpha)$$
set $t = a/b > 0$...
| {
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Proof that $n^3-n$ is a multiple of $3$. I'm struggling with this problem of proof by induction:
For any natural number $n$, prove that $n^3-n$ is a multiple of $3$.
I assumed that $k^3-k=3r$
I want to show that $(k+1)^3-(K+1)=3r$
The final statement is $K^3 +3K^2+2K$
Am I missing something ?
| Use Fermat's little theorem.
Case1: if n is a multiple of 3 then trivially $n^3 -n$ is a multiple of $3$.
Case 2: If $n$ is not a multiple of $3$ you shall get $n^2 \equiv 1 \pmod 3$. Multiply $n$ and get $n^3 -n \equiv 0 \pmod 3$.
Now get your result.
| {
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Why is ($\mathbb R$,usual) not homeomorphic to ($\mathbb R$,discrete)?
*
*Why is ($\mathbb R$,usual) not homeomorphic to ($\mathbb R$,discrete)?
($\mathbb R$,discrete) means $d(x,y) =1$ for any $x\neq y$ and $d(x,y) =0$ for all $x=y$, both $x$ and $y$ are in $\mathbb R$.
*Is ($\mathbb R$,discrete) homeomorphic to (... | Because everything is clopen in $\mathbb{R}$ with the discrete topology.
$(0,1)$ is not closed in the usual.
2. The answer is no. Homeomorphisms are open maps.
3. You need a continuous bijection whose inverse is also continuous.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many four digit numbers are there? Assume that 0 can't be a first digit. I got 9,000. Is that right?
Follow up question: How many of those four digit numbers have no repeated digits?
| Equivalent question: You have a bag with 10 marbles in it labeled 0 through 9. You pick out a marble 4 times and place it back in the bag each time. The first time, there is no marble with a 0. The total combinations of picking and replacing 4 marbles with the no-zero-condition on the first pick is equal to the prod... | {
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$G\times H\cong G$ with $H$ non-trivial Aluffi II.3.4 asks the reader to find groups $G$ and $H$ such that $G\times H\cong G$ but $H$ is not trivial. I believe I have found a solution, but would like someone to check. Also, is there a more elementary example? I tried mucking around with $S^1\times (\Bbb Z/2\Bbb Z)$, wh... | A simple example would be the natural numbers (including zero) under the operation "bitwise exclusive or." This is isomorphic to $$G=\oplus_{i=0}^{\infty} \mathbb Z/\langle 2\rangle$$
| {
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Lines in a p-adic plane The geometry of lines in $\mathbb{R}^2$ is fundamental to mathematics and likewise for lines in $\mathbb{C}^2$ since $\mathbb{C}^2 \cong \mathbb{R}^4$. But is there a good treatment of lines in $\mathbb{Q}_p^2$, i.e. the $p$-adic plane?
For example, suppose $y= \zeta x$ is a $p$-adic line. How ... | Unfortunately, there are no lattices in $p$-adic vector spaces. The reason is that if $v$ is a nonzero vector, then $p^nv$ approaches $0$ in the $p$-adic topology, so $0$ is an accumulation point of every subgroup.
| {
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How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$
We did an example of this in class but the equation had less terms.
I took a note in class that says : if there are linear terms, I have to rotate...
This is what I think I have to do.
*
*Put t... | Not knowing what exactly "canonical form" is, here is what I get.
Translating to get rid of the linear terms:
$$
2(x+2)^2+4(x+2)(y-1/2)+6(y-1/2)^2=\frac{23}{2}\tag{1}
$$
With $P=\dfrac{\sqrt{2+\sqrt2}}{2}\begin{bmatrix}1&1-\sqrt2\\-1+\sqrt2&1\end{bmatrix}=\begin{bmatrix}\cos(\pi/8)&-\sin(\pi/8)\\\sin(\pi/8)&\hphantom{+... | {
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Number of subgroups and elements I have a question that I feel I am going about in a roundabout way, and would like some help on. I am preparing for an exam.
Problem: Let $G$ be a group with $|G|=150.$ Let $H$ be a non-normal subgroup in $G$ with $|H|=25$.
