Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Looking for intuition behind coin-flipping pattern expectation I was discussing the following problem with my son:
Suppose we start flipping a (fair) coin, and write down the sequence; for example it might come out HTTHTHHTTTTH.... I am interested in the expected number of flips to obtain a given pattern. For example... | Suppose we have 4-slot queue. By state we mean the longest tail of the coin sequence that matches the pattern $XXXX$ from the left. If there no matching, we denote the state as $\varnothing$. For instance, the state of the sequence $$TTTTHTHHTTTHTH,$$ given the pattern $XXXX = HTHT$, is $HTH$ and the state for the patt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Product of two ideals doesn't equal the intersection The product of two ideals is defined as the set of all finite sums $\sum f_i g_i$, with $f_i$ an element of $I$, and $g_i$ an element of $J$. I'm trying to think of an example in which $IJ$ does not equal $I \cap J$.
I'm thinking of letting $I = 2\mathbb{Z}$, and $... | Maybe it is helpful for you to realise what really happens for ideals in the integers. You probably know that any ideal in $\mathbb Z$ is of the form $(a)$ for $a\in \mathbb Z$, i.e. is generated by one element. The elements in $(a)$ are all integers which are divisible by $a$.
If we are given two ideals $(a)$ and $(b)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Finding the asymptotic limit of an integral. I'm having trouble finding the asymptotic of the integral
$$ \int^{1}_{0} \ln^\lambda \frac{1}{x} dx$$
as $\lambda \rightarrow + \infty$.
Can anyone help?
Thank you!
| Let
$-\log x=u$ then the integral becomes
$$\int\limits_0^1 {{{\left( { - \log x} \right)}^\lambda }dx} = \int\limits_0^{ + \infty } {{e^{ - u}}{u^\lambda }du} $$
This is Euler's famous Gamma function, which has an asymptotic formula by Stirling
$$\int\limits_0^{ + \infty } {{e^{ - u}}{u^\lambda }du} \sim {\left( {\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Convex functions in integral inequality Let $\mu,\sigma>0$ and define the function $f$ as follows:
$$
f(x) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm \exp\left(-\frac{(x-\mu)^2}{2\sigma ^2}\right)
$$
How can I show that
$$
\int\limits_{-\infty}^\infty x\log|x|f(x)\mathrm dx\geq \underbrace{\left(\int\limits_{-\infty}^\infty ... | Below is a probabilistic and somewhat noncomputational proof.
We ignore the restriction to the normal distribution in what follows below. Instead, we consider a mean-zero random variable $Z$ with a distribution symmetric about zero and set $X = \mu + Z$ for $\mu \in \mathbb R$.
Claim: Let $X$ be described as above such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
The number of elements which are squares in a finite field. Meanwhile reading some introductory notes about the projective special linear group $PSL(2,q)$ wherein $q$ is the cardinal number of the field; I saw:
....in a finite field of order $q$, the number of elements ($≠0$) which are squares is $q-1$ if $q$ is even ... | Another way to prove it, way less elegant than Dustan's but perhaps slightly more elementary: let $$a_1,a_2,...,a_{q-1}$$ be the non zero residues modulo $\,q\,,\,q$ an odd prime . Observe that $\,\,\forall\,i\,,\,\,a_i^2=(q-a_i)^2 \pmod q\,$ , so that all the quadratic residues must be among $$a_1^2\,,\,a_2^2\,,...,a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
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Infinite Degree Algebraic Field Extensions In I. Martin Isaacs Algebra: A Graduate Course, Isaacs uses the field of algebraic numbers $$\mathbb{A}=\{\alpha \in \mathbb{C} \; | \; \alpha \; \text{algebraic over} \; \mathbb{Q}\}$$ as an example of an infinite degree algebraic field extension. I have done a cursory google... | Another simple example is the extension obtained by adjoining all roots of unity.
Since adjoining a primitive $n$-th root of unity gives you an extension of degree $\varphi(n)$ and $\varphi(n)=n-1$ when $n$ is prime, you get algebraic numbers of arbitrarily large degree when you adjoin all roots of unity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/151586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 5,
"answer_id": 1
} |
Flux of a vector field I've been trying to solve a flux integral with Gauss' theorem so a little input would be appreciated.
Problem statement: Find the flux of ${\bf{F}}(x,y,z) = (x,y,z^2)$ upwards through the surface ${\bf r}(u,v) = (u \cos v, u \sin v, u), \hspace{1em} (0 \leq u \leq 2; 0 \leq v \leq \pi)$
OK. I not... | I am not convinced that your integration limits are in order. Domain of integration is the volume below a half cone. So I would proceed as follows
$$2\int_{0}^{\pi}\int_{0}^{2}\int_{0}^{r}\left(z+1\right)rdzdrd\theta=2\int_{0}^{\pi}\int_{0}^{2}\left(\frac{r^{3}}{2}+r^{2}\right)drd\theta=2\int_{0}^{\pi}\left[\left.\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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When is something "obvious"? I try to be a good student but I often find it hard to know when something is "obvious" and when it isn't. Obviously (excuse the pun) I understand that it is specific to the level at which the writer is pitching the statement. My teacher is fond of telling a story that goes along the lines ... | Mathematical statements are only evaluated by individuals. Since individuals differ in mathematical ability, the answer is that "something" is never obvious to everyone or to yourself. The crux of the joke is that it was only obvious to the professor after reflection, which is deliberate irony since there would be no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 10,
"answer_id": 6
} |
Doubly exponential sequence behaviour from inequality I am investigating a strictly decreasing sequence $(a_i)_{i=0}^\infty$ in $(0, 1)$, with $\lim_{i\to\infty}a_i=0$, such that there exist constants $K>1$ and $m\in\mathbb{N}$ such that $$\frac{a_{i-1}^m}{K} \leq a_i \leq K a_{i-1}^m$$ for all $i$. Even though $K>1$, ... | [Edit: Now that the question has been changed a day later, I removed analysis of the old version of the first inequality. Perhaps sometime I will update to fully answer the new question. The following still applies to the second inequality.]
On the other hand, f For the second inequality, $$a_i\leq Ka_{i-1}^m\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all $P$ with $P(x^2)=P(x)^2$ The following problem is from Golan's book on linear algebra, chapter 4. I have posted a proposed answer below.
Problem: Let $F$ be a field. Find all nonzero polynomials $P\in F[x]$ satisfying
$$P(x^2)=[P(x)]^2.$$
| Assume first that $F$ is a field with characteristic not equal to 2. The only ones are 1 and $x^n$, $n\in \mathbb{N}$.
Let $a_n$ denote the coefficient of $x^n$ in $P$. We see immediately that all $a_n=0$ for odd $n>0$. Examining the constant coefficient, we see $a_0=a_0^2\Rightarrow a_0=1$ or $a_0=0$.
Now proceed by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
(Regular) wreath product of nilpotent groups Is the wreath product of two nilpotent groups always nilpotent?
I know the answer is no due to a condition "The regular wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the s... | Following Jug's suggestion: let $\,\,A:=C_3=\langle a\rangle\,,\,B:=C_2=\langle c\rangle\,$ , with the regular action of $\,B\,$ on itself, and form the (regular) wreath product $$A\wr B\cong \left(C_3\times C_3\right)\rtimes_R C_2$$ Take the elements $$\pi=((1,1),c))\,\,,\,\,\sigma=((a,a^2),1)$$It's now easy to check ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sin(x+\frac{\pi}{n})$ converges uniformly to $\sin(x)$. I've just starting learning uniform convergence and understand the formal definition. What I've got so far is:
$|\sin(x+ \frac{\pi}{n}) - \sin(x)| < \epsilon \ \ \ \ \forall x \in \mathbb{R} \ \ \ \ $ for $n \geq N, \epsilon>0$
LHS = $|2\cos(x+\frac{... | Use the fact that the sine function's derivative has absolute value of at most one to see that
$$|\sin(x) - \sin(y)| \le |x - y|.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/152026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Localisation is isomorphic to a quotient of polynomial ring I am having trouble with the following problem.
Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.
I just want a hint.
