Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
$$\tan x+\sec x=2\cos x$$
$$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$
$$\left(\dfrac{\sin x+1}{\cos x}\right)=2\cos x$$
$$\sin x+1=2\cos^2x$$
$$2\cos^2x-\sin x... | \begin{eqnarray}
\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)&=&2\cos x\\
\sin x + 1 &=& 2 \cos^2 x \\
\sin x + 1 &=& 2(1-\sin^2 x)\\
\end{eqnarray}
Then $\sin x = -1$ or $\sin x = 1/2$ and the solutions are $-\pi + 2k \pi, \pi/6 + 2 k\pi$ and $5\pi/6 + 2k \pi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/171792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Is there a proof of Gödel's Incompleteness Theorem without self-referential statements? For the proof of Gödel's Incompleteness Theorem, most versions of proof use basically self-referential statements.
My question is, what if one argues that Gödel's Incompleteness Theorem only matters when a formula makes self-refere... | Roughly speaking, the real theorem is that the ability to express the theory of integer arithmetic implies the ability to express formal logic.
Gödel's incompleteness theorem is really just a corollary of this: once you've proven the technical result, it's a simple matter to use it to construct variations of the Liar's... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 1
} |
limit of an integral with a Lorentzian function We want to calculate the $\lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2} dx $ for a function $f(x)$ such that $f(0)=0$. We are physicist, so the function $f(x)$ is smooth enough!.
After severals trials, we have not been able to calculate it e... | I'll assume that $f$ has compact support (though it's enough to suppose that $f$ decreases very fast). As $f(0)=0$ he have $f(x)=xg(x)$ for some smooth $g$. Let $g=h+k$, where $h$ is even and $k$ is odd. As $k(0)=0$, again $k(x)=xm(x)$ for some smooth $m$.
We have $$\int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\int(2x^2+1)e^{x^2}dx$ $$\int(2x^2+1)e^{x^2}dx$$
The answer of course:
$$\int(2x^2+1)e^{x^2}\,dx=xe^{x^2}+C$$
But what kind of techniques we should use with problem like this ?
| You can expand the integrand, and get
$$2x^2e^{x^2}+e^{x^2}=$$
$$x\cdot 2x e^{x^2}+1\cdot e^{x^2}=$$
Note that $x'=1$ and that $(e^{x^2})'=2xe^{x^2}$ so you get
$$=x\cdot (e^{x^2})'+(x)'\cdot e^{x^2}=(xe^{x^2})'$$
Thus you integral is $xe^{x^2}+C$. Of course, the above is integration by parts in disguise, but it is goo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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An example of bounded linear operator Define $\ell^p = \{ x (= \{ x_n \}_{-\infty}^\infty) \;\; | \;\; \| x \|_{\ell^p} < \infty \} $ with $\| x \|_{\ell^p} = ( \sum_{n=-\infty}^\infty \|x_n \|^p )^{1/p} $ if $ 1 \leqslant p <\infty $, and $ \| x \|_{\ell^p} = \sup _{n} | x_n | $ if $ p = \infty $. Let $k = \{ k_n \}_{... | In the first comment I suggested the following strategy: write $T=\sum_j T_j$, where $T_j$ is a linear operator defined by $T_jx=\{k_jx_{n-j}\}$. You should check that this is indeed correct, i.e., summing $T_j$ over $j$ indeed gives $T$. Next, show that $\|T_j\|=|k_j|$ using the definition of the operator norm. Finall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Exam question: Normals While solving exam questions, I came across this problem:
Let $M = \{(x, y, z) \in \mathbb{R^3}; (x - 5)^5 + y^6 + z^{2010} = 1\}$. Show that for every unit vector, $v \in \mathbb{R^3}$, exists a single vertex $p \in M$ such that $N(p) = v$, where $N(p)$ is the outer surface normal to $M$ at $p$... | It is straightforward, but tedious, to analytically show that the assertion is false. A picture illustrates the problem more clearly.
First, note that if $z=0$, then the normal will also have a zero $z$ component, so we can take $z=0$ and look for issues there. Plotting the contour of $(x-5)^5+y^6 = 1$ gives the pictur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$ how to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$?
I know how to prove
$ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $
in this way:
$ \tan{20^0} = \sqrt{3}.\tan{50^0}.\tan{10^0}$
... | Another approach:
Lets, start by arranging the expression:
$$\tan(20°) \tan(30°) \tan(40°) = \tan(30°) \tan(40°) \tan(20°)$$$$=\tan(30°) \tan(30°+10°) \tan(30° - 10°)$$
Now, we will express $\tan(30° + 10°) $ and $\tan(30° - 10°)$ as the ratio of Prosthaphaeresis Formulas, giving us:
$$\tan(30°) \left( \frac{\tan(3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Is there a "natural" topology on powersets? Let's say a topology T on a set X is natural if the definition of T refers to properties of (or relationships on) the members of X, and artificial otherwise. For example, the order topology on the set of real numbers is natural, while the discrete topology is artificial.
Supp... | Natural is far from well-defined. For example, I don’t see why the discrete topology on $\Bbb R$ is any less natural than the order topology; one just happens to make use of less of the structure of $\Bbb R$ than the other.
That said, the Alexandrov topology on $\wp(X)$ seems pretty natural:
$$\tau=\left\{\mathscr{U}\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
Proof of $(A - B) - C = A - (B \cup C)$ I have several of these types of problems, and it would be great if I can get some help on one so I have a guide on how I can solve these. I tried asking another problem, but it turned out to be a special case problem, so hopefully this one works out normally.
The question is:
... | Working with the Characteristic function of a set makes these problems easy:
$$1_{(A - B) - C}= 1_{A-B} - 1_{A-B}1_C=(1_A- 1_A1_B)-1_A1_C+ 1A1_B1_C \,$$
$$1_{A-(B \cup C)}= 1_{A}- 1_{A}1_{B \cup C}=1_A- 1_A ( 1_B+ 1_C -1_B1_C)\,$$
It is easy now to see that the RHS are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/172292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Trigonometric equation inversion I am trying to invert the following equation to have it with $\theta$ as the subject:
$y = \cos \theta \sin \theta - \cos \theta -\sin \theta$
I tried both standard trig as well as trying to reformulate it as a differential equation (albeit I might have chosen an awkward substitution). ... | Rewrite the equation as $(1-\cos\theta)(1-\sin\theta)=1+y.$
Now make the Weierstrass substitution $t=\tan(\theta/2)$. It is standard that
$\cos\theta=\frac{1-t^2}{1+t^2}$ and $\sin\theta=\frac{2t}{1+t^2}$. So our equation becomes
$$\frac{2t^2}{1+t^2}\cdot \frac{(1-t)^2}{1+t^2}=1+y.$$
Take the square root, and clear d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Coordinate-free method to determine local maxima/minima? If there is a function $f : M \to \mathbb R$ then the critical point is given as a point where
$$d f = 0$$
$df$ being 1-form (btw am I right here?). Is there a coordinate independent formulation of a criteria to determine if this point is a local maximum or minim... | Let $p$ be a critical point for a smooth function $f:M\to \mathbb{R}.$
Let $(x_\,\ldots,x_n)$ be an arbitrary smooth coordinate chart around $p$ on $M.$
From multivariate calculus we know that a sufficient condition for $p$ to be a local maximum (resp. minimum) of $f$ is the positiveness (resp. negativeness) of the Hes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Use the Division Algorithm to show the square of any integer is in the form $3k$ or $3k+1$
Use Division Algorithm to show the square of any int is in the form 3k or 3k+1
What confuses me about this is that I think I am able to show that the square of any integer is in the form $X*k$ where $x$ is any integer. For Exam... | Hint $\ $ Below I give an analogous proof for divisor $5$ (vs. $3),$ exploiting reflection symmetry.