(a) How many elements of order 5 does $G$ have?
(b) How many e... | Your first part is correct $G$ has 6 Sylow 5-subgroups. $H$ is abelian and either cyclic or isomorphic to $C_5 \times C_5$. So ...
| {
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Prove-If n is prime $\mathbb{Z}_n$ is a field. I need to prove that $\mathbb{Z}_n$ is a field if and only if $n$ is prime. And I proved the forward.
But I am not sure how to prove the backward, 'if n is prime $\mathbb{Z}_n$ is a field.
'
What can I assume from the assumption $n$ is prime?
and What do I need to show? ... | Other than showing trivial axioms, you need to show that every element in $Z/pZ$ has a multiplicative inverse. In other words, given $k$, you need to find $m$ such that $k \cdot m \equiv 1 \pmod p$
This, though, is equivalent to showing that there exists $n$ such that $km = pn + 1$. Because $p$ is a prime, we know that... | {
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Indented Path Integration The goal is to show that
$$\int_0^\infty \frac{x^{1/3}\log(x)}{x^2 + 1}dx = \frac{\pi^2}{6}$$
and that
$$\int_0^\infty \frac{x^{1/3}}{x^2 + 1}dx = \frac{\pi}{\sqrt{3}}.$$
So, we start with the function
$$f(z) = \frac{z^{1/3}\log(z)}{z^2 + 1}.$$
Let $c_r$ be the upper semicircle with radius r ... | First, make the substitution $t=\displaystyle\frac1{1+x^2}$ , and rewrite $I_1=\displaystyle\frac14\int_0^1t^{^{-\frac16-\frac12}}\cdot(1-t)^{^{\frac16-\frac12}}\ln{1-t\over t}dt$ and $I_2=\displaystyle\frac12\int_0^1t^{^{-\frac16-\frac12}}\cdot(1-t)^{^{\frac16-\frac12}}dt$ . Then recognize the expression of Euler's Be... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that at least one of the real numbers $a_1 , a_2 , … , a_n$ is greater than or equal to the average of these numbers Prove that at least one of the real numbers $\,a_1 , a_2 , … , a_n$ is
greater than or equal to the average of these numbers. What kind of
proof did you use?
I think I should use contradiction but ... | Let average $g$ and $a_i<g$ for $1\le i\le n$
$$\implies g\cdot n=\sum_{1\le i\le n}a_i<\sum_{1\le i\le n}g=g\cdot n$$
| {
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Find the value of $\cos(2\pi /5)$ using radicals This is homework so if there is another example that can illustrate the technique I would happily accept that as guidance. The only thing I have been able to find is a question asking about $\cos(2\pi/7)$, which I think is a much harder problem.
I dont have the faintest ... | Note that
$2\cdot \dfrac{2\pi}{5} + 3\cdot \dfrac{2\pi}{5} = 2\pi,$
therefore
$\cos(2⋅\dfrac{2\pi}{5})=\cos(3⋅\dfrac{2\pi}{5})$.
Put $\cos(\dfrac{2\pi}{5})=x$. Using the formulas
$\cos2x=2\cos2x−1,\cos 3x=4\cos 3x−3\cos x$,
we have
$4x^3−2x^2−3x+1=0⇔(x−1)(4x^2+2x−1)=0$.
Because $\cos(\dfrac{2\pi}{5})≠1$, we get
$4x^2+2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about finite ring with more than one element; division ring
A finite ring with more than one element and no zero divisors is a division ring. (Hungerford, Algebra, Exercise 6, Chapter 3, Section 1.)
This a problem taken from Hungerford's graduate algebra text. Hungerford defines left and right zero divisors ... |
Lemma: A ring with no nonzero two-sided zero divisors has no nonzero one-sided zero divisors.
Suppose $ab=0$ for nonzero $a,b$. Then $ba\neq 0$. But also $(ba)^2=0$, contradicting the hypothesis. Therefore such $a,b$ do not exist.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that if $a$ and $b$ are positive integers then $(a, b) = (a + b, [a, b])$. Show that if $a$ and $b$ are positive integers then $(a, b)=(a + b, [a, b])$.