Basically, I've been looking for a surjec... | Here's another answer using the universal property in another way (I know it's a bit late, but is it ever too late ?)
As for universal properties in general, the ring satisfying the universal property described by Arturo Magidin in his answer is unique up to isomorphism. Thus to show that $R[x]/(ax-1) \simeq R_D$, it s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 5,
"answer_id": 0
} |
Moment of inertia of an ellipse in 2D I'm trying to compute the moment of inertia of a 2D ellipse about the z axis, centered on the origin, with major/minor axes aligned to the x and y axes. My best guess was to try to compute it as:
$$4\rho \int_0^a \int_0^{\sqrt{b^2(1 - x^2/a^2)}}(x^2 +y^2)\,dydx$$
... I couldn't fi... | Use 'polar' coordinates, as in $\phi(\lambda, \theta) = (\lambda a \cos \theta, \lambda b \sin \theta)$, with $(\lambda, \theta) \in S = (0,1] \times [0,2 \pi]$. It is straightforward to compute the Jacobian determinant as
$$ J_{\phi}(\lambda, \theta) = |\det D\phi(\lambda, \theta)| = \lambda a b.$$
Let $E = \{ (x,y) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A Banach space is reflexive if and only if its dual is reflexive How to show that a Banach space $X$ is reflexive if and only if its dual $X'$ is reflexive?
| Here's a different, more geometric approach that comes from Folland's book, exercise 5.24
Let $\widehat X$, $\widehat{X^*}$ be the natural images of $X$ and $X^*$ in $X^{**}$ and $X^{***}$.
Define $\widehat X^0 = \{F\in X^{***}: F(\widehat x) = 0 \text{ for all } \widehat x \in \widehat X\}$
1) It isn't hard to show t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 4,
"answer_id": 3
} |
Probability problem of 220 people randomly selecting only 12 of 35 exclusive options. There are 220 people and 35 boxes filled with trinkets.
Each person takes one trinket out of a random box.
What is the probability that the 220 people will have grabbed a trinket from exactly 12 different boxes?
I'm trying to calcula... | If the probability that the first $n$ people have chosen from exactly $c$ boxes out of a possible $t$ total boxes [$t=35$ in this case] is $p(n,c,t)$ then $$p(n,c,t)=\frac{c \times p(n-1,c,t)+(t-c+1)\times p(n-1,c-1,t)}{t}$$ starting with $p(0,c,t)=0$ and $p(n,0,t)=0$ except $p(0,0,t)=1$.
Using this gives $p(220,12,35)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Inverse of transformation matrix I am preparing for a computer 3D graphics test and have a sample question which I am unable to solve.
The question is as follows:
For the following 3D transfromation matrix M, find its inverse. Note that M is a composite matrix built from fundamental geometric affine transformations onl... | I know this is old, but the inverse of a transformation matrix is just the inverse of the matrix. For a transformation matrix $M$ which transforms some vector $\mathbf a$ to position $\mathbf v$, then to get a matrix which transforms some vector $\mathbf v$ to $\mathbf a$ we just multiply by $M^{-1}$
$M\cdot \mathbf a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Finding solutions to equation of the form $1+x+x^{2} + \cdots + x^{m} = y^{n}$ Exercise $12$ in Section $1.6$ of Nathanson's : Methods in Number Theory book has the following question.
*
*When is the sum of a geometric progression equal to a power? Equivalently, what are the solutions of the exponential diophan... | I liked your question much. The cardinality of the solutions to the above equation purely depends upon the values of $m,n$.
Let me break your problem into some cases. There are three cases possible.
*
*When $ m = 1 $ and $ n = 1 $ , you know that there are infinitely many solutions .
*When $m=2$ and $n=1$ you kn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of $e^x$. The process went something like this:
$$\begin{align}
(e^x)... | Let say $y=e^h -1$, then $\lim_{h \rightarrow 0} \dfrac{e^h -1}{h} = \lim_{y \rightarrow 0}{\dfrac{y}{\ln{(y+1)}}} = \lim_{y \rightarrow 0} {\dfrac{1}{\dfrac{\ln{(y+1)}}{y}}} = \lim_{y \rightarrow 0}{\dfrac{1}{\ln{(y+1)}^\frac{1}{y}}}$. It is easy to prove that $\lim_{y \rightarrow 0}{(y+1)}^\frac{1}{y} = e$. Then usin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 8,
"answer_id": 2
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Showing the divergence of $ \int_0^{\infty} \frac{1}{1+\sqrt{t}\sin(t)^2} dt$ How can I show the divergence of
$$ \int_0^x \frac{1}{1+\sqrt{t}\sin(t)^2} dt$$
as $x\rightarrow\infty?$
| For $t \gt 0$:
$$
1 + t \ge 1 + \sqrt{t}\sin^2t
$$
Or:
$$
\frac{1}{1 + t} \le \frac{1}{1 + \sqrt{t}\sin^2t}
$$
Now consider:
$$
\int_0^x \frac{dt}{1 + t} \le \int_0^x \frac{dt}{1 + \sqrt{t}\sin^2t}
$$
The LHS diverges as $x \to +\infty$, so the RHS does too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/152649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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What is the Taylor series for $g(x) =\frac{ \sinh(-x^{1/2})}{(-x^{1/2})}$, for $x < 0$?
What is the Taylor series for $$g(x) = \frac{\sinh((-x)^{1/2})}{(-x)^{1/2}}$$, for $x < 0$?
Using the standard Taylor Series:
$$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!}$$
I substituted in $x = x^{1/2}$, since... | As Arturo pointed out in a comment, It has to be $(-x)^{\frac{1}{2}}$ to be defined for $x<0$, then you have:
$$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!}+\dots$$
Substituting $x$ with $(-x)^{\frac{1}{2}}$ we get:
$$\sinh (-x)^{\frac{1}{2}} = (-x)^{\frac{1}{2}} + \frac{({(-x)^{\frac{1}{2}}})^3}{3!} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given an alphabet with 6 non-distinct integers, how many distinct 4-digit integers are there?
How many distinct four-digit integers can one make from the digits
$1$, $3$, $3$, $7$, $7$ and $8$?
I can't really think how to get started with this, the only way I think might work would be to go through all the cases. F... | Distinct numbers with two $3$s and two $7$s: $\binom{4}{2}=6$.
Distinct numbers with two $3$s and one or fewer $7$s: $\binom{4}{2}3\cdot2=36$.
Distinct numbers with two $7$s and one or fewer $3$s: $\binom{4}{2}3\cdot2=36$.
Distinct numbers with one or fewer $7$s and one or fewer $3$s: $4\cdot3\cdot2\cdot1=24$.
Total: $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Quadratic equation related to physics problem - how to proceed? It's a physics-related problem, but it has a nasty equation:
Let the speed of sound be $340\dfrac{m}{s}$, then let a heavy stone fall into the well. How deep is the well when you hear the impact after $2$ seconds?
The formula for the time it takes the st... | Hint: if you insert the values of $g, t$ and $v$ you have a quadratic equation in $s$. Even if you just regard $g, t$ and $v$ as constants, you can plug this into the quadratic formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/152839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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expressing $x^3 /1000 - 100x^2 - 100x + 3$ in big theta Hello can somebody help me in expressing $x^3/1000 - 100x^2 - 100x + 3$ in big theta notation. It looks like of $x^3$ to me, but obviously at $x =0$ obviously this polynomial gives a value of $3$. And multiplying $x^3$ by any constant won't help at all. Is there a... | More generally, given an arbitrary real polynomial $p(x)=a_nx^n+\cdots+a_1x+a_0$ with $a_n>0$, let us denote by $M$ a number greater than all of $p$'s real roots. We have
$$\lim_{x\to\infty}\frac{p(x)}{x^n}=a_n+a_{n-1}(0)+\cdots+a_1(0)+a_0(0)=a_n>0.$$
Now $p$ is continuous and has no roots beyond $M$ so it cannot chang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are isomorphic structures really indistinguishable? I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different with respect to some property and I just wanted to know more about i... | I'm not sure, if this is what you are referring to, but here goes...