Lemma $\ $ Every integer $\rm\:n\:$ has form $\rm\: n = 5\,k \pm r,\:$ for $\rm\:r\in\{0,1,2\},\ k\in\Bbb Z.$
Proof $\ $ By the Division algorithm
$$\rm\begin{eqnarray} n &=&\:\rm 5\,q + \color{#c00}r\ \ \ for\ \ some\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove $ (r\sin A \cos A)^2+(r\sin A \sin A)^2+(r\cos A)^2=r^2$ How would I verify the following trig identity?
$$(r\sin A \cos A)^2+(r\sin A \sin A)^2+(r\cos A)^2=r^2$$
My work thus far is
$$(r^2\cos^2A\sin^2A)+(r^2\sin^2A\sin^2A)+(r^2\cos^2A)$$
But how would I continue? My math skills fail me.
| Just use the distributive property and $\sin^2(x)+\cos^2(x)=1$:
$$
\begin{align}
&(r\sin(A)\cos(A))^2+(r\sin(A)\sin(A))^2+(r\cos(A))^2\\
&=r^2\sin^2(A)(\cos^2(A)+\sin^2(A))+r^2\cos^2(A)\\
&=r^2\sin^2(A)+r^2\cos^2(A)\\
&=r^2(\sin^2(A)+\cos^2(A))\\
&=r^2\tag{1}
\end{align}
$$
This can be generalized to
$$
\begin{align}
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How do I know if a simple pole exists (and how do I find it) for $f(z)$ without expanding the Laurent series first? In general, how do I recognize that a simple pole exists and find it, given some $\Large f(z)$? I want to do this without finding the Laurent series first.
And specifically, for the following function: ... | Here is how you find the roots of $z^4+16=0$,
$$ z^4 = -16 \Rightarrow z^4 = 16\, e^{i\pi} \Rightarrow z^4 = 16\, e^{i\pi+i2k\pi} \Rightarrow z = 2\, e^{\frac{i\pi + i2k\pi}{4} }\, k =0\,,1\,,2\,,3\,.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/172678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
what should be the range of $u$ satisfying following equation. Let us consider that $u\in C^2(\Omega)\cap C^0(\Omega)$ and satisfying the following equation .
$\Delta u=u^3-u , x\in\Omega$ and
$u=0 $ in $\partial \Omega$
$\Omega \subset \mathbb R^n$ and bounded .
I need hints to find out what possible value $u$ c... | We must of course assume $u$ is continuous on the closure of $\Omega$.
Since this is bounded and $u = 0$ on $\partial \Omega$, if $u > 0$ somewhere it must achieve a maximum in $\Omega$. Now $u$ is subharmonic on any part of the domain where $u > 1$, so the maximum principle says ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/172747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
composition of convex function with harmonic function. Consider $u:\Omega \to \mathbb R$ be harmonic and $f$ be convex function. How do i prove that $f\circ u$ is subharmonic?
It seems straight forward : $\Delta (f\circ u (x)) \ge f(\Delta u(x))=0 $.
Is this all to this problem? Is there a better way?
Also $|u|^p$ is s... | Basically you do what Davide Giraudo suggested. We use the definition as given on the EOM
An upper semicontinuous function $v:\Omega \to\mathbb{R}\cup \{-\infty\}$ is called subharmonic if for every $x_0 \in \Omega$ and for every $r > 0$ such that $\overline{B_r(x_0)} \subset \Omega$, that
$$ v(x_0) \leq \frac{1}{|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Can continuity of inverse be omitted from the definition of topological group? According to Wikipedia, a topological group $G$ is a group and a topological space such that
$$ (x,y) \mapsto xy$$ and
$$ x \mapsto x^{-1}$$
are continuous. The second requirement follows from the first one, no? (by taking $y=0, x=-x$ in th... | There is even a term for a group endowed with a topology such that multiplication is continuous (but inversion need not be): a paratopological group.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/172945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 0
} |
Universal Cover of projective plane glued to Möbius band This is the second part of exercise 1.3.21 in page 81 of Hatcher's book Algebraic topology, and the first part is answered here.
Consider the usual cell structure on $\mathbb R P^2$, with one 1-cell and one 2-cell attached via a map of degree 2. Consider the spac... | Let $M$ be the Möbius band and let $D$ be the $2$-cell of $RP^2$. Then $X$ is the result of gluing $M$ to $D$ along a map $\partial D\rightarrow \partial M$ of degree $2$. Hence $\pi_1(X)$ has a single generator $\gamma$, that comes from the inclusion $M\rightarrow X$, and the attached disc $D$ kills $\gamma^4$, hence ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Showing that the closure of the closure of a set is its closure I have the feeling I'm missing something obvious, but here it goes...
I'm trying to prove that for a subset $A$ of a topological space $X$, $\overline{\overline{A}}=\overline{A}$. The inclusion $\overline{\overline{A}} \subseteq \overline{A}$ I can do, bu... | The condition you want to check is
\[
x \in \bar A \quad \Leftrightarrow \quad \text{for each open set $U$ containing $x$, $U \cap A \neq \emptyset$}
\]
This definition implies, among other things, that $A \subset \bar A$. Indeed, with the notation above we always have $x \in U \cap A$. Is it clear why this implies the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
} |
No pandiagonal latin squares with order divisible by 3? I would like to prove the claim that pandiagonal latin squares, which are of form
0 a 2a 3a ... (n-1)a
b b+a b+2a b+3a ... b+ (n-1)a
. . ... | [Edit: replacing the old set of hints with a detailed answer]
Combining things from the question as well as its title and another recent question by the same use tells me that the question
is the following. When $n$ is divisible by three, show that no combination of parameters, $a,b\in R=\mathbb{Z}/n\mathbb{Z}$ in a ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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For what value of k, $x^{2} + 2(k-1)x + k+5$ has at least one positive root? For what value of k, $x^{2} + 2(k-1)x + k+5$ has at least one positive root?
Approach: Case I : Only $1$ positive root, this implies $0$ lies between the roots, so $$f(0)<0$$ and $$D > 0$$
Case II: Both roots positive. It implies $0$ lies beh... | You only care about the larger of the two roots - the sign of the smaller root is irrelevant. So apply the quadratic formula to get the larger root only, which is
$\frac{-2(k-1)+\sqrt{4(k-1)^2-4(k+5)}}{2} = -k+1+\sqrt{k^2-3k-4}$. You need the part inside the square root to be $\geq 0$, so $k$ must be $\geq 4$ or $\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$ Recently, I ran across a product that seems interesting.
Does anyone know how to get to the closed form:
$$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$
I tried using the identity $\cos(x)=\frac{\sin(... | The roots of the polynomial $X^{2n}-1$ are $\omega_j:=\exp\left(\mathrm i\frac{2j\pi}{2n}\right)$, $0\leq j\leq 2n-1$. We can write
\begin{align}
X^{2n}-1&=(X^2-1)\prod_{j=1}^{n-1}\left(X-\exp\left(\mathrm i\frac{2j\pi}{2n}\right)\right)\left(X-\exp\left(-\mathrm i\frac{2j\pi}{2n}\right)\right)\\
&=(X^2-1)\prod_{j=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 0
} |
Moment generating function for the uniform distribution Attempting to calculate the moment generating function for the uniform distrobution I run into ah non-convergent integral.
Building of the definition of the Moment Generating Function
$
M(t) = E[ e^{tx}] = \left\{ \begin{array}{l l}
\sum\limits_x e^{tx} p(x) &\tex... | The limits of integration are not correct. You should integrate from $a$ to $b$ not from $-\infty$ to $+\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/173331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Calculating maximum of function I want to determine the value of a constant $a > 0$ which causes the highest possible value of $f(x) = ax(1-x-a)$.
I have tried deriving the function to find a relation between $x$ and $a$ when $f'(x) = 0$, and found $x = \frac{1-a}{2}$.
I then insert it into the original function: $f(a... | We can resort to some algebra along with the calculus you are using, to see what happens with this function:
$$f(x)=ax(1−x−a)=-ax^2+(a-a^2)x$$
Note that this is a parabola. Since the coefficient of the $x^2$ term is negative, it opens downward so that the maximum value is at the vertex. As you have already solved, the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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Bounded ration of functions If $f(x)$ is a continuous function on $\mathbb R$, and $f$ is doesn't vanish on $\mathbb R$, does this imply that the function $\frac{f\,'(x)}{f(x)}$ be bounded on $\mathbb R$?
