Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding the average value of a cubic function
Let $p(x)=ax^{3}+bx^{2}+cx+d$. There exists real numbers $r$ and $s$(independent of $a,b,c$ and $d$) $0<r<s<1$. For which the average value of $p(x)$ on the interval $[0,1]$ is equal to the average value of $p(r)$ and $p(s)$. Find the product of $rs$ expressed as a fractio... | The question probably misses a condition $a,b,c,d\ne0$.
We have
\begin{align}
p_{\mathrm{avg}} &= \tfrac14 a+\tfrac13 b+\tfrac12 c+d
=p(r)+p(s)
=
\tfrac12 (r^3+s^3)a
+
\tfrac12 (r^2+s^2)b
+
\tfrac12(r+s)c
+d
\end{align}
From this system we have
\begin{align}
\tfrac12 r+\tfrac12s &=\tfrac12
\\
\tfrac12 s^2+\tfrac12 r^2&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Problem concerning eigenspace and rank of some matrix Problem: Let $N,P \in \mathbb{R}^{n \times n}$ be matrices, and let $P \neq 0$. Suppose that $P = NP$ and that $P$ is diagonalizable. Prove then that $N$ has an eigenspace with dimension greater than or equal to the rank of $P$.
Attempt: Let $E_{\lambda_i}$ be an ei... | Take the first $m = \text{rank}(P)$ eigenvalues in $D$ to be nonzero. Then
$$
P = NP \Rightarrow BD = NBD \Rightarrow 1 \cdot (b_jd_j) = N (b_jd_j), \; j=1,\ldots,m
$$
where $b_j$ is the column $j$ of $B$ and so are independent. This says $N$ has an eigenspace associated with $\lambda = 1$ with at least rank $m$. I'm n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings?
Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings? If so, write down an explicit isomorphism. If not, prove they are not.
My Try:
Since $x^2+2$ is irreducible in $\mathbb{F}_5[x]$, and $y^2+y+1... | Hint: $(y + 3)^2 + 2 = y^2 + y + 1$ as elements of $\mathbb F_5[y]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
What does this use of little-o notation mean? I am currently going through the proof of Prime Number Theorem, as given in Hardy and Wright, and in it they define the following constant:
$$\alpha = \limsup_\limits{x \to \infty} \left|V(x) \right|$$
It then states:
$$|V(x)| \leq \alpha + o(1)$$
The book explains that $f(... | The whole point of the big-oh and little-oh notation is to provide a way to write down that a particular function is negligible/is of the same order as another simpler function in a rigorous way. It allows us to deal with possibly disgusting and unpredictable functions by comparing them to much simpler functions of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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What do we call a "function" which is not defined on part of its domain? Before the immediate responses come in, I realize that a properly defined function means that it is defined for every value in its domain.
My question is this: if $f:A\to B$ has the property $f(a)=b_1$ and $f(a)=b_2$, then it is often still called... | It is not possible that $f(a) = b_1$ and $f(a) = b_2$ unless $b_1=b_2$. However, we can consider functions that output sets, so $f(a) = \{b_1,b_2\}$ and $f(c) = \{b_3\}$. If we did that we could think of $f$ as taking multiple values on $a$ but only a single value on $b$. This would be called a multivalued function (ht... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Question on Lebesgue point integration
If $f(x)$ is finite at $x$ and $\lim\limits_{h\to 0}\frac{1}{h}\int_x^{x+h} |f(t)-f(x)|dt = 0$ then $x$ is called a Lebesgue point of function $f$.
a) If $f$ is continuous at $x$ then $x$ is a Lebesgue point.
b) Give an example Lebesgue point is not a continuous point.
c) If $f$ ... | For c), it appears that you have the right idea. Let $r \in \mathbb{Q}$ and $f \in L^1$. From Lebesgue's differentiation theorem applied to the function $|f(t) - r|$, we have that for ae $x \in [a,b]$ that
$\frac{1}{h} \int_x^{x+h} |f(t) - r| dt \to |f(x) - r|$ as $h \to 0$.
In particular, let $E_r$ be the set of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Cohomology ring of $\mathbb RP^n$ with integral coefficient. I know cup product structure on $H^*(\mathbb{R}P^n;\mathbb{Z}_2)= \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$. How to get $H^*(\mathbb{R}P^n;\mathbb{Z})$ from this? I have two cochain complexes for two coefficient rings.
Now my question is what will be the induced ... | This is a duplicate of this question, the answer to which is given in a comment. See also this website, which was the first thing I found when I searched for "cohomology of projective space".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Continuous surjective mapping from half-open interval to the reals / the Cantor set 1) Is there a continuous surjective mapping of $(0, 1]$ to $\mathbb{R}$ ?
To me it seems not, but haven't been able to prove why it is false though. I was thinking of finding compact sets contained in (0, 1], and having the right endpoi... | For 1) consider $\frac{1}{x} \sin\frac{1}{x}$
2) is false since the image of a connected set under a continuous map is connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why are the Kuratowski closure axioms so interesting? I never really understood the relevance of the Kuratowski closure axioms. My problem is this:
A user2520938 closure operator is an assignment $cl:\mathcal{P}(X)\to\mathcal{P}(X)$ s.t.
*
*$cl(X)=X$ and $cl(\emptyset)=\emptyset$
*For any collection of set... | A Kuratowski closure operator $f:\mathcal P(X)\to\mathcal P(X)$ has the property that there is a (unique) topology $\tau$ on $X$ such that $f$ is the closure operator for that topology, i.e., for every subset $U$ of $X$, $f(U)$ is the $\tau$-closure of $U$.
A "user2520938 closure operator" $f:\mathcal P(X)\to\mathcal P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Change of variable leads to contradiction for an elementary integral. Integral $I_1(\alpha,\beta)=\int_0^\infty t^\alpha \exp(-i t^{\beta}) ~dt$ converges for $-1<\alpha<\beta-1$. By introducting $u = t^\beta$ the integral is reduced to $\frac{1}{\beta}\int_0^\infty u^{\frac{\alpha+1}{\beta}-1} \exp(-i u)~du$ and the c... | You can't plug $-\beta$ into your formula in place of $\beta$, since your formula only works for $\beta\gt0$. Working with a negative $\beta$ inside the exponential gives different results.
If $\beta\gt0$, then substituting $t\mapsto t^{1/\beta}$ gives
$$
\begin{align}
\int_0^\infty t^\alpha\exp\!\left(it^\beta\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Metric Isometry is always smooth? Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which induces the topology on $M$. Let $f:(M,d) \rightarrow (M,d) $ be an isometry (in the sense of metric spaces). Is it true that $f$ must be smooth?
(if the metric $d$ is induced by some Riemannian metric $g$ then the answer... | No. Let $(X,d')$ be a metric space and let $f \colon X \rightarrow X$ be a homeomorphism. You can define a new metric on $X$ (which you might call the pullback metric) by $d(x,y) := d'(f(x),f(y))$. The topology induced by $d$ is the same as the topology induced by $d'$ and $\varphi \colon X \rightarrow X$ is an isometr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
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Help with trig identities to solve an AIME geometry question Quadrilateral $ABCD$ has side lengths $AB = 20$, $BC = 15$, $CD = 7$, and $AD = 24$, with diagonal length $AC = 25$. If we write $\angle ACB = \alpha$ and $\angle ABD = \beta$, then $\tan (\alpha + \beta)$ can be expressed in the form $-m/n$, where $m$ and $n... | A meta observation. The comments discussion under the question shows how to solve this by recognizing right triangles in the picture, so that tangents of the two angles are known (and from that, the tangent of the sum of those angles).
The meta point is that had the lengths been different, we would only know $\cos \a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluate the limit $\lim_{x \to 0} \left(\frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$
Evaluate the limit $$\lim_{x \to 0}\left( \frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$$
My attempt
So we have $$\frac{1}{x^{2}}-\frac{\cos^{2}x}{\sin^{2}x}$$
$$=\frac{\sin^2 x-x^2\cos^2 x}{x^2\sin^2 x}$$
$$=\frac{x^2}{\sin^2 x}\cdot\... | Using L'Hospital's rule (since direct evaluation gives $\bigl(\frac{0}{0}\bigr)$ ), we have the following:
$$\lim_{x \to 0} \frac{\cos x-\cos x +x\sin x}{3x^2}= \lim_{x \to 0} \frac{\sin x}{3x}.$$
We take the derivative of the numerator and denominator again:
$$\lim_{x \to 0} \frac{\cos x}{3} = \frac{1}{3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Algebra and Analysis I'm a math major at university and my tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble understanding the difference between them.
