Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Good places to start for Calculus? I am a student, and due to my school's decision to not teach Calculus in high school (They said we'd learn it in college, but that's a year and a half away for me), I have to learn it myself. I am trying to get a summer internship as a Bioinformatics intern, and I would like to have p... | MIT OpenCourseWare is useful. Also, check out Paul's Online Notes:
http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Finding the upper and lower limit of the following sequence.
$\{s_n\}$ is defined by $$s_1 = 0; s_{2m}=\frac{s_{2m-1}}{2}; s_{2m+1}= {1\over 2} + s_{2m}$$
The following is what I tried to do.
The sequence is $$\{0,0,\frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{3}{8},\frac{7}{8},\frac{7}{16},\cdots \}$$
So the even terms... | Shouldn't it be $E_i = \frac{1}{2} - 2^{-i}$ and $O_i = 1 - 2^{1-i}$? That way $E_i = 0, \frac{1}{4}, \frac{3}{8}...$ and $E_i = 0, \frac{1}{2}, \frac{3}{4}...$, which seems to be what you want.
Your conclusion looks fine, but you might want to derive the even and odd terms more rigorously. For example, the even terms ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Without using L'Hospital's rule, I want to find a limit of the following.
Given a series with $a_n = \sqrt{n+1}-\sqrt n$ , determine whether it converges or diverges.
The ratio test was inconclusive because the limit equaled 1.
So I tried to use the root test. So the problem was reduced to finding $$\lim_{n \to \inft... | Hint: what is the $n^\text{th}$ partial sum of your series?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$.
If $X_1,X_2\sim \text{Normal} (0,1)$, then find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$.
I solve problems where transformation is given and I need to find the distribution. But here I need to find the transformation. I have no idea how to proceed. Pleas... | Why not use the probability integral transform?
Note that if
$$
F(x)
= \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-u^2/2} du
$$
then $F(X_i) \sim U(0,1)$. So you could take $f(x) = 2*F(x) - 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Integral of $\int \frac{1+\sin(2x)}{\operatorname{tg}(2x)}dx$ I'm trying to find the $F(x)$ of this function but I don't find how to do it, I need some hints about the solution.
I know that $\sin(2x) = 2\sin(x)\cos(x)$ its help me? It's good way to set $2x$ as $t$?
$$\int \frac{1+\sin(2x)}{\operatorname{tg}(2x)}dx$$
E... | Effective hint:
Let $\int R(\sin x,\cos x)dx$ wherein $R$ is a rational function respect to $\sin x$ and $\cos x$. If $$R(-\sin x, -\cos x)\equiv R(\sin x, \cos x) $$ then $t=\tan x, t=\cot x$ is a good substitution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On Polar Sets with respect to Continuous Seminorms In the following, $X$ is a Hausdorff locally convex topological vector space and $X'$ is the topological dual of $X$. If $p$ is a continuous seminorm on $X$ then we shall designate by $U_p$ the "$p$-unit ball", i.e,
$$U_p=\{x\in X: p(x)\le 1\}.$$
The polar set of $U_p$... | Assume that $p(x)=0$. Then for all $\lambda>0$, $\lambda x\in U_p$ hence $|f(\lambda x)|\leqslant 1$ and $f(x)=0$ whenver $f\in U_p^0$.
If $p(x)\neq 0$, then considering $\frac 1{p(x)}x$, we get $\geqslant$ direction. For the other one, take $f(a\cdot x):=a\cdot p(x)$ for $a\in\Bbb R$; then $|f(v)|\leqslant p(v)$ for ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Applications of computation on very large groups I have been studying computational group theory and I am reading and trying to implement these algorithms. But what that is actually bothering me is, what is the practical advantage of computing all properties of extremely large groups, moreover it is a hard problem?
It ... | The existence of several of the large finite simple sporadic groups, such as the Lyons group and the Baby Monster was originally proved using big computer calculations (although I think they all now have computer-free existence proofs).
Many of the properties of individual simple groups, such as their maximal subgroup... | {
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Endomorphim Ring of Abelian Groups In the paper
"Über die Abelschen Gruppen mit nullteilerfreiem Endomorphismenring."
Szele considers the problem of describing all abelian groups with endomorphism ring contaning no zero-divisors. He proved that there is no such group among the mixed groups. While $C(p)$ and $C(p^\inf... | Kulikov proved that an indecomposable abelian group is either torsion-free or $C(p^k)$ for some $k=0,1,\dots,\infty$. A direct summand creates a zero divisor in the endomorphism ring: Let $G = A \oplus B$ and define $e(a,b) = (a,0)$. Then $e^2=e$ and $e(1-e) = 0$. However $1-e$ is the endomorphism that takes $(a,b)$ to... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Where can I learn about the lattice of partitions? A set $P \subseteq \mathcal{P}(X)$ is a partition of $X$ if and only if all of the following conditions hold:
*
*$\emptyset \notin P$
*For all $x,y \in P$, if $x \neq y$ then $x \cap y = \emptyset$.
*$\bigcup P = X$
I have read many times that the partitions of ... | George Grätzers book General Lattice Theory has a section IV.4 on partition lattices,
see page 250 of this result of Google books search. A more recent version of the book is called Lattice Theory: Foundation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative of $\left(x^x\right)^x$ I am asked to find the derivative of $\left(x^x\right)^x$. So I said let $$y=(x^x)^x \Rightarrow \ln y=x\ln x^x \Rightarrow \ln y = x^2 \ln x.$$Differentiating both sides, $$\frac{dy}{dx}=y(2x\ln x+x)=x^{x^2+1}(2\ln x+1).$$
Now I checked this answer with Wolfram Alpha and I get that ... | If $y=(x^x)^x$ then $\ln y = x\ln(x^x) = x^2\ln x$. Then apply the product rule:
$$ \frac{1}{y} \frac{dy}{dx} = 2x\ln x + \frac{x^2}{x} = 2x\ln x + x$$
Hence $y' = y(2x\ln x + x) = (x^x)^x(2x\ln x + x).$
This looks a little different to your expression, but note that $\ln(x^x) \equiv x\ln x$.
| {
"language": "en",
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If $\mid \lambda_i\mid=1$ and $\mu_i^2=\lambda_i$, then $\mid \mu_i\mid=1$? If $|\lambda_i|=1$ and $\mu_i^2=\lambda_i$, then $|\mu_i|=1$?
$|\mu_i|=|\sqrt\lambda_i|=\sqrt |\lambda_i|=1$. Is that possible?
| Yes, that is correct. Or, either you could write $1=|\lambda_i|=|{\mu_i}^2|=|\mu_i|^2$, and use $|\mu_i|\ge 0$ to arrive to the unique solution $|\mu_i|=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408649",
"timestamp": "2023-03-29T00:00:00",
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Prove that $U$ is a self adjoint unitary operator Let $W$ be the finite dimensional subspace of an inner product space $V$ and $V=W\oplus W^\perp $.
Define $U:V \rightarrow V$ by $U(v_1+v_2)=v_1-v_2$ where $v_1\in W$ and $v_2 \in W^\perp$.
Prove that $U$ is a self adjoint unitary operator.
I know I have to show that ... | $\langle U(x),U(x)\rangle = \langle U(v_1+v_2) , U(v_1+v_2)\rangle = \langle v_1 - v_2, v_1 - v_2\rangle = \langle v_1,v_1\rangle + \langle v_2,v_2\rangle = \langle x,x\rangle$
where last two equalities comes frome the fact that $\langle v_1,v_2\rangle = 0$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Which one is bigger: $\;35{,}043 × 25{,}430\,$ or $\,35{,}430 × 25{,}043\;$?
