Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Which sequences converge in the topological space $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$? Which sequences converges in the topological space $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$?
That i have tried:
For each $(p,q)\in \mathbb{N}\times\mathbb... | $(1,1), (1,2), (1,3), (1,4), (1,5), \ldots$ converges to $(2,1)$.
| {
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Some examples of local and nonlocal properties Today I learned that continuity at a point is a local property. Concretely, if $f: \mathbb R \to \mathbb R$ is continuous on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is continuous on $ \mathbb R$.
Uniform convergence on the other hand is not a local property: if $g_n ... | Uniform continuity is also not local. A continuous function is uniformly continuous on compact sets. For example, the function $f(x)=x^2$ is uniformly continuous on any finite interval $[-K,K]$ but not on the whole real line.
We want to theck the definition of uniform continuity, i.e. for each $\varepsilon>0$ there ex... | {
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Factor irreducible polynomial in Z[x] and R[x] I've got a couple of problems from an old exam in abstract algebra that I have difficulty in understanding.
1) Write the polynomial $2x^3 - 10$ as a product of irreducible elements in $\mathbb{Z}[x]$, and list the irreducible elements in this factorization.
2) Write the po... | First of all, we notice that the polynomial does not have integer or rational roots (since thos would be one of -5, 5, -1, 1).
The polynomial is not irreducible in $Z[x]$. From the definition of irreducible polynomials: for $f(x), a(x), b(x)$ that belong to $R[x]$, $R$ ring, if $f(x)$ is not invertible, it is called i... | {
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Minimization of log-sum-exponential function subject to constraints. I would like to minimize the following function:
$f(x)=log(e^{-x_1}+..+e^{-x_n})$
Subject to:
$\sum_{i=1}^{n}{x_i}=1$
$0 \leq x_i \leq 1$
So far I have discovered the following: If all the $x_i$'s are equal, $f(x)=max(x_i)+log(n)$, but I have not been... | log is increasing, so the minimum occurs at the same place as the minimum of $\sum e^{-x_i}$; and $x\mapsto e^{-x}$ is convex, so $\sum e^{-x_i} \ge ne^{-\frac1n\sum x_i} = ne^{-1/n}$, with equality when all $x_i$ are equal.
| {
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Prove or disprove inequality $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}$. Let $a$, $b$ and $c$ be real numbers greater than $0$. Prove inequality $$\displaystyle{\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}}.$$
| By AM-GM and C-S we obtain:
$$\frac{a^4+b^4+c^4}{2abc}=\sum_{cyc}\frac{a^3}{2bc}\geq\sum_{cyc}\frac{2a^3}{(b+c)^2}=$$
$$=\frac{2}{a+b+c}\sum_{cyc}\frac{a^3}{(b+c)^2}\sum_{cyc}a\geq\frac{2}{a+b+c}\left(\sum_{cyc}\frac{a^2}{b+c}\right)^2=$$
$$=\frac{2}{a+b+c}\cdot\sum_{cyc}\frac{a^2}{b+c}\cdot\sum_{cyc}\frac{a^2}{b+c}\ge... | {
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"url": "https://math.stackexchange.com/questions/781405",
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Finding indempotents in a quotient ring
I am trying to find the nontrivial indempotents in the ring $\mathbb{Z_3}[x]/(x^2+x+1)$.
We can clearly see that $0,1$ are indempotents. I want to prove they are the only ones. Thus I am wondering if there is a way other than just brute force to show this. Currently I am down ... | Any element of the ring can be written (uniquely) in the form $ax+b$. If such an element is an idempotent, then $(ax+b)^2=ax+b$. Expand this, use the relation $x^2 = - x -1$, and equate coefficients to get a system of two equations for $a$ and $b$. If that system has a solution in $\mathbb{Z}_3$ then you have found ... | {
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Understanding proof that $\pi$ is irrational Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part:
Since $n!f(x)$ has integral coefficients and terms in $x$ of degree
not less than $n$, $f(x)$ and its derivatives (...) have integral
values for $x=0$; also for $x=\pi=\frac{a}... | The derivatives $f^{(i)}(x)$ have constant term $0$ for $i<n$ since each term of $f(x)$ has degree at least $n$, and thus $f^{(i)}(0)=0$. For $i\ge n$, each term will have a multiplier of $i!$ in front of it, and $n!\mid i!$, so the constant term is an integer.
| {
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Representation of dense Subset let $\mathcal B \subset \mathcal A$ a dense subset of a C*-algebra $\mathcal A$.
I have a representation for $\mathcal B$. Can I then conclude that this is somehow also a representation for $\mathcal A$?
By a representation for $\mathcal B$, I mean that I have a Hilbertspace $H$ and a *-... | Since your representation is a bounded linear map, it extends to $\mathcal A$. Then, using the density, you prove that it is also a $*$-homomorphism. Of course, as Yurii mentioned, this works if $\mathcal B$ is a subalgebra; if it is not, it is not really clear what "representation" would be.
| {
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Intersection curve between a circle and a plane - Stokes theorem What is the intersection curve between the circle
$$x^2+y^2=1$$
and the plane
$$x+y+z=0$$
If i am not wrong, I should solve the equation system
\begin{align}
x^2+y^2-1=0 \\
x+y+z=0
\end{align}
But I don't get the right curve. If i solve x in the first equ... | From $x+y+z=0$ we have $x=-(y+z)$, so $$x^2+y^2=(y+z)^2+y^2=1$$You can parametrize this equation by setting $y=\cos\theta$ and $y+z=\sin\theta$, i.e. $z=\sin\theta-\cos\theta$. Then our intersection is $$\boxed{\langle-\sin\theta,\cos\theta,\sin\theta-\cos\theta\rangle,\,\theta\in[0,2\pi)}$$
You could also have paramet... | {
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Derivative of $\ z=v^{3}u^{5} $ by chain rule and substitution Let $\ z=u^{3}v^{5} $ where $\ u=x+y, v=x-y $ Find $\ \frac{dz}{dy} $
For that I just did
$$\ \frac{dz}{dy}=\frac{dz}{du}\frac{du}{dy}+\frac{dz}{dv}\frac{dv}{dy} $$
And I got: $$\ 3(x+y)^{2}(y-5)^{5}+5(x+y)^{3}(x-y)^{4}$$ Is this right?
This is the solution... | $$\frac{dz}{du}=3u^2v^5$$ and $$\frac{du}{dy}=1$$ because for $\frac{du}{dy}$ you treat $x$ as a constant. Now you substitute the expressions for $u$ and $v$.
The second part is calculated similarly.
| {
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Solving two varibles system equation above $\mathbb{C}$ A bit emmbarrassed to ask this newbie question:
Let:
$$(1+i)x + y = 2$$
$$(1-i)x + iy = 0$$
Multiplying the first equation by $(-i)$ and summing the two equations, we have:
$$(2-2i)x + 2i = 0$$
How to get the final result of: $$x = {1 \over 2} - {1\over 2}i$$?... | If $(2-2i)x+2i=0$, then
$$x=\frac{-2i}{2-2i}=\frac{-i}{1-i}=\frac{-i(1+i)}{2}=\frac{1-i}{2}$$
Using the fact that $(1-i)(1+i)=|1-i|^2=2$
More generally, it's useful to remember
$$\frac{1}{a+ib}=\frac{a-ib}{a^2+b^2}$$
Or
$$\frac1z=\frac{\bar{z}}{|z|^2}$$
| {
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2D grid which is topologically equivalent to a sphere? I admit that my knowledge of topology is limited to the idea that a mug and doughnut are homomorphic since you can morph one into the other with a continuous deformation. I am a game dev working on a game which is played on a 2D grid. We were talking about how to w... | Depending on your criteria, this can either be done or not done. If the edges of the grid are labelled clockwise with $a,b,c,d$ then by gluing $a$ to $b$ and $c$ to $d$, you are left with something which is topologically a sphere, but its metric properties are not those of the usual sphere (constant positive curvature)... | {
"language": "en",
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Finding the Norm of an element in a field extension If I have a field extension of $\mathbb{Q}$ given by $\mathbb{Q}(\alpha)$ and the only thing I know about the primitive element $\alpha$ is it's minimal polynomial $p(x) = a_0 + a_1x + ... + x^n$ such that $p(\alpha) = 0$ how can I find the norm of an element $\beta \... | Think of the linear map from $\mathbb Q(\alpha)$ into itself given by
$$T_\beta(\gamma) = \beta \gamma $$
the norm of $\beta$ is just the determinant of the matrix associated to this linear map, for instance if you write
$$ \beta \alpha^j = a_{0j} + a_{1j}\alpha + \dots + a_{(n-1)j} \alpha^{n-1}\quad \quad j = ... | {
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If I pick a random rearrangement (in an equally likely manner) of RIVENDELL, what is the probabiltiy that it will start with R? If I pick a random rearrangement (in an equally likely manner) of RIVENDELL, what is the probability that it will start with R?