What are the actual differences between algebra and analysis, and how do I tell which is for me?
| At an elementary level, algebra has shorter proofs; and sometimes a formula says all, whereas in geometry or topology a picture can say a lot. At a higher level, these areas begin to merge. For example, algebraic number theory very often uses analytic tools (see the "analytic class number formula", to give an example),... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 2
} |
Quadratic formula - check my simplificaiton I am trying to solve this equation using the quadratic formula:
$$x^2 + 4x -1 = 0$$
I start by substituting the values into the quadratic formula:
$$x = {-(4) \pm \sqrt {(4)^2 - 4(1)(-1)} \over 2}$$
which becomes
$$x = {-4 \pm \sqrt{20} \over 2}$$
This is the answer the textb... | Note that
$$\frac{B+C}{A}=\frac{B}{A}+\frac{C}{A}$$
$$x=\frac{-4\pm 2\sqrt{5}}{2}=\frac{-4}{2}\pm\frac{2\sqrt{5}}{2}=-2\pm\sqrt{5}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Prove a matrix is invertible The $2 \times 2$ matrix ${A}$ satisfies
$A^2 - 4 {A} - 7{I} = {0},$
where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible.
What is the best way to do this?
| $${1\over7}(A-4I)A={1\over 7}(A^2-4A)={1\over 7}{7I}=I\\
A{1\over7}(A-4I)={1\over 7}(A^2-4A)={1\over 7}{7I}=I$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How do people perform mental arithmetic for complicated expressions?
This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky.
The problem on the blackboard is:
$$
\dfrac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365}
$$
The answer is easy us... | If you know your squares out to $14$ (which students used to memorize) and do some simple three-digit arithmetic in your head, you can see that
$$100+121+144=365$$
and
$$169+196=365$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "183",
"answer_count": 20,
"answer_id": 17
} |
Algebraic Aproach For this word problem How can the followin question be solved algebraically?
A certain dealership has a total of 100 vehicles consisting of cars and trucks. 1/2 of the cars are used and 1/3 of the trucks are used. If there are 42 used vehicles used altogether, how many trucks are there?
| $
\newcommand{\xu}{x^{\text{used}}}
\newcommand{\yu}{y^{\text{used}}}
\newcommand{\xn}{x^{\text{new}}}
\newcommand{\yn}{y^{\text{new}}}
$
The key in converting text problems into algebraic expressions is to write an expression for every sentence or phrase which contains quantifiable information.
For example, consider... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Perpendicular Bisector of Made from Two Points For a National Board Exam Review:
Find the equation of the perpendicular bisector of the line joining
(4,0) and (-6, -3)
Answer is 20x + 6y + 29 = 0
I dont know where I went wrong. This is supposed to be very easy:
Find slope between two points:
$${ m=\frac{y^2 - y^1}... | Notice, the mid=point of the line joining $(4, 0)$ & $(-6, -3)$ is given as $$\left(\frac{4+(-6)}{2}, \frac{0+(-3)}{2}\right)\equiv \left(-1, -\frac{3}{2}\right)$$ The slope of the perpendicular bisector
$$=\frac{-1}{\text{slope of line joining}\ (4, 0)\ \text{&}\ (-6, -3)}$$
$$=\frac{-1}{\frac{-3-0}{-6-4}}=-\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
evaluate the integral Evaluate the integral from:
$$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx$$
The way I approach this problem is
$$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx = \frac{1}{2}\int_{-\infty}^{\infty} \frac{z \cdot e^{i2z}}{(z - i\sqrt{3})(i+i\sqrt{3})}dz$$
and
$$ \text{Res}_{i\sqrt3}(f(z)) ... | Here is another approach:
$$f(a)=\int_0^{\infty} \frac{x \cdot \sin(ax)}{x^2+3}dx$$ take a Laplace transform with respect to $a$ to obtain
\begin{align}
\mathcal{L}(f(a))&=\int_0^{\infty} \frac{x^2}{(x^2+3)(x^2+s^2)}dx\\
&=\frac{\pi}{2\sqrt3+2s}
\end{align}
now take an inverse Laplace to obtain
$$f(a)=\frac{\pi}{2} e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
A question from Titchmarsh's The Theory of the Riemann Zeta-Function. On pages 35-36 here, we have that the integral
$$\frac{1}{2i\sqrt{y}}\int_{1/2-i\infty}^{1/2+i\infty}\phi(s-1/2)\phi(1/2-s)(s-1)\Gamma(1+s/2)\pi^{-s/2}\zeta(s)y^sds$$
equals for $\phi(s)=1$ to:
$$\frac{1}{i\sqrt{y}}\sum_{n=1}^\infty \int_{2-i\infty}^... | We move the contour to the line $\operatorname{Re}(s)=2$ then a residue at $s=1$ appears, but this residue is zero. Replacing $\phi(s)=1$
$$
\begin{align*}
I&:=\frac{1}{2i\sqrt{y}}\int_{2-i\infty}^{2+i\infty}(s-1)\Gamma(1+s/2)\pi^{-s/2}\zeta(s)y^sds\\
&=\frac{1}{2i\sqrt{y}}\sum_{n\geq 1}\int_{2-i\infty}^{2+i\infty}(s-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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loop invariant fibonacci fibonacci(int n)
if n < 0
return 0;
else if n = 0
return 0;
else if n = 1
return 1;
else x = 1;
y = 0;
for i from 2 to n {
t = x;
x = x + y;
y = t;
}
}
return x;
I'm trying to find a loop invariant for the above algorithm, but am not sure where to start. I know that the invaria... | Let $f(n)$ be the $n^{th}$ term of the Fibonacci sequence.
A correct invariant would be, for example:
" At the end of an iteration , $y=f(i-1)$ and $x=f(i)$ "
It is true for the first iteration ($i=2, y=1, x=1$), and stays true during the loop due to the definition of $f$ ($f(n)=f(n-1)+f(n-2)$ for $n > 1$).
Hence, at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Calculating Rotation from centroid I have a polygon as such:
where the green polygon is the rotated polygon and the purple is the extent of the polygon.
Is there a way to calculate the angle of rotation of the green polygon from the extent?
I am using python to do this as well, so if you have it in python code, that w... | Rotated from what? Are we to assume that the lower left edge of the rectangle was initially at or parallel to the lower edge of the "extent"? If so then:
Set up a coordinate system with $(0, 0)$ at the lower left corner of the "extent", the $x$-axis along the lower edge, and the y-axis along the left edge. Let the $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Which pair of ratios form a proportion? My daughter is stuck on this question and we're not sure what the answer is. I'm rusty with math skills, so I don't understand how to help her, or what approaches she's tried. Please help us!
10.5/12 and 2/5
13/7 and 7/13
9/30 and 1.5/5
7/5 and 10/8
| Two numbers $a,b$ are proportional when there is an integer $A \neq 0$ and such that $a\cdot A = b$ or $ a = A \cdot b$.
You can check that the only pair of numbers that you have listed satysfying the above requirement are $\frac{9}{30}$ and $\frac{1.5}{5}$ because $\frac{9}{30} = 1 \cdot \frac{1.5}{5}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Question related to elliptical angles Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ be an ellipse and $AB$ be a chord. Elliptical angle of A is $\alpha$ and elliptical angle of B is $\beta$. AB chord cuts the major axis at a point C. Distance of C from center of ellipse is $d$. Then the value of $\tan \frac{\alpha}{2}\tan \... | Notice, we have the coordinates of the points $A$ & $B$ as follows $$A\equiv(a\cos\alpha, b\sin \alpha)$$
$$B\equiv(a\cos\beta, b\sin \beta)$$ Hence, the equation of the chord AB
$$y-b\sin\beta=\frac{b\sin\alpha-b\sin \beta}{a\cos\alpha-a\cos \beta}(x-a\cos \beta)$$
$$y-b\sin\beta=\frac{b}{a}\frac{\sin\alpha-\sin \beta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
maximum value of $a+b+c$ given $a^2+b^2+c^2=48$? How can i get maximum value of this $a+b+c$ given $a^2+b^2+c^2=48$ by not using AM,GM and lagrange multipliers .
| The Cauchy-Schwarz-inequality yields
$$|a + b + c|^2 \le (a^2 + b^2 + c^2)\cdot 3 = 144$$
and therefore $a + b + c \le 12$. Plugging in $a = b = c = 4$ shows that this value is actually the maximum.