Which of the two quantities is greater?
Quantity A: $\;\;35{,}043 × 25{,}430$
Quantity B: $\;\;35{,}430 × 25{,}043$
What is the best and quickest way to get the answer without using calculation, I mean using bird's eye view?
| Hint: Compare $a\times b$ with
$$(a+x)\times (b-x)=ab-ax+bx-x^2=ab-x(a-b)-x^2$$
keeping in mind that in your question, $a> b$ and $x>0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
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Solving $f_n=\exp(f_{n-1})$ : Where is my mistake? I was trying to solve the recurrence $f_n=\exp(f_{n-1})$.
My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$.
The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$.
if $f(m)=g(m)>0$ for some real $m$ then for $n>m$ we would have $g(n)>f(n)$.... |
The problem is that the primitive $\displaystyle\int_\cdot^x\frac{\mathrm dt}{\mathrm e^t-t}$ does not converge to infinity when $x\to+\infty$.
The comparison between $(f_n)$ and $g$ reads
$$
\int_{f_1}^{f_n}\frac{\mathrm dt}{\mathrm e^t-t}\leqslant n-1,
$$
for every $n\geqslant1$. When $n\to\infty$, the LHS converge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exercise 3.15 [Atiyah/Macdonald] I have a question regarding a claim in Atiyah, Macdonald. A is a commutative ring with $1$, $F$ is the free $A$-module $A^n$. Assume that $A$ is local with residue field $k = A/\mathfrak m$, and assume we are given a surjective map $\phi: F\to F$ with kernel $N$. Then why is the followi... | A general principle in homological algebra is the following:
Every ses of chain complexes gives rise to a LES in homology.
One can apply this principle to many situations, in our case it can be used to show that every ses of $A$ - modules gives rise to a LES in Tor. The LES in your situation is exactly
$$\ldots \to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/408881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Strictly convex sets If $S\subseteq \mathbb{R} ^2$ is closed and convex, we say $S$ is strictly convex if for any $x,y\in Bd(S)$ we have that the segment $\overline{xy} \not\subseteq Bd(S)$.
Show that if $S$ is compact, convex and constant width then $S$ is strictly convex.
Any hint? Than you.
| The idea of celtschk works just fine. Suppose that the line $L$ meets $\partial S$ along a line segment. Let $a\in S$ be a point that maximizes the distance from $L$ among all points in $S$. This distance, say $w$, is the width of $S$. Let $b$ any point of $L\cap \partial S$ which is not the orthogonal projection of $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/408947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all positive integers $x$ such that $13 \mid (x^2 + 1)$ I was able to solve this by hand to get $x = 5$ and $x =8$. I didn't know if there were more solutions, so I just verified it by WolframAlpha. I set up the congruence relation $x^2 \equiv -1 \mod13$ and just literally just multiplied out. This lead me to two ... | Starting with $2,$ the minimum natural number $>1$ co-prime with $13,$
$2^1=2,2^2=4,2^3=8,2^4=16\equiv3,2^5=32\equiv6,2^6=64\equiv-1\pmod{13}$
As $2^6=(2^3)^2,$ so $2^3=8$ is a solution of $x^2\equiv-1\pmod{13}$
Now, observe that $x^2\equiv a\pmod m\iff (-x)^2\equiv a$
So, $8^2\equiv-1\pmod {13}\iff(-8)^2\equiv-1$
Now,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $f(a)$ is divisible by either $101$ or $107$ for each $a\in\Bbb{Z}$, then $f(a)$ is divisible by at least one of them for all $a$ I've been struggling with this problem for a while, I really don't know where to start:
Let $f(x) \in \mathbb{Z}[X]$ be a polynomial such that for every value of $a \in \mathbb{Z}$, $f(a... | If neither of the statements "$f(x)$ is always divisible by $101$" or "$f(x)$ is always divisible by $107$" is true, then there exist $a,b\in{\bf Z}$ so that $107\nmid f(a)$ and $101\nmid f(b)$. It follows from hypotheses that
$$\begin{cases} f(a)\equiv 0\bmod 101 \\ f(a)\not\equiv0\bmod 107\end{cases}\qquad \begin{cas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How could we see that $\{n_k\}_k$ converges $\infty$? Let $x \in \Bbb R\setminus \Bbb Q$ and the sequence $\{\frac {m_k} {n_k}\}_k$ concerges to $x$. The question is from this comment by Ilya:
How could we see that $\{n_k\}_k$ converges $\infty$?
Thanks for your help.
| Let $M$ be fixed. Show that there exists such $k_0$ that
$$n_k>M, k\geq k_0.$$
By assuming the contradiction, we'll get such subsequance $\{n_{k_j}\}_j$ that
$n_{k_j} \leq M$ for all $j\geq 1$. Note that
$$\frac{m_{k_j}}{n_{k_j}}\to x.$$ Since such fractions can written as
$$\frac{m_{k_j}}{n_{k_j}}=\frac{A_{k_j}}{M!},$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the greatest common divisor of the polynomials: a) $X^m-1$ and $X^n-1$ $\in$ $Q[X]$
b) $X^m+a^m$ and $X^n+a^n$ $\in$ $Q[X]$
where $a$ $\in$ $Q$, $m,n$ $\in$ $N^*$
I will appreciate any explanations! THanks
| Let $n=mq+r$ with $0\leq r<m $ then
$$x^n-1= (x^m)^q x^r-1=\left((x^m)^q-1\right)x^r+(x^r-1)=(x^m-1)\left(\sum_{k=0}^{q-1}x^{mk}\right)x^r+(x^r-1)$$
and
$$\deg(x^r-1)<\deg(x^m-1)$$
hence by doing the Euclidean algorithm in parallel for the integers and the polynomials, we find
$$(x^n-1)\wedge(x^m-1)=x^{n\wedge m}-1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/409201",
"timestamp": "2023-03-29T00:00:00",
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If $T\colon V\to V$ is linear then$\text{ Im}(T) = \ker(T)$ implies $T^2 = 0$ I'm trying to show that if $V$ is finite dimensional and $T\colon V\to V$ is linear then$\text{ Im}(T) = \ker(T)$ implies $T^2 = 0$.
I've tried taking a $v$ in the kernel and then since it's in the kernel we know its in the image so there is ... | Hint: Note that $T^2$ should be read as $T\circ T$. You wish to show that $(\forall v\in V)((T\circ T)(v)=0)$. I think you can do this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/409259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving $(A \land B) \to C$ and $A \to (B \to C)$ are equivalent
Prove that $(A \land B) \rightarrow C$ is equivalent to $A \rightarrow (B \rightarrow C)$ in two ways: by semantics and syntax.
Can somebody give hints or answer to solve it?
| Semantically you can just consider two cases. 1) Suppose A is true, and 2) Suppose A is false. Since all atomic propositions in classical logic are either true or false, but not both, this method will work.
Syntactically, we'll need to know the proof system (the axioms and the rules of inference for your system) to k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Insertion sort proof I am reading Algorithm design manual by Skiena.It gives proof of Insertion sort by Induction.
I am giving the proof described in the below.