The function $1/f(x)$ will be bounded because $f$ doesn't vanish, and I guess that the derivative will reduce the g... | In fact for any positive continuous function $g(x)$ on $\mathbb R$ and any $y_0 \ne 0$, the differential equation $\dfrac{dy}{dx} = g(x) y$ with initial condition $y(0) = y_0$ has a
unique solution $y = f(x) = y_0 \exp\left(\int_0^x g(t)\ dt\right)$, and this is a nonvanishing continuous function with $f'(x)/f(x) = g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
simple integration question I've tried but I cannot tell whether the following is true or not. Let $f:[0,1]\rightarrow \mathbb{R}$ be a nondecreasing and continuous function. Is it true that I can find a Lebesgue integrable function $h$ such that
$$
f(x)=f(0)+\int_{0}^{x}h(x)dx
$$
such that $f'=h$ almost everywhere?
A... | The Cantor function (call it $f$) is a counterexample. It is monotone, continuous, non-constant and has a zero derivative a.e.
If the above integral formula holds, then you have $f'(x) = h(x)$ at every Lebesgue point of $f$, which is a.e. $x \in [0,1]$. Since $f'(x) = 0$ a.e., we have that $h$ is essentially zero. Sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Question about direct sum of Noetherian modules is Noetherian Here is a corollary from Atiyah-Macdonald:
Question 1: The corollary states that finite direct sums of Noetherian modules are Noetherian. But they prove that countably infinite sums are Noetherian, right? (so they prove something stronger)
Question 2: I hav... | As pointed out by others, the submodules of $M=M_1\oplus M_2$ are not necessarily direct sums of submodules of $M_1$ and $M_2$. Nevertheless, you always have the exact sequence
$$0\rightarrow M_1\rightarrow M\rightarrow M_2\rightarrow 0$$
Then $M$ is Noetherian if (and only if) $M_1$ and $M_2$ are Noetherian. One dire... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Check if two 3D vectors are linearly dependent I would like to determine with code (c++ for example) if two 3D vectors are linearly dependent.
I know that if I could determine that the expression
$ v_1 = k · v_2 $is true then they are linearly dependent; they are linearly independent otherwise.
I've tried to construct ... | Here is the portion of the code you need:
if((x1*y2 - x2*y1) != 0 || (x1*z2 - x2*z1) != 0 || (y1*z2 - y2*z1) != 0)
{
//Here you have independent vectors
}
else
{
//Here the vectors are linearly dependent
}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/173710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Computing a sum of binomial coefficients: $\sum_{i=0}^m \binom{N-i}{m-i}$ Does anyone know a better expression than the current one for this sum?
$$
\sum_{i=0}^m \binom{N-i}{m-i}, \quad 0 \le m \le N.
$$
It would help me compute a lot of things and make equations a lot cleaner in the context where it appears (as some... | Here's my version of the proof (and where it arose in the context). You have a cyclic graph of size $n$, which is essentially $n$ vertices and a loop passing through them, so that you can think of them as the roots of unity on a circle (I don't want to draw because it would take a lot of time, but it really helps).
You... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
checking whether certain numbers have an integral square root I was wondering if it is possible to find out all $n \geq 1$, such that $3n^2+2n$ has an integral square root, that is, there exists $a \in \mathbb{N}$ such that $3n^2+2n = a^2$
Also, similarly for $(n+1)(3n+1)$.
Thanks for any help!
| $3n^2+2n=a^2$, $9n^2+6n=3a^2$, $9n^2+6n+1=3a^2+1$, $(3n+1)^2=3a^2+1$, $u^2=3a^2+1$ (where $u=3n+1$), $u^2-3a^2=1$, and that's an instance of Pell's equation, and you'll find tons of information on solving those in intro Number Theory textbooks, and on the web, probably even here on m.se.
Try similar manipulations for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functional and Linear Functional may I know what is the distinction between functional analysis and linear functional analysis? I do a search online and came to a conclusion that linear functional analysis is not functional analysis and am getting confused by them. When I look at the book Linear Functional Analysis pub... | It is mostly a matter of taste. Traditionally functional analysis has dealt mostly with linear operators, whereas authors would typically title their books nonlinear functional analysis if they consider the theory of differentiable (or continuous or more general) mappings between Banach (or more general function) space... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What is the formula to find the count of (sets of) maps between sets that equal the identity map? Given two finite sets $A$ and $B$, what is the formula to find the number of pairs of mappings $f, g$ such that $g \circ f = 1_A$?
| A function $f\colon A\to B$ has a left inverse, $g\colon B\to A$ such that $g\circ f = 1_A$, if and only if $f$ is one-to-one.
Given a one-to-one function $f\colon A\to B$, then the values of a left inverse $g\colon B\to A$ of $f$ on $f(A)$ are uniquely determined; the values on $B-f(A)$ are free.
I'm going to assume ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there "one way" integrals? If we suppose that we can start with any function we like, can we work "backwards" and differentiate the function to create an integral that is hard to solve?
To define the question better, let's say we start with a function of our choosing, $f(x)$. We can then differentiate the function... | Here is an example (by Harold Davenport) of a function that is hard to integrate:
$$\int{2t^6+4t^5+7t^4-3t^3-t^2-8t-8\over
(2t^2-1)^2\sqrt{t^4+4t^3+2t^2+1}}\,dt\ .$$
The primitive is given by (check it!)
$$\eqalign{&-2\log(t^2+4t+2)-\log(\sqrt2+2t)\cr&+ \log\left(\sqrt2
t+2\sqrt2-2\sqrt{t^4+4t^3+2t^2+1}\,\right)\cr
&-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
What is the difference between normal and perpendicular? What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?
| If normal is 90 degree to the surface, that means normal is used in 3D. Perpendicular in that case is more in 2D referring two same entities (line-line) with 90 degree angle between them. Normal in this case could refer 2 different entities (line-surface) making this term valid for 3D case (I definitely remember hearin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 6,
"answer_id": 4
} |
Need to find the recurrence equation for coloring a 1 by n chessboard So the question asks me to find the number of ways H[n] to color a 1 by n chessboard with 3 colors - red, blue and white such that the number of red squares is even and number of blue squares is at least one. I am doing it in this way -
1.If the fir... | Letting $H[n]$ be the number of ways to color the chessboard with an even number of red squares and at least one blue square, and letting $G[n]$ be the very closely related number of ways to color the chessboard with an odd number of red squares with at least one blue square, we notice that $H[n]+G[n]$ is the number of... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof of the divisibility rule of 17.
Rule: Subtract 5 times the last digit from the rest of the number, if the
result is divisible by 17 then the number is also divisible by 17.
How does this rule work? Please give the proof.
| Let
$$n=\sum_{k=0}^N 10^k a_k$$
be the number we want to test for divisibility, where the $a_k$s are the digits in the decimal expansion of $n$. We form the second number $m$ by the process you describe, $$m = \sum_{k=1}^N 10^{k-1} a_k - 5 a_0 = \frac{n-a_0}{10}- 5 a_0 $$
Now suppose $17|m$. Then there exists a natu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 4
} |
Expectation and median (Jensen’s inequality) of spacial functions Let’s have a 1-Lipschitz function $f:S^n \to \mathbb{R}$, where $S^n$ is equipped with the geodesic distance $d$ and with the uniform measure $\mu$.
How can I show that such an $f$ satisfies Jensen’s inequality:
$(\int_{S^n} f d\mu)^2 \leq {\int_{S^n} f^... | The first inequality is called Cauchy-Schwarz inequality, rather than Jensen inequality. Its proof is simple and very general: one considers $g=(f-a)^2$ with $a=\int\limits_{S_n}f\,\mathrm d\mu$. Then $g\geqslant0$ everywhere hence $\int\limits_{S_n}g\,\mathrm d\mu\geqslant0$. Expanding this by linearity and using the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
orientability of riemann surface Could any one tell me about the orientability of riemann surfaces?
well, Holomorphic maps between two open sets of complex plane preserves orientation of the plane,I mean conformal property of holomorphic map implies that the local angles are preserved and in particular right angles are... | Holomorphic maps not only preserve nonoriented angles but oriented angles as well. Note that the map $z\mapsto\bar z$ preserves nonoriented angles, in particular nonoriented right angles, but it does not preserve the clockwise or anticlockwise orientation of small circles.