I was thinking that since $[a, b]=LCM(a, b)=\frac{ab}{(a, b)}$ that if $d= (a + b, [a, b])$, then $d|[a,b]$ and thus $d|(a, b)$ since $(a, b)|[a, b]$
Then I would... | Another way : Let $(a,b)=d$ and $\displaystyle \frac aA=\frac bB=d\implies (A,B)=1$
So, $\displaystyle(a+b, [a,b])=(d(A+B), dAB)=d(A+B,AB)$
Now, if $D$ divides both $ A+B, AB; D$ will divide $A(A+B)-AB=A^2$ and $D$ will divide $B(A+B)-AB=B^2$
$\displaystyle\implies D$ will divide $(A^2,B^2)=(A,B)^2=1\implies D=1$
| {
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Find an estimation (using polar coordinates) Consider the function
$$
f(x,y):=\lVert x\rVert^{1-n}\ln(\lVert x\rVert)(\arctan(\lVert x-y\rVert))^{-\alpha},~~0<\alpha<n,~~n>1,~~(x,y)\in\Omega\times\Omega,~~~\Omega\subset\mathbb{R}^n
$$
with $x\neq y$.
I am searching for an estimation
$$
\lvert f(x,y)\rvert\leq\frac{\lve... | Easy Peasy. The new function ||x|| ln ||x|| is continuous (although negative near he orign), so it is bounded on $\Omega$ if $\Omega$ is bounded, so you can just use your old estimate times the bound on xlnx.
| {
"language": "en",
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What are the techniques to find the sum to infinity of a series? Given a series, what are the techniques to find a formula that sums the series to infinity?
I only know the method of multiplying the series with a factor and then taking their difference (like here). But today, I found out that we can also try to find su... | There are many techniques, but unfortunately there is no general method which is guaranteed to succeed, hence the difficulty (perhaps impossibility) of finding things like a closed formula for $\zeta(3)$.
The techniques that I can recall using in the past include:
*
*Using known power series expansions (including th... | {
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Real analysis question about limits? If $0 < x < 1$, show that $x^n → 0$ as $n → ∞$ .
I'm thinking it has something to do with rational numbers (declaring $x = p/q$) and then using exponent laws to show that $x^n = p^n/q^n$, and since $q > p$ ... but I'm not sure where to go with this. Any help is appreciated. Thank... | Hints:
$$0<x<1\implies x>x^2>x^3>\ldots >x^n\;,\;\;\forall\,n\in\Bbb N$$
$$a_n>0\;\;\forall\,n\in\Bbb N\implies \lim_{n\to\infty}a_n=\infty\iff \lim_{n\to\infty}\frac1{a_n}=0$$
| {
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Evaluation of a line integral using Green's Theorem where P, Q, and partial derivatives of P & Q are not continuous
How can the author evaluate the below?
$$\oint_{C'}\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$$
Doesn't this contradict Theorem 9.12.1? P(0, 0) is undefined on region $R_2$ (corresponding to $C'$).
| The author is correct that he is able to do this, but he does not justify himself very well. It is actually true that if two paths are homotopic (see wikipedia) through a region where the vector field is curl free, then the integrals over both paths will be equal.
I will draw some pictures to show you this is true onl... | {
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Does the orientation you evaluate line integrals matter?
If instead of evaluating the above line integral in counter-clockwise direction, I evaluate it via the clockwise direction, would that change the answer? What if I evaluate $C_1$ and $C_3$ in the counter-clockwise direction, but I evaluate $C_2$ in the clockwise... | Direction does not matter for the line integral of a function, but here you are dealing with a work integral (i.e. the integral of a vector field along the curve). In the latter case, orientation does matter.