Consider the correctness of insertion sort, which we introduced at the beginning
of this chapter. The reason it is correct can be shown inductively:
*
*The... | The algorithm will have the property that at each iteration, the array will consist of two subarrays: the left subarray will always be sorted, so at each iteration our array will look like
$$
\langle\; \text{(a sorted array)}, \fbox{current element},\text{(the other elements)}\;\rangle
$$
We work from left to right, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every principal ideal domain satisfies ACCP. Every principal ideal domain $D$ satisfies ACCP (ascending chain condition on principal ideals)
Proof. Let $(a_1) ⊆ (a_2) ⊆ (a_3) ⊆ · · ·$ be a chain of principal ideals in $D$. It can be easily verified that $I = \displaystyle{∪_{i∈N} (a_i)}$ is an ideal of $D$. Since $D$ i... | Your proof is right but you can let t = max(i,j) and any k > t.
| {
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"timestamp": "2023-03-29T00:00:00",
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How many resulting regions if we partition $\mathbb{R}^m$ with $n$ hyperplanes? This is a generalization of this question. So in $\mathbb{R}^2$, the problem is illustrated like so:
Here, $n = 3$ lines divides $\mathbb{R}^2$ into $N_2=7$ regions. For general $n$ in the case of $\mathbb{R}^2$, the number of regions $N_2... | Denote this number as $A(m, n)$. We will prove $A(m, n) = A(m, n-1) + A(m-1, n-1)$.
Consider removing one of the hyperplanes, the maximum number is $A(m, n-1)$. Then, we add the hyperplane back. The number of regions on the hyperplane is the same as the number of newly-added regions. Since this hyperplane is $m-1$ dim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Recurrence relations: How many numbers between 1 and 10,000,000 don't have the string 12 or 21 so the question is (to be solved with recurrence relations: How many numbers between 1 and 10,000,000 don't have the string 12 or 21?
So my solution: $a_n=10a_{n-1}-2a_{n-2}$. The $10a_{n-1}$ represents the number of strings ... | We look at a slightly different problem, from which your question can be answered.
Call a digit string good if it does not have $12$ or $21$ in it. Let $a_n$ be the number of good strings of length $n$. Let $b_n$ be the number of good strings of length $n$ that end with a $1$ or a $2$, Then $a_n-b_n$ is the number of... | {
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Can every real number be represented by a (possibly infinite) decimal? Does every real number have a representation within our decimal system? The reason I ask is because, from beginning a mathematics undergraduate degree a lot of 'mathematical facts' I had previously assumed have been consistently modified, or altoget... | Irrational numbers were known to the ancient Greeks, as I expect you know. But it took humankind another 2000 years to come up with a satisfactory definition of them. This was mainly because nobody realised that a satisfactory definition was lacking.
Once humankind realised this, various suggestions were proposed. One ... | {
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"timestamp": "2023-03-29T00:00:00",
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Existence and uniqueness of God Over lunch, my math professor teasingly gave this argument
God by definition is perfect. Non-existence would be an imperfection, therefore God exists. Non-uniqueness would be an imperfection, therefore God is unique.
I have thought about it, please critique from mathematical/logical po... | Existence is not a predicate. You may want to read Gödel's onthological proof, which you can find on Wikipedia.
Equally good is the claim that uniqueness is imperfection, since something which is perfect cannot be scarce and unique. Therefore God is inconsistent..?
| {
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"url": "https://math.stackexchange.com/questions/409735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to prove " $¬\forall x P(x)$ I have a step but can't figure out the rest. I have been trying to understand for hours and the slides don't help. I know that since I have "not P" that there is a case where not All(x) has P... but how do I show this logically?
1. $\forall x (P(x) → Q(x))$ Given
2. $¬Q(x)$ ... | First, you want to instantiate your quantified statement with a witness, say $x$:
So from $(1)$ we get $$\;P(x) \rightarrow Q(x) \tag{$1\dagger$}$$
Then from $(1\dagger)$ with $(2)$ $\lnot Q(x)$, by modus tollens, you can correctly infer $(3)$: $\lnot P(x)$.
So, from $(3)$ you can affirm the existence of an $x$ such ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Linear Algebra dependent Eigenvectors Proof Problem statement:
Let $n \ge 2 $ be an integer. Suppose that A is an $n \times n$ matrix and that $\lambda_1$, $\lambda_2$ are eigenvalues of A with corresponding eigenvectors $v_1$, $v_2$ respectively. Prove that if $v_1$, $v_2$ are linearly dependent then $\lambda_1 = \l... | You know that $$Av_1=\lambda_1v_1\\Av_2=\lambda_2v_2$$
If $v_1,v_2$ are linearly dependent, then $v_1=\mu v_2$ for some scalar $\mu$. Putting this in the first equation,
$$A(\mu v_2) = \lambda_1(\mu v_2) \implies Av_2 = \lambda_1 v_2$$
This gives $\lambda_1=\lambda_2$ as desired.
I think your idea is on the right track... | {
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Prove inequality: $74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 +37\sqrt 2$ without calculus
Let $a,b,c,d \in \mathbb R$ such that $a^2 + b^2 + 1 = 2(a+b),
c^2 + d^2 + 6^2 = 12(c+d)$, prove inequality without calculus (or
langrange multiplier): $$74 - 37\sqrt 2 \le a+b+6(c+d) \le 74
+37\sqrt 2$$
The original problem is ... | Hint: You can split this problem to find max and min of $a+b$ and $c+d$.
| {
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supremum of an array of a convex functions If $\{J_n\}$ is an array of a convex functions on a convex set $U$ and $G(u)=\sup J_i(u), u\in U$, how to show that $G(u)$ is convex too?
I've done this, but I am not sure about properties of a supremum.
Since $U$ is convex, $\alpha x +(1-\alpha) y)\in U$ for all $x,y\in U... | It seems that you assume that your $J_n$ are convex real-valued functions. One can prove that their pointwise supremum is a convex without assuming that the common domain $U$ is convex, or even that the set of indices $n$ is finite.
A function $J_n$ is convex iff its epigraph is a convex set. The epigraph of the suprem... | {
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Maximum value of a product How to write the number $60$ as $\displaystyle\sum^{6}_{i=1} x_i$ such that $\displaystyle\prod^{6}_{i=1} x_i$ has maximum value?
Thanks to everyone :)
Is there a way to solve this using Lagrange multipliers?
| (of course the $x_i$ must be positive, otherwise the product may be as great as you want)
Hint: if you have $x_i \ne x_j$, substitute both with their arithmetic mean.
| {
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CDF of the distance of two random points on (0,1) Let $Y_1 \sim U(0,1)$ and $Y_2 \sim U(0,1)$.
Let $X = |Y_1 - Y_2|$.
Now the solution for the CDF in my book looks like this:
$P(X < t) = P(|Y_1 - Y_2| < t) = P(Y_2 - t < Y_1 < Y_2 + t) = 1-(1-t)^2$
They give this result without explanation. How do they come up with the ... | I want to change notation. Call the random variable called $Y_1$ in the problem by the name $X$. Call the random variable called $Y_2$ in the problem by the name $Y$. And finally, call the random variable called $X$ in
the problem by the name $T$. Trust me, these name changes are a good idea!
We need to assume that ... | {
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probability of sum of two integers less than an integer Two integers [not necessarily distinct] are chosen from the set {1,2,3,...,n}. What is the probability that their sum is <=k?
My approach is as follows. Let a and b be two integers. First we calculate the probability of the sum of a+b being equal to x [1<=x<=n]. W... | Notice if $k\le 1$ the probability is $0$, and if $k\ge 2n$ the probability is $1$, so let's assume $2\le k\le 2n-1$. For some $i$ satisfying $2\le i\le 2n-1$, how many ways can we choose $2$ numbers to add up to $i$? If $i\le n+1$, there are $i-1$ ways. If $i\ge n+2$, there are $2n-i+1$ ways.
Now, suppose $k\le n+1... | {
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Test for convergence of improper integrals $\int_{0}^{1}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$ and $\int_{1}^{\infty}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$ I need to test if, integrals below, either converge or diverge:
1) $\displaystyle\int_{0}^{1}\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}dx$
2) $\displaystyle\int_{1}^{\infty}\frac{\... | A related problem.