Alternatively, you could use the convexity of the function $x \mapsto \sqrt{x}$. By Jensen's inequality we have
$$a + b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
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Solving Second Order PDE with Dirac Delta I want to find the functional form of the Green function G(x,t) for a parabolic differential equation:
$$ \frac{\partial{}G(x,t)}{\partial{}t}=a\frac{\partial{}^2G(x,t)}{\partial{}x^2}+\delta(t)\delta(x)$$
Then, I'd like to write the general solution of the heat equation:
$$\fr... | You can separate the dimensions. You start by trying to find a function $G(x,t)$ such that
*
*$G(x,t)$ satisfies the homogeneous heat equation in all of space and time, except formally at $x=t=0$;
*At $t=0$, $G(x,t)$ is zero forall non-zero values of $x$;
*For any nonzero real $(a,b)$, $\int_{-|a|}^{|b} G(x,0) d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
$\lim _{ x\to 1 }{ \frac { \sqrt { x } +\sqrt [ 3 ]{ x } -2 }{ x-1 } } $ I have been trying to algebraically solve this limit problem without using L'Hospital's rule but was whatsoever unsuccessful:
$$\lim _{ x\to 1 }{ \frac { \sqrt { x } +\sqrt [ 3 ]{ x } -2 }{ x-1 } } $$
Thanks in advance!
| Or without using L'Hospital, you can do this:
$$
\lim _{ x\rightarrow 1 }{ \frac { \sqrt { x } +\sqrt [ 3 ]{ x } -2 }{ x-1 } } \quad =\quad \lim _{ x\rightarrow 1 }{ \frac { \sqrt { x } -1+\sqrt [ 3 ]{ x } -1 }{ x-1 } } \\ \qquad \qquad \qquad \qquad \qquad \quad =\quad \quad \lim _{ x\rightarrow 1 }{ \frac { \sqrt {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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some basic cardinal arithmetic on $\text{cf}(\aleph_{\omega_1})$ I'm reading The Joy of Sets by K. Devlin, by self-study. I've just seen a statement $\text{cf}(\aleph_{\omega_1})=\omega_1$ without proof, but I think this is slightly harder to prove than more obvious one, $\text{cf}(\aleph_{\omega})=\omega$.
Concretely... | This is a consequence of a more general fact:
Suppose $\gamma$ is a limit ordinal and I have a sequence of ordinals $\langle \alpha_{\eta}:\eta<\theta\rangle$, with each $\alpha_\eta<\omega_\gamma$. Then this induces a sequence of ordinals $\langle \beta_\eta: \eta<\theta\rangle$, each $<\gamma$, as follows: $\beta_\et... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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Cohomology of Grassmannian of 2-planes in $\mathbb C^4$ The cohomology of the Grassmannian of 2-planes in $\mathbb C^4$ can be deduced from computations in section 4.D of Hatcher's book, using the Leray-Hirsch theorem for fiber bundles.
However, I was told this specific case is not that hard, because this Grassmannian ... | This is a slight adaptation of the construction for real Grassmannians in chapter 6 of Milnor and Stasheff's Characteristic Classes. In particular, the cohomology is trivial to compute, since cells appear only in even dimensions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Paul Garret's Proof of the Cayley-Hamilton Theorem I am trying to understand the proof of Cayley-Hamilton Theorem given in Paul Garrett's notes.
We have a finite dimensional vector space $V$ over a field $k$ and are given a linear operator $T\in \mathcal L(V)$.
The proof on pg 431 in the above link starts out as:
The m... | $V$ is a $k[T]$-module.
$V \otimes $ Something is a ($k[T] \otimes$ Something) - module.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are some usual norms for matrices? I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix?
If so, that would that imply matrices also form some sort of vector space?
| Let $A=(a_{i,j})\in\mathcal M_{n,m}(\Bbb C)$. These some norms on $\mathcal M_{n,m}(\Bbb C)$ which are equivalent
*
*$$\|A\|=\sum_{i,j}|a_{i,j}|$$
*$$\|A\|^2=\sum_{i,j}|a_{i,j}|^2$$
*$$\|A\|=\max_{i}\sum_{j}|a_{i,j}|$$
*$$\|A\|=\max_{j}\sum_{i}|a_{i,j}|$$
and of course we can see an analogy with the norms define... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Is the chance of a variable also a parameter for a probability distribution? I'm new to statistics and I'm a bit confused about the concepts of 'chance of a variable' and 'parameters of a probability distribition'.
Is chance also a parameter? And if so: can computing the chance of a variable be considered as an estimat... | Maybe you are new enough to statistics that you've only thought about normal and binomial distributions.
BINOMIAL. A binomial random variable $X$ counts the number of Successes in a simple experiment. Perhaps $X$ is the number of times you get a 6 when you roll a die twice. There are two parameters of the distribution... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solve the equation $x(\log \log k - \log x) = \log k$ I want to solve this equation by expressing $x$ in function of $k$. Is it possible?
Thanks.
| differentiating both sides with respect to x and taking $$loglogk = p$$
$$d(px)/dx - d(xlogx)/dx = 0$$
$$p - x*d(logx)/dx - logx = 0$$
$$p - 1 - logx = 0$$
$$logx = p - 1$$
$$x = e^p/e$$
$$x = (logk)/e$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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AP in Chessboard The natural numbers $1,2,...,n^2$ are arranged in a $nXn$ chessboard. In how many ways can we arrange the numbers such that the numbers on every row and every column are in arithmetic progression?
I know that $1$ has to be put in one of the corners. I have also observed that the given criteria is maint... | If $n \ge 2$ the $8$ are the only possibilities. First prove that the condition of arithmetic progressions means that if cell $(0,0)$ is the top-left, which must contain $1$, and cell $(1,0)$ contains $1+a$ and cell $(0,1)$ contains $1+b$ then cell $(x,y)$ must contain $1+ax+by$ for any positive integers $x,y < n$. Cle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I compute the gradient vector of pixels in an image? I'm trying to find the curvature of the features in an image and I was advised to calculate the gradient vector of pixels. So if the matrix below are the values from a grayscale image, how would I go about calculating the gradient vector for the pixel with the... | Suppose the image is continuous and differentiable in $x$ and $y$. Then $I(x,y)$ is the value of the pixel at each $(x,y)$, i.e. $I: \mathbb{R}^2 \mapsto \mathbb{R}$. Recall that the gradient at a point $(u,v)$ is:
$$
\nabla I(u,v) = \begin{bmatrix} \frac{\partial I}{\partial x}(u,v) \\ \frac{\partial I}{\partial y}(u... | {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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Alternative area of a triangle formula The problem is as follows:
There is a triangle $ABC$ and I need to show that it's area is: $$\frac{1}{2} c^2 \frac{\sin A \sin B}{\sin (A+B)}$$
Since there is a half in front I decided that base*height is equivalent to $c^2 \frac{\sin A \sin B}{\sin (A+B)}$. So I made an assumptio... | $$\triangle =\dfrac12bc\sin A=\dfrac12(2R\sin B)c\sin A\cdot\dfrac c{2R\sin C}$$
Now $A+B=\pi-C\implies\sin(A+B)=\sin(\pi-C)=?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How can I solve this integral equation by converting it to a differential equation Let we have the following integral equation :$$y(x)=e^{-x}cos(x)-\int_{0}^{x}e^{-x+t}cos(x)y(t)dt$$
How can I solve this integral equation by converting it to a differential equation
| Given $\displaystyle y(x)=e^{-x}\cos(x)-\cos(x) \cdot \int_{0}^{x}e^{-x+t}y(t)dt...........(1)$
Using Differentitation Under Integral Sign.
Given $\displaystyle \frac{d}{dx}\left(y(x)\right) = \frac{d}{dx}\left[e^{-x}\cdot \cos x\right] - \frac{d}{dx}\left[\cos x \cdot \int_{0}^{x}e^{-x+t} \cdot y(t)dt\right]$
We Get $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral of $\frac{1}{x^2+x+1}$ and$\frac{1}{x^2-x+1}$ How to integrate two very similar integrals. I am looking for the simplest approach to that, it cannot be sophisticated too much as level of the textbook this was taken from is not very high. $$\int \frac{1}{x^2+x+1} dx$$ and$$\int \frac{1}{x^2-x+1} dx$$
| For the first one write $x^2 + x + 1 = (x + 1/2)^2 + 3/4$ and the second one $x^2 - x + 1 = (x - 1/2)^2 + 3/4$ these both are arctan
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Which numbers are square modulo 9? How can I prove that n = $1,4,7,9$ for every integer k such that $k^2 = n$ (mod9)?
| You would better state it using $0$ instead of $9$, i.e., $\forall{k}:k^2\equiv0,1,4,7\pmod9$.