Now a Riemann surface $S$ is covered by local ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
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Decomposition of a prime number in a cyclotomic field Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$.
Let $A$ be the ring of algebraic integers in $\mathbb{Q}(\zeta)$.
Let $p$ be a prime number such that $p \neq l$.
Let $f$ be the smallest positive integer such that $p^f... | Once you know what the ring of integers in $\mathbf{Q}(\zeta)$ is, you know that the factorization of a rational prime $p$ therein is determined by the factorization of the minimal polynomial of $\zeta$ over $\mathbf{Q}$, which is the cyclotomic polynomial $\Phi_\ell$, mod $p$. So you basically just need to determine t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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The number $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is always an integer For each $n$ consider the expression $$\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$$
I am trying to prove by induction t... | Hint $\rm\quad \phi^{\:n+1}\!-\:\bar\phi^{\:n+1} =\ (\phi+\bar\phi)\ (\phi^n-\:\bar\phi^n)\ -\ \phi\:\bar\phi\,\ (\phi^{\:n-1}\!-\:\bar\phi^{\:n-1})$
Therefore, upon substituting $\rm\ \phi+\bar\phi\ =\ 1\ =\, -\phi\bar\phi\ $ and dividing by $\:\phi-\bar\phi = \sqrt 5\:$ we deduce that $\rm\:f_{n+1} = f_n + f_{n-1}.\:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Area of Triangle inside another Triangle I am stumped on the following question:
In the figure below $AD=4$ , $AB=3$ , and $CD=9$. What is the area of Triangle AEC?
I need to solve this using trigonometric ratios however if trig. ratios makes this problem a lot easier to solve I would be interested in looking ... | $AE+ED=AD=4$, and $ECD$ and $EAB$ are similar (right?), so $AE$ is to $AB$ as $ED$ is to $CD$. Can you take it from there?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/174496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For Maths Major, advice for perfect book to learn Algorithms and Date Structures Purpose: Self-Learning, NOT for course or exam
Prerequisite: Done a course in basic data structures and algorithms, but too basic, not many things.
Major: Bachelor, Mathematics
My Opinion: Prefer more compact, mathematical, rigorous book o... | Knuth is an interesting case, it's certainly fairly rigorous, but I would never recommend it as a text to learn from. It's one of those ones that is great once you already know the answer, but fairly terrible if you want to learn something. On top of this, the content is now rather archaically presented and the pseudoc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Methods to solve differential equations We are given the equation
$$\frac{1}{f(x)} \cdot \frac{d\left(f(x)\right)}{dx} = x^3.$$
To solve it, "multiply by $dx$" and integrate:
$\frac{x^4}{4} + C = \ln \left( f(x) \right)$
But $dx$ is not a number, what does it mean when I multiply by $dx$, what am I doing, why does it... | FIRST QUESTION
This is answered by studying the fundamental theorem of calculus, which basically (in this context) says that if on an interval $(a,b)$
\begin{equation}
\frac{dF}{dx} = f
\end{equation}
then,
\begin{equation}
\int^{b}_{a} f\left(x\right) dx = F(b) - F(a)
\end{equation}
where the integral is the limit of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Decomposition of a prime number $p \neq l$ in the quadratic subfield of a cyclotomic number field of an odd prime order $l$ Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$.
Let $K = \mathbb{Q}(\zeta)$.
Let $A$ be the ring of algebraic integers in $K$.
Let $G$ be the Galoi... | This is most easily established by decomposition group calculations; it is a special case of the more general result proved here (which is an answer to OP's linked question).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/174848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Does $z ^k+ z^{-k}$ belong to $\Bbb Z[z + z^{-1}]$? Let $z$ be a non-zero element of $\mathbb{C}$.
Does $z^k + z^{-k}$ belong to $\mathbb{Z}[z + z^{-1}]$ for every positive integer $k$?
Motivation:
I came up with this problem from the following question.
Maximal real subfield of $\mathbb{Q}(\zeta )$
| Yes. It holds for $k=1$; it also holds for $k=2$, since
$$z^2+z^{-2} = (z+z^{-1})^2 - 2\in\mathbb{Z}[z+z^{-1}].$$
Assume that $z^k+z^{-k}$ lie in $\mathbb{Z}[z+z^{-1}]$ for $1\leq k\lt n$. Then, if $n$ is odd, we have:
$$\begin{align*}
z^n+z^{-n} &= (z+z^{-1})^n - \sum_{i=1}^{\lfloor n/2\rfloor}\binom{n}{i}(z^{n-i}z^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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$\mathbb{CP}^1$ is compact? $\mathbb{CP}^1$ is the set of all one dimensional subspaces of $\mathbb{C}^2$, if $(z,w)\in \mathbb{C}^2$ be non zero , then its span is a point in $\mathbb{CP}^1$.let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and $[z:w]=[\lambda z:\lambda w],\lambda\in\mat... | The complex projective line is the union of the two open subsets
$$
D_i=\{[z_0:z_1]\text{ such that }z_i\neq0\},\quad i=0, 1.
$$
Each of them is homeomorphic to $\Bbb C$, hence to the open unit disk.
Compacteness follows from the fact that it is homeomorphic to the sphere $S^2$. You can see this as a simple elaboration... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about $p$-adic numbers and $p$-adic integers I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks.
Let $p$ be a prime. Then we define the ring of $p$-adic integers to be
$$ \mathbb Z_p = \{ \sum_{k=m}^\infty a_k p^k \mid m \in \mathbb... | I want to emphasize that $\mathbb Z_p$ is not just $\mathbb F_p[[X]]$ in disguise, though the two rings share many properties. For example, in the $3$-adics one has
\[
(2 \cdot 1) + (2 \cdot 1) = 1 \cdot 1 + 1 \cdot 3 \neq 1 \cdot 1.
\]
I know three ways of constructing $\mathbb Z_p$ and they're all pretty useful. It s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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A Trivial Question - Increasing by doubling a number when its negative
The question is : if $x=y-\frac{50}{y}$ , where x and y are both > $0$ . If the value of y is doubled in the equation above the value of x will a)Decrease b)remain same c)increase four fold d)double e)Increase to more than double - Ans)e
Here i... | Let $x_1=y-\frac{50}{y}$ and $x_2=2y-\frac{50}{2y}$. Then $x_2-x_1=(2y-y)-(\frac{50}{2y}-\frac{50}{y})=y+\frac{50}{2y}=y-\frac{50}{y}+(\frac{150}{2y})=x_1+(\frac{150}{2y})$
$\implies x_2-2x_1=(\frac{150}{2y})\gt 0$ (as $y\gt 0$)$\implies x_2\gt 2x_1\implies x_2/x_1\gt 2$ which is option (e).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
optimality of 2 as a coefficient in a continued fraction theorem I'm giving some lectures on continued fractions to high school and college students, and I discussed the standard theorem that, for a real number $\alpha$ and integers $p$ and $q$ with $q \not= 0$, if $|\alpha-p/q| < 1/(2q^2)$ then $p/q$ is a convergent i... | 2 is optimal. Let $\alpha=[a,2,\beta]$, where $\beta$ is a (large) irrational, and let $p/q=[a,1]=(a+1)/1$. Then $p/q$ is not a convergent to $\alpha$, and $${p\over q}-\alpha={1\over2-{1\over \beta+1}}$$ which is ${1\over(2-\epsilon)q^2}$ since $q=1$.
More complicated examples can be constructed. I think this works,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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If $p$ is an element of $\overline E$ but not a limit point of $E$, then why is there a $p' \in E$ such that $d(p, p') < \varepsilon$? I don't understand one of the steps of the proof of Theorem 3.10(a) in Baby Rudin. Here's the theorem and the proof up to where I'm stuck:
Relevant Definitions
The closure of the subset... | Basically Rudin needs to write a triangle inequality with two additional arbitrary points $p',q'$.