1) The integral diverges since as $x\sim 0$
$$\frac{\sqrt{x}}{(1+x)\ln^3(1+x)}\sim \frac{\sqrt{x}}{(1)(x^3) }\sim \frac{1}{x^{5/2}}.$$
Note:
$$ \ln(1+x) = x - \frac{x^2}{2} + \dots. $$
2) For the second integral, just replace $x \leftrightarrow 1/x $, so the integrand will behave as $x\sim 0$ as
$$ \... | {
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Under What Conditions and Why can Move Operator under Integral? Given a function space $V$ of some subset of real-valued functions on the real line, linear operator $L: V \rightarrow V$, and $f,g \in V$, define
$$ h(t) = \int_{\mathbb{R}}f(u)g(u-t)du $$
Further, assume $h \in V$. Is the below true? $$L(h(t)) = \in... | An arbitrary operator cannot be moved into the convolution. For example, if $Lh=\psi h$ for some nonconstant function $\psi$, then $$\psi(t) \int_{\mathbb R} f(u) g(u-t) \,du \ne \int_{\mathbb R} f(u) g(u-t) \psi(u-t) \,du $$
for general $f,g$.
However, the identity is true for translation-invariant operators, i.e., ... | {
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Theorems' names that don't credit the right people The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) appear in the theorem name.
For instance the Cantor Schröder Bernstei... | Nobody's mentioned the Pythagorean theorem yet?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 17
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Inner Product Spaces : $N(T^{\star}\circ T) = N(T)$ (A PROOF) Let $T$ be a linear operator on an inner product space. I really just want a hint as to how prove that $N(T^{\dagger}\circ T) = N(T)$, where "$^\dagger$" stands for the conjugate transpose.
Just as an aside, how should I read to myself the following symboli... | Hint: Let $V$ denote your inner product space. Clearly $N(T)\subseteq N(T^* T)$, so you really want to show that $N(T^* T)\subseteq N(T)$. Suppose $x\in N(T^* T)$. Then $T^* Tx = 0$, so we have $\langle T^* Tx, y\rangle = 0$ for all $y\in V$. Can you see where to go from here?
| {
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Concept about series test I have five kind of test here
1. Divergent test
2. Ratio test
3. Integral test
4. Comparison test
5. Alternating Series test
And a few questions here.
1. Are test 1,2,3,4 only available for Positive Series? and alternate series test is only for alternating series?
2. To show $\sum_{n=1}^{\in... | I assume that for (1) you mean the theorem that says that if the $n^\text{th}$ term does not approach 0 as $n \to \infty$ then the series diverges. This test does not require the terms to be positive, so you can apply it to show that the series $\sum_{n=1}^\infty (-1)^n$ diverges.
The ratio test does not require the t... | {
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Online Model Theory Classes Since "model theory" is kind of too general naming, I have encountered with lots of irrelevant results (like mathematical modelling etc.) when I searched for some videos on the special mathematical logic branch "model theory".
So, do you know/have you ever seen any online lecture videos on ... | If you are french fluent, here are the lecture notes of Tuna Altinel course :
http://math.univ-lyon1.fr/~altinel/Master/m11415.html
| {
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Is this kind of space metrizable? It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156.
If the closure of every discrete subset of a space is compact then the whole space is compact.
The proof can be seen here by Brian M. ... | $\newcommand{\cl}{\operatorname{cl}}$The first conjecture is true at least for $T_1$ spaces.
If $X$ is $T_1$ and not countably compact, then $X$ has an infinite closed discrete subspace, which is obviously not countably compact. Thus, if every discrete subspace of a $T_1$ space $X$ is countably compact, so is $X$.
It’... | {
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Prove $x^2+y^2+z^2 \ge 14$ with constraints Let $0<x\le y \le z,\ z\ge 3,\ y+z \ge 5,\ x+y+z = 6.$ Prove the inequalities:
$I)\ x^2 + y^2 + z^2 \ge 14$
$II)\ \sqrt x + \sqrt y + \sqrt z \le 1 + \sqrt 2 + \sqrt 3$
My teacher said the method that can solve problem I can be use to solve problem II. But I don't know what m... | Hint:
$$x^2+y^2+z^2 \ge 14 = 1^2+2^2+3^2\iff (x-1)(x+1)+(y-2)(y+2)+(z-3)(z+3) \ge 0$$
$$\iff (z-3)[(z+3)-(y+2)] + (y+z-5)[(y+2)-(x+1)] + (a+b+c-6)(a+1) \ge 0$$ (alway true)
| {
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Arctangent integral How come this is correct: $$\int \dfrac{3}{(3x)^2 + 1} dx = \arctan (3x) + C$$
I learned that
$$\int \dfrac{1}{x^2+1} = \arctan(x) + C$$
But I don't see how you can get the above one from the other. The $1$ in the denominator especially confuses me.
| We can say even more in the general case: if a function $\;f\;$ is derivable , then
$$\int \frac{f'(x)}{1+f(x)^2}dx=\arctan(f(x)) + K(=\;\text{a constant})$$
which you can quickly verify by differentiating applying the chain rule.
In your particular case we simply have $\;f(x)=3x\;$ ...
| {
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Need help with $\int \dfrac{2x}{4x^2+1}$ We want$$\int \dfrac{2x}{4x^2+1}$$
I only know that $\ln(4x^2 + 1)$ would have to be in the mix, but what am I supposed to do with the $2x$ in the numerator?
| Again, as in your past question, there's a general case here: if $\,f\,$ is derivable then
$$\int\frac{f'(x)}{f(x)}dx=\log|f(x)|+K$$
Here, we have
$$\frac{2x}{4x^2+1}=\frac14\frac{(4x^2+1)'}{4x^2+1}\ldots$$
| {
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Integral defined as a limit using regular partitions Definition. Given a function $f$ defined on $[a,b]$, let $$\xi_k \in [x_{k-1},x_k],\quad k=1,\ldots,n$$ where $$ x_k=a+k\frac{b-a}n, \quad k=0,\ldots,n \; .$$
One says that $f$ is integrable on $[a,b]$ if the limit $$\lim_{n\to\infty}\frac{b-a}n\sum_{k=1}^n f(\xi_k)$... | HINT:
Take Two cases:
1.When $c$ is a tag of a sub-interval $[x_{k},x_{k+1}]$ of $\dot{P}$,where $\dot{P}$ is your tagged partition ${(I_{i},t_{i})}_{i=1}^{n}$,such that $I_{i}=[x_{i},x_{i+1}]$
2.When $c$ is an end-point of a sub-interval of $\dot{P}$.
| {
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Name of this "cut 'n slide" fractal?
Can you identify this fractal--if in fact is has a name--based either upon its look or on the method of its generation? It's created in this short video.