There you go:
*
*$k\equiv0\pmod9 \implies k^2\equiv0^2\equiv0\pmod9$
*$k\equiv1\pmod9 \implies k^2\equiv1^2\equiv1\pmod9$
*$k\equiv2\pmod9 \implies k^2\equiv2^2\equiv4\pmod9$
*$k\equiv3\pmod9 \implies k^2\equiv3^2\equi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Coin - Probability explanation Let's say we flip a fair coin $1$ time. The probability of obtaining at least one heads is $50\%$.
However let's say we flip the coin $2$ times. The probability of obtaining at least one heads becomes $75\%$.
I can't seem to wrap my head around why our probability of obtaining at least ... | If you are going to flip the coin twice, you know that if do fail to obtain a head on the first try, there will still be another chance, so how can the probability of obtaining at least one head in two flips be any thing but greater than the probability of obtaining one head on the first flip?
$$\begin{align} & \; \mat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding roots of the polynomial $x^4+x^3+x^2+x+1$ In general, how could one find the roots of a polynomial like $x^4+x^3+x^2+x^1+1$? I need to find the complex roots of this polynomial and show that $\mathbb{Q (\omega)}$ is its splitting field, but I have no idea of how to proceed in this question. Thanks in advance.
| Since $(x-1)(x^4+x^3+x^2+x+1)=x^5-1$, the roots are the fifth roots of $1$, excluding $1$. Note that the set of fifth roots of $1$ is a group of prime order, so it is cyclic and any element is a generator. Thus, if $\omega$ is any of the roots of the polynomials, the full set of roots is given by $\omega,\omega^2,\omeg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that a sequence whose second difference is a nonzero constant is quadratic. For example, if {$a_0, a_1, a_2, a_3, ...$} is the sequence,
the first difference is {$a_1-a_0, a_2-a_1, a_3-a_2, ...$},
and the second difference is {$(a_2-a_1)-(a_1-a_0), (a_3-a_2)-(a_2-a_1), ...$}.
I think that using facts from up to... | Let $d$ denote the constant second difference. Moreover, let
\begin{align*}
c\equiv&\,a_1-a_0-\frac{d}{2},\\
\end{align*}
Claim: $a_n=(d/2)n^2+cn+a_0$ for all $n\in\{0,1,2\ldots\}$.
Proof: The claim is obviously true for $n=0$. For $n=1$, $$\frac{d}{2}\times n^2+cn+a_0=\frac{d}{2}+\left(a_1-a_0-\frac{d}{2}\right)+a_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395094",
"timestamp": "2023-03-29T00:00:00",
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Closed plus finite dimensional in a TVS If $E$ is a topological vector space (TVS), $F_1$ a closed subspace of $E$, and $F_2$ a finite dimensional subspace of $E$, such that $F_1 \cap F_2=\{0\}$, is $F_1+F_2$ necessarily closed? If yes, are the projection from $F_1+F_2$ onto $F_1$ and $F_2$ respectively, continuous?
| Yes If we suppose that $E$ is a normed space , the sum of a closed subspace $Y$ and a finite dimensional space $F$ is closed, in fact :
suppose that a sequence $(y_n+f_n) \subset Y+F$ converge vers $x\in X$, Let $P$ the (continuous Why? ) projection from $Y+F$ to $F$, the sequence $(y_n+f_n)$ is bounded (because it con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the standard definition of vector wrong? The definition of a vector is usually something like "a quantity that has both a magnitude and a direction". But, in the context of, say, economics rather than physics, does this definition make sense? Let's say we are working in 4 dimensional space? Lastly, how does this ... | That is not a very general definition of a vector, no. But it is hardly "wrong" for that reason.
Direction and magnitude are intimately tied to spatial intuition.
The magnitude referred to is usually a "length" measured by a real number, but there are vector spaces where that idea makes no sense.
The same goes for "di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$ $\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$
I tried to solve it.
$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx=\int\limits_{0}^{\pi/2}\frac{4+2\cos x}{(2+\cos x)^2}-\frac{3}{(2+\cos x)^2}dx=\int\limits_{0}^{\pi/2}\frac{2}{2+\cos x}-\frac... | You may observe that: $$\frac{2 \cos (x)+1}{(\cos (x)+2)^2}=\frac{\cos (x)}{\cos (x)+2}+\frac{\sin ^2(x)}{(\cos (x)+2)^2}=\frac{\frac{d}{dx}\sin(x)}{\cos (x)+2}-\frac{\sin(x)\frac{d}{dx}(\cos x +2)}{(\cos (x)+2)^2}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $W$ is a symplectic matrix i.e. $W^T J W=J$ Good day,
This is my first question, I hope all information is given. If not, feel free to ask.
Currently I am reading "New Trends for Hamiltonian Systems and Celestial Mechanics" by E. A. Lacomba & J. Libre, 1994. On page 325 it says the following:
$\rightarrow$ Le... | Preliminaries: Let us see how the product of two complex $n\times n$ matrices $A$ and $B$ looks like in terms of their real and imaginary parts
$$
A=A_r+iA_i,\quad B=B_r+iB_i\quad\Rightarrow\quad AB=A_rB_r-A_iB_i+i(A_rB_i+A_iB_r).
$$
It means that the complex multiplication for matrices works the same as for numbers as... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does $\cos (\pi/5)$ belong to $\mathbb{Q} (\sin(\pi/5))$? I need to know if $$\cos(\pi/5) \in \mathbb{Q} (\sin(\pi/5))?$$
I can compute explicitly such $\cos$ and $\sin$, but I have some difficulties how to deduce from this an answer.
| Hint : show that $c=\frac{3}{2}-2s^2$, where $c=\cos(\frac{\pi}{5})$
and $s=\sin(\frac{\pi}{5})$. You can check it using Euler's identities for example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395622",
"timestamp": "2023-03-29T00:00:00",
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Given any base for a second countable space, is every open set the countable union of basic open sets? Second countability implies the existence of a countable topological base.
Does it also imply that any open set in a second countable space itself is a countable union of basic open sets?
I mean, in a second countable... | Given any second-countable space $X$ and any base $\mathcal B$ for $X$, every open $U \subseteq X$ is the union of a countable subfamily of $\mathcal B$.
First fix a countable base $\mathcal D$ for $X$, and let $U \subseteq X$ be open.
Note that there is a $\mathcal B_U \subseteq \mathcal B$ such that $U = \bigcup \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Area of shaded region circle help
Find the area of the shaded region
Area of the sector is $240^\circ$ or $\frac{4\pi}{3}$
Next find $\frac{b\cdot h}{2}$ which is $\frac{2\cdot2}{2}$ which is $2$.
Then subtract the former from the latter: $\frac{4\pi}{3} - 2$
Therefore the answer is $~2.189$?
Is this correct?
| Here, aperture angle $\theta=120^\circ=\frac{2\pi}{3}$
Area of shaded portion $$=\text{(area of the sector)}-\text{(area of isosceles triangle)}$$
$$=\frac{1}{2}(\theta)(r^2)-\frac{1}{2}(r^2)\sin\theta$$
$$=\frac{1}{2}\frac{2\pi}{3}(2)^2-\frac{1}{2}(2)^2\sin\frac{2\pi}{3}$$
$$=2.456739397$$
Edit:
Note: Area of an iso... | {
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"timestamp": "2023-03-29T00:00:00",
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What function looks like $\overbrace{}$? Here's my awesome drawing:
Basically it's a function that takes a high (or infinite) value at $0$, then falls off logarithmically for a while before falling off exponentially.
It doesn't need to be symmetric, it would be ok if the negative $x$ values were
reversed in sign or ... | How about $\frac{x^2-1}{x^2(x-2)(x+2)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Step functions on $[a,b]$ are dense in $\mathcal C^0([a,b])$.
Let $\|f \|=\sup_{[a,b]}|f|$. We consider ($\mathcal C^0([a,b]),\|\cdot \|)$ and $(\mathcal E([a,b]),\|\cdot \|)$ where $\mathcal E([a,b])$ is a set of the step functions on $[a,b]$. I have to show that $\mathcal E([a,b])$ is dense in $\mathcal C^0([a,b])$.... | Yes. Uniform continuity means that you can use a finite number of balls to cover the domain, and in each of those balls, the values of the functions are $\varepsilon$-close. Hence, you can build a step function that is $\varepsilon$-close to the given function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
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Integration of $\sin(\theta)$ I hope I'm not asking a silly question.