The statement of Green's Theorem includes (or it should, to make sense) the orientation required for the equal... | {
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Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$ I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe it will converge only for $0 < n < 1$, a... | Firstly, to determine the range of $n$ for which the integral converges, just look at the singularity at $0.$ To integrate, and put the pesky $n$ where it is easier to deal with, make the substitution $u=\log x.$ This will transform your integral to $\int_{-\infty}^{\infty} u \exp(-(n-1) u)/(1+\exp u) du,$ which should... | {
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How to define a finite objects with parametric equations I never had seen parametric equations, but while trying to learn line integrals through Wikipedia, quickly found these equations are remarkable. Some can represent things for which more normal equations or functions are needed, if at all possible.
However, the p... | You do this by defining an interval for the parameter.
Take a simple line: $x = y = z = t.$ The (infinite) line implies that the parameter $t$ can take on any value.
The line segment from $(-1, -1, -1)$ to $(2, 2, 2)$ means restricting $t$ to the interval $[-1, 2]$.
| {
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Why do we have to do the same things to both sides of an equation? Forgive me in advanced if this is a trivial question.
This convention makes perfect sense to me intuitively, but is there any rigorous underpinning to it?
I'm beginning to read through an abstract algebra textbook, and soon after establishing what a rin... | Here's a short example of doing the same thing to both sides of an equation, and a justification for each step, using only substitution and the field axioms. The magic happens in the first two steps.
Statement:
\begin{align*}
a+c=b+c \Rightarrow a=b
\end{align*}
Proof:
\begin{align*}
(a+c)+(-c)&=(a+c)+(-c) \tag{propert... | {
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How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
Show that $2730\mid n^{13}-n,\;\;\forall n\in\mathbb{N}$
I tried, $2730=13\cdot5\cdot7\cdot3\cdot2$
We have $13\mid n^{13}-n$, by Fermat's Little Theorem.
We have $2\mid n^{13}-n$, by if $n$ even then $n^{13}-n$ too is even; if $n$ is odd $n^{13}-n$ is od... | HINT:
$$n^{13} \equiv n^5 \cdot n^5 \cdot n^3 \equiv n \cdot n \cdot n^3 \equiv n^5 \equiv n \pmod 5$$
$$n^{13} \equiv n^6 \cdot n^7 \equiv n \pmod 7$$
Also you've missed $3$ as prime factor. But that should be easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/596074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Determinant without computing How can I show without computing the determinant that the equation is true?
$$\det \begin{pmatrix}
b1 + c1 & c1 + a1 & a1 + b1\\
b2 + c2 & c2 + a2 & a2 + b2\\
b3 + c3 & c3 + a3 & a3 + b3
\end{pmatrix} = 2 \det \begin{pmatrix}
a1 & b1 & c1\\
a2 & b2 & c2\\
a3 & b3 & c3
\end{pmatrix}$$
| Hints:
(1) Determinant is a multilinear function. For example
$$\det\begin{pmatrix}a+b&c&d\\a'+b'&c'&d'\\a''+b''&c''&d''\end{pmatrix}=\det\begin{pmatrix}a&c&d\\a'&c'&d'\\a''&c''&d''\end{pmatrix}+\det\begin{pmatrix}b&c&d\\b'&c'&d'\\b''&c''&d''\end{pmatrix}$$
and likewise for each column/row.
(2) The determinant is an al... | {
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what are the eigenvalues in orthgonal matrix, How to explain? what are the possible eigenvalues of an orthogonal matrix?
I got the answer key which says its 1 and -1 but it doesn't explain well
| Here is an approach. Recalling the fact
if $\lambda$ is the eigenvalue of $A$, then $\lambda^{-1}$ is the eigenvalue of $A^{-1}$,
we have
$$ \langle Ax,x \rangle = \langle x, A^T x \rangle = \langle x, A^{-1} x \rangle $$
$$ \implies \langle \lambda x,x \rangle = \langle x, \frac{1}{\lambda} x \rangle $$
$$ \impl... | {
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Easy GRE question: Statistics I'm not sure how to set this statistics problem when they give me a group of arbitrary values. Can someone help?