It looks similar to a dragon fractal, but I don't think they are the same. Help, please?
| That is the twindragon. It is a two-dimensional self-similar set. That is it is composed of two smaller copies of itself scaled by the factor $\sqrt{2}$ as shown here:
Using this self-similarity, one can construct a tiling of the plane with fractal boundary. Analysis of the fractal dimension of the boundary is also... | {
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How to evaluate double integrals over a region? Evaluate the double integral $\iint_D(1/x)dA$, where D is the region bounded by the circles $x^2+y^2=1$ and $x^2+y^2=2x$
Alright so first I converted to polar coordinates:
$$ x^2 + y^2 = 1 \ \Rightarrow \ r = 1 \ \ , \ \
x^2 + y^2 = 2x \ \Rightarrow \ r^2 = 2r \cos θ \ \R... | Of course, you can do the problem by using the polar coordinates. If it's understood correctly, you would want to find the right limits for double integrals. I made a plot of the region as follows:
The red colored part is our $D$. So:
$$r|_1^{2\cos\theta},\theta|_{-\pi/3}^{\pi/3}$$
| {
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putting a complex structure on a graph I am studying Riemann Surfaces, and an example that comes up in two of my references, as a preamble to smooth affine plane curves, is the following:
Let $D$ be a domain in the complex plane, and let $g$ be holomorphic on $D$; giving the graph the subspace topology, and letting the... | Suppose $g$ is a continuous complex-valued function on $D$. Then the set $\Omega=\{(z,g(z))\in\mathbb C^2: z\in D\}$, which gets the subspace topology from $\mathbb C^2$, is homeomorphic to $D$ via $z\mapsto (z,g(z))$. By declaring this homeomorphism to be an isomorphism of complex structures, we can make $\Omega$ a c... | {
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Ideal of smooth function on a manifold vanishing at a point I'm trying to prove the following lemma: let $M$ be a smooth manifold and consider the algebra $C^{\infty}(M)$ of smooth functions $f\colon M \to \mathbb{R}$. Given $x_0 \in M$, consider the ideals
$$\mathfrak{m}_{x_0} := \{f\in C^{\infty}(M) : f(x_0)=0\},$$
$... | $\forall i \in \{1,2,\dots,n\}$, let $g_i(x_1, \dots, x_n)=\int_0^1\frac{\partial f}{\partial x_i}(tx_1, \dots, tx_n)dt$, it is easy to verify that $f=\sum_{i=1}^nx_ig_i$
| {
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I found out that $p^n$ only has the factors ${p^{n-1}, p^{n-2}, \ldots p^0=1}$, is there a reason why? So I've known this for a while, and only finally thought to ask about it.. so, any prime number ($p$) to a power $n$ has the factors $\{p^{n-1},\ p^{n-2},\ ...\ p^1,\ p^0 = 1\}$
So, e.g., $5^4 = 625$, its factors are:... | Let $ab=p^n$. Consider the prime factorization of the two terms on the left hand side. If any prime other than $p$ appears on the left, say $q$, then it appears as an overall factor and so we construct a prime factorization of $ab$ that contains a $q$. But then the right hand side has only $p$ as prime factors. Since t... | {
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property of equality The property of equality says:
"The equals sign in an equation is like a scale: both sides, left and right, must be the same in order for the scale to stay in balance and the equation to be true."
So for example in the following equation, I want to isolate the x variable. So I cross-multiply both s... | You did not multiple it by two but $\frac{2}{2}=1$ instead.
| {
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What am I missing here? That's an idiot question, but I'm missing something here. If $x'= Ax$ and $A$ is linear operator in $\mathbb{R}^n$, then $x'_i = \sum_j a_{ij} x_j$ such that $[A]_{ij} =a_{ij} = \frac{\partial x'_i}{\partial x_j}$, therefore $\frac{\partial}{\partial x_i'} = \sum_j \frac{\partial x_j}{\partial x... | When you wrote $\frac{\partial}{\partial x_i'} = \frac{\partial}{\partial \sum_j a_{ij} x_j} = \sum_j a_{ij} \frac{\partial}{\partial x_j} $ the $a_{ij}$ somehow climbed from the denominator to the numerator
| {
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"source": "stackexchange",
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Solving elementary row operations So I am faced with the following:
$$
\begin{align}
x_1 + 4x_2 - 2x_3 +8x_4 &=12\\
x_2 - 7x_3 +2x_4 &=-4\\
5x_3 -x_4 &=7\\
x_3 +3x_4 &=-5
\end{align}$$
How should I approach this problem? In other words, what is the next elementary row operation that I should attempt in order to solve i... | HINT:
Use Elimination/ Substitution or Cross Multiplication to solve for $x_3,x_4$ from the last two simultaneous equation.
Putting the values of $x_3,x_4$ in the second equation, you will get $x_2$
Putting the values of $x_2,x_3,x_4$ in the first equation you will get $x_1$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Show that Total Orders does not have the finite model property I am not sure whether my answer to this problem is correct. I would be grateful if anyone could correct my mistakes or help me to find the correct solutions.
The problem:
Show that Total Orders does not have the finite model property by finding a sentenc... | Looks like a good candidate to me. As you say, this clearly holds in every finite model of our theory, but infinite counterexamples exist, like $\mathbb{Z}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of $\lim\limits_{x\to 0}\left(\frac{1+x\sin(2x)-\cos(x)}{\sin^2(x)}\right)$ I want to evaluate this limit and I faced with one issue.
for this post I set $L`$ as L'Hôpital's rule
$$\lim\limits_{x\to 0}\left(\frac{1+x\sin(2x)-\cos(x)}{\sin^2(x)}\right)$$
Solution One:
$$\lim\limits_{x\to 0}\left(\frac{1+x\sin(2x)-... | As mentioned your first solution is incorrect. the reason is $$\displaystyle lim_{x\to0}\frac{2xcos(2x)}{sin(2x)}\neq 0$$ you can activate agin l'hospital:
$$lim_{x\to0}\frac{2xcos(2x)}{sin(2x)}=lim_{x\to0}\frac{2cos(2x)-2xsin(2x)}{2cos(2x)}=lim_{x\to0} 1-2xtg(2x)=1+0=1$$, so now after we conclude this limit, in the fi... | {
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Proof for Lemma about convex hull I have to prove a Lemma: "For the set B of all convex combinations of arbitrary finite number of points from set A, $co (A)=B$"
I started by showing $B\subset co(A)$ first.
$B$ contains all convex combinations of arbitrary finite number of points from A.
Let $x=\alpha_1 x_1 +...+\al... | The conclusion of your question is not correct. You are right in whatever you have done. The set $B$ may not even be convex, so how can it be equal to co($A$)?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/411914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Problem related to GCD I was solving a question on GCD. The question was calculate to the value of $$\gcd(n,m)$$
where $$n = a+b$$$$m = (a+b)^2 - 2^k(ab)$$
$$\gcd(a,b)=1$$
Till now I have solved that when $n$ is odd, the $\gcd(n,m)=1$.
So I would like to get a hint or direction to proceed for the case when $n$ is ev... | Key idea: $ $ employ $\bigg\lbrace\begin{eqnarray}\rm Euclidean\ Algorithm\ \color{#f0f}{(EA)}\!: &&\rm\ (a\!+\!b,x) = (a\!+\!b,\,x\ \,mod\,\ a\!+\!b)\\ \rm and\ \ Euclid's\ Lemma\ \color{blue}{(EL)}\!: &&\rm\ (a,\,b\,x)\ =\ (a,x)\ \ \,if\,\ \ (a,b)=1\end{eqnarray}$
$\begin{eqnarray}\rm So\ \ f \in \Bbb Z[x,y]\Rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Sum of exponential of normal random variables Suppose
$X_i \sim N(0,1)$ (independent, identical normal distributions)
Then by Law of large number,
$$
\sqrt{1-\delta} \frac{1}{n}\sum_i^\infty e^{\frac{\delta}{2}X_i^2} \rightarrow \sqrt{1-\delta} \int e^{\frac{\delta}{2}x^2}\frac{1}{\sqrt{2\pi}}e^{\frac{1}{2}x^2}dx =1
... | Note that
$$ \Bbb{E} \exp \left\{ \tfrac{1}{2}\delta X_{i}^{2} \right\} = \frac{1}{\sqrt{1-\delta}}
\quad \text{and} \quad
\Bbb{V} \exp \left\{ \tfrac{1}{2}\delta X_{i}^{2} \right\} = \frac{1}{(1-\delta)^{3/2}} < \infty. $$
Then the right form of the (strong) law of large number would be
$$ \frac{\sqrt{1-\delta}}{n} \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How does the Hahn-Banach theorem implies the existence of weak solution? I came across the following question when I read chapter 17 of Hormander's book "Tha Analysis of Linear Partial Differential Operators", and the theorem is
Let $a_{jk}(x)$ be Lipschitz continuous in an open set $X\subset\mathbb{R}^n$, $a_{ij}=a_{j... | Define the functional:
$$k(M Lw)=\int_{X} fw$$
where $L$ is the differential opeartor.