We can integrate $\sin(\theta)$ simply by the following identity:
$$\int_0^\frac{\pi}{2} \sin\theta\ \mathsf d\theta = \left[-\cos\theta \vphantom{\frac 1 1} \right]_0^\frac{\pi}{2}=1.$$
But how can we do this by summation ?
For example, $$\int_0^{... | $$\cdots\approx\frac{\pi}{ 200}\sum_{k=1}^{100}\sin\left(\frac{k}{100}\right)$$
In general:
if $b-a$ small,
$$\int_a^b f(x)dx\approx\frac{b-a}{100}\sum_{k=1}^{100}f\left(a+k\frac{(b-a)}{100}\right)$$
since $$\frac{b-a}{n}\sum_{k=1}^nf\left(a+k\frac{b-a}{n}\right)\underset{n\to\infty }{\longrightarrow} \int_a^b f(x)dx.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Composition of homotopy classes with self-maps of spheres Are there some general rules/formulas on the relation between the homotopy class $[f]\in \pi_i(S^n)$ and the homotopy class of the
composition
$S^i\stackrel{a}{\to} S^i\stackrel{f}{\to}S^n\stackrel{b}{\to}S^n$
where $a,b$ are maps of degree $d_a,d_b$ respective... | (this is not a full answer, but for comments it's too long)
For given (smooth) map $f:S^3\to S^2$ if you take two non-critical values $p,q\in S^2$, then linking number of $f^{-1}(p)$ with $f^{-1}(q)$ equals to degree of $f$. Maybe, something similar occurs in case $i=2n-1$ for all $n$.
And when you take a suspension of... | {
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Correctly calculating permutations and combinations without duplicate patterns Given 16 balls each numbered 1 through 16, and 5 glass tubes numbered 1 through 5; how many ways are there to slot all 16 balls into the glass tubes, selected one at a time, with the only condition that each slot should always have at least ... | We assume that order of balls in the tubes matters.
Line up the $16$ balls in some order. There are $16!$ ways to do this. There are $15$ interball gaps. We choose $4$ of them to place a separator into in the usual Stars and Bars style. This can be done in $\binom{15}{4}$ ways.
Thus the total number of ways is $16!\b... | {
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"url": "https://math.stackexchange.com/questions/1396273",
"timestamp": "2023-03-29T00:00:00",
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Fraction field of $F[X,Y](f)$ isomorphic to $F(X)[Y]/(f)$ Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$.
Then, by Gauss's lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field (where $(f)$ is the ideal generated by $f$).
I'm looking for a s... |
Let $A$ be a UFD, and $f\in A[Y]$ irreducible with $\deg f\ge 1$. Then the field of fractions of $A[Y]/(f)$ is $K[Y]/(f)$, where $K$ is the field of fractions of $A$.
Set $S=A-\{0\}$. Then $K[Y]/(f)=S^{-1}A[Y]/S^{-1}(f)\simeq S^{-1}(A[Y]/(f))$, so $K[Y]/(f)$ is a ring of fractions of the integral domain $A[Y]/(f)$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Angle between segments resting against a circle Motivation:
A couple of days ago, when I was solving this question, I had to consider a configuration like this
Now, I didn't intentionally make those two yellow bars stand at what appears to be a $90^{\circ}$-angle but it struck me as an interesting situation, so much s... | Set Cartesian coordinates on the plane so that the center of the circle
is $(0,0)$ and the black line $\overline{AB}$ is parallel to the $x$-axis.
The coordinates of $A$ are $(-(a - b), c)$ and the distance $OA$ is
$\sqrt{(a - b)^2 + c^2}$. Therefore
\begin{align}
\angle OAB & = \arcsin\left( \frac{\sqrt{(a - b)^2 + c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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two variable nonhomogeneous inequality Let $$x\ge 0,y\ge 0,x\neq 1,y \neq 1$$Prove the inequality
$$\dfrac {x}{(y-1)^2} +\dfrac {y}{(x-1)^2} \ge \dfrac {x+y-1}{(x-1)(y-1)} $$
| hint: The endpoints case you can handle with ease, for more general case that: $x, y > 1\to x(x-1)^2 +y(y-1)^2 \geq (x+y-1)(xy-(x+y-1))\iff x(x^2-2x+1)+y(y^2-2y+1)\geq xy(x+y-1)-(x+y-1)^2\iff (x^3+y^3)-2(x^2+y^2)+(x+y)\geq xy(x+y-1)-(x^2+y^2+1+2xy-2x-2y)\iff f(x,y)=x^3+y^3-(x^2+y^2)+1+3xy-(x+y) -xy(x+y)\geq 0$. Taking ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Compute $\int_M \omega$
Let $M=\{(x,y,z): z=x^2+y^2, z<1\}$ be a smooth 2-manifold in $\Bbb{R}^3$. Let $\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in \Omega^2(\Bbb{R}^3)$. Compute $$\int_M \omega.$$
I parametrised $M$ (up to a null-set) with $g(r,\theta)=(r\cos \theta, r\sin \theta,r)$ where $(r,\theta)\in (0,1)\t... | If $f(x,y)=(x,y,x^2+y^2)$ then $$ \int_M \omega = \int_{D} f^\ast \omega $$ where $D$ is a unit disk. Hence $$ \int_D (-x^2-y^2)dxdy = \int_D (-r^2) rdrd\theta = 2\pi \frac{-r^4}{4}\bigg|_0^1 = -\frac{\pi}{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Get the right "upper" Sequence I want to find out the limit for the sequence $\frac{n!}{n^n}$
using the squeeze theorem.
My idea was:
$\frac{n}{n^n} \leq \frac{n!}{n^n} \leq \frac{n!+1}{n^n} $
So the limits of the smaller sequence and the bigger sequence are 0, thus
the limit of the original sequence is 0.
But is the ... | $$\frac {n!}{n^n} =\frac {n \times (n-1)\times \dots \times 2 \times 1}{n^n}\le \frac {\overbrace {n\times n \times \dots \times n}^{{n-1}\text {times}}\times 1}{n^n}=\frac {n^{n-1}}{n^n}=\frac1n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Using the $\ln(\cdot)$ for $(1-e^{-x})$ The given function:
$$B= A(1-(e^{-x}))$$
Now, I want to 'destroy' the e-function by taking the logarithm of it.
First, since
$\ln(ab) = \ln(a) + \ln(b)$ we get that $\ln(b) = \ln(a) + \ln(1-e^{-x})$.
Using $\ln(1) =0$, $\ln(b) = \ln(A) +x$.
Is this the right way to handle the f... | No it is not true that $\ln(a+b)=\ln a+\ln b$ which you used to split up $\ln(1-e^{-x})$
A better approach is to first solve for the $e^{-x}$
$$e^{-x} = 1-\frac{B}{A}$$
Now take the logarithm.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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If $t=\tanh\frac{x}{2}$, prove that $\sinh x = \frac{2t}{1-t^2}$ and $\cosh x = \frac{1+t^2}{1-t^2}$.
If $t=\tanh\frac{x}{2}$, prove that $\sinh x = \frac{2t}{1-t^2}$ and $\cosh x = \frac{1+t^2}{1-t^2}$. Hence solve the equation $7\sinh x + 20 \cosh x = 24$.
I have tried starting by writing out $\tanh\frac{x}{2}$ in ... | Note that
(i) tanh($x$) = sinh(x) / cosh($x$)
(ii) 1 - tanh$^2(x$) = 1/ cosh$^2(x$)
(iii) sinh($x + y$) = sinh($x$)cosh($y$) + sinh($y$)cosh($x$)
and apply these to the term
$\frac{2t}{1 - t^2}$ with $t =$ tan($x/2$) to directly obtain your result for sinh($x$). The formula for cosh$(x)$ follows then directly by applyi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proofs that Dirichlet's function is not differentiable Define $f: (0,1) \to (0,1)$ by
$f(x)=
\begin{cases}
\frac{1}{q}, & \text{if $x=\frac{p}{q}$ in lowest terms with $p,q \in \mathbb{N}$} \\
0, & \text{if $x$ is irrational}
\end{cases}
$
The version of this proof I found from Spivak's Calculus is, for irrational $a... | Fix an irrational $x\in(0,1)$. Suppose $f'(x)$ exists; then $f'(x)=0$. For each prime $q$, pick $r_q$ to be a multiple of $1/q$ satisfying $|x-r_q|\leq 1/q$. Then $|f(x)-f(r_q)|/|x-r_q|\geq 1$. So $|f'(x)|\geq 1$,a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration by Parts Question: Integrate $x^3e^x$ Evaluate $$\int x^3e^x \mathrm{d}x$$
I tried to use integration by parts to do this and I let $u = x^3$ and $\mathrm{d}v = e^x \mathrm{d}x$. So I get $$\int x^3e^x \mathrm{d}x= \int x^3e^x \mathrm{d}x- \int e^x \cdot 3x^2\mathrm{d}x$$
How do I do this if my answer has t... | $$x^n\int e^x\ dx\ne\int x^ne^x\ dx$$
$$I_n=\int x^ne^x\ dx=x^n\int e^x\ dx-\int\left[\dfrac{d(x^n)}{dx}\int e^x\ dx\right]dx$$
$$\implies I_n=x^ne^x-nI_{n-1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 3
} |
Characterization of uniform integrability of random variables Let $\{X_n \}$ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P)$.