There are questions that are easy to decide in one structure, but much more difficult in another isomorphic structure. The discrete logarithm problem comes to mind. The additive group
$G_1=\mathbf{Z}_{502}$ is generated by $5$, and to a given $x\in G_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
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A good reference to begin analytic number theory I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about analytic number theory. I'd like to know if there would be a book that I could ... | I'm quite partial to Apostol's books, and although I haven't read them (yet) his analytic number theory books have an excellent reputation.
Introduction to Analytic Number Theory (Difficult undergraduate level)
Modular Functions and Dirichlet Series in Number Theory (can be considered a continuation of the book above)
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 8,
"answer_id": 1
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harmonic function question Let $u$ and $v$ be real-valued harmonic functions on $U=\{z:|z|<1\}$. Let $A=\{z\in U:u(z)=v(z)\}$. Suppose $A$ contains a nonempty open set. Prove $A=U$.
Here is what I have so far: Let $h=u-v$. Then $h$ is harmonic. Let $X$ be the set of all $z$ such that $h(z)=0$ in some open neighbor... | Each real harmonic function $h$ on a simply connected domain defines unique up to the constant holomorphic function $f\in\mathcal{O}(U)$ such that
$$
\mathrm{Im}(f)=h
$$
$$
\mathrm{Re}(f)=
\int\limits_{(x_0,y_0)}^{(x,y)}\left(\frac{\partial h}{\partial y}dx-\frac{\partial h}{\partial x}dy\right)+C
$$
If $h=0$ on some... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is $\lim_{(x,y)\to(0,0)} \frac{(x^3+y^3)}{(x^2-y^2)}$? In class, we were simply given that this limit is undefined since along the paths $y=\pm x$, the function is undefined.
Am I right to think that this should be the case for any function, where the denominator is $x^2-y^2$, regardless of what the numerator is?
... | For your function, in the domain of $f$ (so $x\ne \pm y)$, to compute the limit you can set $x=r\cos\theta, y=r\sin\theta$, and plug it it. You get $\lim\limits_{r\to 0} \frac{r^3(cos^3\theta+sin^3\theta)}{r^2(cos^2\theta-sin^2\theta)} =\lim\limits_{r\to 0} \frac{r(cos^3\theta+sin^3\theta)}{(cos^2\theta-sin^2\theta)}$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces Looking for the proof of Eberlein-Smulian Theorem.
Searching for the proof is what I break with this morning. Some of my friends recommend Haim Brezis (Functional Analysis, Sobolev Spaces and Partial
Differential Equations). After I search the book, I onl... | Kôsaku Yosida, Functional Analysis, Springer 1980, Chapter V, Appendix, section 4. (This appears to be the 6th edition).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why doesn't this find the mid point? I saw a simple question and decided to try an alternate method to see if I could get the same answer; however, it didn't work out how I had expected.
Given $A(4, 4, 2)~$ and $~B(6, 1, 0)$, find the coordinates of the
midpoint $M$ of the line $AB$.
I realize that this is quite ea... | That's right...your calculation doesn't take into account position in any way. You are going half the distance from $A$ to $B$, but starting at the origin, not at $A$. Try $A+\frac{1}{2}\vec{AB}$
EDIT: It occured to me that I should point out: $$A+\frac{1}{2}\vec{AB}=A+\frac{1}{2}(B-A)=\frac{1}{2}(A+B)$$
| {
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Probability of a baseball team winning next 2 games Given their previous performance, the probability of a particular baseball team winning any given game is 4/5.
The probability that the team will win their next 2 games is...
I'm confused on how to start this question. Any help is appreciated.
| Probability of a particular baseball team winning any given game is 4/5.
Probability that the team will win their next 2 games is probability of winning 1st match $*$ probability of winning 2nd match.
$$P = (4/5) * (4/5)$$
$$P = 16/25$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Intermediate fields of cyclotomic splitting fields and the polynomials they split Consider the splitting field $K$ over $\mathbb Q$ of the cyclotomic polynomial $f(x)=1+x+x^2 +x^3 +x^4 +x^5 +x^6$. Find the lattice of subfields of K and for each subfield $F$ find polynomial $g(x) \in \mathbb Z[x]$ such that $F$ is the s... | Somehow, the theme of symmetrization often doesn't come across very clearly in many expositions of Galois theory. Here is a basic definition:
Definition. Let $F$ be a field, and let $G$ be a finite group of automorphisms of $F$. The symmetrization function $\phi_G\colon F\to F$ associated to $G$ is defined by the for... | {
"language": "en",
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"source": "stackexchange",
"question_score": "7",
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How does trigonometric substitution work? I have read my book, watched the MIT lecture and read Paul's Online Notes (which was pretty much worthless, no explanations just examples) and I have no idea what is going on with this at all.
I understand that if I need to find something like $$\int \frac { \sqrt{9-x^2}}{x^2}d... | There are some basic trigonometric identities which is not hard to memorise, one of the easiest and most important being $\,\,\cos^2x+\sin^2x=1\,\,$ , also known as the Trigonometric Pythagoras Theorem.
From here we get $\,1-\sin^2x=\cos^2x\,$ , so (watch the algebra!)$$\sqrt{9-x^2}=\sqrt{9\left(1-\left(\frac{x}{3}\rig... | {
"language": "en",
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"source": "stackexchange",
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A surjective homomorphism between finite free modules of the same rank I know a proof of the following theorem using determinants.
For some reason, I'd like to know a proof without using them.
Theorem
Let $A$ be a commutative ring.
Let $E$ and $F$ be finite free modules of the same rank over $A$.
Let $f:E → F$ b... | This answer is not complete. See the comments below.
The modules $E$ and $F$ being free of finite rank $n$ over $A$ means that they each have a finite basis over $A$. Take $y \in F$, and since $f$ is surjective some $x \in E$ maps to $y$. Pick a basis for $\langle e_1, \dots, e_n \rangle$ of $E$ over $A$, so $x = a_1e... | {
"language": "en",
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"source": "stackexchange",
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How to solve $x_j y_j = \sum_{i=1}^N x_i$ I have N equations and am having trouble with finding a solution.
$$\left\{\begin{matrix}
x_1 y_1 = \sum_{i=1}^N x_i\\
x_2 y_2 = \sum_{i=1}^N x_i\\
\vdots\\
x_N y_N = \sum_{i=1}^N x_i
\end{matrix}\right.$$
where $x_i$, ($i = 1, 2, \cdots, N$) is an unknown and $y_i$, ($i = 1, 2... | *
*$x_i= 0$ $\forall i$ is always a solution.
2 Suppose that $y_i \ne 0$ $\forall i$. Then, $x_1 = \frac{1}{y_1} \sum x_i$ and summing over all indexes we get $\sum x_i = \sum \frac{1}{y_i} \sum x_i$ So we must either have $\sum x_i = 0$ or $\sum \frac{1}{y_i} = 1$
2.a The case $\sum x_i = 0$ gives only the trivial... | {
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Simplify integral of inverse of derivative. I need to simplify function $g(x)$ which I describe below.
Let $F(y)$ be the inverse of $f'(\cdot)$ i.e. $F = \left( f'\right)^{-1}$ and $f(x): \mathbb{R} \to \mathbb{R}$, then $$g(x) =\int_a^x F(y)dy$$ Is it possible to simplify $g(x)$?
| Let $t = F(y)$. Then we get that $y = F^{-1}(t) = f'(t)$. Hence, $dy = f''(t) dt$. Hence, we get that
\begin{align}
g(x) & = \int_{F(a)}^{F(x)} t f''(t) dt\\
& = \left. \left(t f'(t) - f(t) \right) \right \rvert_{F(a)}^{F(x)}\\
& = F(x) f'(F(x)) - f(F(x)) - (F(a) f'(F(a)) - f(F(a)))\\
& = xF(x) - f(F(x)) - aF(a) + f(F... | {
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Probability of getting two consecutive 7s without getting a 6 when two dice are rolled Two dice are rolled at a time, for many time until either A or B wins. A wins if we get two consecutive 7s and B wins if we get one 6 at any time.
what is the probability of A winning the game??
| Let $p_A$ and $p_B$ the winning probabilities for $A$ and $B$, and $p_6$ and $p_7$ the probabilities to roll 6 and 7.