$d(p,q) \leq d(p,p') + d(p',q) $
$ d(p,p') + d(p',q) \leq d(p,p') + d(p',q') + d(q',q)$
He assumes $E$ is non-empty, so we might pick points $p'$ and $q'$ for this inequality, even if $p = p'$ and $q = q'$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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The Star Trek Problem in Williams's Book This problem is from the book Probability with martingales by Williams. It's numbered as Exercise 12.3 on page 236. It can be stated as follows:
The control system on the starship has gone wonky. All that one can do is to set a distance to be traveled. The spaceship will then m... | in the book it seems to me that the sum terminates when they get home, so if they get home the sum is finite. williams also hints: it should be clear what thm to use here. i am sure he is referring to levy's extension of the borel-cantelli lemma, $\frac 1 {R_n^2}$ is the conditional probability of getting home at ti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine invertible and inverses in $(\mathbb Z_8, \ast)$ Let $\ast$ be defined in $\mathbb Z_8$ as follows:
$$\begin{aligned} a \ast b = a +b+2ab\end{aligned}$$
Determine all the invertible elements in $(\mathbb Z_8, \ast)$ and determine, if possibile, the inverse of the class $[4]$ in $(\mathbb Z_8, \ast)$.
Identit... | From your post:
Determine all the invertible elements in $(\mathbb Z_8, \ast)$
You have shown that the inverse of $a$ is $\alpha \equiv_8 -\frac{a}{(2a+1)}$. I would go a step further and list out each element which has an inverse since you only have eight elements to check.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The longest sum of consecutive primes that add to a prime less than 1,000,000
In Project Euler problem $50,$ the goal is to find the longest sum of consecutive primes that add to a prime less than $1,000,000. $
I have an efficient algorithm to generate a set of primes between $0$ and $N.$
My first algorithm to try th... | Small speed up hint: the prime is clearly going to be odd, so you can rule out about half of the potential sums that way.
If you're looking for the largest such sum, checking things like $2+3,5+7+11,\cdots$ is a waste of time. Structure it differently so that you spend all your time dealing with larger sums. You don't ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
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Show matrix $A+5B$ has an inverse with integer entries given the following conditions Let $A$ and $B$ be 2×2 matrices with integer entries such that
each of $A$, $A + B$, $A + 2B$, $A + 3B$, $A + 4B$ has an inverse with integer
entries. Show that the same is true for $A + 5B$.
| First note that a matrix $X$ with integer entries is invertible and its inverse has integer entries if and only if $\det(X)=\pm1$.
Let $P(x)=\det(A+xB)$. Then $P(x)$ is a polynomial of degree at most $4$, with integer coefficients, and $P(0),P(1),P(2),P(3),P(4) \in \{ \pm 1 \}$.
Claim: $P(0)=P(1)=P(3)=P(4)$.
Proof: It ... | {
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"timestamp": "2023-03-29T00:00:00",
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$G$ be a group acting holomorphically on $X$
could any one tell me why such expression for $g(z)$, specially I dont get what and why is $a_n(g)$? notations confusing me and also I am not understanding properly, and why $g$ is automorphism? where does $a_i$ belong? what is the role of $G$ in that expression of power se... | The element $g$ gives rise to a function $g: X \to X$. Since $g$ is a function, it has a Taylor expansion at the point $p$ in the coordinate $z$. We could write
$$
g(z) = \sum_{n=0}^\infty a_nz^n.
$$
for some complex numbers $a_n$. However, we are going to want to do this for all the elements of the group. If there's a... | {
"language": "en",
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"answer_id": 0
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A Couple of Normal Bundle Questions We are working through old qualifying exams to study. There were two questions concerning normal bundles that have stumped us:
$1$. Let $f:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}$ be smooth and have $0$ as a regular value. Let $M=f^{-1}(0)$.
(a) Show that $M$ has a non-vanishing... | Some hints:
*
*(a) consider $\nabla f$. $\quad$(b)Show that $TM\oplus \epsilon^1\cong T\mathbb{R}^{n+1}|_M$ is a trivial bundle, then analyze $T(M\times S^1)$.
*Try to use homotopy to construct an orientation of $TN|_M$. Let $F:M\times[0,1]\rightarrow N$ be a smooth homotopy map s.t. $F_0$ is embedding, $F_1$ is ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is this statement stronger than the Collatz conjecture?
$n$,$k$, $m$, $u$ $\in$ $\Bbb N$;
Let's see the following sequence:
$x_0=n$; $x_m=3x_{m-1}+1$.
I am afraid I am a complete noob, but I cannot (dis)prove that the following implies the Collatz conjecture:
$\forall n\exists k,u:x_k=2^u$
Could you help me in th... | Call your statement S. Then: (1.) S does not hold. (2.) S implies the Collatz conjecture (and S also implies any conjecture you like). (3.) I fail to see how Collatz conjecture should imply S. (4.) If indeed Collatz conjecture implies S, then Collatz conjecture does not hold (and this will make the headlines...).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question about proof of $A[X] \otimes_A A[Y] \cong A[X, Y] $ As far as I understand universal properties, one can prove $A[X] \otimes_A A[Y] \cong A[X, Y] $ where $A$ is a commutative unital ring in two ways:
(i) by showing that $A[X,Y]$ satisfies the universal property of $A[X] \otimes_A A[Y] $
(ii) by using the unive... | Define $l(X^i Y^j) := b'(X^i, Y^j)$ and continue this to an $A$-linear map $A[X,Y] \to N$ by
$$l\left(\sum_{i,j} a_{ij} X^i Y^j\right) = \sum_{i,j} a_{ij} b'(X^i,Y^j).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
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How to prove that if $m = 10t+k $ and $67|t - 20k$ then 67|m? m, t, k are Natural numbers.
How can I prove that if $m = 10t+k $ and $67|t - 20k$ then 67|m ?
| We have
$$10t+k=10t -200k+201k=10(t-20k)+(3)(67)k.$$
The result now follows from the fact that $67\mid(t-20k)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Distance in the Poincare Disk model of hyperbolic geometry I am trying to understand the Poincare Disk model of a hyperbolic geometry and how to measure distances. I found the equation for the distance between two points on the disk as:
$d^2 = (dx^2 + dy^2) / (1-x^2-y^2)^2$
Given two points on the disk, I am assuming... | The point is $dx$ and $dy$ in the formula (the one with the 4 in is right btw) don't represent the difference in the $x$ and $y$ co-ordinates, but rather the 'infinitesimal' distance at the point $(x,y)$, so to actually find the distance between two points we have to do some integration.
So the idea is that at the poin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
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Formal notation for function equality Is there a short-hand notation for
$$
f(x) = 1 \quad \forall x
$$
?
I've seen $f\equiv 1$ being used before, but found some some might (mis)interpret that as $f:=1$ (in the sense of definition), i.e., $f$ being the number $1$.
| Go ahead and use $\equiv$. To prevent any possibility of misunderstanding, you can use the word "constant" the first time this notation appears. As in "let $f$ be the constant function $f\equiv1$..."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/175989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prime-base products: Plot For each $n \in \mathbb{N}$, let $f(n)$ map $n$ to the product of the primes that divide $n$.
So for $n=112$, $n=2^4 \cdot 7^1$, $f(n)= 2 \cdot 7 = 14$.
For $n=1000 = 2^3 \cdot 3^3$, $f(1000)=6$.