Then, $\{X_n\}$ is uniformly integrable if $$\lim_{M \to \infty} \sup_n \int_{|X_n| > M } |X_n| = 0 \tag{1}$$
From this, I know that $$\displaystyle \sup_n E(|X... | No, the converse is, in general, not true. Just consider $((0,1),\mathcal{B}(0,1))$ endowed with Lebesgue measure and the sequence of random variables $$X_n(x) := 2n \cdot 1_{(0,1/n)}(x).$$ Then $$\mathbb{E}(|X_n|) = 2$$ for all $n \in \mathbb{N}$, but $$\limsup_{M \to \infty} \sup_{n \in \mathbb{N}} \int_{|X_n|>M} |X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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solution of variable coefficient equation Consider the equation $$u_x + yu_y = 0$$
and I know that this PDE has solution $u(x,y) = f(e^{-x}y)$
Can someone help me to derive this PDE to get the solution? Thank you
| Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$
$\dfrac{dy}{dt}=y$ , letting $y(0)=y_0$ , we have $y=y_0e^t=y_0e^x$
$\dfrac{du}{ds}=0$ , letting $u(0)=f(y_0)$ , we have $u(x,y)=f(y_0)=f(e^{-x}y)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Find the number of real roots of $1+x/1!+x^2/2!+x^3/3! + \ldots + x^6/6! =0$.
Find the number of real roots of $1+x/1!+x^2/2!+x^3/3! + \ldots + x^6/6! =0$.
Attempts so far:
Used Descartes signs stuff so possible number of real roots is $6,4,2,0$
tried differentiating the equation $4$ times and got an equation with no... | I think the notion that the fourth derivative having no real roots proves that the polynomial itself has four real roots is your problem. Can you explain your reasoning a bit more? I mean to say, $x^6+x^4+1$ clearly has no real roots (it is everywhere positive), but its fourth derivative $360x^2+24$ has no real roots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 10,
"answer_id": 7
} |
showing an inequality not using stirling formula I don't know how to show that
$$
\frac{k^k}{k!}\leq e^{k}
$$
without using Stirling's approximation. I want to show it directly. I guess I need some inequality to achieve this but I don't know.
| A somewhat different approach:
In the pmf for a Poisson$(k)$ $P(X=k) = \frac{e^{-k} k^k}{k!}$ but $P(X=k)<1$ (since it's a probability).
The result follows by multiplying by $e^k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 3
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Division in Summations Suppose $a_n=\dfrac{2^n}{n(n+2)}$ and $b_n=\dfrac{3^n}{5n+18}$.
I need to find the value of:
$$\displaystyle\sum_{n=1}^{\infty}\dfrac{a_n}{b_n}$$
I think this problem is meant for me to compute each sum differently and then divide. Is this some property of summations that we need to utilize here... | $$\begin{equation}
\begin{split}
\sum_{n=1}^\infty \frac{a_n}{b_n}
&
= \sum_{n=1}^\infty \frac{2^n(5n+18)}{3^nn(n+2)} \\
&
= \sum_{n=1}^\infty \left[ \left(\frac{2}{3}\right)^n \left(\frac{5}{n+2} + 9\left(\frac{1}{n} - \frac{1}{n+2} \right) \right) \right] \\
&
= \sum_{n=1}^\infty \left[ \left(\frac{2}{3}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Jacobson radical of $\mathbb{F}_{2}(t)[x]/(x^4-t^2)$ Let $\mathbb{F}_{2}$ be the field of two elements. Let $R=\mathbb{F}_{2}(t)[x]/(x^4-t^2)$.
Why is $R/J(R)$ equal to $\mathbb{F}_{2}(t)[x]/(t-x^2)$? here $J(R)$ denotes the Jacobson radical of $R$.
| The Jacobson radical of $R$ is, by definition, the intersection of maximal ideals of $R$. In our case $R$ is local with the maximal ideal $(x^2+t)/(x^2+t)^2$. If we want to see this as an $R$-module, then we can conclude that it is isomorphic to $R/(x^2+t)$, but I can't see how is this isomorphic to $\mathbb F_2(t)[x]/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Why does basic algebra provide one value for $x$ when there should be two? I have the equation $x^2=x$.
If I divide $x$ from both sides I get $x=1$.
Yet clearly $x$ can also equal $0$.
What step in this process is wrong? It seems to me that there's only one step. And isn't dividing the same thing from both sides a vali... | The given quadratic equation $x^2=x$ will have two real roots given as follows $$x^2-x=0$$ $$x(x-1)=0$$ $$x=0\ \ \ \ \textrm{or}\ \ \ x-1=0$$
$$x=0\ \ \ \ \textrm{or}\ \ \ x=1$$ Hence, we get $x=0$ or $x=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Prove that if $A$, $B$ are countable, then $A \times B$ is countable? Is $A\times B$ referring to the axis here? So an $X$ and $Y$ coordinate plane?
$A$ is countable, therefore a bijection occurs from $A \rightarrow \mathbb{N}$.
$B$ is countable, therefore a bijection occurs from $B \rightarrow \mathbb{N}$.
If these st... | Let $f:A \to \mathbb{N}$ be a bijection from $A$ to $\mathbb{N}$
Let $g:B \to \mathbb{N}$ be a bijection from $B$ to $\mathbb{N}$
Define $h:A \times B \to \mathbb{N}$ as follows:
$h(x,y)=2^{f(x)}\cdot3^{g(y)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this proof about the countability of $\Bbb Q \times \Bbb Q \times \cdots \times \Bbb Q$ sound? If $\Bbb{Q}$ is countable, prove that the set $\Bbb{Q}^n$ for $n = 2,3,...$ is countable.
Base case: $n = 2 \rightarrow \Bbb{Q}^2 = \Bbb{Q}\times\Bbb{Q}$ which, by Proposition 4.5 (see bottom of question), is countable.
As... | As you wrote in a comment, the proposition you are trying to use doesn't apply directly to $\Bbb Q^k\times \Bbb Q$.
However, if we know that $D$ is countable and $E$ is some other countable set, then there is a bijection $f: D \to E$, and the map from $D \times D$ to $D \times E$ given by $(x,y) \mapsto (x,f(y))$ is a ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Group $G$ acting on $\Omega$ such that each $\alpha \in \Omega$ has unique $p$-element fixing $\alpha$.
Let $G$ be a group acting on a set $\Omega$ and let $p$ be a prime. Suppose that for each $\alpha \in\Omega$ there is a $p$-element $x \in G$ such that $\alpha$ is the only point fixed by $x$. If $\Omega$ is finite,... | First suppose that $\Omega$ is finite, and so $G/K$ is finite, where $K$ is the kernel of the action, so we can assume that $K=1$ and $G$ is finite.
For any $\alpha \in \Omega$, there is a nontrivial $p$-subgroup $Q(\alpha)$ of $G$ with the unique fixed point $\alpha$. Let $P(\alpha)$ be a Sylow $p$-subgroup of $G$ con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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computing the cubed root of a complex number... I do know how to calculate the cubed root of a complex number....like if I'm given that $x^3=p$, where $p$ is a complex number, then $$x= r^{1/3}\left(\cos\left(\frac{2k\pi+m}{3}\right) + i\sin \left(\frac{2k\pi+m}{3}\right)\right)$$ where $p$ is $r\left(\cos m +i\sin m\r... | HINT: use that $$x^3-a=(x-a^{1/3})(x^2+xa^{1/3}+a^{2/3})$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that $\sum\frac{(\log n)^2}{n^3}$ converges This question is from Serge Lang's textbook, in a chapter that comes before the ratio and integral tests are introduced, so those can't be used. I've already proved that $\sum\frac{\log n}{n^3}$ converges and have an inkling that this result may be useful, but I can't f... | Well then use $\ln(n)^2 = 4\ln(\sqrt{n})^2 < 4\sqrt{n}^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Find a basis for $U\cap V$
Let $$a = (0,2,3,-1)^T \quad b=(0,2,7,-2)^T \quad c = (0,-2,1,0)^T \quad u = (1,2,0,1)^T\quad v = (2,2,1,2)^T$$
Let $U= \langle a,b,c \rangle, V = \langle u,v\rangle$
Then a) find a basis for $U$ and $V$, b) find a basis for $U\cap V$.