Now, by regarding the different possibilities for the first roll (and if one starts with 7, also the second roll), we find:
$$p_A=p_6 \cdot 0 + p_7 (p_6 \cdot 0 +p_7+(1-p_6-p_7)p_A) + (1-p_6-p_7)p_A.$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Simple geometric proof for Snell's law of refraction Snell's law of refraction can be derived from Fermat's principle
that light travels paths that minimize the time using simple
calculus. Since Snell's law only involves sines I wonder whether
this minimum problem has a simple geometric solution.
| Perhaps this will help, if you are looking at a non Calculus approach.
Consider two parallel rays $A$ and $B$ coming through the medium $1$ (say air) to the medium $2$ (say water). Upon arrival at the interface $\mathcal{L}$ between the two media (air and water), they continue their parallel course in the directions $U... | {
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"source": "stackexchange",
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Derivative of a linear transformation. We define derivatives of functions as linear transformations of $R^n \to R^m$. Now talking about the derivative of such linear transformation ,
as we know if $x \in R^n$ , then
$A(x+h)-A(x)=A(h)$, because of linearity of $A$, which implies that $A'(x)=A$ where , $A'$ is derivativ... | $A'$, where $A$ is seen as a linear /map/, has a derivative $A$, where $A$ is now seen as a (constant) matrix..
| {
"language": "en",
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"source": "stackexchange",
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Prove: The weak closure of the unit sphere is the unit ball. I want to prove that in an infinite dimensional normed space $X$, the weak closure of the unit sphere $S=\{ x\in X : \| x \| = 1 \}$ is the unit ball $B=\{ x\in X : \| x \| \leq 1 \}$.
$\\$
Here is my attempt with what I know:
I know that the weak closure of ... | With the same notations in you question: Notice that if $x_i^*(x) = 0$ for all $i$, then $x \in U$, and therefore the intersection of the kernels $\bigcap_{i=1}^n \mathrm{ker}(x_i^*)$ is in $U$. Since the codimension of $\mathrm{ker}(x^*_i)$ is at most $1$, then the intersection has codimension at most $n$ (exercise: p... | {
"language": "en",
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History of Modern Mathematics Available on the Internet I have been meaning to ask this question for some time, and have been spurred to do so by Georges Elencwajg's fantastic answer to this question and the link contained therein.
In my free time I enjoy reading historical accounts of "recent" mathematics (where, to m... | Babois's thesis on the birth of the cohomology of groups .
Beaulieu on Bourbaki
Brechenmacher on the history of matrices
Demazure's eulogy of Henri Cartan
Serre's eulogy of Henri Cartan
Dolgachev on Cremona and algebraic cubic surfaces
The Hirzebruch-Atiyah correspondence on $K$-theory
Krömer's the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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Integral of$\int_0^1 x\sqrt{2- \sqrt{1-x^2}}dx$ I have no idea how to do this, it seems so complex I do not know what to do.
$$\int_0^1 x\sqrt{2- \sqrt{1-x^2}}dx$$
I tried to do double trig identity substitution but that did not seem to work.
| Here is how I would do it, and for simplicity I would simply look at the indefinite integral.
First make the substitution $u = x^2$ so that $du = 2xdx$. We get: $$\frac{1}{2} \int \sqrt{2-\sqrt{1-u}} du$$ Then, make the substitution $v = 1-u$ so that $dv = -du$. We get: $$-\frac{1}{2} \int \sqrt{2 - \sqrt{v}} dv$$ Then... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Sum with binomial coefficients: $\sum_{k=1}^m \frac{1}{k}{m \choose k} $ I got this sum, in some work related to another question:
$$S_m=\sum_{k=1}^m \frac{1}{k}{m \choose k} $$
Are there any known results about this (bounds, asymptotics)?
| Consider a random task as follows. First, one chooses a nonempty subset $X$ of $\{1,2,\ldots,m\}$, each with equal probability. Then, one uniformly randomly selects an element $n$ of $X$. The event of interest is when $n=\max(X)$.
Fix $k\in\{1,2,\ldots,m\}$. The probability that $|X|=k$ is $\frac{1}{2^m-1}\,\binom{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 7,
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probability of a horse winning a race. Lets suppose ten horses are participating in a race and each horse has equal chance
of winning the race. I am required to find the following:
(a) the probability that horse A wins the race followed by horse B.
(b) the probability that horse C becomes either first or second in t... | The answer to A is 1/90 because of non replacement method. Horse A has 1/10 chance to be 1st and horse b would then be 1 of 9 with a chance to be 2nd. Multiply 1/10 times 1/9 gets 1/90th
The answer to b is different as its an addition problemas in he has a chance to be first and/or 2nd so 1/10 plus 1/9...common denomi... | {
"language": "en",
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"source": "stackexchange",
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Subring of polynomials Let $k$ be a field and $A=k[X^3,X^5] \subseteq k[X]$.
Prove that:
a. $A$ is a Noetherian domain.
b. $A$ is not integrally closed.
c. $dim(A)=?$ (the Krull dimension).
I suppose that the first follows from $A$ being a subring of $k[X]$, but I don't know about the rest.
Thank you in advance.
| a) Not every subring of a noetherian ring is noetherian (there are plenty of counterexamples), so this doesn't work here. Instead, use Hilbert's Basis Theorem.
b) The element $X^2 = \frac{X^5}{X^3}$ is in $\mathrm{Quot}(A)$. Try to show that it is integral over $A$, but not in $A$.
c) The dimension is the transcendence... | {
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Piecewise functions: Got an example of a real world piecewise function? Looking for something beyond a contrived textbook problem concerning jelly beans or equations that do not represent anything concrete. Not just a piecewise function for its own sake. Anyone?
| As Wim mentions in the comments, piecewise polynomials are used a fair bit in applications. In designing profiles and shapes for cars, airplanes, and other such devices, one usually uses pieces of Bézier or B-spline curves (or surfaces) during the modeling process, for subsequent machining. In fact, the continuity/smoo... | {
"language": "en",
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"source": "stackexchange",
"question_score": "23",
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"answer_id": 4
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Numerical solution of the Laplace equation on circular domain I was solving Laplace equation in MATLAB numerically. However I have problems when the domain is not rectangular.
The equation is as follows:
$$
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0
$$
domain is circular
$$
x^2 + y^2 < 16
$... | Perhaps begin by rewriting the problem in polar coordintates:
$$\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}}=0$$
$$r^2<16$$
$$\left. u(r,\theta)\right|_D=r^4\cos^2\theta\sin^2\theta$$
| {
"language": "en",
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Reflections generating isometry group I was reading an article and it states that every isometry of the upper half plane model of the hyperbolic plane is a composition of reflections in hyperbolic lines, but does not seem to explain why this is true. Could anyone offer any insight? Thanks.
| An isometry $\phi:M\to N$ between connected Riemannian manifolds $M$ and $N$ is completely determined by its value at a single point $p$ and its differential at $d\phi_p$.
Take any isometry $\phi$ of $\mathbb{H}^2$. Connect $i$ and $\phi(i)$ by a (unique) shortest geodesic and let $C$ be a perpendicular bisector of th... | {
"language": "en",
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Is Koch snowflake a continuous curve? For Koch snowflake, does there exits a continuous map from $[0,1]$ to it?
The actural construction of the map may be impossible, but how to claim the existence of such a continuous map? Or can we conside the limit of a sequence of continuous map, but this sequence of continuous map... | Consider the snowflake curve as the limit of the curves $(\gamma_n)_{n\in \mathbb N}$, in the usual way, starting with $\gamma_0$ which is just a equilateral triangle of side length 1. Then each $\gamma_n$ is piecewise linear, consisting of $3\cdot 4^n$ pieces of length $3^{-n}$ each; for definiteness let us imagine th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"8 Dice arranged as a Cube" Face-Sum Equals 14 Problem I found this here:
Sum Problem
Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same.
$\hskip2.7in$
Here is one of 20 736 solutions with the sum 14.
You find more at the German magazine "Bild der Wissenschaft ... | No, 14 is not the only possibility.