This is a bounded functional thanks to estimate you are assuming, also notice that you use the estimate to check that $k$ is well defined .
Then thanks to the Hahn-Banach theorem you can extendthe domain of this functional to the wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Proving that the $n$th derivative satisfies $(x^n\ln x)^{(n)} = n!(\ln x+1+\frac12+\cdots+\frac1n)$ Question:
Prove that $(x^n\ln x)^{(n)} = n!(\ln x+1+\frac 12 + ... + \frac 1n)$
What I tried:
Using Leibnitz's theorem, with $f=x^n$ and $g=\ln x$.
So
$$f^{(j)}=n\cdots(n-j+1)x^{n-j} , g^{(j)}=(-1)^{j+1} \dfrac 1{x^{n-j... | Hint: Try using induction. Suppose $(x^n\ln x)^{(n)} = n!\left(\ln x+\frac{1}{1}+\cdots\frac{1}{n}\right)$, then
$$\begin{align}{}
(x^{n+1}\ln x)^{(n+1)} & = \left(\frac{\mathrm{d}}{\mathrm{d}x}\left[x^{n+1} \ln x\right]\right)^{(n)} \\
&= \left((n+1)x^n\ln x + x^n\right)^{(n)} \\
&= (n+1)(x^n\ln x)^{(n)} + (x^n)^{(n)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Barbalat's lemma for Stability Analysis Good day,
We have:
Lyapunov-Like Lemma: If a scaler function V(t, x) satisfies the
following conditions:
*
*$V(t,x)$ is lower bounded
*$\dot{V}(t,x)$ is negative semi-definite
*$\dot{V}(t,x)$ is uniformly continuous in time
then $\dot{V}(t,x) \to 0$ as $t \to \infty $.
No... | Since $\dot{V} = -2e^2 \leq 0$, from the Lyapunov stability theory, one concludes that the system states $(e,\theta)$ are bounded.
Observe above that $\dot{V}$ is a function of only one state ($e$); if it were a function of the two states $(e,\theta)$ and $\dot{V} < 0$ (except in $e=\theta = 0$, in that case $V=0$), o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$ Why is this true?
$(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
| In classical logic the following equivalence is logically valid:
$$
\exists x (\varphi\Rightarrow\psi)\Longleftrightarrow(\forall x\varphi\Rightarrow\psi)
$$
providing that $x$ is a variable not free in $\psi$. So the formula in question is logically equivalent to $\forall xP(x)\Rightarrow\forall yP(y)$.
Looking at the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 14,
"answer_id": 2
} |
Finding the root of a degree $5$ polynomial $\textbf{Question}$: which of the following $\textbf{cannot}$ be a root of a polynomial in $x$ of the form $9x^5+ax^3+b$, where $a$ and $b$ are integers?
A) $-9$
B) $-5$
C) $\dfrac{1}{4}$
D) $\dfrac{1}{3}$
E) $9$
I thought about this question for a bit now and can anyone p... | Use the rational root theorem, and note that the denominator of one of the options given does not divide $9$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/412490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
At least one vertex of a tetrahedron projects to the interior of the opposite triangle How i can give a fast proof of the following fact:
Given four points on $\mathbb{R}^3$ not contained in a plane we can choose one such that its projection to the plane passing through the others is in the triangle generated by the th... | Here is a graphical supplement (that I cannot place into a comment) to the excellent answer by @achille hui .
I have taken the case $\eta=0.4.$ with normals in red.
The (complicated) name of this polyhedron is "tetragonal disphenoid" (https://en.wikipedia.org/wiki/Disphenoid).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/412553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
X,Y are independent exponentially distributed then what is the distribution of X/(X+Y) Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together.
The exercise goes as this:
If X,Y are independent exponen... | In other words, for each $a > 0$, you want to compute $P\left(\frac{X}{X+Y} < a \right)$.
Outline: Find the joint density of $(X,Y)$, and integrate it over the subset of the plane $\left\{ (x,y) : \frac{x}{x+y} < a \right\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/412615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Let $R$ be a ring with $1$. a nonzero proper ideal $I$ of $R$ is a maximal ideal iff the $R/I$ is a simple ring. Let $R$ be a ring with $1$. Prove that a nonzero proper ideal $I$ of $R$ is a maximal ideal if and only if the quotient ring $R/I$ is a simple ring.
My attempt:-
$I$ is maximal $\iff$ $R/I$ is a field. $\if... | You won't necessarily get a field in the quotient without commutativity, but you have a decent notion, nonetheless.
The rightmost equivalence is just the definition of simple ring.
If $I$ isn't maximal, then there is a proper ideal $J$ of $R$ with $I\subsetneq J$. Show that $J/I$ is a non-trivial ideal of $R/I$. Thus, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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When a group algebra (semigroup algebra) is an Artinian algebra?
When a group algebra (semigroup algebra) is an Artinian algebra?
We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group algebra (semigroup algebra) is an Artinian algebra if the grou... | A result of E. Zelmanov (Zel'manov), Semigroup algebras with identities,
(Russian) Sib. Mat. Zh. 18, 787-798 (1977):
Assume that $kS$ is right artinian. Then $S$ is a finite semigroup. The converse holds if $S$ is monoid.
See this assertion in Jan Okniński,
Semigroup algebras.Pure and Applied Mathematics, 138 (1990), ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why why the natural density of the set $\{ n \mid n \equiv n_{0}\mod{m}\}$ is $\frac{1}{m}$. The natural density of a set $S$ is defined by $\displaystyle\lim_{x \to{+}\infty}{\frac{\left | \{ n\le x \mid n\in S \} \right |}{x}}$.
This is maybe a silly question, but I got a confusion with this definition. And I reall... | I think you should add the condition $n\geq 0$ here.
Let $n_{0}=pm+k$, where $0\leq k< m$. We know that $x\to\infty$. Let $x=rm+t$, where $0\leq t< m$.
Then $\frac {n|n\equiv n_{0}\text { and } n\in S}{x}=\frac{r+1}{rm+t}$ if $t\geq k$, and $\frac {r}{rm+t}$ if $t<k$.
In either case, as $x\to \infty$, the fractions ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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2 questions regarding solutions for $\sqrt{a+b} - (a-b)^2 = 0$ Here's two questions derived from the following question:
$\quad\begin{matrix}
\text{Is there more than one solution to the following statement?} \\
\!\sqrt{a+b} - (a-b)^2 = 0
\end{matrix}$
$\color{Blue}{(1)\!\!:\;}$How would one (dis)prove this? I.e. In w... | If you are looking for real solutions, then note that $a+b$ and $a-b$ are just arbitrary numbers, with $a + b \ge 0$. This is because the system
$$
\begin{cases}
u = a + b\\
v = a - b
\end{cases}
$$
has a unique solution
\begin{cases}
a = \frac{u+v}{2}\\
\\
b = \frac{u-v}{2}.