The probability that I pick R is 1/9, but there are repetitions ... | You don't need to.
Imagine two ways to rearrange the letters:
If you shuffle "real" letters (from Scrabble, for example), then you don't need to take repetitions into account, since your letter R is unique (just suppose you number your letters, in order to distinguish them).
If you take randomly a word from all possibl... | {
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integral of $\cos{x}\cos{(\sin{x})}dx$ I set $u=\cos{x}$, $du=-\sin{x}dx$, $dx=\frac{1}{-\sin{x}}$
$u\cos{(\sin{x})}dx$
$u [-\sin{(\sin{x})}\cos{x}(1)]+C$
$\frac{-(\sin{(\sin{x}})).(\cos{x})}{-sin{x}} + C$
tentative answer: $\sin{(\cos{x})} + C$
Another source says that the answer is $sin{(\sin{x})} + C$, and I wanted ... | Hint:
$$(\sin(\cos(x)))'=-\sin(x)\cos(\cos(x)),$$
$$(\sin(\sin(x)))'=\cos(x)\cos(\sin(x)).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/782643",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational
Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational.
The sum of a rational and irrational number is always irrational, that much I know - thus, if $n$ is a perfect square, we are finished.
However, is it not po... | By a simple 1-line proof: $\,\Bbb Q(\sqrt n\! +\! \sqrt 2) = \Bbb Q(\sqrt n,\sqrt 2),\, $ so $\,\sqrt n\! +\! \sqrt 2\in\Bbb Q\,\Rightarrow\sqrt 2\in \Bbb Q\,\Rightarrow\!\Leftarrow$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/782806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dedekind Sum Integrality Result Can we prove the following is always an integer?
$$6b\sum_{k=1}^bk\left\{\frac{ka}{b}\right\}$$
where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part operator.
UPDATE:
Through the calculation of several identities and going through heavy casework, I have proven the above formula; ... | Let $a,b \in \Bbb{R}$. Then each can be writ $a = a_i + a_f$, sim. for $b$, where $a_i = $ integer part, $a_f$ = fractional part. Then $ab = a_i b_i + a_i b_f + a_f b_i + a_f b_f$. Notice that if $(c \geq 0) \in \Bbb{Z}$ then $\{c + a\} = \{a\}$. Thus $\{ab\} = \{a_ib_f + a_f b_i + a_f b_f\}$. Notice that the frac... | {
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Proof that the combination formula actually gives you the number of combinations Ok, there's no problem in defining a binomial coefficient the way it this:
$$\binom {a} {b} = \frac{a!}{b!(a-b)!}$$
I can also prove to myself that if I have $n$ elements, like: $\{a_1, a_2, \ldots, a_n\}$ then the ways I can permute this,... | Another way is :
we want to know coefficient of $x^p $ in $(1+x)^n$.
$(1+x)(1+x)(1+x)...$
Clearly, we can choose any $p$ brackets to multiply $x$ and rest to be multiplied with $p$. Hence, coefficient of $x^p$ is $n\choose p$
Also, let's go back to the world of combinatorics. Suppose for some reason, you want to selec... | {
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$l^2+m^2=n^2$ $\implies$ $lm$ is always a multiple of 3 when $l,m,n,$ are positive integers. Let $l,m,n$ be any three positive integers such that $l^2+m^2=n^2$
Then prove that $lm$ is always a multiple of 3.
| Here is another approach. The general solution of this equation is:
$l = x^2 - y^2$, $m = 2xy$, $n = x^2 + y^2$. So: $l\cdot m = 2\cdot (x^2 - y^2)\cdot x\cdot y = 2\cdot (x - y)\cdot (x + y) \cdot x\cdot y$. From this, we have some cases to consdier:
*
*$3|x$ or $3|y$ then $3|l\cdot m$.
*$3 \not|x$ and $3\not|y$ t... | {
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Integrate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$ integrate $$\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
I've started by dividing this into two integrals:
$$\int_0^{1/2} \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
and
$$\int_{1/2}^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
Then I'm trying to find a primitive to
$$\int \frac... | $$x(1-x)=\frac{-(4x^2-4x)}4=\frac{1-(2x-1)^2}4$$
Set $2x-1=\sin\theta$
$$\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
$$=2\int_{-\frac\pi2}^\frac\pi2\frac{\cos\theta}{\cos\theta}\frac{d\theta}2$$
$$=\frac\pi2-\left(-\frac\pi2\right)$$
| {
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Matrix decomposition definition Wikipedia says "In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems."
But in my opin... | I have tried answering this question, It was long so I am putting it as link. I am not a mathematician but programmer, please provide some inputs.
| {
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Sequence of polynomials converging uniformly to $\frac 1z$ on semicircle in $\mathbb{C}$ I wish to construct a sequence of polynomials that converges uniformly on the semicircle $\{z: |z| = 1 , Re(z) \geq 0 \}$ to the function $\frac 1z$. Any help with this would be really appreciated, as I not sure even where to begin... | If there were a disk $D_R(a)$ containing the semicircle $K = \{ z : \lvert z\rvert = 1, \operatorname{Re} z \geqslant 0\}$ on which $f(z) = \frac{1}{z}$ is holomorphic, we could just use the Taylor polynomials of $f$ with centre $a$.
But, disks are convex, hence every disk containing $K$ also contains the origin, so it... | {
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How did Euler realize $x^4-4x^3+2x^2+4x+4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$? How did Euler find this factorization?
$$\small x^4 − 4x^3 + 2x^2 + 4x + 4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$$
where $\alpha = \sqrt{4+2\sqrt{7}}$
I know that he had s... | I like this rather old question. Here is a yet another possible way Euler could have taken:
Note that $\displaystyle x^4+ax^2+b$ can be factorized easily if $\displaystyle a^2-4b\geq 0$. If, however, $\displaystyle a^2-4b\leq 0$, then
\begin{align}
x^4+ax^2+b&=(x^2+\sqrt{b})^2-(x\sqrt{2\sqrt{b}-a})^2\\
&=(x^2+\sqrt{b}... | {
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How to prove that a function is continuous? Could you give me some hint how to solve this question:
Suppose $f$ is a differentiable function for all $0<x<1$,$f(0)=1,f'(x)>0$ in the given interval.
It is obvious that $f$ is continuous for all $0<x<1$, but is it continuous at $x=0$ ?
Thanks.
| First, knowing the definition of a continuous function helps.
Quoting wikipedia here,
A function f(x) is continuous at point c if the limit of f(x) as x approaches c is f(c).
You have some facts stated earlier:
*
*f(x) is differentiable in 0 < x < 1
*f(0) = 1
*f'(x)>0 in (0,1)
We should check the limit of f(x) a... | {
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No. of equilateral triangles required to completely fill a bigger equilateral triangle $\triangle ABC$ is equilateral with side length=2.1cm
Smaller equilateral triangles with side length=1cm are placed over $\triangle ABC$ so that it is fully covered. Find the minimum number of such small triangles.
I am not getting i... | This is not answer to the question.