For example:
Arrange the dice, so that you only see 1,2 and 3 pips and all the 2's are on the upper and lower face of the big cube. This gives you face sum 8.
Please ask your other questions as separate questions if you are still interested.
| {
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"source": "stackexchange",
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Computing the length of a finite group Can someone suggest a GAP or MAGMA command (or code) to obtain the length $l(G)$ of a finite group $G$, i.e. the maximum length of a strictly descending chain of subgroups in $G$?
Thanks in advance.
| Just to get you started, here is a very short recursive Magma function to compute
this. You could do something similar in GAP. Of course, it will only work in reasonable time for small groups. On my computer it took about 10 seconds to do $A_8$. To do better you would need to do something more complicated like working ... | {
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Find the area of overlap of two triangles Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically determine the area (possibly $0$) of their overlap; call it $T_{common}$.
We have a ... | Sorry about the comment -- I hit the return key prematurely.
This isn't really an answer (except in the negative sense).
The common (overlap) area is a function of the coordinates of the 6 points, so it's a mapping from $R^{12}$ into $R$. Think about one of the points moving around, while the other 5 are fixed. When t... | {
"language": "en",
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"source": "stackexchange",
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Operators from $\ell^\infty$ into $c_0$ I have the following question related to $\ell^\infty(\mathbb{N}).$ How can I construct a bounded, linear operator from $\ell^\infty(\mathbb{N})$ into $c_0(\mathbb{N})$ which is non-compact?
It is clear that $\ell^\infty$ is a Grothendieck space with Dunford-Pettis property, henc... | A bounded operator $T:\ell_\infty\rightarrow c_0$ has the form $Tx=(x_n^*(x))$ for some weak$^*$ null sequence $(x_n^*)$ in $\ell_\infty^*$. A set $K\subset c_0$ is relatively compact if and only if there is a $x\in c_0$ such that $|k_n|\le |x_n|$ for all $k\in K$ and all $n\ge1$. From these two facts, it follows tha... | {
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How to "rotate" a polar equation? Take a simple polar equation like r = θ/2 that graphs out to:
But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is there a way to easily shift the plot?
| A way to think about this is is that you want to shift all $\theta$ to $\theta'=\theta +\delta$, where $\delta$ is the amount by which you want to rotate. This question has a significance if you want to rotate some equation which is a function of theta. In the case $r=\theta$ that becomes $r=\theta+\delta$.
Of course... | {
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$\mid\theta-\frac{a}{b}\mid< \frac{1}{b^{1.0000001}}$, question related to the dirichlet theorem The question is:
A certain real number $\theta$ has the following property: There exist infinitely many rational numbers $\frac{a}{b}$(in reduced form) such that:
$$\mid\theta-\frac{a}{b}\mid< \frac{1}{b^{1.0000001}}$$
Prov... | Hint: Let $\theta=\frac{p}{q}$, where $p$ and $q$ are relatively prime. Look at
$$\left|\frac{p}{q}-\frac{a}{b}\right|.\tag{$1$}$$
Bring to the common denominator $bq$. Then if the top is non-zero, it is $\ge 1$, and therefore Expression $(1)$ is $\ge \frac{1}{bq}$.
But if $b$ is large enough, then $bq<b^{1.0000001}... | {
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Indefinite integral of secant cubed $\int \sec^3 x\>dx$ I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't know how to integrate it myself. I have been trying some substitutions... | We have an odd power of cosine. So there is a mechanical procedure for doing the integration. Multiply top and bottom by $\cos x$. The bottom is now $\cos^4 x$, which is $(1-\sin^2 x)^2$. So we want to find
$$\int \frac{\cos x\,dx}{(1-\sin^2 x)^2}.$$
After the natural substitution $t=\sin x$, we arrive at
$$\int \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154900",
"timestamp": "2023-03-29T00:00:00",
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Can a function with suport on a finite interval have a Fourier transform which is zero on a finite interval? If $f$ has support on $[-x_0,x_0]$ can its Fourier transform $\tilde{f}$ be zero on $[-p_0,p_0]$? If so, what is the maximum admissible product $x_0p_0$?
| Let's assume that $f$ is not identically zero.
In this answer, it is shown that if $f$ has compact support, then $\hat{f}$ is entire. A non-zero entire function cannot be zero on a set with a limit point. Thus, if $f=0$ on $[-p_0,p_0]$, then $p_0=0$, and therefore, the maximum of $x_0p_0=0$.
| {
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Need help with the integral $\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$ I'm having a problem resolving the following integral, spent almost all day trying.
Any help would be appreciated.
$$\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$$
| Hint:
*
*Multiply the numerator and denomiator by $\sec^{2}(x)$. Then you have $$\int \frac{2 \tan{x} +3}{5\sin^{2}(x)+4} \times \frac{\sec^{2}(x)}{\sec^{2}{x}} \ dx$$
*Now, the denomiator becomes $5\tan^{2}(x) + 4\cdot \bigl(1+\tan^{2}(x)\bigr)$, and you can put $t=\tan{x}$.
*So your new integral in terms of $t$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Counting matrices over $\mathbb{Z}/2\mathbb{Z}$ with conditions on rows and columns I want to solve the following seemingly combinatorial problem, but I don't know where to start.
How many matrices in $\mathrm{Mat}_{M,N}(\mathbb{Z}_2)$ are there such that the sum of entries in each row and the sum of entries in each co... | If you consider the entries of the matrices as unknowns, you have $N\cdot M$ unknowns and $N+M$ linear equations.
If you think a little bit, you find out that these equations are not independent, but you get linearly independent equations if you omit one of them.
So the solution space has the dimension $NM-N-M+1$, henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to prove that the sum and product of two algebraic numbers is algebraic? Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \cdots + a_n \alpha^n = 0$, then dividing by... | Okay, I'm giving a second answer because this one is clearly distinct from the first one. Recall that finding a polynomial over which $\alpha+\beta$ or $\alpha \beta$ is a root of $p(x) \in F[x]$ is equivalent to finding the eigenvalue of a square matrix over $F$ (living in some algebraic extension of $F$), since you c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 5,
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Natural question about weak convergence. Let $u_k, u \in H^{1}(\Omega)$ such that $u_k \rightharpoonup u$ (weak convergence) in $H^{1}(\Omega)$. Is true that $u_{k}^{+}\rightharpoonup u^{+}$ in $\{u\geqslant 0\}$? You can do hypothesis on $\Omega$ if you need.
| I get the idea from Richard's answer.
Let $\Omega:=(0,2\pi)$ and $u_k(x):=\frac{\cos(kx)}{k+1}$. Then $\{u_k\}$ converges weakly to $0$ in $H^1(\Omega)$ (as it's bounded in $H^1(\Omega)$, and $\int_{\Omega}(u_k\varphi+u'_k\varphi')dx\to 0$ for all test function $\varphi$).
Assume that $\{u_k^+\}$ converges weakly to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to solve this quartic equation? For the quartic equation:
$$x^4 - x^3 + 4x^2 + 3x + 5 = 0$$
I tried Ferrari so far and a few others but I just can't get its complex solutions. I know it has no real solutions.
| $$x^4 - x^3 + 4x^2 + \underbrace{3x}_{4x-x} + \overbrace{5}^{4+1} = \\\color{red}{x^4-x^3}+4x^2+4x+4\color{red}{-x+1}\\={x^4-x^3}-x+1+4(x^2+x+1)\\={x^3(x-1)}-(x-1)+4(x^2+x+1)\\=(x-1)(x^3-1)+4(x^2+x+1)\\=(x-1)(x-1)(x^2+x+1)+4(x^2+x+1)\\=(x^2+x+1)(x-1)^2+4)=(x^2+x+1)(x^2-2x+5)$$
As for the roots, I assume you could solve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Effective cardinality Consider $X,Y \subseteq \mathbb{N}$.
We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$.
We say that $X \equiv_c Y$ iff there exist a bijective computable function between $X$ and $Y$.