\end{cases}
In the variables $u,v$ the gene... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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For $0I have problems with computing the following limit:
Given a sequence $0<q_n<1$ such that $\lim\limits_{n\to\infty} q_n =q < 1$, prove that for a fixed $k \in \mathbb N$, $\lim\limits_{n\to\infty} n^k q_n^n= 0$.
I know how to prove this, but I can't do it without using L'Hôpital's Rule. Does someone have an elem... | Note that $$\frac{(n+1)^k}{n^k}=1+kn^{-1}+{k\choose2}n^{-2}+\ldots+n^{-k}\to1$$
as $n\to\infty$, hence for any $s$ with $1<s<\frac1q$ (possible because $q<1$) we can find $a$ with $n^k<a\cdot s^n$.
Select $r$ with $q<r<\frac1s$ (possible because $s<\frac1q$). Then
For almost all $n$, we have $q_n<r$, hence
$$n^kq_n^n<... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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A group that has a finite number of subgroups is finite I have to show that a group that has a finite number of subgroups is finite. For starters, i'm not sure why this is true, i was thinking what if i have 2 subgroups, one that is infinite and the other one that might or not be finite, that means that the group isn't... | Consider only the cyclic subgroups. No one of them can be infinite, because an infinite cyclic group has infinitely many subgroups. So every cyclic subgroup is finite, and the group is the finite set-theoretic union of these finite cyclic subgroups.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Estimation of the number of prime numbers in a $b^x$ to $b^{x + 1}$ interval This is a question I have put to myself a long time ago, although only now am I posting it. The thing is, though there is an infinity of prime numbers, they become more and more scarce the further you go.
So back then, I decided to make an ine... | The prime number theorem is what you need. A rough statement is that if $\pi(x)$ is the number of primes $p \leq x$, then
$$
\pi(x) \sim \frac{x}{\ln(x)}
$$
Here "$\sim$" denotes "is asymptotically equal to".
A corollary of the prime number theorem is that, for $1\ll y\ll x$, then $\pi(x)-\pi(x-y) \sim y/\ln(x)$. So ye... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Graph with no even cycles has each edge in at most one cycle? As the title says, I am trying to show that if $G$ is a finite simple graph with no even cycles, then each edge is in at most one cycle.
I'm trying to do this by contradiction: let $e$ be an edge of $G$, and for contradiction suppose that $e$ was in two dist... | Here is the rough idea:
Suppose $C_1$ and $C_2$ overlap in at least two disjoint paths. If we follow $C_1$ along the end of one path to the beginning of the next path, and then follows $C_2$ back to the end of the first path, we obtain a cycle $C_3$. Since this cycle must have odd length, the parity of the two parts ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given basis spanning the vector space I am learning Linear algebra nowadays. I had a small doubt and I know it's an easy one. But still I am not able to get it. Recently I came across a statement saying "((1,2),(3,5)) is a basis of $ F^2 $ ".
According to the statement a linear combination of the vectors in the list,... | Note that if you can get to $(1,0)$ and $(0,1)$ you can get anywhere. So, can you find linear combinations $a(1,2) + b(3,5)$ that get you to these two vectors? Note that you can expand the equation, say, $a(1,2) + b(3,5) = (1,0)$ into two equations with two unknowns by looking at each coordinate separately.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Why does this summation of ones give this answer? I saw this in a book and I don't understand it.
Suppose we have nonnegative integers $0 = k_0<k_1<...<k_m$ - why is it that
$$\sum\limits_{j=k_i+1}^{k_{i+1}}1=k_{i+1}-k_i?$$
| Because $1+1+\cdots+1=n$ if there are $n$ ones.
So $j$ going from $k_i+1$ to $k_{i+1}$ is the same as going from $1$ to $k_{i+1}-k_i$ since the summand doesn't depend on $j$. There are $k_{i+1}-k_i$ ones in the list.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
CINEMA : Mathematicians I know that a similar question has been asked about mathematics documentaries in general, but I would like some recommendations on films specifically about various mathematicians (male and or female).
What would be nice is if you'd recommend something about not just the famous ones but also the... | The real man portrayed in A Beautiful Mind was not only an economist but a mathematician who published original discoveries in math. There is also a recent film bio of Alvin Turing. Sorry I forget the title.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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the following inequality is true, but I can't prove it The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find it is true, but I can't prove it. Thanks for your answer.
| Sorry I didn't check this sooner: the problem was cross-posted at
mathoverflow and I eventually was able to give a version of David Speyer's
analysis that makes it feasible to compute many more coefficient of the
asymptotic expansion (I reached the $d^{-13}$ term, and could go further),
and later to give error bounds g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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limit of evaluated automorphisms in a Banach algebra Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible elements such that $U_n\to 0$ as $n\to\infty$. Define $\sigma_n\in\o... | This limit need not exist. For example, let's work in $M_2(\mathbb R)$.
If
$$
U_n=
\left(
\begin{array}{cc}
\frac{1}{n} & 0 \\
0 & \frac{1}{n^2}
\end{array}
\right),
$$
then $\Vert U_n \Vert \to 0$ as $n \to \infty$, and
$$
U_n^{-1}=
\left(
\begin{array}{cc}
{n} & 0 \\
0 & {n^2}
\end{array}
\right).
$$
If we now let
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Is this notation for Stokes' theorem? I'm trying to figure out what $\iint_R \nabla\times\vec{F}\cdot d\textbf{S}$ means. I have a feeling that it has something to do with the classical Stokes' theorem. The Stokes' theorem that I have says
$$
\int\limits_C W_{\vec{F}} = \iint\limits_S \Phi_{\nabla\times\vec{F}}
$$
wher... | It seems to me that the integrals $$\int\limits_C W_{\vec{F}}~~~~\text{and}~~~~\oint_{\mathfrak{C}}\vec{F}\cdot d\textbf{r}$$ have the same meanings. I don't know the notation $ \Phi_{\nabla\times\vec{F}}$, but if it means as $$\textbf{curl F}\cdot \hat{\textbf{N}} ~dS$$ then your answer is Yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong?
$$(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$$
Thanks
| To Prove:
$$(a-b)^3 + (b-c)^3 + (c-a)^3 =3(a-b)(b-c)(c-a)$$
we know, $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$
so, $$(a-b)^3 + (b-c)^3 = (a -c)((a-b)^2 - (a-b)(b-c) + (b-c)^2)$$
now, $$(a-b)^3 + (b-c)^3 + (c-a)^3 = (a -c)((a-b)^2 - (a-b)(b-c) + (b-c)^2) + (c-a)^3 =
(c-a)(-(a-b)^2 + (a-b)(b-c)- (b-c)^2 +(c-a)^2)$$
now, $(c-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 12,
"answer_id": 11
} |
Is there a dimension which extends to negative or even irrational numbers? Just elaborating on the question:
We all use to natural numbers as dimensions: 1 stands for a length, 2 for area, 3 for volume and so on. Hausdorff–Besicovitch extends dimensions to any positive real number.
So my question: is there any dimensio... | The infinite lattice is a fractal of negative dimension: if you scale the infinite lattice on a line 2x, it becomes 2x less dense, thus 2 scaled lattices compose one non-scaled. If you take a lattice or on a plane, scaling 2x makes it 4x less dense so that 4 scaled lattices compose one non-scaled, etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Period of derivative is the period of the original function Let $f:I\to\mathbb R$ be a differentiable and periodic function with prime/minimum period $T$ (it is $T$-periodic) that is, $f(x+T) = f(x)$ for all $x\in I$. It is clear that
$$
f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\to 0} \frac{f(x+T+h) - f(x+T... | One solution is to note that $f(x)$ has an associated Fourier series, and since the derivative of a sinusoid of any frequency is another sinusoid of the same frequency, we deduce that the Fourier series of the derivative will have all the same sinusoidal terms as the original.