I just post a rough sketch showing a way of how the 6 equilateral triangles can be arranged to cover the original.
| {
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Meromorphic on unit disc with absolute value 1 on the circle is a rational function. Let $f$ be a meromorphic function on the open unit disk such that $f$ has a continuous extension to the boundary circle.Suppose $f$ has only poles in the open unit disc and suppose $|f(z)|=1$ for all $z$ with $|z|=1$.Prove that $f$ is ... | Since the closed unit disk is compact, $f$ can have only finitely many zeros and poles in the unit disk. Let $k$ be the order of $f$ in $0$, that is, $f(z) = z^k\cdot g(z)$ where $g$ is holomorphic in a neighbourhood of $0$ with $g(0) \neq 0$. Let $\zeta_1,\dotsc, \zeta_k$ be the zeros of $f$ in the punctured unit disk... | {
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Finding order of group intersection Let $G$ be cyclic group, and $H_1, H_2$ subgroups. $|H_1|=15$, and $|H_2|=25$ Find $|H_1 \cap H_2|$.
So this is the solution we were presented at recital:
$|H_1|$ and $|H_2|$ divides $|G|$, so $|G|=lcm(15,25)=75k$, $k \in \mathbb N$. Since $G$ is cyclic, so are its subgroups, so $H_1... | $\;G\;$ cyclic and finite (why?) , so it has one unique subgroup of each order dividing its order.
$$|H_1\cap H_2|\;\mid \;15\,,\,25\implies |H_1\cap H_2|=1,5$$
But there's a subgroup of $\;G\;$ of order $\;5\;$, and since any subgroup of a cyclic one is cyclic, this subgroup of order $\;5\;$ is a subgroup both of $\;... | {
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Is there $f$ in $\operatorname{Hom}(\mathbb{Q},\mathbb{Q})$ with kernel $\mathbb{Z}$? Is there a group homomorphism from $\mathbb{Q}$ (the group of rationals) to $\mathbb{Q}$ whose kernel is $\mathbb{Z}$?
| The hint in in the same vein as that given by Tobias in the comments above, but a bit more explicit. Consider where $1/2$ would be sent to, given that $1/2 + 1/2 = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/783926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$2^n+1 =xy \implies (2^a|(x-1) \iff 2^a|(y-1))$ I'd like my proof to be verified of the following exercise from Niven's The Theory of Numbers.
Section 1.1 Problem 52: Suppose $2^n+1=xy$, where $x$ and $y$ are integers $>1$ and $n>0$. Show that $2^a|(x-1)$ if and only if $2^a|(y-1)$.
Proof: Suppose $2^a|(x-1)$. Then $(x... | Let $\displaystyle x=A2^a+1,y=B2^{a+c}+1$ where positive integers $A,B$ are odd and $c\ge0$
$\displaystyle\implies xy=AB2^{2a+c}+A2^a+B2^{a+c}+1=2^n+1$
$\displaystyle\implies AB2^{a+c}+A+B2^c=2^{n-a}$ which is even as $n>a$ as $x,y>1$
but $\displaystyle AB2^{a+c}+A+B2^c$ is odd if if $\displaystyle c>0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/784012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Numerical evaluation of polynomials in Chebyshev basis I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and -1, where Chebyshev polynomials change rapidly, is also required... | Evaluating polynomials of arbitrarily large degree in a Chebyshev basis is practical, and provably numerically stable, using a barycentric interpolation formula. In this case, extended precision isn't needed, even for order 1,000,000 polynomials. See the first section of this paper and the references, or here (Myth #2)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Proof metric space with distance function
Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii).
Let $X$ be the Set of all complex sequences.
$$
d((a_n),(b_n)) := \sum^\infty_{i=0} \frac{1}{2^{i+1}}\frac{\left | a_i-b_i \right |}{1+\left | a_i-b_i \right|}... | Add on Daniel Fisher's comment
When $f(t)= \frac{t}{1+t}>0$ is concave for $t>0$, then $$ f(|a-b|)
+f(|b-c|)\geq f(|a-b| + |b-c|)\geq f(|a-c|)$$ where $a,\ b,\ c$ are
points in a metric space $(X,d=|\ |)$. Hence $(X,\frac{d}{1+d})$ is
a metric space.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that [F(a,b):F] is finite Suppose F $\subset$ L is a field extension, a, b $\in$ L are algebraic over F. Prove that [F(a, b): F] is finite.
Unfortunately I don't even know where to begin with this one, other than establishing the tower of extensions:
F $\subset$ F(a) $\subset$ F(a, b)
What does $F(a)$ even look l... | Hints:
*
*What does it mean that $a$ is algebraic over $F$?
(Note that $F(a)=\{u+v\cdot a\,\mid\,u,v\in F\}$ only if the degree of $a$ over $F$ is $2$ [or if $a\in F$ already].)
*Is $b$ also algebraic over $F(a)$?
| {
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Solve: $\tan2x=1$ Are there any errors in my work? Thanks in advance! (Sorry for the bad format. I'm still new to this)
$\tan2x=1$
$\frac{2\tan x}{1-\tan^2x}=1$
$2\tan x=1-\tan^2x$
$0=1-\tan^2x-2\tan x$
$0 =-\tan^2x-2\tan x +1$
$0=\tan^2x+2\tan x-1$
$\frac{-(2)\sqrt{2^2-4(1)(-1)}}{2(1)}$
$x=0.4142, x=2.4142$
$\tan^{-... | $$
\begin{align}
\tan 2x&=1\\
\tan 2x&=\tan(180^\circ n+45^\circ)\quad\Rightarrow\quad n\in\mathbb{Z}\\
2x&=180^\circ n+45^\circ\\
x&=90^\circ n+22.5^\circ.
\end{align}
$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basic Combinatorics I have a basic combinatorics question I am unsure how to complete, the question is as follows:
A company has 9 people in Office A, 6 in Office B and 3 in Office C. A new team of 6 people is to be formed.
How many ways can the new team be formed if:
a) The team includes two members from each office
b... | a) There are $\binom{9}{2}$, $\binom{6}{2}$, and $\binom{3}{2}$ ways to choose $2$ persons from office $A, B, C$ respectively. So there are $\binom{9}{2}\cdot \binom{6}{2}\cdot \binom{3}{2}$ ways of choosing $6$-person teams with $2$ members from each office.
b) If $A$ has $2$ members, then the other $4$ members are ch... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does the field of mathematical biology study? I like math and bio and I want to study both. There is a subject called mathematical biology. What is it? What does a mathematical biologist do? What institutions have good mathematical biology programs?
| Mathematical biology asks many different questions. An intro course will look at things like population genetics (the study of the dynamics of gene propagation in populations) and basic bioinformatics. But any use of mathematical models in biology is in this field, and it can get fairly deeply mathematical. René Tho... | {
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Show the following is a subspace and find its dimension If $V=K^{2009}$ where $K$ is a field. Show $W=\{(a,b,a,b,a,b,...)|a,b \in K \}$ is a subspace and find $dim_{K}W$.
My Attempt;
For $W$ to be a subspace of $V$ two propeties must hold;
Closure by Additivity
Closure by Scalar Multiplication
For the first assume t... | Hint: consider the span of the vectors $(1,0,1,0,1 \dots)$ and $(0,1,0,1,0, \dots)$.
| {
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Basic question on Implication Could anyone conceive of any predicates and Universe ( in mathematics, in the world, etc ) where we should use $\exists x ( P(x) \to Q(x) )$, and not necessarily $\forall x ( P(x) \to Q(x) )$ ?
I was thinking of some sittuation where the property of P implies property of Q for at least som... | It is worth remarking that a claim of the form $\exists x ( P(x) \to Q(x) )$ is typically unlikely to be interestingly informative and worth saying.
Why so?
Well, $\exists x ( P(x) \to Q(x) )$ is true so long as $P(a) \to Q(a)$ is true for some case where $a$ newly dubs an element of the domain. But that material cond... | {
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Real-valued Discrete Fourier Transform I have sequence of $N$ real numbers: $\mathbf{x} = (x_0, x_1, \ldots, x_{N-1})$.