A group of 20 values has a mean of 85 and a median of 80. A different group of 30 values has a mean of 75 and a median of 72.
a) What is the mean of the 50 values?
b) What is th... | The formula to find mean is
Mean=
(total number of values) ÷ (quantity of values)
So, as we know the mean and the quantity values of the first group is, 85 and 20. (It is given)
So we can substitute the values into the formula and get the (total number of values)
Same goes to the other group that contains 30 values... | {
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Visualizing Non-Zero Closed-Loop Line Integrals Via Sheets? How do I visualize $\dfrac{xdy-ydx}{x^2+y^2}$?
In other words, if I visualize a differential forms in terms of sheets:
and am aware of the subtleties of this geometric interpretation as regards integrability (i.e. contact structures and the like):
then since... | First, it is important to note that those are oriented sheets: they have two sides, one painted red and another blue. When a curve crosses a sheet, the red-blue crossing counts as $1$ while the blue-red crossing counts as $-1$.
Picture a stack of vertical planes passing through the $z$-axis, like a book that opens 3... | {
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Finding no-self-intersecting path in geometric graphs Is there a polynomial algorithm to determine whether there exists no-self-intersecting path between given vertices $s$ and $t$ in a geometric graph $G$?
Geometric graph is an image of a graph on a plane where vertices are represented as points and edges are drawn as... | The second problem is a variant of problem called Path avoiding forbidden pairs (PAFP) (see Computers and intractability by Garey and Johnson, 1979, page 203).
I could not find any mention of the main problem, though, so I prooved it is NP-complete. I uploaded the paper to my Github (sorry, only in Russian).
| {
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Finding the limit of $\left(\frac{n}{n+1}\right)^n$
Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$
I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$.
Also, when factoring: $$n^{1/n}\left(\frac{1}{1+\frac1n}\right)^n$$ it looks like it goes to ... | $$ \left(\dfrac{n}{n+1}\right)^n = \frac {1}{\left(\dfrac{n+1}{n}\right)^n}=\frac {1}{\left(1+\dfrac{1}{n}\right)^n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/596771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable Suppose that $A \in M_{n\times n}(\Bbb F)$ has two distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ and that $\dim (E_{\lambda_1})=n-1$ show that $A$ is diagnolizable.
| Hint: Since $\dim(E_{\lambda _1})=n-1$, there exist $v_1, \ldots , v_{n-1}$ linearly independent eigenvectors of $\lambda _1$. Let $v_n$ be an eigenvector of $\lambda _2$. Now consider the $n\times n$ matrix $P$ whose $i^{\text{th}}$column is $v_i$. The invertibility of $P$ follows from this.
Can you take it from here?... | {
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Proving a relation with induction I have a problem:
Let $p_n$ be the $n:th$ prime number ($p_1=2, p_2=3, p_3=5$ and so on). With induction, show that $p_{n+2}>3n$ for each integer $n\geq1$.
I can't figure this out because the primes are confusing me, making me unable to show the inductive step.
| Proof by induction.
Let's start with the case $n=1$:
$p_{1+2}=p_3=5>3*1=3$
So the equality is true for $n=1$.
Now, we assume that the equality is true for the case $n-1$, that is,
$p_{n-1+2}=p_{n+1}>3(n-1)=3n-3$
and let's see that it is also true for n:
$p_{n+2}\geq p_{n+1}+2>3(n-1)+2=3n-1$; so $p_{n+2}\geq3n$
Now,you ... | {
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Finding new probability density function with change of variable Y=sqrt(X) Say we have a given distribution, such as X~No(a, b). I am trying to find the pdf and mean for $Y=\sqrt{X}$.
I know the steps for finding the PDF, but since Y can only take on positive values, then the new PDF is only valid for Y>0.
Then how d... | If $Y=\sqrt{|X|}$, then, for every $y\gt0$,
$$
f_Y(y)=2y\cdot(f_X(y^2)+f(-y^2)).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/597010",
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Area of a square in polar coordinates? I was attempting, for the exercise of it, to find the area of the a simple square with an infinite number of infinitesimal circle sectors. Let us say this square is $[5 x 5]$.
Alas, it's been proving as awkward to compute this as it sounds. First, I did the integral in rectangul... | The area of a circular sector of radius $r$ and angle $d\theta$ is $\pi r^2 \frac{d\theta}{2\pi} = \frac{1}{2} r^2 d\theta$.