I have solved a) and found a basis $\{a,b\}$ for $U$ an... | Now you have two ways of expressing your vector $x$ in terms of a single arbitrary value $r$. Since $x$ was arbitrary any element of the intersection has this form. So use one of the two ways to find a fixed vector $w$ such that your arbitrary $x = rw$ for some $r$. As a check to your work so far, you should get the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Order of a corrector-predictor method Given an explicit method:
$$ x_{i+1} = x_i+ h \Phi(t_i,x_i,h) $$
as predictor method and an implicit method:
$$ x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1},h) $$
as corrector method, it follows that
$$
x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1}^*, h), \quad
(x^*_{i+1} = x_i+\Phi(t_i,x_i,h))... | Pure predictor and pure corrector schemes
$$
\begin{array}{rlrlc}
x(t_{i+1}) &= x(t_i) + h \Phi(t_i, x(t_i), h) + h \tau_p(t_i,h),& \tau_p(h) &= \max_i |\tau_p(t_i,h)| & (1)\\
x(t_{i+1}) &= x(t_i) + h \Psi(t_i, x(t_i), x(t_{i+1}), h) + h \tau_c(t_i,h),& \tau_c(h) &= \max_i |\tau_c(t_i,h)| & (2)
\end{array}
$$
And the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Determining if a map on a space is continuous by checking on a dense subset I recently read a proof of something my calculus teacher had told me, namely that the set of continuous maps $f: \mathbb{R} \to \mathbb{R}$ had the cardinality of $\mathbb{R}$. The proof was simple enough: There must be at least that many, as i... | It is necessary and sufficient that for any Cauchy sequence $\{x_n\}$ of rational numbers, the sequence of $\{f(x_n)\}$ is also Cauchy.
Since the real numbers are the completion of the rational numbers, if a sequence in the rational numbers converges (i.e., is Cauchy), then it converges to a real number, and you need t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Derivative of integral $\int_0^{\infty} e^{-x \cosh t} dt$ I am given the integral $y = \int_0^{\infty}e^{-x\cosh t} dt$ and wish to show that this integral solves the modified Bessell equation: $x^2y'' + xy' -x^2y = 0.$
To do this I need to calculate the derivatives $y'$ and $y''$. I suspect integration by parts is ne... | You can check the equation as follows:
\begin{align*}
x^2(y'' - y)
&= x \int_{0}^{\infty} x (\cosh^2 t - 1) e^{-x \cosh t} \, dt \\
&= x \int_{0}^{\infty} x \sinh^2 t \, e^{-x \cosh t} \, dt \\
&= x \left[ -\sinh t \, e^{-x \cosh t} \right]_{0}^{\infty} + x \int_{0}^{\infty} \cosh t \, e^{-x\cosh t} \, dt \\
&= -xy'.
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$\int_{0}^{\infty}\frac{\ln x dx}{x^2+2x+2}$ $$\int_{0}^{\infty}\frac{\ln x .dx}{x^2+2x+2}$$
$$\int_{0}^{\infty}\frac{\ln x .dx}{x^2+2x+2}=\int_{0}^{\infty}\frac{\ln x .dx}{(x+1)^2+1}\\
=\ln x\int_{0}^{\infty}\frac{1}{(x+1)^2+1}-\int_{0}^{\infty}\frac{1}{x}\frac{1}{(x+1)^2+1}dx$$ and then lost track,answer is $\frac{\p... | $\bf{Another\; Solution::}$ Let $$\displaystyle I = \int_{0}^{\infty}\frac{\ln x}{x^2+2x+2}dx.$$
Put $\displaystyle x = \frac{2}{t}\;,$ Then $\displaystyle dx = -\frac{2}{t^2}dt$ and Changing Limits, We get
$$\displaystyle I = \int_{\infty}^{0}\frac{\ln\left(\frac{2}{t}\right)}{\frac{4}{t^2}+\frac{4}{t}+2}\cdot -\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Evaluate $\int e^x \sin^2 x \mathrm{d}x$ Is the following evaluation of correct?
\begin{align*} \int e^x \sin^2 x \mathrm{d}x &= e^x \sin^2 x -2\int e^x \sin x \cos x \mathrm{d}x \\ &= e^x \sin^2 x -2e^x \sin x \cos x + 2 \int e^x (\cos^2 x - \sin^2x) \mathrm{d}x \\ &= e^x \sin^2 x -2e^x \sin x \cos x + 2 \int e^x (1 -... | $$\int \left(e^x\sin^2(x)\right)\text{d}x =$$
$$\int \left(e^x\left(\frac{1}{2}(1-\cos(2x))\right)\right)\text{d}x =$$
$$\frac{1}{2}\int \left(e^x-e^x\cos(2x)\right)\text{d}x =$$
$$\frac{1}{2} \left(\int \left(e^x\right) \text{d}x-\int \left(e^x\cos(2x)\right) \text{d}x\right) =$$
$$\frac{1}{2} \left(\int e^x \text{d}x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1398965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 5
} |
Left cosets of $A_6$ in $S_6$ Which may be the all of left cosets of $A_6$ in $S_6$?
$\{A_6,(156)A_6\},\{A_6,(34)A_6\},\{A_6,(42)(35)A_6\}, \{A_6,(46523)A_6\}$ or $\{A_6\}$
I dont understand why the answer is $\{A_6,(34)A_6\}$ .I know if we multiply $A_6$ elements in $S_6$ with $A_6$ we get $A_6$ itself. why multiplyi... | $A_{6}$ is the group of all even permutations of $6$ elements. If you multiply that set by an even permutation, you get $A_{6}$ back, but if you multiply by any odd permutation you get every odd permutation (you need to prove that every distinct even permutation is sent to a distinct odd permutation). So the problem is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
For which $n \in \mathbb{N}$ $f(x) = x^{2n}+x^n+1$ is irreducible in $\mathbb{F}_2[x]$? I have $$f(x) = x^{2n}+x^n+1 \in \mathbb{F}_2[x].$$ When is this polynomial irreducible? It is obvious that for even $n$ this polynomial is reducible.
But I don't have any idea about odd $n$.
| Before we delve into $\Bbb{F}_2[x]$ let's consider the irreducibility of $f(x)$ in $\Bbb{Z}[x]$.
Because
$$
f(x)(x^n-1)=x^{3n}-1,
$$
the zeros of $f$ are roots of unity of order dividing $3n$.This makes us consider the cyclotomic polynomial $\Phi_{3n}(x)\in \Bbb{Z}[x]$. Its degree is given by the Euler (totient) funct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Understanding the Group Structure A Group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions
*
*Closure
*Associativity
*Existence of Identity
*Existence of Inverse
Intuitively I under... | Groups abstract symmetries of a set, that is, bijections $X \to X$ of a set $X$:
*
*Function composition is associative.
*Every bijection has an inverse, which is also a bijection.
*The identity map is a bijection.
Closure just allows us to consider sets of bijections that are smaller than the full group of all bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Can we make a subgroup of a group by selecting exactly one element from each distinct left cosets of a subgroup of the given group? Let $G$ be a group and $H$ be a subgroup of $G$ ; can we select exactly one element from each distinct left coset of $H$ such that the set of all those elements form a subgroup of $G$ ? Ho... | If $H$ is a retract of $G$ or a kernel of a retraction of $G$ then it is possible (in other words: if $H$ is a semidirect factor of $G$). But I am not sure if that exhausts all possibilities.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Theorem 3.22 from baby Rudin $\sum a_n$ converges if and only if for every $\varepsilon >0$ there is an integer $N$ such that $$\left|\sum_{k=n}^{m}a_k\right|\leqslant \varepsilon$$ if $m\geqslant n\geqslant N$.