Can you show me some examples in which the two concepts disagree?
| The structure with the computable equivalence (which is defined as $A\leq_T B$ and $B \leq_T A$) is called Turing degrees and has a very rich structure (unlike the usual bijection).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/155373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Evaluating $ \int_1^{\infty} \frac{\{t\} (\{t\} - 1)}{t^2} dt$ I am interested in a proof of the following.
$$ \int_1^{\infty} \dfrac{\{t\} (\{t\} - 1)}{t^2} dt = \log \left(\dfrac{2 \pi}{e^2}\right)$$
where $\{t\}$ is the fractional part of $t$.
I obtained a circuitous proof for the above integral. I'm curious about ... | Let's consider the following way involving some known results of celebre integrals with fractional parts:
$$ \int_1^{\infty} \dfrac{\{t\} (\{t\} - 1)}{t^2} dt = \int_1^{\infty} \dfrac{\{t\}^2}{t^2} dt - \int_1^{\infty} \dfrac{\{t\}}{t^2} dt = \int_0^1 \left\{\frac{1}{t}\right\}^2 dt- \int_0^1 \left\{\frac{1}{t}\right\}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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$\{1,1\}=\{1\}$, origin of this convention Is there any book that explicitly contain the convention that a representation of the set that contain repeated element is the same as the one without repeated elements?
Like $\{1,1,2,3\} = \{1,2,3\}$.
I have looked over a few books and it didn't mention such thing. (Wikipedia... | I took a quick look through some of the likelier candidates on my shelves. The following introductory discrete math texts all explicitly point out, with at least one example, that neither the order of listing nor the number of times an element is listed makes any difference to the identity of a set:
*
*Winfried K. G... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Real life application of Gaussian Elimination I would normally use Gaussian Elimination to solve a linear system. If we have more unknowns than equations we end up with an infinite number of solutions. Are there any real life applications of these infinite solutions? I can think of solving puzzles like Sudoku but are t... | One important application is this: Given the corner points of a convex hull $\{\mathbf v_1,\cdots,\mathbf v_m \}$ in $n$ dimensions, s.t. $m > n+1$ and a point $\mathbf c$ inside the convex hull, find an enclosing simplex of $\mathbf c$(of size $r \le n+1$). To solve the the problem, one can find a solution to $\alpha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $4^{2n} + 10n -1$ is a multiple of 25 Prove that if $n$ is a positive integer then $4^{2n} + 10n - 1$ is a multiple of $25$
I see that proof by induction would be the logical thing here so I start with trying $n=1$ and it is fine. Then assume statement is true and substitute $n$ by $n+1$ so I have the follo... | $\rm\displaystyle 25\ |\ 10n\!-\!(1\!-\!4^{2n}) \iff 5\ |\ 2n - \frac{1-(-4)^{2n}}{5}.\ $ Now via $\rm\ \dfrac{1-x^k}{1-x}\, =\, 1\!+\!x\!+\cdots+x^{k-1}\ $
$\rm\displaystyle we\ easily\ calculate\ that, \ mod\ 5\!:\, \frac{1-(-4)^{2n}}{1-(-4)\ \ \,}\, =\, 1\!+\!1\!+\cdots + 1^{2n-1} \equiv\, 2n\ \ $ by $\rm\: -4\equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Simplify these expressions with radical sign 2 My question is
1) Rationalize the denominator:
$$\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$
My answer is:
$$\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{18}$$
My question is
2) $$\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$$
My answer is: $$\frac... | $\begin{eqnarray*}
(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{12}+\sqrt{18}-\sqrt{30}) & = & (\sqrt{2}+\sqrt{3}+\sqrt{5})(2\sqrt{3}+3\sqrt{2}-\sqrt{2}\sqrt{3}\sqrt{5})\\& = & 12,
\end{eqnarray*}$
if you expand out the terms, so your first answer is incorrect. The denominator should be $12$.
$\begin{eqnarray*}
(\sqrt{2}+\sqrt{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Constructing arithmetic progressions It is known that in the sequence of primes there exists arithmetic progressions of primes of arbitrary length. This was proved by Ben Green and Terence Tao in 2006. However the proof given is a nonconstructive one.
I know the following theorem from Burton gives some criteria on how ... | This may not answer the question, but I would like to point out that more recent work of Green and Tao have proven even stronger results.
Specifically, Green and Tao give exact asymptotics for the number of solutions to systems of linear equations in the prime numbers, and their paper Linear Equations in the Primes w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 2
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$\bar{\partial}$-Poincaré lemma This is $\bar{\partial}$-Poincaré lemma: Given a holomorphic funtion $f:U\subset \mathbb{C} \to \mathbb{C}$ ,locally on $U$ there is a holomorphic function $g$ such that : $$\frac{\partial g}{\partial \bar z}=f$$
The author says that this is a local statement so we may assume $f$ with co... | I don't have the book, and thus I can't check the statement.
However, I believe that the statement holds for smooth $f$.
Basically we want to construct/find $g$ as the following integral:
$$g(z) = \frac{1}{2 \pi i}\int_{w\in \mathbb{C}} \frac{f(w)}{z-w} d\overline{w}\wedge dw$$
In order to do this, $f$ must be defin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that for all $\lambda \geq 1~$ $\frac{\lambda^n}{e^\lambda} < \frac{C}{\lambda^2}$
Show that for any $n \in \mathbb N$ there exists $C_n > 0$ such that for all $\lambda \geq 1$
$$ \frac{\lambda^n}{e^\lambda} < \frac{C_n}{\lambda^2}$$
I can see that both sides of the inequality have a limit of $0$ as $\lambda \ri... | The function $\lambda \mapsto \lambda^{n+2}$ is strictly increasing for positive $\lambda$ and also $e^{\lambda} > \lambda$. Combining this you get
$$e^{\lambda} = \left( e^{\frac{\lambda}{n+2}} \right)^{n+2} > \left( \frac{\lambda}{n + 2} \right)^{n+2}$$
for all positive $\lambda$ and therefore
$$\frac{\lambda^n}{e^\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why are bump functions compactly supported? Smooth and compactly supported functions are called bump functions. They play an important role in mathematics and physics.
In $\mathbb{R}^n$ and $\mathbb{C}^n$, a set is compact if and only if it is closed and bounded.
It is clear why we like to work with functions that have... | *
*On spaces such as open intervals and (more generally) domains in $\mathbb R^n$, compactness of support tells us much more than its boundedness. Any function $f\colon (0,1)\to\mathbb R$ has bounded support, since the space $(0,1)$ itself is bounded. But if the support is compact, that means that $f$ vanishes near $0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How do I show that this function is always $> 0$
Show that $$f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +
\frac{x^4}{4!} > 0 ~~~ \forall_x \in \mathbb{R}$$
I can show that the first 3 terms are $> 0$ for all $x$:
$(x+1)^2 + 1 > 0$
But, I'm having trouble with the last two terms. I tried to show that the followin... | Hint:
$$f(x) = \frac{1}{4} + \frac{(x + 3/2)^2}{3} +\frac{x^2(x+2)^2}{24}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Derivative of an implicit function I am asked to take the derivative of the following equation for $y$:
$$y = x + xe^y$$
However, I get lost. I thought that it would be
$$\begin{align}
& y' = 1 + e^y + xy'e^y\\
& y'(1 - xe^y) = 1 + e^y\\
& y' = \frac{1+e^y}{1-xe^y}
\end{align}$$
However, the text book gives me a differ... | You can simplify things as follows:
$$y' = \frac{1+e^y}{1-xe^y} = \frac{x+xe^y}{x(1-xe^y)} = \frac{y}{x(1-y+x)}$$
Here in the last step we used $y=x+xe^y$ and $xe^y=y-x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover. I really need help with this question:
Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite sub... | Let $(a_n)$ be a sequence in the metric space $M$ that doesn't have any convergent subsequence. The set $\{a_n\}$ consists of isolated points (that is, it doesn't have any accumulation points; otherwise you could take a convergent subsequence), and it's infinite (because if it wasn't, one of the points would repeat inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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In center-excenter configuration in a right angled triangle My question is:
Given triangle $ABC$, where angle $C=90°$.
Prove that the set $\{ s , s-a , s-b , s-c \}$ is identical to $\{ r , r_1 , r_2 , r_3 \}$.