Discrete Fourier Transform (DFT) is defined as
$$
X_k = \sum_{n=0}^{N-1} x_n e^{-i 2\pi k \frac{n}{N}}, \quad (k=0,1,\ldots,N-1).
$$
Coefficients $X_0,X_1,\ldots,X_{N-1}$ are complex-valued at all.
H... | The condition $\overline{X_k} = X_k$ can be written as
$$ \sum_{n=0}^{N-1} \bar{x}_n e^{ i 2\pi k \frac{n}{N}} = \sum_{n=0}^{N-1} x_n e^{-i 2\pi k \frac{n}{N}} \tag{1}$$
The change of index $n \mapsto N -n$ turns the left sum in (1) into
$$ \sum_{n=1}^{N } \bar{x}_{N-n} e^{ - i 2\pi k \frac{n}{N}} \tag{2}$$
Com... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The analytic spread of an ideal Let $(R,\mathbb{m})$ be a Noetherian local ring and $I$ an ideal of $R$. Let $t$ be an indeterminate over $R$. The analytic spread $l(I)$ of $I$ is defined to be the Krull dimension of the ring $R[It]/\mathbb{m}R[It]$. Let $x$ be another indeterminate over $R$, then $\mathbb{m}R[x]$ is a... | Let $A=R[It]/\mathfrak{m}R[It]$. Let $k=R/\mathfrak{m}$.
Let $B=R[x]_{\mathfrak{m}R[x]}$, $J=IB$, $C=B[Jt]/\mathfrak{m}B[Jt]$.
We need to show that the Krull dimensions of $A$ and $C$ are same.
Then $B/\mathfrak{m}B=k(x)$, $A=R/\mathfrak{m}\otimes_R R[It]$ and
$C=B/\mathfrak{m}B\otimes_BB[Jt]$.
Since $B$ is flat over ... | {
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Find radical of hermitian form $\langle , \rangle _A$
Determine the radical of the hermitian form $\langle , \rangle _A$ over the field $\mathbb{C}^3$, where $$A = \begin{pmatrix} 1 & -i & -i \\ i & 2 & 1 \\ i & 1 & 1 \end{pmatrix}$$
Would it be sufficient to calulate the kernel of the matrix $A$ or is there another ... | I suppose you define $\langle , \rangle _A$ as $\langle \vec x , \vec y \rangle _A = \vec x^TA\overline{\vec y} $. I emphasize this as it is sometimes (particularly in physics) defined as $\langle \vec x , \vec y \rangle _A = \vec x^HA\vec y$.
The (left) radical is defined as $L_A=\{\vec x \in \mathbb C^3 | (\forall ... | {
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Statistics example 5 Joint distribution of random variables X and Y is
$$f (x, y) = e^{-(x+y)} $$ for $$ 0<x,y<∞$$ and 0 otherwise.
Determine the distribution densities of random variables U = X + Y and V = X/X + Y.
| There are several ways to do this:
a) CDF method: find $P(X+Y < u)$ by integration
b) you can notice that $X,Y$ are independent Exponential, and you probably studied what the distribution of their sum is...
c) Jacobian method: see e.g. here
http://web.eecs.umich.edu/~aey/eecs501/lectures/jacobian.pdf
| {
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General bibliography for the work of Grothendieck I'm reading the first volume of Scharlau's Grothendieck biography (eagerly anticipating the other two/three volumes) and the Grothendieck-Serre correspondence as part of a historical-philosophical side project. I find myself regularly digging up papers of his, and somet... | You can find here a collection of all the minor (in length) works by Grothendieck with links
Works
Included partial transcriptions of unpublished works
| {
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Does $\sum_{n=0}^{\infty}\frac{4^n}{4^{n+1}}$ Diverge Or Converge? I am told it diverge, however surely;
$$\frac{4^n}{4^{n+1}} = \frac{4^n}{4\cdot 4^n} = \frac{1}{4}$$
| Since
$$\lim_{n \to \infty} \frac{4^n}{4^{n+1}} = \frac{1}{4}\neq 0$$
the general term test says that this series is divergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/785290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Krull dimension of a direct limit of modules Suppose that $\left\{M_{\lambda}\right\}$ is a directed system of $R$-modules, all of them with finite Krull dimension, $n$. Is it true that $\dim\varinjlim M_{\lambda}\leq\sup\left\{\dim{M_{\lambda}}\right\}$?
Thank you.
| Set $M := \varinjlim_{\lambda \in \Lambda} M_{\lambda}$. Let $M'_{\lambda} \subset M$ be the image of the canonical maps $M_{\lambda} \to M$; then $\operatorname{Supp} M'_{\lambda} \subseteq \operatorname{Supp} M_{\lambda}$. Thus after replacing the $M_{\lambda}$ by $M'_{\lambda}$, we reduce to the case when $M_{\lambd... | {
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Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.
Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = hcf(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.
We know that if $a|c$ and $b|c$ then $a\cdot b\cdot s=c$ (for som... | Let us put it this way: Define a', b' so that a=a'd and b=b'd, where d=GCD(a,b). Then a|c means c = a'ds for some s, and b|c means c = b'dt for some t. But that doesn't mean ab|c because we could have a'|t and b'|s, i.e c = a'b'du for some u. I.e in dividing c, a and b are sharing d between them.
Of course, if c = a... | {
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Can one work with any classes of numbers in a proof of number theory? Can one work with any classes of numbers, like natural, integer, rational, real and complex, in a proof of number theory, as long as the result tells something about the integers ? Or should the result be proven using integer-operations only (we can ... | Why on earth would you impose such restrictions? There's only one thing you should require of a proof: that it be logically valid. The end.
Not only is there absolutely no reason to apply a restriction like that, but modern number theory very frequently steps outside the realm of ordinary integers. Analaytic number the... | {
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Riemann surface associated with complete analytic function of $(z^2-1)^{1/3}$ I'm trying to define an analytic function on '$\mathbb{C}$' of the form $f(z)=(z^2-1)^{1/3}$, i.e. I first remove two semi-infinite rays $l_1$ and $l_2$, one going from $1$ to $\infty$ along the positive reals, one from $-1$ to $\infty$ along... | A more direct way, without any glueing, is to look directly at the locus $E$ of $y^3=z^2-1$ in $\mathbf C^2$. This equation defines a smooth affine curve, and the projection $(y,z) \mapsto z$ is locally an isomorphism away from the points $(0,1)$ and $(0,-1)$. Moreover this projection is generically three-sheeted. On $... | {
"language": "en",
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Why is the fundamental matrix of a linear system of ODEs always invertible? Why does $\phi^{-1}(0)$ exist, where $\phi(t)$ is the fundamental matrix of the system $\dot{x}=A(t)x$, $x \in \mathbb{R}^n$? I am not able to figure this out.
| Consider the ODE $\dot x = - A(T-t) x$. Its fundamental solution $\psi(t)$ satisfies $\psi(T) \phi(T) = \phi(T) \psi(T) = \text{Identity}$. (In other words, I am solving the equation $\dot x = A(t) x$ backwards in time.)
| {
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Limit of a function with square roots I've got the following limit to solve:
$$\lim_{s\to 1} \frac{\sqrt{s}-s^2}{1-\sqrt{s}}$$
I was taught to multiply by the conjugate to get rid of roots, but that doesn't help, or at least I don't know what to do once I do it. I can't find a way to make the denominator not be zero wh... | $$\frac{\sqrt{s} - s^2}{1 - \sqrt{s}} = \frac{\sqrt s \left(1 - s^{3/2}\right)}{\left(1 - s^{3/2}\right)\left(1 + s^{3/2}\right)} = \frac{\sqrt s}{1 + s^{3/2}}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Convexity and Jensen's Inequality for simple functions Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are able to boost $\varphi$'s convexity property up to look like Jens... | You are basically done. You just need to apply the inductive hypothesis on your last term.
| {
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"timestamp": "2023-03-29T00:00:00",
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Zero variance Random variables I am a probability theory beginner. The expression for the variance of a random variable $x$ (of a random process is
$$\sigma^2 = E(x^2) - (\mu_{x})^2$$
If $E(x^2) = (\mu_{x})^2$, then $\sigma^2 = 0$. Can this happen ? Can a random variable have a density function whose variance (the sec... | The variance
$$
E(X^2)-E(X)^2=E(X-E(X))^2
$$
is equal to $0$ if and only if $X$ is equal to $E(X)$ in all of its support. This can only happen if $X$ is equal to some constant with probability $1$ (known as a degenerate distribution).
| {
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Integration and differentiation of an approximation to a function - order of approximation For my research I am working with approximations to functions which I then integrate or differentiate and I am wondering how this affects the order of approximation.