In particular, by taking $m=n$ above inequality becomes $$|a_n|\leqslant \varepsilon \quad(n\geqslant N).$$... | The mistake is that
$$
\sum_{k=m}^m a_k = a_m
$$
and you are saying nothing about the series $\sum_k a_k$. You are only looking at the single term $a_m$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$ Problem statement: Find the derivative of
$$f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy$$
and find an ordinary differential equation that $f$ solves. Find the solution to this ordinary differential equation to determine an ... | Using Differentiation Under the Integral Sign, we get
$$
\begin{align}
f(x)&=\int_{-\infty}^\infty\frac{e^{-xy^2}}{1+y^2}\,\mathrm{d}y\tag{1}\\
f'(x)&=\int_{-\infty}^\infty\frac{-y^2e^{-xy^2}}{1+y^2}\,\mathrm{d}y\tag{2}\\
f(x)-f'(x)&=\int_{-\infty}^\infty e^{-xy^2}\,\mathrm{d}y\tag{3}\\
&=\sqrt{\frac\pi{x}}\tag{4}
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Trig limit in Spivak's Calculus $$\lim_{x\rightarrow 1} (x-1)^3 \sin\frac{1}{(1-x)^3} = 0$$
To prove that this is true, the chapter on limits has things like $\lim_{x\rightarrow a}(f\cdot g)(x) = \lim_{x\rightarrow a}f(x)\cdot \lim_{x\rightarrow a} g(x)$, when both limits exist. Now, in the case of the problem, $\lim_{... | $$
\underbrace{-|x-1|^3}_A \quad \le \quad \underbrace{|x-1|^3\cdot\sin(\cdots)}_B \quad \le \quad \underbrace{|x-1|^3}_C.
$$
You don't need to know the nature of the function of $x$ inside the sine function in order to deduce the inequalities $A\le B\le C$. If $A$ and $C$ both approach $0$, then so does $B$. Most ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why do we not have to prove definitions? I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? It seems like some definitions should have a foundation based on proof. How si... | Frequently, a definition is given, and then an example or proof follows to show that whatever has been defined actually exists. Some authors will also attempt to motivate a definition before they give it: for example, by studying the symmetries of triangles and squares and how those symmetries are related to each other... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "71",
"answer_count": 12,
"answer_id": 2
} |
Find all integers such that $2 < x < 2014$ and $2015|(x^2-x)$ Find all integers, $x$, such that $2 < x < 2014$ and $2015|(x^2-x)$.
I factored it and now I know that $x > 45$ and I have found one solution so far: $(156)(155)= (2015)(12)$. It's just that I don't think I'm approaching it the right way.
I'm new to this kin... | You need a couple of pieces of information to complete this.
First of all, a property of prime numbers: If $p$ is a prime number such that $p ~|~ ab$, then $p~|~a$ or $p~|~b$. Also $2015=5 \cdot 13\cdot 31$.
Now, if $2015~|~N$, then $5~|~N$. Since $2015~|~x(x-1)$, $5~|~x$ or $5~|~(x-1)$; thus $x\equiv0\pmod5$ or $x\equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
A k-lipschitz function Let $f:M\to \mathbb{R}$ a k-lipschitz function, i.e, $\vert f(x)-f(y)\vert\leq kd(x,y)$, for any $x,y\in M$. Show that $f(x)=\displaystyle \inf_{y\in M}[f(y)+kd(x,y)]=\displaystyle\sup_{y\in M}[f(y)-k d(x,y)]$, for all $x\in M$.
Any hint pls!. Regards
| Hint: what is $f\left(y\right)+kd\left(x,y\right)$ at $y=x$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Dimension of quotient ring
What is the dimension of the following quotient ring, $\mathbb{Z}[x,y,z]/\langle xy+2, z+4 \rangle$, where $\mathbb{Z}$ is the ring of integers?
I realized this is isomorphic to $\mathbb{Z}[x,-2/x]$. How does $-2/x$ affect the dimension since the ring is $\mathbb{Z}$.
| Let $R=\mathbb Z[X,Y]/(XY+2)$. We have $\dim R\le2$. Furthermore, since $x$ is a non-zero divisor on $R$ we have $\dim R\ge\dim R/(x)+1=2$. (Note that $R/(x)\simeq(\mathbb Z/2\mathbb Z)[Y]$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate $\iint{f d\mu dv}$ and $\iint{f dv d\mu}$ The purpose of this problem is to show that in Fubini-Tonelli theorem, the condition $f \in L^{+}(X \times Y)$ or $f \in L^1$ is necessary. Here is the problem:
Let $X = Y = \mathbb{N}$, $\mathcal{M} = \mathcal{N} = \mathcal{P}(\mathbb{N})$, $\mu = v =$ counting mea... | We have that $\iint{fd\mu dv}=\int\left(\int f d\mu\right) dv$. And for each fixed $n$, $f(m,n)$ has only two nonzero values which are of equal measure. Thus for each fixed $n\in Y$:
$$\int_{m\in X} f(m,n) d\mu=f(n,n)+f(n+1,n)+0=1-1=0.$$
So we get $\iint{fd\mu dv}=0$. And for each fixed $m\in X$ for $m\neq 1$:
$$\int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Confusion with the eccentricity of ellipse Confusion with the eccentricity of ellipse. On wikipedia I got the following in the directrix section of ellipse.
Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right, in which the ellipse ... | Hint: If the eccentricity $e$ & the major axis $2a$ of an ellipse are known then we have the following
*
*Distance of each focus from the center of ellipse
$$=\text{(semi-major axis)}\times \text{(eccentricity of ellipse)}=\color{red}{ae}$$
*Distance of each directrix from the center of ellipse
$$=\frac{\text{semi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Showing 2 vector spaces are isomorphic. I am trying to understand how to show two vector spaces are isomorphic. You do this by showing there is an isomorphism that can be mapped between the two spaces.
What I don't understand is my lecturer's way of showing that the isomorphism is one-to-one?
It seems to me that he o... | In general, showing a map $f:A \to B$ is one-to-one involves showing that for any $x,y \in A$ such that $x \ne y$, we have $f(x) \ne f(y)$. However, an isomorphism between vector spaces is by definition a linear map, and your lecturer should have shown (maybe much earlier) that a linear map is one-to-one if and only if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that Cauchy's function is infinitely differentiable Show that
$$f(x)=
\begin{cases}
exp(-\frac{1}{x^2}), & \text{if $x\gt 0$} \\[2ex]
0, & \text{if $x\le 0$ }
\end{cases}$$
is infinitely differentiable.
Clearly $f^{(n)}(x)=0$ for all $x\lt 0$ and $f_{-}^{(n)}(0)=0$. Also since the derivatives of $exp(-1/x^2)$ pr... | Hint: Use induction and l'Hopital. Consider $\lim_{x\rightarrow 0^+}\frac{e^{-1/x^2}}{x^2}$. Let $y=1/x$, so this becomes $\lim_{y\rightarrow\infty}\frac{e^{-y^2}}{y^{-2}}=\lim_{y\rightarrow\infty}\frac{y^2}{e^{y^2}}$. After one step of l'Hopital, you have your answer. This approach can be generalized for any $n\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is a usual order relation? I've just started learning about relations and now I'm at partial order relations and total order relations; essentially, I'm trying to convey that I'm very much a beginner to this relations stuff.
My textbook includes the following remark:
The usual order relation on the real line $ \... | The usual ordering on $\Bbb{R}$ can be uniquely defined as the total order $\lt$ on $\Bbb{R}$ that satisfies the following two properties:
(Let $a,b,c \in \Bbb{R}$)
1) $c \gt 0 \iff c$ is positive.
2) $a \lt b, \ \ 0 \lt c \implies ac \lt bc$
Proof that this defines uniquely $\lt$: Suppose that $\lt'$ also satisfies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Given a graph G, if X(G) = k, and G is not complete, must we have a k-colouring with two vertices distance 2 that have the same colour? As the title asks, If given a graph, $G$, with chromatic number $k$, and $G$ is not complete must there exist a $k$-colouring of $G$, $f$, where there are two vertices $x,y$ such that ... | I'll give an alternative proof to Leen's.
Suppose we have a graph $G$ where $\chi(G) =k$ and $G$ is not complete. We will proceed by showing that using kempe chains we can always get two vertices distance $2$ with the same colour.
Now take any $k$-colouring of $G$, $\alpha$. Since $G$ is not complete, there is at least... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What method of numerical integration is this? I am trying to update some old code that finds the area under a curve from $17$ evenly spaced discrete data points. I'd like to update it to calculate from $65$ data points. I'd like to use the same methodology, so I'm trying to determine the method used to approximate it. ... | This appears to be the composite Simpson rule: https://en.wikipedia.org/wiki/Simpson%27s_rule#Composite_Simpson.27s_rule
A summary: start with a partition of $[a,b]$ into $N$ subintervals of equal length, and add a point in the middle of each of the subintervals. Now you have $2N+1$ evaluation points. (Note that this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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