$s=$semiperimeter, $r_1,r_2,r_3$ are the ex-radii.
Any help to solve this would be greatly appreciated.
| This problem have some interesting fact behind so I use pure geometry method to solve it to show these facts:
first we draw a picture.
$\triangle ABC$, $I$ is incenter, $A0,B0,C0$ are the tangent points of incircle. $R1,R2,R3$ is ex-circle center,their tangent points are $A1,B1,C1,A2,B2,C2,A3,B3,C3$.
FOR in-circle, w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that 16, 1156, 111556, 11115556, 1111155556… are squares. I'm 16 years old, and I'm studying for my exam maths coming this monday. In the chapter "sequences and series", there is this exercise:
Prove that a positive integer formed by $k$ times digit 1, followed by $(k-1)$
times digit 5 and ending on one 6, is ... | Multiply one of these numbers by $9$, and you get $100...00400...004$, which is $100...002^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 10,
"answer_id": 2
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Formula to estimate sum to nearly correct : $\sum_{n=1}^\infty\frac{(-1)^n}{n^3}$ Estimate the sum correct to three decimal places :
$$\sum_{n=1}^\infty\frac{(-1)^n}{n^3}$$
This problem is in my homework. I find that n = 22 when use Maple to solve this. (with some programming) But, in my homework, teacher said find th... | For alternating sums $\sum(-1)^n a_n$ with $a_n> 0 $ strictly decreasing there is a simple means to estimate the remainder $\sum^\infty_{k=n} (-1)^k a_k$. You can just use $a_{n-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Evaluating $\int \frac{dx}{x^2 - 2x} dx$ $$\int \frac{dx}{x^2 - 2x}$$
I know that I have to complete the square so the problem becomes.
$$\int \frac{dx}{(x - 1)^2 -1}dx$$
Then I set up my A B and C stuff
$$\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{-1}$$
With that I find $A = -1, B = -1$ and $C = 0$ which I know is w... | Added: "I know that I have to complete the square" is ambiguous. I interpreted it as meaning that the OP thought that completing the square was necessary to solve the problem.
Completing the square is not a universal tool. To find the integral efficiently, you certainly do not need to complete the square.
The simples... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combination of cards
From a deck of 52 cards, how many five card poker hands can be formed
if there is a pair (two of the cards are the same number, and none of
the other cards are the same number)?
I believe you can pick out the first card by ${_4}C_2$, as there are 4 cards which would be the same number (as the... | We will use the notation $\binom{n}{r}$, which is more common among mathematicians, where you write ${}_nC_r$.
The kind of card that we have a pair of can be chosen in $\binom{13}{1}$ ways. For each choice of kind, the actual cards can be chosen in $\binom{4}{2}$ ways.
(By kind we mean things like Ace, or $7$, or Qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Index notation clarification Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw something of the form $T_i^j$ which seems to work not entirely differently from what I was used t... | Mostly it's just a matter of the author's preference.
The staggered index notation $T^i{}_j$ works great in conjunction with the Einstein summation convention, where one of the rules is that an index that is summed over must appear once as a subscript and once as a superscript. Usually the index of an ordinary vector's... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Set theory puzzles - chess players and mathematicians I'm looking at "Basic Set Theory" by A. Shen. The very first 2 problems are: 1) can the oldest mathematician among chess players and the oldest chess player among mathematicians be 2 different people? and 2) can the best mathematician among chess players and the be... | (1) Think of it in terms of sets. Let $M$ be the set of mathematicians, $C$ the set of chess players. Both are asking for the oldest person in $C\cap M$.
(2) Absolutely fantastic reasoning, though perhaps less simply set-theoretically described.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Solving for $c$ in $a(b + cd) \equiv 0 \mod e$ If I have a modulo operation like this:
$$(a ( b+cd ) ) \equiv 0 \pmod{e},$$
How can I derive $c$ in term of other variables present here. I.e. What function $f$ can be used such that:
$$c = f (a,b,d,e) $$
And what is the implication of a mod operation's result being $0$ ... | EDIT
My original solution to this problem was, in hindsight, radically overcomplicated. I have it below the modified post.
We have $a(b + dx) \equiv 0 \mod e$, or equivalently $ab + adx \equiv 0$, or equivalently $(ad)x \equiv -ab \mod e$. This is a very well understood subset of the problem on solving $ax \equiv b \mo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others For p = 2, we have,
$\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$
It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then,
$\begin{align} &5 \sum_{n=1}^\infty[\zeta(5n)-1] = 6+\gamma+z_5^{... | $$
\begin{align}
\sum_{n=1}^\infty\left[\zeta(pn)-1\right] & = \sum_{n=1}^\infty \sum_{k=2}^\infty \frac{1}{k^{pn}} \\
& = \sum_{k=2}^\infty \sum_{n=1}^\infty (k^{-p})^n \\
& = \sum_{k=2}^\infty \frac{1}{k^p-1}
\end{align}
$$
Let $\omega_p = e^{2\pi i/p} = z_p^2$, then we can decompose $1/(k^p-1)$ into partial fraction... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $\log_{-2}{4}$ complex? With the logarithm being the inverse of the exponential function, it follows that $ \log_{-2}{4}$ should equal $2$, since $(-2)^2=4$. The change of base law, however, implies that $\log_{-2}{4}=\frac{\log{4}}{\log{-2}}$, which is a complex number. Why does this occur when there is a real ... | The exponential function is not invertible on the complexes. Correspondingly, the complex logarithm is not a function, it is a multi-valued function. For example, $\log(e)$ is not $1$ -- instead it is the set of all values $1 + 2 \pi \mathbf{i} n$ over all integers $n$.
How are you defining $\log_a(b)$? If you are defi... | {
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"answer_id": 1
} |
"Negative" versus "Minus" As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and
not "minus $0.8$" to denote $-0.8$?
The so called "textbook answer" regarding this question reads:
A number and its opposite are called additive inverses of each other because their sum is zero,... | As a retired teacher, I can say that I tried very hard for many years to get my students to use the term "negative" instead of "minus", but after so many years of trying, I was finally happy if they could understand the concept, and stopped worrying so much about whether they used the correct terminology!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "55",
"answer_count": 26,
"answer_id": 9
} |
Which of the following are Dense in $\mathbb{R}^2$? Which of the following sets are dense in $\mathbb R^2$ with respect to the usual topology.
*
*$\{ (x, y)\in\mathbb R^2 : x\in\mathbb N\}$
*$\{ (x, y)\in\mathbb R^2 : x+y\in\mathbb Q\}$
*$\{ (x, y)\in\mathbb R^2 : x^2 + y^2 = 5\}$
*$\{ (x, y)\in\mathbb R^2 : xy\n... | *
*No. It's a bunch of parallel lines. These are vertical and the go through the integer points on the $x$-axis.
*No. It's a circle.
*Yes. It's the plane with the $x$ and $y$ axes excised.
*Interesting. It is a union of parallel lines with slope -1 and $y$-intercept at the various rationals. It's dense in the plan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Matrix commutator question Here's a nice question I heard on IRC, courtesy of "tmyklebu."
Let $A$, $B$, and $C$ be $2\times 2$ complex matrices. Define the commutator $[X,Y]=XY-YX$ for any matrices $X$ and $Y$. Prove
$$[[A,B]^2,C]=0.$$
| Here's a better argument (not posted at midnight...) which shows that the result holds over any field: we don't need the matrices to be complex.
As in the other answer, the trace of $[A,B]$ is $0$. Therefore, the characteristic polynomial of $[A,B]$ is $x^2+\det[A,B]$. By the Cayley-Hamilton Theorem,
$$[A,B]^2 = -\det[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Calculating statistic for multiple runs I have s imple, general question regarding calculating statistic for N runs of the same experiment. Suppose I would like to calculate mean of values returned by some Test. Each run of the test generates $ \langle x_1 ... x_n \rangle$ , possibly of different length. Let's say the ... | $\def\E{{\rm E}}\def\V{{\rm Var}}$Say you have $M$ runs of lengths $n_1,\dots,n_M$. Denote the $j$th value in the $i$th run by $X^i_j$, and let the $X^i_j$ be independent and identically distributed, with mean $\mu$ and variance $\sigma^2$.