Consider as a minimal example the case of $e^x$ for which integ... | Suppose that $f(x)$ has a Taylor series expansion about $x=0$ with a radius
of convergence $r>0$. For convenience we set $f(0)=1$.
We write
$$
f(x)=1+xf^{(1)}(0)+\frac{x^{2}}{2}f^{(2)}(0)+\mathcal{O}%
(x^{3})=1+xf^{(1)}(0)+\frac{x^{2}}{2}f^{(2)}(0)+g(x),
$$
where, in a neighbourhood of $0$,
$$
|x^{-3}g(x)|<c.
$$
Then, ... | {
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Trigonometry problem cosine identity Let $\cos^6\theta = a_6\cos6\theta+a_5\cos5\theta+a_4\cos4\theta+a_3\cos3\theta+a_2\cos2\theta+a_1\cos\theta+a_0$. Then $a_0$ is
(A) $0$ (B) $\frac{1}{32}$ (C) $\frac{15}{32}$ (D) $\frac{10}{32}$
Any hints on how to approach this?
| To solve this question, you have a choice of using
(1) Top level – Through Fourier Analysis
(2) Advanced level – By complex number approach
(3) Intermediate level – Via compound angle formulas
(4) Elementary level (the most painful method) – Use brute force substitution
[Classifying into these levels is purely accordi... | {
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To solve $ \frac {dy}{dx}=\frac 1{\sqrt{x^2+y^2}}$ How do we solve the differential equation $ \dfrac {dy}{dx}=\dfrac 1{\sqrt{x^2+y^2}}$ ?
| Since this problem is not amenable to the standard repertoire of 1st order techniques, we might use some asymptotic analysis to understand the qualitative features of this ODE. Firstly, the derivative is always positive and bounded as $x$ or $y$ or both approach $\pm\infty$. The limiting differential equation in all of... | {
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the presentation of $SL(2,\mathbb{Z})$ There is a natural presentation $SL(2,\mathbb{Z})\hookrightarrow GL(2,\mathbb{R})$, are there other presentations in real dimension 2? Or there is a classification of all the presentation of $SL(2,\mathbb{Z})\to GL(2,\mathbb{R})$? Thanks in advance.
| To supplement the answer of @Dietrich Burde, the representations $PSL(2,\mathbb{Z}) = \mathbb{Z}/2 * \mathbb{Z}/3 \to PSL(2,\mathbb{R})$ correspond bijectively to ordered pairs of elements $X,Y \in PSL(2,\mathbb{R})$ such that $X$ has order $1$ or $2$ and $Y$ has order $1$ or $3$; equivalently, $X$ is the identity or h... | {
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How many different right triangles are possible with the shorter side of odd length? I was trying to solve this problem but unable to figure it out completely.
I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one help me?
Here is the problem link.
... | If $a$ is odd and $a^2+b^2=c^2$ then $a^2=c^2-b^2=(c+b)(c-b)$. So $c-b$ must be among the smaller factors of $a^2$. The number of solutions is thus $\frac12$ times the number of divisors of $a^2$.
| {
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Find the product $(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$ Let $1,$ $a_i$ for $1 \leq i \leq 6$ be the different roots of $x^7-1$.
Then find the product:
$(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$
I don't know how to proceed.
| Hint. Factorising,
$$x^7-1=(x-1)(x-a_1)\cdots(x-a_6)\ .$$
Dividing by $x-1$ gives
$$x^6+x^5+\cdots+1=(x-a_1)\cdots(x-a_6)\ .$$
I'm sure you can finish the problem from here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Abelian groups of order n.
Is there a number $n$ such that there are exactly 1 million abelian groups of order $n$?
Can anyone please explain. I would yes because numbers are infinitive, and so any number n can be expressed as a direct product of cyclic groups of order n.
Can anyone please help me understand.
Thank... | If $n=\prod_{i=1}^rp_i^{k_i}$, then the number of distinct abelian groups of order $n$ is given by
$$
\prod_{i=1}^rp(k_i),
$$
where $p(k)$ denotes the number of partitions of $k$. Now $10^6=p(k_1)\cdots p(k_r)$ has to be solved. But $p(2)=2$ and $p(4)=5$, so we can solve it.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Why is this polynomial a monomial? Let $p$ be a polynomial of degree $n$ such that $|p(z)| = 1$ for all $|z| = 1$.
Why is it that $p(z) = az^n$ for some $|a| = 1$?
I've noticed that we could easily prove this by induction if we could show that 0 was a root of $p$. My guess is that Rouche's theorem and/or the Maximum... | ...some comments that don't necessarily go anywhere (because I need to go).
*
*define $p(e^{i\theta})=e^{iP(\theta)}$ with $P:\mathbb{R}\rightarrow \mathbb{R}$, $2\pi$-periodic and of course $p((e^{i\theta})^n=p(e^{in\theta}))=e^{iP(n(\theta))}$
*let $p(e^{i\theta})=\sum_{\ell=0}^n\alpha_\ell e^{i\ell\theta}$ and t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Elevator Probability Question There are four people in an elevator, four floors in the building, and each person exits at random. Find the probability that:
a) all exit at different floors
b) all exit at the same floor
c) two get off at one floor and two get off at another
For a) I found $4!$ ways for the passengers... | Your calculation for $c)$ is correct, though, as the comments and answers show, you seem not to have expressed it in a readily comprehensible manner.
I would express what I understand your argument to be as: There are $3$ ways to split the $4$ people into $2$ groups. Then we can treat those two groups like $2$ people a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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$\sup A = \inf B$ implies $\forall\varepsilon>0.\exists a\in A, b\in B. b-a<\varepsilon$ Let $A, B$ two sets such that $\sup A = \inf B$. Is it right that:
$$
\forall \varepsilon > 0. \exists a\in A, b\in B. b-a<\varepsilon \quad ?
$$
The question doesn't mention the sets are densed, but that was probably the intention... | You're correct, but being a bit silly.
The thing you are being silly about is the "denseness" thing.
This is unnecessary; definition of the supremum and infemum imply that for any
positive $\epsilon$,
there is an element at most $\epsilon$ away from the supremum (or infemum) in the set.
(This element may, of course, be... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Wolf cabbage and goat using dijkstra. A farmer has to cross a river with a wolf, a goat and a cabbage.
He has a boat, but in the boat he can take just one thing.
He cannot let the goat alone with the wolf or the goat with the cabbage. It’s obvious why.
What is the solution?
Ok So I know the two solutions and I arrived ... | Let $W, G, C$ denote wolf, goat, cabbage.
Let $F$ be the farmer.
Here are all the possible states:
\begin{align*}
\newcommand{\sep}{\; \mid \;}
&1 & WGCF &\sep &\text{(start)} \\
&2 & &\sep WGCF &\text{(goal)} \\
&3 & WCF &\sep G \\
&4 & G &\sep WCF \\
&5 & WGF &\sep C \\
&6 & C &\sep WGF \\
&7 & GCF &\sep W \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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What is Derivative of order more than one? So, Its easy to geometrically interpret the first order derivative in a graph by drawing a tangent to the curve of any function showing derivative as same the slope of the line but how can we draw a second order derivative on the same graph or how can we visualise this derivat... | The second derivative of a function at a point will tell you about how the slope of the tangent line tends to change in a neighborhood of that point.