The right side of the square ($0 < \theta < \pi/4$) is the line $x = 5$, which in polar coordinates is $r = \frac{5}{\cos \theta}$ (not $\sin$).
Putting this together, the integrand should be $\f... | {
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Compute the minimum distance between the centre to the curve $xy=4$. I wish to solve the following problem:
Compute the minimum distance between the center to the curve $xy=4$.
But I don't know where to start from?
| You need to minimize the function
$$d= \sqrt{x^2 +y^2} = \sqrt{x^2+ 16/x^2}.$$
Which we got by considering the distance from a point on the curve to the origin. Now, use the derivative test. See related problem.
| {
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Least prime of the form $38^n+31$ I search the least n such that
$$38^n+31$$
is prime.
I checked the $n$ upto $3000$ and found none, so the least prime of that form must have more than $4000$ digits. I am content with a probable prime, it need not be a proven prime.
| This is not a proof, but does not conveniently fit into a comment.
I'll take into account that $n=4k$ is required, otherwise $38^n+31$ will be divisible by $3$ or $5$ as pointed in the comments.
Now, if we treat the primes as "pseudorandom" in the sense that any large number $n$ has a likelihood $1/\ln(n)$ of being pri... | {
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Inner product doubt I have a doubt about a problem involving inner product spaces. The exercise is:
Given the subspace generated by the vectors $ (1,1,1) $ and $ (1,-1,0) $, find the orthogonal subspace and give a basis.
Now, what I understood of this problem is that I have to find the orthogonal complement. But they d... | The orthogonal component depends on the inner product. However, typically the inner product that is chosen (assuming you are dealing with a vector space over real numbers) is the following one:
$$\langle(x_1,x_2,x_3), (y_1,y_2,y_3) \rangle = x_1y_1 + x_2y_2 + x_3y_3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/597511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding a 2x2 Matrix raised to the power of 1000 Let $A= \pmatrix{1&4\\ 3&2}$. Find $A^{1000}$.
Does this problem have to do with eigenvalues or is there another formula that is specific to 2x2 matrices?
| Perform an eigenvalue decomposition of $A$, we then get
$$A =
\begin{bmatrix}
-4/5 & -1/\sqrt2\\
3/5 & -1/\sqrt2
\end{bmatrix}
\begin{bmatrix}
-2 & 0\\
0 & 5
\end{bmatrix}
\begin{bmatrix}
-4/5 & -1/\sqrt2\\
3/5 & -1/\sqrt2
\end{bmatrix}^{-1}
=VDV^{-1}
$$
where $V = \begin{bmatrix}
-4/5 & -1/\sqrt2\\
3/5 & -1/\sqrt2
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/597602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 3
} |
Let $n=2m$, where $m$ is odd. How many elements of order $2$ does the group $D_n/Z(D_n)$ have?
Let $n=2m$, where $m$ is odd. How many elements of order $2$ does the group $D_n/Z(D_n)$ have?
I don't how to begin this proof. All I have so far is that Dn/Z(Dn) should have one element of order 2.
| SO I have that the Dn/Z(Dn)={R0,R180}. I also figured out that the order of Dn=(m+1)(n+1)=(mn)+2n+m+1. But I don't know where to go from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/597662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Exponential distribution function. I am having trouble with this question as well. Any help or hints will be really appreciated.
On a TV game show contestants have to complete a particular task. Suppose that the time take for a typical contestant to complete the task is a random variable with an exponential distributio... | For the first question, try integrating that distribution from 3 to $\infty$. This is the probability that the contestant takes 3 minutes or more to complete the task.
For the second question, I'm going to assume that it is considered a failure if the contestant takes more than 3 minutes to complete a task. In this cas... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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On Pr(X>Y) when X and Y are independent normal Let X∼N(6,1) and Y∼N(7,1) be two independent normal variables. Find Pr(X>Y). the answer is 0.2389 but I do not know how to do it.I have tried adding them and subtracting but i am still clueless.
| We have $Pr(X>Y)=Pr(X-Y>0)$. Because $X$ and $-Y$ are independent Gaussians, their sum is normal (mean the sum of the means and variance the sum of the variances). Therefore $X+(-Y)\sim N(6-7,1+1)=N(-1,2)$.