Continuing in this manner, I arrive at the following plot:
Essentially: I would appre... | I cannot tell how much you know. If a number $n$ is squarefree then $f(n) = n.$ This is the most frequent case, as $6/\pi^2$ of natural numbers are squarefree. Next most frequent are 4 times an odd squarefree number, in which case $f(n) = n/2,$ a line of slope $1/2,$ as I think you had figured out. These numbers are no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Identity related to binomial distribution? While writing a (non-math) paper I came across the following apparent identity:
$N \cdot \mathop \sum \limits_{i = 1}^N \frac{1}{i}\left( {\begin{array}{*{20}{c}}
{N - 1}\\
{i - 1}
\end{array}} \right){p^{i - 1}}{\left( {1 - p} \right)^{N - i}} = \frac{{1 - {{\left( {1 - p} \r... | $$
\begin{align}
N\sum_{i=1}^N\dfrac1i\binom{N-1}{i-1}p^{i-1}(1-p)^{N-i}
&=\frac{(1-p)^N}{p}\sum_{i=1}^N\binom{N}{i}\left(\frac{p}{1-p}\right)^i\tag{1}\\
&=\frac{(1-p)^N}{p}\left[\left(1+\frac{p}{1-p}\right)^N-1\right]\tag{2}\\
&=\frac{(1-p)^N}{p}\left[\frac1{(1-p)^N}-1\right]\tag{3}\\
&=\frac{1-(1-p)^N}{p}\tag{4}
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Convergence of the sequence $f_{1}\left(x\right)=\sqrt{x} $ and $f_{n+1}\left(x\right)=\sqrt{x+f_{n}\left(x\right)} $
Let $\left\{ f_{n}\right\} $ denote the set of functions on
$[0,\infty) $ given by $f_{1}\left(x\right)=\sqrt{x} $ and
$f_{n+1}\left(x\right)=\sqrt{x+f_{n}\left(x\right)} $ for $n\ge1 $.
Prove t... | You know that the sequence is increasing. Now notice that if $f(x)$ is the positive solution of $y^2=x+y$, we have $f_n(x)<f(x) \ \forall n \in \mathbb{N}$. In fact, notice that this hold for $n=1$ and assuming for $n-1$ we have
$$f^{2}_{n+1}(x) = x +f_n(x) < x+f(x) = f^{2}(x).$$ Then there exist $\lim_n f_n(x)$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Proof by contradiction that $n!$ is not $O(2^n)$ I am having issues with this proof: Prove by contradiction that $n! \ne O(2^n)$. From what I understand, we are supposed to use a previous proof (which successfully proved that $2^n = O(n!)$) to find the contradiction.
Here is my working so far:
Assume $n! = O(2^n)$. The... | In
Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$,
they have obtained
$$
n!\geq \left(\frac{n}{e}\right)^n.
$$
Hence
$$
\lim_{n\rightarrow\infty}\frac{n!}{2^n}\geq\lim_{n\rightarrow\infty}\left(\frac{n}{2e}\right)^n=\infty.
$$
Suppose that $n!=O(2^n)$. Then there exist $C>0$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
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Continuous maps between compact manifolds are homotopic to smooth ones
If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$.
Seems to be fairly basic, but I can't find a proof. It might be necessary to ... | It is proved as Proposition 17.8 on page 213 in Bott, Tu, Differential Forms in Algebraic Topology. For the necessary Whitney embedding theorem, they refer to deRham, Differential Manifolds.
This Whitney Approximation on Manifolds is proved as Proposition 10.21 on page 257 in Lee, Introduction to Smooth Manifolds.
T... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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What is a lift? What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit $$\lim_{\longleftarrow} G/G_n$$
I wasn't sure how to construct the map so I asked this question to which I got an answer but ... | The term lift is merely meant to mean the following. Given a surjective map $G\to G'$ a lift of an element $x\in G'$ is a choice of $y\in G$ such that $y\mapsto x$ under this map.
In the linked answer you have $(x_n)\in \lim G/G_n$. Thus the notation has $x_n\in G/G_n$. The elements $y_n$ are just chosen preimages of $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find endomorphism of $\mathbb{R}^3$ such that $\operatorname{Im}(f) \subset \operatorname{Ker}(f)$ I have a problem:
Let's find an endomorphism of $\mathbb{R}^3$ such that $\operatorname{Im}(f) \subset \operatorname{Ker}(f)$.
How would you do it?
The endomorphism must be not null.
| Observe that such operator has the property that $f(f(\vec x))=0$, since $f(\vec x)\in\ker f$. Any matrix $M$ in $M_3(\mathbb R)$ such that $M^2=0$ will give you such operator.
For example, then, $f(x_1,x_2,x_3)=(x_3,0,0)$, for the canonical basis the matrix looks like this:$$\begin{pmatrix} 0&0&1\\0&0&0\\0&0&0\end{pma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Alternating sum of binomial coefficients
Calculate the sum:
$$ \sum_{k=0}^n (-1)^k {n+1\choose k+1} $$
I don't know if I'm so tired or what, but I can't calculate this sum. The result is supposed to be $1$ but I always get something else...
| Another way to see it: prove that
$$\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}\,\,\,,\,\,\text{so}$$
$$\sum_{k=0}^n(-1)^k\binom{n+1}{k+1}=\sum_{k=0}^n(-1)^k\cdot 1^{n-k}\binom{n}{k}+\sum_{k=0}^n(-1)^k\cdot 1^{n-k}\binom{n}{k+1}=$$
$$=(1+(-1))^n-\sum_{k=0}^n(-1)^{k+1}\cdot 1^{n-k-1}\binom{n}{k+1}=0-(1-1)^n+1=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/176622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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A simple question of finite number of basis Let $V$ be a vector space. Define
" A set $\beta$ is a basis of $V $" as "(1) $\beta$ is linearly independent set, and (2) $\beta$ spans $V$ "
On this definition, I want to show that "if $V$ has a basis (call it $\beta$) then $\beta$ is a finite set."
In my definition, I ha... | I take it $V$ is finite-dimensional? For instance, the (real) vector space of polynomials over $\mathbb{R}$ is infinite-dimensional $-$ in this space, no basis is finite. And for finite-dimensional spaces, it's worth noting that the dimension of a vector space $V$ is defined as being the size of a basis of $V$, so if $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $\iint_{D} f(x,y)(1 - x^2 - y^2) ~dx ~dy = \pi/2$ Suppose $f(x,y)$ is a bounded harmonic function in the unit disk
$D = \{z = x + iy : |z| < 1 \} $
and $f(0,0) = 1$. Show that
$$\iint_{D} f(x,y)(1 - x^2 - y^2) ~dx ~dy = \frac{\pi}{2}.$$
I'm studying for a prelim this August and I haven't taken Complex in a long ti... | Harmonic functions have the mean value property, that is
$$
\frac1{2\pi}\int_0^{2\pi}f(z+re^{i\phi})\,\mathrm{d}\phi=f(z)
$$
If we write the integral in polar coordinates
$$
\begin{align}
\iint_Df(x,y)(1-x^2-y^2)\,\mathrm{d}x\,\mathrm{d}y
&=\int_0^1\int_0^{2\pi}f(re^{i\phi})(1-r^2)\,r\,\mathrm{d}\phi\,\mathrm{d}r\\
&=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Why does the $2$'s and $1$'s complement subtraction works? The algorithm for $2$'s complement and $1$'s complement subtraction is tad simple:
$1.$ Find the $1$'s or $2$'s complement of the subtrahend.
$2.$ Add it with minuend.
$3.$ If there is no carry then then the answer is in it's complement form we have take the ... | We consider $k$-bit integers with an extra $k+1$ bit, that is associated with the sign. If the "sign-bit" is 0 then we treat this number as usual as a positive number in binary representation. Otherwise this number is negative (details follow).
Let us start with the 2-complement. To compute the 2-complement of an numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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Fourier transform of the derivative - insufficient hypotheses? An exercise in Carlos ISNARD's Introdução à medida e integração:
Show that if $f$ and $f'$ $\in\mathscr{L}^1(\mathbb{R},\lambda,\mathbb{C})$ and $\lim_{x\to\pm\infty}f(x)=0$ then $\hat{(f')}(\zeta)=i\zeta\hat{f}(\zeta)$.
($\lambda$ is the Lebesgue measure... | We can find a sequence of $C^1$ functions with compact support $\{f_j\}$ such that $$\lVert f-f_j\rVert_{L^1}+\lVert f'-f_j'\rVert_{L^1}\leq \frac 1j.$$
This is a standard result about Sobolev spaces. Actually more is true: the smooth functions with compact support are dense in $W^{1,1}(\Bbb R)$, but that's not needed ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is smallest possible integer k such that $1575 \times k$ is perfect square? I wanted to know how to solve this question:
What is smallest possible integer $k$ such that $1575 \times k$ is a perfect square?
a) 7, b) 9, c) 15, d) 25, e) 63. The answer is 7.
Since this was a multiple choice question, I guess I co... | If $1575\times k$ is a perfect square, we can write it as
$\prod_k p_k^2$. Since $1$ is not among the choices given, let's assume that $1575$ is not a square from the beginning.