If the slope of the tangent line increases, then the graph of the function is convex in a neighborhood of that point. If the slope of the tangent line decreases, then the... | {
"language": "en",
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"question_score": "2",
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Showing that the curves defined in the $xy$ plane by $u(x,y)=1$ and $v(x,y)=1$ cross at right angles at the origin. Suppose $f$ is an entire function with $f(0)=1+i$. Let $u(x,y)=Re(f(x+iy))$ and $v(x,y)=Im(f(x+iy))$.
A) Show that the function $u$ is a harmonic function of $x$ and $y$.
B) Show that the curves defined ... | For $A$, use the Cauchy Riemann Eq : $u_x = v_y$ , $u_y = - v_x$ So
$$ u_{xx} = v_{yx} = (v_{x})_y = - ( u _y )_y \implies u_{xx} + u_{yy} = 0$$
For $B$, the curves with $u=1$ mean $f(x,y) = 1 + i v(x,y)$, curves with $v=1$ mean $f(x,y) = u(x,y) + i$. What can you say about these curves if $f(0,0) =1 + i$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is distributivity "the only way to reconcile addition and multiplication" Today my prof called distributivity "amazing". I asked him why he thought so, and he replied "it's the only way to reconcile addition and multiplication." It was a tangential question, so I didn't ask him to elaborate, despite having no idea ... | Let $\mathbf{N}$ denote the structure $(\mathbb{N},+,\times,0,1).$
Then $(\mathbb{N},+,0)$ satisfies the following identities.
*
*$+$ is associative
*$+$ is commutative
*$0$ is left-neutral for $+$
*$0$ is right-neutral for $+$
In fact, it turns out that all the identities that hold for $(\mathbb{N},+,0)$ can b... | {
"language": "en",
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Prove that $A\subseteq B\Longleftrightarrow A\cap B = A$ In set theory logic mathematics. How would i do the proof for: $A\subseteq B\Longleftrightarrow A\cap B = A$
| First Part:
Suppose A⊆B. Then if for any x belonging to A, then x belongs to B.
Now suppose that x belongs to (A∩B). So, x belongs to A, and x belongs to B also. Thus, x belongs to B. Since x comes as arbitrary, for any x if x belongs to (A∩B), x belongs to A also.
Suppose that x belongs to A. Since A⊆B, x belo... | {
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"source": "stackexchange",
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Arrangement of Numbers to Get a Common Sum I'm having trouble with a math problem. I need to arrange 6 numbers on a certain diagram:
At every intersection of two circles, I have to put one of these six numbers: 4, 5, 5, 6, 6, or 7. The sum of all of the numbers on each circle must be the same. Is it possible to arrang... | Edit: I misunderstood the question. Anyway, I'm keeping my answer here in case it provides additional ideas...
The sum of the numbers in each circle is 17.
| {
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"timestamp": "2023-03-29T00:00:00",
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annihilator method confusion I have a final in the morning and I am extremely confused on the annihilator method.
I have been googling different explanations all night and I just dont get it at all. I am looking at an example:
$$\ddot{y}+6\dot{y}+y=e^{(3x)}-\sin(x)$$
now I get that the annihilator of the $e$ term is $(... | $(D-\lambda)$ annihilates $e^{\lambda t}$, whether $\lambda$ is real or complex. Normally, if $\lambda$ is complex, you want to use $(D-\lambda)(D-\overline{\lambda})$ so that the resulting operator will annihilate $e^{\Re\lambda t}\cos(\Im\lambda t)$ and $e^{\Re\lambda t}\sin(\Im\lambda t)$, where $\overline{\lambda}$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Computability: why any m-degree a is denumerable? The problem printed in Cutland 9-2.9-6 is wrong, it should be countable, not denumerable
m-degree is an equivalence class of the relation $\equiv_m$(many-one equivalent).
Question:
Why any m-degree a is denumerable?
My thoughts:
Denote m-degree of A as $d_m(A)=\{B:A\e... | So first, according to your definition, $d_m(A)$ is a set of subset of $\mathbb{N}$. So I think it is confusing to write $d_m(\emptyset)=\emptyset$. Instead you should write $d_m(\emptyset)=\{\emptyset\}$. (And $d_m(\mathbb{N})=\{\mathbb{N}\}$).
Now, for any $X$, the set $\{Y\ :\ X \geq_m Y\}$ is countable, essentially... | {
"language": "en",
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"source": "stackexchange",
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Trying to prove that there are no p and q such that $|\sqrt5 - p/q| < 1/(7q^2)$. Like the title says, I'm having trouble proving that there are no integers p and q such that
$|\sqrt5 - p/q| < 1/(7q^2)$. I was given the hint that $|(q\sqrt5 - p)(q\sqrt5 + p)| \geq 1$, but I don't quite know how that helps...
Thanks!
| I think you are my classmate, since we had submitted our homework a few hours ago, I would like to share my solution. Because we are studying continued fractions now, so my solution is based on it.
First, we know if we can find some $(p, q)$ pair, and $q \not= 0$, by Theorem 12.18 or Corollary 12.18.1 in the textbook(E... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finitely presentable objects After introducing the notion of finitely presentable object as an object $A$ such that ${\sf Hom}(A, -)$ preserves directed colimits, an "explicit" form of it is given:
$A$ is finitely presentable iff $(B, \bar{b}_i)$ is the colimiting cone for a directed diagram $(B_i, b_{ij}$, then for ev... | The point is that the directed colimit of $\hom(A,B_i)$'s in $\Bbb{Set}$ is the following quotient set:
$${\coprod_i\hom(A,B_i)}\ \ /\sim$$
where $f_i\sim f_j$ for some $f_i\in\hom(A,B_i)$ and $f_j\in\hom(A,B_j)$, if
$$\exists k\ge i,j:\ b_{ik}\circ f_i=b_{jk}\circ f_j\,.$$
Try to use this to get to the explicit form... | {
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Find minimum value of the expression x^2 +y^2 subject to conditions Find the values of $x,y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$.
Tried Cauchy-Schwarz and AM - GM , unable to do.
| Hint: take a look at the picture below, and all the problem will vanish...
In fact the picture shows the circle of equation $(x+5)^2 +(y-12)^2 =14$, and the line passing trough its centre and the origin. The question asks the minimum length of the segment whose extremities are the origin and a point on the circumferen... | {
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Random (Union Find) Spanning Tree, probability of resulting with two unconnected halves before the last step? Let N be a large even positive integer.
We start with a set of singletons from {1}, {2} ... to {N}.
In each step we randomly pick two integers and merge the sets that contain them. We continue until only one se... | Number of ways to break the set $\{1,2,\dots ,n\}$ into singletons (exactly the reverse process of what is asked) is
$$a_n=\prod_{k=2}^{n}\binom{k}{2}=\frac{n!(n-1)!}{2^{n-1}}$$
corresponds with this sequence (as David Callan's comment on the page suggests)
This comes from the recursion: $a_n=\binom{n}{2}a_{n-1}$ and a... | {
"language": "en",
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Let $f: V_3 \rightarrow V_3$ be the function such that $p(X) \mapsto p''(X)$, calculate the eigenvalues of f Let $V_3$ be the vector space of all polynomials of degree less than or equal to 3. The linear map $f: V_3 \rightarrow V_3$ is given by $p(X) \mapsto p''(X)$. Calculate the eigenvalues of f.
First of all I'm not... | You only need to delete the part $ "\; = \lambda\; "$ from $det(A-\lambda I) = (-\lambda)^4 = \lambda$ and the remaining is correct. You can also notice that $A^2=0$ which means that $A$ is a nilpotent matrix so the only possible eigenvalue is $0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$ So guys, how can I evaluate and prove that $$\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0.$$ Any ideas are welcomed.