In your first approach you calculate
$$\mu_1 = \frac{1}{n_1+\cdots n_M} \sum_{i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Limits of Subsequences If $s=\{s_n\}$ and $t=\{t_n\}$ are two nonzero decreasing sequences converging to 0, such that $s_n ≤t_n$ for all $n$. Can we find subsequences $s ′$ of $s$ and $t ′$ of $t$ such that $\lim \frac{s'}{t'}=0$ , i.e., $s ′$ decreases more rapidly than $t ′$ ?
| Yes, we can. So we have two positive decreasing sequences with $s_n \leq t_n$, and $s_n \to 0, t_n \to 0$.
Then we can let $t' \equiv t$. As $\{t_n\}$ is positive, $t_1 > 0$. As $s_n \to 0$, there is some $k$ s.t. $s_k < t_1/1$. Similarly, there is some $l > k$ s.t. $s_l < t_2/2$. Continuing in this fashion, we see tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Recurrence relation $T_{k+1} = 2T_k + 2$ I have a series of number in binary system as following:
0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, ...
I want to understand : Is there a general seri for my series?
I found this series has a formula as following:
(Number * 2) + 2
but i don't know this formula is corre... | $T_{k+1} = 2T_k + 2$. Adding $2$ to both sides, we get that $$\left(T_{k+1}+2 \right) = 2 T_k + 4 = 2 \left( T_k + 2\right)$$
Calling $T_k+2 = u_k$, we get that $u_{k+1} = 2u_k$. Hence, $u_{k+1} = 2^{k+1}u_0$. This gives us $$\left(T_{k}+2 \right) = 2^k \left( T_0 + 2\right) \implies T_k = 2^{k+1} - 2 +2^kT_0$$
Since, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $\frac{n}{\sigma(n)} > (1-\frac{1}{p_1})(1-\frac{1}{p_2})\cdots(1-\frac{1}{p_r})$ If $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ then show that :
$$1>\frac{n}{ \sigma (n)} > \left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right)\cdots\cdots\left(1-\frac{1}{p_r}\right)$$
I ha... | Note that the function $\dfrac{n}{\sigma(n)}$ is multiplicative. Hence, if $n = p_1^{k_1}p_2^{k_2} \ldots p_m^{k_m}$, then we have that $$\dfrac{n}{\sigma(n)} = \dfrac{p_1^{k_1}}{\sigma \left(p_1^{k_1} \right)} \dfrac{p_2^{k_2}}{\sigma \left(p_2^{k_2} \right)} \ldots \dfrac{p_m^{k_m}}{\sigma \left(p_m^{k_m} \right)}$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
hausdorff, intersection of all closed sets Can you please help me with this question?
Let's $X$ be a topological space.
Show that these two following conditions are equivalent :
*
*$X$ is hausdorff
*for all $x\in X$ intersection of all closed sets containing the neighborhoods of $x$ it's $\{x\}$.
Thanks... | HINTS:
*
*If $x$ and $y$ are distinct points in a Hausdorff space, they have disjoint open nbhds $V_x$ and $V_y$, and $X\setminus V_y$ is a closed set containing $V_x$.
*If $F$ is a closed set containing an open nbhd $V$ of $x$, then $V$ and $X\setminus F$ are disjoint open sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
A space is normal iff every pair of disjoint closed subsets have disjoint closed neighbourhoods.
A space is normal iff every pair of disjoint closed subsets have disjoint closed neighbourhoods.
Given space $X$ and two disjoint closed subsets $A$ and $B$.
I have shown necessity: If X is normal then by Urysohn's lemma ... | If $A$ and $B$ have disjoint closed neighborhoods $U$ and $V$, then by definition of neighborhood we know that $A\subseteq \mathrm{int}(U)$ and $B\subseteq\mathrm{int}(V)$. Now, the interior of $U$ is an open neighborhood of $A$, the interior of $V$ is an open neighborhood of $B$, and so $A$ and $B$ have disjoint open ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why do $\mathbb{C}$ and $\mathbb{H}$ generate all of $M_2(\mathbb{C})$? For this question, I'm identifying the quaternions $\mathbb{H}$ as a subring of $M_2(\mathbb{C})$, so I view them as the set of matrices of form
$$
\begin{pmatrix}
a & b \\ -\bar{b} & \bar{a}
\end{pmatrix}.
$$
I'm also viewing $\mathbb{C}$ as the ... | Hint: Use linear combinations of
$$
jk=\pmatrix{i&0\cr0&-i\cr}\qquad\text{and the scalar}\qquad \pmatrix{i&0\cr 0&i\cr}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
a function that maps half planes Define
$H^{+}=\{z:y>0\}$
$H^{-}=\{z:y<0\}$
$L^{+}=\{z:x>0\}$
$L^{-}=\{z:x<0\}$
$f(z)=\frac{z}{3z+1}$ maps which portion onto which from above and vice-versa? I will be glad if any one tell me how to handle this type of problem? by inspection?
| HINTS
*
*Fractional transformations/Möbius transformations take circles and lines to circles and lines, i.e. they are 'circilinear.' They also preserve connected regions.
*If you find out what happens to the boundaries, you'll know almost everything (except for in which side of the boundary the image resides); in o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Nearest matrix in doubly stochastic matrix set Suppose $\mathcal{D}_N$ denote an $N\times N$ doubly stochastic matrix, given any element $M\in \mathcal{D}_N$ , the singular value decomposition for $M$ is $$ M=USV'$$
where $U$ and $V$ are two $N\times N$ orthogonal matrix and $S$ is a $N \times N$ diagonal matrix
Let $... | I didn't exactly get your question. But the solution for the optimization problem you are looking is always a permutation matrix. This follows from the birkhoff's theorem. The birkhoff's theorem states that every doubly stochastic matrix is a convex combination of the permutation matrices. Hence, permutation matrices f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Multiplicative Selfinverse in Fields I assume there are only two multiplicative self inverse in each field with characteristice bigger than $2$ (the field is finite but I think it holds in general). In a field $F$ with $\operatorname{char}(F)>2$ a multiplicative self inverse $a \in F$ is an element such that
$$ a \cdot... | Hint $\rm\ x^2\! =\! 1\!\iff\! (x\!-\!1)(x\!+\!1) = 0\! \iff\! x = \pm1,\:$ by $\rm\:ab=0\:\Rightarrow\: a=0\:\ or\:\ b=0\:$ in a field.
This may fail if the latter property fails, i.e. if nontrivial zero-divisors exist. Consider, for example, $\rm\ x^2 = 1\:$ has $4$ roots $\rm\:x = \pm1, \pm 3\:$ in $\rm\:\mathbb Z/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Evaluation of $\lim\limits_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$ One of the previous posts made me think of the following question: Is it possible to evaluate this limit without L'Hopital and Taylor?
$$\lim_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$$
| Here is a different approach. Let $$L = \lim_{x \to 0} \dfrac{\tan(x) - x}{x^3}$$
Replacing $x$ by $2y$, we get that
\begin{align}
L & = \lim_{y \to 0} \dfrac{\tan(2y) - 2y}{(2y)^3} = \lim_{y \to 0} \dfrac{\dfrac{2 \tan(y)}{1 - \tan^2(y)} - 2y}{(2y)^3}\\
& = \lim_{y \to 0} \dfrac{\dfrac{2 \tan(y)}{1 - \tan^2(y)} - 2 \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Combining a radical and simplifying? How would I combine and simplify the following radical:
$$\sqrt {\frac{A^2}{2}} - \sqrt \frac{A^2}{8}$$
| $$\sqrt {\frac{A^2}{2}} - \sqrt \frac{A^2}{8}\\=\frac{|A|}{\sqrt 2}-\frac{|A|}{2\sqrt 2}\\=\frac{2|A|-|A|}{2\sqrt 2}\\=\frac{|A|}{2\sqrt 2}\frac{\sqrt 2}{\sqrt 2}\\=\frac{\sqrt 2|A|}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
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