Therefore $9=3^2$ and $25=5^2$ are ruled out, since multiplying a non square with a square doesn't make it a square. $63=9\cdot 7$ is ruled ou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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I want to prove $ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $ How can I prove this integral diverges?
$$ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $$
| $$
\int_{0}^{\infty}\frac{e^{-x}}{x}= \int_{0}^{1}\frac{e^{-x}}{x}+\int_{1}^{\infty}\frac{e^{-x}}{x} \\
> \int_{0}^{1}\frac{e^{-x}}{x} \\
> e^{-1}\int_{0}^{1}\frac{1}{x}
$$
which diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/177010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What's the meaning of the unit bivector i? I'm reading the Oersted Medal Lecture by David Hestenes to improve my understanding of Geometric Algebra and its applications in Physics. I understand he does not start from a mathematical "clean slate", but I don't care for that. I want to understand what he's saying and what... | The unit bivector represents the 2D subspaces. In 2D Euclidean GA, there are 4 coordinates:
*
*1 scalar coordinate
*2 vector coordinates
*1 bivector coordinate
A "vector" is then (s, e1, e2, e1^e2).
The unit bivector is frequently drawn as a parallelogram, but that's just a visualization aid. It conceptually m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does "nullity" have a potentially conflicting or confusing usage? In Linear Algebra and Its Applications, David Lay writes, "the dimension of the null space is sometimes called the nullity of A, though we will not use the term." He then goes on to specify "The Rank Theorem" as "rank A + dim Nul A = n" instead of callin... | While choices of terminology is often a matter of taste (I would not know why the author should prefer to say $\operatorname{Nul}A$ instead of $\ker A$), there is at least a mathematical reason why "rank" is more important than "nullity": is it connected to the matrix/linear map in a way where source and destination sp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding $E(N)$ in this question suppose $X_1,X_2,\ldots$ is sequence of independent random variables of $U(0,1)$ if
$N=\min\{n>0 :X_{(n:n)}-X_{(1:n)}>\alpha , 0<\alpha<1\}$ that $X_{(1:n)}$ is smallest order statistic and
$X_{(n:n)}$ is largest order statistic. how can find $E(N)$
| Let $m_n=\min\{X_k\,;\,1\leqslant k\leqslant n\}=X_{(1:n)}$ and $M_n=\max\{X_k\,;\,1\leqslant k\leqslant n\}=X_{(n:n)}$. As explained in comments, $(m_n,M_n)$ has density $n(n-1)(y-x)^{n-2}\cdot[0\lt x\lt y\lt1]$ hence $M_n-m_n$ has density $n(n-1)z^{n-2}(1-z)\cdot[0\lt z\lt1]$.
For every $n\geqslant2$, $[N\gt n]=[M_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What digit appears in unit place when $2^{320}$ is multiplied out Is there a way to answer the following preferably without a calculator
What digit appears in unit place when $2^{320}$ is multiplied out ? a)$0$ b)$2$ c)$4$ d)$6$ e)$8$
---- Ans(d)
| $\rm mod\ 5\!:\, \color{#0A0}{2^4}\equiv \color{#C00}1\Rightarrow2^{320}\equiv (\color{#0A0}{2^4})^{80}\equiv \color{#C00}1^{80}\!\equiv \color{#C00}1.\,$ The only choice $\:\equiv \color{#C00}1\!\pmod 5\:$ is $6,\: $ in d).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/177312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots$ converges to $\frac 1 2 $ Show that
$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots = \frac{1}{2}.$$
I'm not exactly sure what to do here, it seems awfully similar to Zeno's paradox.
If the series continues infin... | Solution as per David Mitra's hint in a comment.
Write the given series as a telescoping series and evaluate its sum:
$$\begin{eqnarray*}
S &=&\frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+\cdots \\
&=&\sum_{n=1}^{\infty }\frac{1}{\left( 2n-1\right) \left( 2n+1\right) } \\
&=&\sum_{n=1}^{\infty }\left( \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combinatorics question: Prove 2 people at a party know the same amount of people I recently had an assignment and got this question wrong, was wondering what I left out.
Prove that at a party where there are at least two people, there are two people who know the same number of other people there. Be sure to use the var... | First of all, you left out saying that you were proceding by induction, and you left out establishing a base case for the induction.
But let's look at case 2. Suppose that of the original two people, one knows 17 of the 43 people at the party, the other knows 29. Now the newcomer knows the first of the two, but not th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the $\tau$ symbol in the Bourbaki text? I'm reading the first book by Bourbaki (set theory) and he introduces this logical symbol $\tau$ to later define the quantifiers with it. It is such that if $A$ is an assembly possibly contianing $x$ (term, variable?), then $\tau_xA$ does not contain it.
Is there a refer... | Adrian Mathias offers the following explanation here:
Bourbaki use the Hilbert operator but write it as $\tau$ rather than $\varepsilon$, which latter is visually too close to the sign $\in$ for the membership relation. Bourbaki use the word assemblage, or in their English translation, assembly, to mean a finite seque... | {
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When was $\pi$ first suggested to be irrational? When was $\pi$ first suggested to be irrational?
According to Wikipedia, this was proved in the 18th century.
Who first claimed / suggested (but not necessarily proved) that $\pi$ is irrational?
I found a passage in Maimonides's Mishna commentary (written circa 1168, Eir... | From a non-wiki source:
Archimedes [1], in the third century B.C. used regular polygons inscribed and circumscribed to a circle to approximate : the more sides a polygon has, the closer to the circle it becomes and therefore the ratio between the polygon's area between the square of the radius yields approximations to... | {
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"url": "https://math.stackexchange.com/questions/177546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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analytically calculate value of convolution at certain point i'm a computer science student and i'm trying to analytically find the value of the convolution between an ideal step-edge and either a gaussian function or a first order derivative of a gaussian function.
In other words, given an ideal step edge with amplitu... | We can write $i(t):=B+A\chi_{[t_0,+\infty)}$. Using the properties of convolution, we can see that
\begin{align}
o(t_0)&=B+A\int_{-\infty}^{+\infty}g(t)\chi_{[t_0,+\infty)}(t_0-t)dt\\
&=B+A\int_{-\infty}^0\frac 1{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)dx.
\end{align}
The integral can be express... | {
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"timestamp": "2023-03-29T00:00:00",
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If a player is 50% as good as I am at a game, how many games will it be before she finally wins one game? This is a real life problem. I play Foosball with my colleague who hasn't beaten me so far. I have won 18 in a row. She is about 50% as good as I am (the average margin of victory is 10-5 for me). Mathematically sp... | A reasonable model for such a game is that each goal goes to you with probability $p$ and to her with probability $1-p$. We can calculate $p$ from the average number of goals scored against $10$, and then calculate the fraction of games won by each player from $p$.
The probability that she scores $k$ goals before you s... | {
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"url": "https://math.stackexchange.com/questions/177725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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showing almost equal function are actually equal I am trying to show that if $f$ and $g$ are continuous functions on $[a, b]$ and if $f=g$ a.e. on $[a, b]$, then, in fact, $f=g$ on $[a, b]$. Also would a similar assertion be true if $[a, b]$ was replaced by a general measurable set $E$ ?
Some thoughts towards the proof... | The set $\{x\mid f(x)\neq g(x)\}$ is open (it's $(f-g)^{—1}(\Bbb R\setminus\{0\})$), and of measure $0$. It's necessarily empty, otherwise it would contain an open interval.
| {
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"source": "stackexchange",
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Prove whether a relation is an equivalence relation Define a relation $R$ on $\mathbb{Z}$ by $R = \{(a,b)|a≤b+2\}$.
(a) Prove or disprove: $R$ is reflexive.
(b) Prove or disprove: $R$ is symmetric.
(c) Prove or disprove: $R$ is transitive.
For (a), I know that $R$ is reflexive because if you substitute $\alpha$ into t... | You are right, the attempt to prove transitivity doesn't work. But your calculation should point towards a counterexample.