$n!!$ is the double factorial, as explained in this wolfram post.
| $\lim\limits_{n\to \infty}\dfrac{(2n-1)!!}{(2n)!!}=0\quad$ and $\quad\lim\limits_{n\to \infty}\dfrac{(2n+1)!!}{(2n)!!}=\infty.~$ Where it really gets interesting, however,
is when we attempt to evaluate their product. Their polar-opposite tendencies will cancel each
other out, yielding $~\lim\limits_{n\to \infty}\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/788096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Symmetric chain decomposition of cartesian product
Let $A,B$ be finite posets. Define '$\leq$' for the cartesian product
$A\times B$ as $(a,b) \leq (a',b') \Leftrightarrow a \leq a' \wedge b
\leq b' \forall a \in A, b \in B $.
Show that there is a symmetric chain decomposition (partition into
symmetric chains)... | Do this in a few steps:
(1) Take the assumed symmetric chain decompositions of A and B, say A_1, A_2, . . . , A_s for
A and B_1, B_2, . . . , B_t for B and create the s x t chain products A_i x B_j ;
(2) Argue that each of the chain products is a "symmetric" and "saturated" subposet of A x B;
(3) Prove that a product o... | {
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Showing $2$ is not definable in $(\mathbb{Q},+)$. As stated, I'm to show that $2$ is not definable in $(\mathbb{Q},+)$.
I tried proving it by contradiction by showing that if $2$ were definable, then we could define $\mathbb{N}$ and multiplication over $\mathbb{N}$, which would be impossible because the automorphism $... | Hint: Show that any definable element in a structure $M$ is fixed by all automorphisms of $M$. Can you find an automorphism of $\mathbb{Q}$ which does not fix $2$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same? Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form?
Positive defined quadratic form is
A(x,y), that all it's corner minors are posit... | Let $z=Ax$. $A$ is positive definite and hence also invertible. Thus, it follows that $
x=A^{-1}z.
$
Premultiply by $z'$ to get $z'x=z'A^{-1}z$. But, $z'=x'A'$ so what you have is actually
$$
x'A'x=z'A^{-1}z>0
$$
since $A=A'$ and $x'A'x>0$ by the nature of semi definite matrices.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/788356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Finding roots of function Consider the function $$f(x)=(2x-9) \cdot 2 \cdot e^{\frac{x^3}{3}-9x+ \frac{46}{3}}$$ Now, the only root to this function is $x=9/2$
I find it quite easy to find this exact root, I will start by saying. Usually i would solve this kind of problem using a CAS. But I would like to know if any of... | $$e^{\text{anything}} \ne 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/788444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing ${Mf(z) \over 10M - f(z)}$ is bounded if $\text{Re}(f(z)) < M$ Data: Let $f(z)$ be analytic on $\Omega - \{a\}$. Suppose further that $\text{Re}(f(z)) < M$ for some $M \in \mathbb{R}_{> 0}$.
Define $g(z)$ as follows:
$$
g(z) = {Mf(z) \over 10M - f(z)}
$$
Since $\text{Re}(f(z)) < M$, we have that the denominato... | @Robert Israel
We have that
$$
|g(z)| = {|Mf(z)| \over |10M - f(z)|} = \underbrace{|-M| + {|10M^2| \over |10M - f(z)|}}_{\text{via (*) below}} \le {10M^2 \over 9M} + M = {10 \over 9} M + M < 3M
$$
via
\begin{equation*}
\tag{*}
{M f(z) \over 10M - f(z)} = {-M(10M - f(z)) + 10M^2 \over 10M - f(z)} = -M + {10M^2 \ove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/788534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Recurrence Relation for binary sequences
How can I find the recurrence relation with a) no block of 2 consecutive 0's and b)no block of 3 consecutive 0's.
Please help me understand this material,
detailed explanation will be much appreciated,
Thanks
| We deal with no $3$ consecutive $0$. The same approach will work for no $2$ consecutive $0$, but is simpler.
Let $a_n$ be the number of binary strings of length $n$ with no $3$ consecutive $0$. Call such a string a good string. Let $n\gt 3$.
A good $n$-string with $n\gt 3$ can be of three types:
Type 1: ends in a $1$.
... | {
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"timestamp": "2023-03-29T00:00:00",
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Why is calculus focused on functions? A curve of a hyperbolic spiral for example is not a graph of a function. But the concept of continuity and finding its slope for example, which are in calculus applies to it. So why is calculus typically phrased in terms of functions only?
Is it for sake of simplicity of didactics,... | I don't know why you were downvoted. If you were in my calculus class, I would be very happy that you question the material like this and I would encourage it.
One possible answer is that elementary calculus is almost exclusively preoccupied with local phenomena: what does a given object look like in a tiny neighborhoo... | {
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"url": "https://math.stackexchange.com/questions/788731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Integral equation solve using Laplace transform How can I solve this integral equation using Laplace transform?
$${\int\limits_0^{\infty}\ }\frac{e^{-t}(1-\cos t)}{t}\operatorname d\!t$$
Knowing that $$ \mathcal{L}\{\cos t\} = \frac{s}{s^2+1} $$
I think I can start by taking limits:
$$\lim_{b \rightarrow \infty} {\int... | To be honest I'm not sure what you meant by 'apply the shortcut of ...', but one way to do it is by writing
$$1/t = \int_0^{\infty}e^{-tx}\, dx $$
so that
$$\int_0^{\infty} \frac{e^{-t}(1-\cos t)}{t} dt = \int_0^{\infty} e^{-t}(1-\cos t) \int_0^{\infty} e^{-tx} \, dx \, dt \\
= \int_0^{\infty} \int_0^{\infty} e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/788827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Which Brownian motion property is the most important? Which Brownian motion property is the most important?
A standard Brownian motion is a stochastic process $(W_t, t\geqslant 0)$ indexed by nonnegative real numbers t with the following properties:
*
*$W_0=0$;
*With probability 1, the function $t \to W_t$ is conti... | I would just exclude $W_0$. The initial value is quite irrelevant and is typically chosen to be zero only for normalization purposes. If you compare it to discrete-time, the Brownian motion is the equivalent of an iid $N(0,\sigma)$ process.
Then, it depends on the context, whether you care most about it being continuo... | {
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For what interval does this power series converge and for what interval does it determine a differentiable function?
For what range of values of $x$ does $\sum_{n=1}^{\infty } \dfrac{1}{n}(1+\sin x)^n$ converge?
Find with proof an interval on which it determines a differentiable function of $x$ and show that the de... | Also,
$$
\sum_{n=1}^{\infty } \dfrac{1}{n}(1+\sin x)^n=-\ln(1-(1+\sin(x))=-\ln(-\sin(x))
$$
which again is valid for $\sin(x)<0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/789013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is this notation? Cyclic group $\mathbb{Z}^*_8$ $\mathbb{Z}^*_8$
As I understand it - $\mathbb{Z}_8$ is the group of integers under addition modulo 8.
So am I correct in thinking its elements are: $\{0,1,2,3,4,5,6,7\}$?
I thought the $*$ meant excluding zero, so I was confused to learn that the elements of $\mathb... | $\mathbb{Z_8}^*$ denotes the multiplicative group of $\mathbb{Z_8}$ as Mark Bennet has said.
You can show that $x \in \mathbb{Z_n}$ has a multiplicative inverse if and only if $(x,n)=1$. The proof is based on a special case of Bezout's theorem that states $(x,n)=1$ if and only if $\exists a,b \in \mathbb{Z}: ax+bn = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Factorization of Polynomials. Irreducible polynomial (basic question) One of the first examples says that:
Let $f(x) = 2x^2 +4$.
*
*$f(x)$ is reducible over $\mathbb{Z}$
*$f(x)$ is irreducible over $\mathbb{Q}$
*$f(x)$ is irreducible over $\mathbb{R}$
*$f(x)$ is reducible over $\mathbb{C}$
Why?
For the first o... | For $\mathbb{Z}$, you are right: it is reducible because you can factor by $2$, which is not invertible.