Depending on what calculational tools you have available to you, you may be able to evaluate $P(N(-1,2)>0)$ dir... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
} |
system of linear congruences when moduli are not coprime $\begin{cases}x\equiv 1 \pmod{3}\\
x\equiv 2 \pmod{5}\\
x\equiv 3 \pmod{7}\\
x\equiv 4 \pmod{9}\\
x\equiv 5 \pmod{11}\end{cases}$
I am supposed to solve the system using the Chinese remainder theorem but $(3,5,7,9,11)\neq 1$
How can I transform the system so th... | Solving the first two equations simultaneously you get X = 7(mod 15).
Solving the third and fourth simultaneously you get X = 31(mod 63).
Solving these two results simultaneously you get X = 157(mod 315).
Solving this result with the fifth equation simultaneously,
you get the final answer X = 1732(mod 3465).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Primes and perfect square on numbers I have two equations, which may play an important role in my further studies on theory of numbers.
1) How many pairs of (A, B, x) we can make in $A^x + B ^x = prime$? Here $x$ is $> 2$ and A, B are positive integers.
2) can we find a number(s) with one hundred 0′s, one hundred 1′s... | For the second question the answer is no.
The number you want is $222\cdots 111\cdots 000\cdots$ which is divisible by $3$ (because the sum of its digits is $2\cdot 100+1\cdot 100+0\cdot 100=300$ divisible by $3$)
But,it is not divisible by $9$ (because also the sum of its digits is not)
There for it must not be a squa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equations and inequalities as parameters: proving that an equation holds. I have $x-y=3$ and $y\le1$ and $x\ge\frac12$.
I proved that $\sqrt{(2x-1)^2}+\sqrt{(2y-2)^2}=7$ and that $-\frac52\le y\le 1$ and $\frac12\le x\le4$.
How can I prove that $|x+y-5|+|x+y+2|=7$?
| ok let us consider following cases :
first side is just $-x-y+5$,second case is $x+y+2$,so sum is $5+2=7$
to be more deeply,let us take such situations
*
*$y=1$
2.$x=4$
we have
$|x+y-5]+|x+y+2|=x+y-5+x+y+2=2*x+2*y-3=2*(x+y)-3
=2*(5)-3=7$
can you continue from this?
just consider this situation when $x<0$ th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Assume $X$ and $Y$ be nonempty subsets of $R$ such that $xWe have that both sets are nonempty, by the completeness axiom, both sup and inf exist for both sets. Since $x < y$, $y$ is an upper bound for $X$ and hence by def. of sup, sup$X \le y$. So $\sup X$ is a lower bound for $Y$ and thus by def. of inf, $\sup X \leq ... | That is basically a good proof, but could use some clarifications. In particular you should try to distinguish more clearly between numbers and sets.
*
*sup and inf do not necessarily exist for a nonempty set of real numbers. However, it is true that a nonempty set which has an upper bound has a sup, and similarly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Compact hypersurface of $\mathbb{R}^{n+1}$ with positive curvature is diffeomorphic to $S^n$. I have a compact hypersurface $M$ of $\mathbb{R}^{n+1}$ with positive curvature. I need to show that it is diffeomorphic to $S^n$.
The hint is to consider the shape operator $A_{\nu_p} x$, where $\nu$ is a smooth unit normal v... | The shape operator is selfadjoint as an endomorphism of the tangent space. Therefore it can be diagonalized over $\mathbb{R}$. The hypothesis of positive (sectional) curvature then implies that all the eigenvalues are positive. Therefore the determinant is positive and in particular nonzero. It follows that the Gauss ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Grammar outside the Chomsky Hierarchy This grammar describes a language that may fall outside the Chomsky Hierarchy (CH):
\begin{array}{l}
S \to abAbba \\
A \to abA \mid bbaB \\
B \to aab \\
\lambda \to Aab \mid aB \\
\end{array}
Going down the list, it's not context-free, it's not regular, etc. Getting all the way t... | The last rule of your grammar does not satisfy the requirements of a formal grammar (every rule in formal grammar must have at least one non-terminal symbol on the left-hand side), so it clearly cannot belong to any hierarchy of formal grammars.