Make $a \le b+2$ in an extreme way, by letting $b=a-2$. Also, make $b\le g+2$ in the same extreme way. Then $a \le g+2$ will fail. Perhaps work with an explicit $a$, like $47$.
| {
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"url": "https://math.stackexchange.com/questions/177877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Difference between "space" and "mathematical structure"? I am trying to understand the difference between a "space" and a "mathematical structure".
I have found the following definition for mathematical structure:
A mathematical structure is a set (or sometimes several sets) with various associated mathematical object... | I'm not sure I must be right. Different people have different ideas. Here I just talk about my idea for your question. In my opinion, they are same: the set with some relation betwen the elements of the set. Calling it a space or calling it just a mathematical structure is just a kind of people's habit.
| {
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"url": "https://math.stackexchange.com/questions/177937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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"answer_id": 1
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Applications of Bounds of the Sum of Inverse Prime Divisors $\sum_{p \mid n} \frac{1}{p}$ Question: What interesting or notable problems involve the additive function $\sum_{p \mid n} \frac{1}{p}$ and depend on sharp bounds of this function?
I'm aware of at least one: A certain upper bound of the sum implies the ABC c... | As it turns out, your proposed inequality is false - in fact false for any $\epsilon>0$, even huge $\epsilon$.
The following approximation to the twin primes conjecture was proved by Chen's method: there exist infinitely many primes $b$ such that $c=b+2$ is either prime or the product of two primes. With $a=2$, this gi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are the two statements concening number theory correct? Statement 1: any integer no less than four can be factorized as a linear combination of two and three.
Statement 2: any integer no less than six can be factorized as a linear combination of three, four and five.
I tried for many numbers, it seems the above two sta... | The question could be generalized, but there is a trivial solution in the given cases.
*
*Any even number can be written as $2n$. For every odd number $x > 1$, there exists an even number $y$ such that $y+3 = x$.
*Likewise, numbers divisible by $4$ can be written as $4n$. All other numbers over $2$ are $4n + 5$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that for any $x \in \mathbb N$ such that $xProve that every positive integer $x$ with $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
| Hint: Note that $x = m (n-1)! + r$ where $0 \le m < n$ and $0 \le r < (n-1)!$. Use induction.
EDIT: Oops, this is wrong: as Steven Stadnicki noted, $m (n-1)!$ doesn't necessarily divide $n!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/178122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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Accidents of small $n$ In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that there is no general algebraic solution for polynomials of degree $n$ except when $n \leq 4$.
It se... | $\mathbb{R}^n$ has a unique smooth structure except when $n=4$. Furthermore, the cardinality of [diffeomorphism classes of] smooth manifolds that are homeomorphic but not diffeomorphic to $\mathbb{R}^4$ is $\mathfrak{c}=2^{\aleph_0}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/178183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 20,
"answer_id": 4
} |
Euler's theorem for powers of 2 According to Euler's theorem,
$$x^{\varphi({2^k})} \equiv 1 \mod 2^k$$
for each $k>0$ and each odd $x$. Obviously, number of positive integers less than or equal to $2^k$ that are relatively prime to $2^k$ is
$$\varphi({2^k}) = 2^{k-1}$$
so it follows that
$$x^{{2^{k-1}}} \equiv 1 \mod ... | An $x$ such that $\phi(m)$ is the least positive integer $k$ for which $x^k \equiv 1 \mod m$ is called a primitive root mod $m$. The positive integers that have primitive roots are
$2, 4, p^n$ and $2 p^n$ for odd primes $p$ and positive integers $n$. In particular you are correct that there is no primitive root for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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$\sigma$-algebra of $\theta$-invariant sets in ergodicity theorem for stationary processes Applying Birkhoff's ergodic theorem to a stationary process (a stochastic process with invariant transformation $\theta$, the shift-operator - $\theta(x_1, x_2, x_3,\dotsc) = (x_2, x_3, \dotsc)$) one has a result of the form
$ \... | The $T$-invariant $\sigma$-field is generated by the (generally nonmeasurable) partition into the orbits of $T$. See https://mathoverflow.net/questions/88268/partition-into-the-orbits-of-a-dynamical-system
This gives a somewhat geometric view of the invariant $\sigma$-field. You should also studied the related notion ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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'Linux' math program with interactive terminal? Are there any open source math programs out there that have an interactive terminal and that work on linux?
So for example you could enter two matrices and specify an operation such as multiply and it would then return the answer or a error message specifying why an answe... | Sage is basically a Python program/interpreter that aims to be an open-source mathematical suite (ala Mathematica and Magma etc.). There are many algorithms implemented as a direct part of Sage, as well as wrapping many other open-source mathematics packages, all into a single interface (the user never has to tell whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 14,
"answer_id": 2
} |
Randomly selecting a natural number In the answer to these questions:
*
*Probability of picking a random natural number,
*Given two randomly chosen natural numbers, what is the probability that the second is greater than the first?
it is stated that one cannot pick a natural number randomly.
However, in this ques... | It really depends on what you mean by the "probability of randomly selecting n natural numbers with property $P$". While you cannot pick random natural number, you can speak of uniform distribution.
For the last problem, the probability is calculated, and is to be understood as the limit when $N \to \infty$ from the "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$
I am trying to find all solutions to
(1) $y^3 = x^2 + x + 1$, where $x,y$ are integers $> 1$
I have attempted to do this using...I think they are called 'quadratic integers'. It would be great if someone could verify the steps and sugge... | In this answer to a related thread, I outline a very elementary solution to this problem. There are probably even simpler ones, but I thought you might appreciate seeing that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/178499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Is every monomial over the UNIT OPEN BALL bounded by its L^{2} norm? Let $m\geq 2$ and $B^{m}\subset \mathbb{R}^{m}$ be the unit OPEN ball . For any fixed multi-index $\alpha\in\mathbb{N}^{m}$ with $|\alpha|=n$ large and $x\in B^{m}$
$$|x^{\alpha}|^{2}\leq \int_{B^{m}}|y^{\alpha}|^{2}dy\,??$$
| No. For a counterexample, take $\alpha=(n,0,\ldots,0)$. Obviously, $\max_{S^m}|x^\alpha|=1$, but an easy calculation shows
$$
\int_{S^m}|y^\alpha|^2{\mathrm{d}}\sigma(y) \to 0,
$$
as $n\to\infty$.
For the updated question, that involves the open unit ball, the answer is the same. With the same counterexample, we have
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Invertible elements in the ring $K[x,y,z,t]/(xy+zt-1)$ I would like to know how big is the set of invertible elements in the ring $$R=K[x,y,z,t]/(xy+zt-1),$$ where $K$ is any field. In particular whether any invertible element is a (edit: scalar) multiple of $1$, or there is something else. Any help is greatly apprecia... | Let $R=K[X,Y,Z,T]/(XY+ZT-1)$. In the following we denote by $x,y,z,t$ the residue classes of $X,Y,Z,T$ modulo the ideal $(XY+ZT-1)$. Let $f\in R$ invertible. Then its image in $R[x^{-1}]$ is also invertible. But $R[x^{-1}]=K[x,z,t][x^{-1}]$ and $x$, $z$, $t$ are algebraically independent over $K$. Thus $f=cx^n$ with $c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Assuming that $(a, b) = 2$, prove that $(a + b, a − b) = 1$ or $2$ Statement to be proved: Assuming that $(a, b) = 2$, prove that
$(a + b, a − b) = 1$ or $2$.
I was thinking that $(a,b)=\gcd(a,b)$ and tried to prove the statement above, only to realise that it is not true.
$(6,2)=2$
but $(8,4)=4$, seemingly contradicti... | if $d\mid(a+b, a-b) \Rightarrow d|(a+b)$ and $d\mid(a-b)$
$\Rightarrow d\mid(a+b) \pm (a-b) \Rightarrow d\mid2a$ and $d\mid2b \Rightarrow d\mid(2a,2b) \Rightarrow d\mid2(a,b)$
This is true for any common divisor of (a+b) and (a-b).
If d becomes (a+b, a-b), (a,b)|(a±b)
as for any integers $P$, $Q$, $(a,b)\mid(P.a+Q.b)$... | {
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"url": "https://math.stackexchange.com/questions/178714",
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