Over fields of characteristic $0$ (for example $\mathbb{Q},\mathbb{R},\mathbb{C}$), $2$ is invertible, so this is not a decomposition. In fact, every non-zero polynomial of degree $0$ is invertible. Hence, if you w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does bounded and continuous implies Lipschitz? If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
| No. Let
$$
f(x)=\left\{ \begin{array}{ll}
0&, x\leq 0\\
\sqrt{x} &,x\in [0,1]\\
-x^2+2x &, x\in [1,2]\\
0&, x\geq 2.
\end{array}\right.
$$
$f$ is continuous, bounded and integrable, but it is not Lipschitz, since $f|_{[0,1]}$ is not Lipschitz.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/789262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Bad proof that if $a + b + ab = 2020$ then $a+b=88$ Can you prove this:
Let $a,b \in \mathbb{N}$. If $a + b + ab = 2020$ then $a+b=88$.
This is the attempt given:
$\frac{2020-88}{a b}=1$
$a+b=88$
Substituting for $b$ using the $2$nd equation.
$2020-88 = a (88-a)$
That is a quadratic that is easily solved and gets $a ... | Your proof is not valid, assuming I understood you correctly.
This is the attempt given:
$\frac{2020-88}{a b}=1$
$a+b=88$
The problem is you started out by assuming $a + b = 88$. You just assumed what you wanted to prove!
Here is the question again:
Can you prove this:
if $a,b$ are positive integers,
and if $a + b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Probability, cells, and balls I have a Problem, I am trying to build a program that solves the game Minesweeper.
and I'm trying to find the probability of a bomb or a ball in each cell.
And I got stuck in a particular situation :
Let's say I have 5 cells in a row, I know that in the first 3 cells there is 2 balls and ... | Since you are writing a computer program anyway, you can just enumerate the possibilities for where balls might be located and calculate the probability that each cell contains a ball by taking the number of configurations that satisfy your constraints that have a ball in that cell, and dividing by the total number of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to interpret probabilities from different time-intervals I have trained different models for prediction bankruptcy 1 year prior bankruptcy, 2 years and 3 years prior and so on. When I use the models on a single sample and I for example get following results:
*
*$0.4$
*$0.8$
*$0.5$
So can I say that this com... | Based on the numbers you have indicated ($.4$, $.8$, $.5$), it seems your model is pretty flawed (else, please state precisely what it is your model is calculating). For, it calculates the probability of bankruptcy within the next three years to be $1.7$, which is larger than $100 \%$. Maybe you just made these numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Intuition - Linear Congruence Theorem
Let a and b be integers (not both 0) with greatest common divisor d.
Then an
integer $c = ax + by$ for some $x, y \in Z$ $\iff d|c$.
In particular, d is the least positive integer of the form ax +by.
Is there intuition? Or illustration? I keep forgetting which variable is su... | You have two integers $a$ and $b$ (not both $0$) with the greatest common divisor $d$. It means that $d|a$ and $d|b$. As we know, if $d|a$ and $d|b$ then $d|ax+by$ for all $x,y\in\mathbb{Z}$. Furthermore, one wonders what the set $$S=\{ax+by:x,y\in\mathbb{Z}\}.$$
Of course, $d$ divides each element in $S$ since is one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Linear independance of (binary) vectors here is what i want to do.
I have 4 vectors (in $\mathbb{Z}^{12}_{2}$), lets say
$v_1 = (1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0)$
$v_2 = (1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0)$
$v_3 = (1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0)$
$v_4 = (1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1)$
These vectors are s... | You can show that a collection of vectors is linearly independent iff there is a unique way to write each element of its span as a linear combination of those vectors. (The trick is that the difference of two duplicates is zero...) This is why searching for duplicates will tell you if the vectors are independent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/789730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Proving these three crazy limit implications I have this question:
$$\lim_{x\to p^{+}}f(x) = L \neq0, \lim_{x\to p^{+}}g(x)=0$$
and exists a $r>0$ such that $g(x)\neq0$ for all $x \in (p, p+r)$.
In these conditions, show that:
$$\lim_{x\to p^{+}}\frac{f(x)}{g(x)} = +\infty \mbox{ or } \lim_{x\to p^{+}}\frac{f(x)}{g(x)}... | The three "or" statements in the "to prove" can basically be summarized as $$\lim_{x\to p^{+}}\frac{f(x)}{g(x)}\quad\text{does not exist}$$
This suggests a proof by contradiction. Suppose that the limit does exist and equals $N$. Then we can make the fraction arbitrarily close to $N$. $$\left|\frac{f(x)}{g(x)}\right|<... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there such thing as an imaginary (imaginary number)? In other words... is there such a thing that is to imaginary numbers what imaginary numbers are to real numbers? And could this be expressed as a "complex" type number? If a complex number is in the form x + yi, I guess this would be in the form of x + yi + zj?
Do... | Right. For some flavor, the complex numbers are perfectly represented by matrices of this pattern:
$$
\left( \begin{array}{rr}
a & b \\
-b & a
\end{array}
\right) ,
$$
with $a,b \in \mathbb R.$
In particular,
$$ 1 \rightarrow
\left( \begin{array}{rr}
1 & 0 \\
0 & 1
\end{array}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/789932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
$\bar{\mathbb{Z}}\cap \mathbb{Q}\left[\sqrt{-3}\right] = \mathbb{Z}\left[\omega\right]$ How do you go about proving $\bar{\mathbb{Z}}\cap \mathbb{Q}\left[\sqrt{-3}\right] = \mathbb{Z}\left[\omega\right]$, where $\omega$ is $\frac{-1+\sqrt{-3}}{2}$? I have tried to approach it number theoretically, but no luck so far.
I... | One of the containments is easy, since $\mathbb Z [\omega]$ is a subring of both $\bar{\mathbb Z}$ and $\mathbb Q [\sqrt {-3}]$. So you want to prove the other identity.
Now, an element of $\bar{\mathbb Z}$ is a number satisfying a monic polynomial equation, i.e. one of the form (where the $a_i$ are integer coefficient... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Polynomial interpolation using derivatives at some points Given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)$, we can interpolate a polynomial of degree 4 using Lagrange method.
But, when we are given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y'''_5)$, how can we interpolate the same degre... | One method, though I am unsure of the accuracy would be:
$y'''_5\approx \frac{y_5-3y_4-3y_3+y_2}{(x_5-x_4)(x_4-x_3)(x_3-x_2)}$ You can rearrange this to find an approximate $y_5$, and then you can use lagrange polynomials as normal, either using all data points for a degree $5$ approximation, or just use $4$ of them fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is the radius of the largest $k$-dimensional ball that fits in an $n$ dimensional unit hypercube? This question is adapted from another question on the 2008 Putnam test which asks specifically for the case when $n = 4$ and $k = 2$. The answer is $\dfrac{1}{2}\sqrt{\dfrac{n}{k}}$ but I am looking for a proof or oth... | Like you said, it's easiest to consider the cube $[-1,1]^n$. I'll try to generalize the argument made in the link you submitted. We can assume by symmetry that the center of the sphere is the origin. Consider a $k-1$ sphere in $k-1$ spherical coordinates $\theta_i$ for $i=1,k-1$ with the range of $\theta_{k-1}$ being $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
About proving that the Continuum Hypothesis is independent of ZFC In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the existence of a "basic" ctm) two different ctm's, where... | Yes, you are right. However there are two ways around this.
*
*We can use Boolean-valued models. These are definable classes, and we can show that for a statement $\varphi$, if there is a complete Boolean algebra $B$ such that in the Boolean-valued model $V^B$, the truth value of $\varphi$ is not $1_B$ then $\varphi$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Classify the nonabelian groups of order $16p$, where $p$ is a prime number I need to classify the nonabelian groups of order $16p$, $p$ is a prime number. Is there any classification of groups of order $16p$?
| It's a (lengthy) exercise. Denote by $C_p$ the cyclic group of order $p$. If $p>7$, then a simple verification (not based on the classification) shows that no 2-group of order $\le 16$ has an automorphism of order $p$. It follows that if $G$ has $C_p$ as a quotient, then $G$ is direct product of a group of order 16 an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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