That being said, the language produced by your grammar (under the intuitiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Concentric and Tangent Ellipse from 2 Hyperbolas Find the equation of the ellipse that is concentric and tangent to the following hyperbolas:
$$\begin{align}
-2x^2 + 9y^2 - 20x - 108y + 256 &= 0 \\
x^2 - 4y^2 + 10x + 48y - 219 &= 0
\end{align}$$
I did the math for both equations and the center is the same: $(-5,6)$.
... | Not a complete solution, but this approach will work:
An ellipse with centre at the point $(-5,6)$ would be
$$\frac{(x+5)^2}{a^2}+\frac{(y-6)^2}{b^2} = 1$$
Now change to a new set of axes ($u, v$) parallel to the $x,y$ axes, but with origin at the point $(-5,6)$. In other words, put $u = x+5$ and $v = y-6$.
Referred t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A curious problem about Lebesgue measure. The Problem:
Let $(B(x_{m},0.5))_{m}$ be a sequence of disjoint open discs in $\mathbb{R}^{2}$ centered in $x_{m}$ and with radius 0.5. Let $\psi(n)$ be the number of these discs contained in the open disc $B(0,n)$ (that is, the disc centered in (0,0) and with radius $n$).
Prov... | The idea is to rescale the disks contained in $B(0,n)$ by $n^{-1}$, thus obtaining a set contained in $B(0,1)$ with measure bounded from below. Some care must be taken to produce a sequence of such sets without double-counting disks.
By the way, it suffices to assume that $$\limsup \frac{\psi(n)}{n^{2}} = k > 0$$ ins... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
isomorphism, integers of mod $n$. Hello I think this is true, but I'm not sure.
Setup:
If $n = p_{1}\cdot p_{2} \cdots p_{n}$ where $p_{i}$ prime for all $i\in\lbrace 1,\dots,n\rbrace$.
Define the ring $A = p_{j}\mathbb{Z}/n\mathbb{Z}$.
Question:
Is $A$ isomorphic to $\mathbb{Z}/(n/p_{j})\mathbb{Z}$? And why?
| Hint: Does $\mathbb Z/(n/p_i)\mathbb Z$ have a multiplicative unit? Does $p_i\mathbb Z/n\mathbb Z$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/598713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $f(x)=\pm x + c$ Using Differentiation Suppose $|f(x)-f(y)|=|x-y|$ for all $x,y \in R$. Prove there is a constant $c$ such that $f(x)=\pm x+c$. Suppose $|f(x)-f(y)|=|x-y|$ for all $x,y \in R$. Prove there is a constant $c$ such that $f(x)=\pm x+c$. What I did was make $|\frac{f(x)-f(y)}{x-y}|=1$. Thus, $f$ is d... | You cannot conclude from $\left|\frac{f(x)-f(y)}{x-y}\right|=1$ that $\lim_{y\to x}\frac{f(x)-f(y)}{x-y}=1$. In fact, even the existence of the limit is not that obvious.
You can solve the problem without differentiation for example by considering $f(0),f(1), f(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/598767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Complex Integration of DTFT Question
A discrete-time signal $u \in \mathcal{l}^2(\mathcal{Z})$ has DTFT
\begin{equation}
\hat{u}(\omega) = \frac{5+3\cos(\omega)}{17+8\cos(\omega)}
\end{equation}
Use complex integration to find $u(k)$ for $k\in\mathcal{Z}$
My Attempt
If I'm not mistaken, to find $u[k]$, I should find t... | Using the residue theorem (complex integration) on integrals of rational functions of sines and cosines is commonplace. The trick is to let $z=e^{i \omega}$. Then $d\omega = -i dz/z$ and the integral may be expressed as a complex integral over the unit circle:
$$\begin{align}u(k) &= -\frac{i}{2 \pi} \oint_{|z|=1} \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/598854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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