Thus, the derivative must have the same ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 1
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help me in trace of following proposition In a paper an author proved the following proposition
Please help me in trace proof of following proposition
Proposition: let $f$ be a homeomorphism of a connected topological manifold $M$ with fixed point set $F$. then either $(1)$
$f$ is invariant on each component of $M-F$... | For the first question, here is a sketch that $F$ is a submanifold under the assumption that the group $G$ acting on $M$ is finite:
Consider the map $M \to \prod_{g \in G} M, m \mapsto (gm)_{g \in G}$. This is smooth and should be a local homeomorphism, hence its regular. The diagonal $\{(m,\ldots,m) | m \in M\}$ of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find Gross from Net and Percentage I would like know if a simple calculation exists that would allow me to determine how much money I need to gross to receive a certain net amount. For example, if my tax rate was 30%, and my goal was to take home 700, I would need to have a Gross salary of $1000.
| Suppose your tax rate is $r$, written in percent. If you want your net to be $N$, then we want a gross of:
$$G=\frac{100N}{100-r}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Ratio between trigonometric sums: $\sum_{n=1}^{44} \cos n^\circ/\sum_{n=1}^{44} \sin n^\circ$ What is the value of this trigonometric sum ratio: $$\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ} = \quad ?$$
The answer is given as $$\frac{\displaystyle\sum_{n=1}^{44} \cos n^... | The last line in the argument you give could say
$$
\sum_{n=1}^{44} \cos\left(\frac{\pi}{180}n\right)\,\Delta n \approx \int_1^{44} \cos n^\circ\, dn.
$$
Thus the Riemann sum approximates the integral. The value of $\Delta n$ in this case is $1$, and if it were anything but $1$, it would still cancel from the numerato... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Algebra simplification in mathematical induction . I was proving some mathematical induction problems and came through an algebra expression that shows as follows:
$$\frac{k(k+1)(2k+1)}{6} + (k + 1)^2$$
The final answer is supposed to be:
$$\frac{(k+1)(k+2)(2k+3)}{6}$$
I walked through every possible expansion; I com... | First, let's write the expression as a sum of fractions with a common denominator.
$$\dfrac{k(k+1)(2k+1)}{6} + (k + 1)^2 = \dfrac{k(k+1)(2k+1)}{6} + \dfrac{6(k+1)^2}{6}\tag{1}$$
Now expand $6(k+1)^2 = 6k^2 + 12k + 6\tag{2}$ and expand
$k(k+1)(2k+1) = k(2k^2 + 3k + 1) = 2k^3 + 3k^2 + k\tag{3}$
So now, $(1)$ becomes $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
What leads us to believe that 2+2 is equal to 4? My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability.
To test us, he asked us to write down our justifications for why do we accept as true that 2 plus 2 i... | I've always liked this approach, that a naming precedes a counting.
===============================================
================================================
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/414269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability puzzle - the 3 cannons (Apologies if this is the wrong venue to ask such a question, but I don't understand how to arrive at a solution to this math puzzle).
Three cannons are fighting each other.
Cannon A hits 1/2 of the time. Cannon B hits 1/3 the time. Cannon C hits 1/6 of the time.
Each cannon fires at... | A has the greatest chance of survival.
Consider the three possible scenarios for the first round:
On the first trial, define $a$ as the probability that A gets knocked out, $b$ is the probability that B gets knocked out, and $c$ is the probability that C gets knocked out.
Since both B and C are firing at A, the probabi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Stirling Binomial Polynomial Let $\{\cdot\}$ denote Stirling Numbers of the second kind. Let $(\cdot)$ denote the usual binomial coefficients. It is known that $$\sum_{j=k}^n {n\choose j} \left\{\begin{matrix} j \\ k \end{matrix}\right\} = \left\{\begin{matrix} n+1 \\ k+1 \end{matrix}\right\}.$$ Note: The indexes for $... | It appears we can give another derivation of the closed form by @vadim123 for the
sum $$q_n = \sum_{j=k}^n m^j {n\choose j} {j \brace k}$$
using the bivariate generating function of the Stirling numbers of the
second kind. This computation illustrates generating function techniques as presented in Wilf's generatingfu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Initial and terminal objects in $\textbf{FinVect}_R$ I am teaching myself category theory and I am having difficulties identifying the initial and terminal object of the category of $\textbf{FinVect}_R$.
I was thinking that because it is finite vectors then the initial and terminal should be the same object ( since the... | I believe the terminal and initial object are both the zero-dimensional vector space $0$.
There is one map from $0$ to any other vector space $V$, since we must send $0$ to $0_V$. This follows from the definition of a linear map, and linear maps are the morphisms in this category. So $0$ is the initial object.
Similar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Which are functions of bounded variations? Let $f, g : [0, 1] \to \mathbb{R}$ be defined as follows:
$f(x) = x^2 \sin (1/x)$ if $x = 0$, $f(0)=0$
$g(x) = \sqrt{x} \sin (1/x)$ if $x = 0, g(0) = 0$.
Which are functions of bounded variations?Every polynomial in a compact interval is of BV?
Could any one just tell me what ... | Yes, bounded derivative implies BV. I explained this in your older question which condition says that $f$ is necessarily bounded variation. Since $f$ has bounded derivative, it is in BV.
A function with unbounded derivative could also be in BV, for example $\sqrt{x}$ on $[0,1]$ is BV because it's monotone. More genera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$ Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$
We can prove using the Beta-Function identity that
$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\frac... | I hope it is not too late. Define
\begin{eqnarray}
I(a)=\int_0^\infty\frac{\log(1+ax^4)}{(1+x^2)^2}dx.
\end{eqnarray}
Then
\begin{eqnarray}
I'(a)&=&\int_0^\infty \frac{x^4}{(1+ax^4)(1+x^2)^2}dx\\
&=&\frac{1}{(1+a)^2}\int_0^\infty\left(-\frac{2}{1+x^2}+\frac{1+a}{(1+x^2)^2}+\frac{1-a+2ax^2}{1+a x^4}\right)dx\\
&=&\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 6,
"answer_id": 0
} |
The principal ideal $(x(x^2+1))$ equals its radical. Let $\mathbb R$ be the reals and $\mathbb R[x]$ be the polynomial ring of one variable with real coefficients. Let $I$ be the principal ideal $(x(x^2+1))$. I want to prove that the ideal of the ideals variety is not the same as its radical, that is, $I(V(I))\not=\tex... | This $x(x^2 + 1)$ is the factorisation of that polynomial into irreducibles (over $\mathbb{R}$). Once you have such a factorisation, the radical is the factorisation with no powers. So it is radical.
EDIT — I am referencing this:
Let $f \in k[x]$ be a polynomial, and suppose that $f = f_1^{\alpha_1} \cdots f_n^{\alpha_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
If a graph with $n$ vertices and $n$ edges there must a cycle? How to prove this question? If a graph with $n$ vertices and $n$ edges it must contain a cycle?
| Here's is an approach which does not use induction:
Let $G$ be a graph with $n$ vertices and $n$ edges.
Keep removing vertices of degree $1$ from $G$ until no such removal is possible, and let $G'$ denote the resulting graph. Note that in each removal, we're removing exactly $1$ vertex and $1$ edge, so $G'$ cannot be e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 1
} |
How many samples are required to estimate the frequency of occurrence of an output (out of 8 different outputs)? I have $N$ marbles and to each of them corresponds a 1 or 2 or 3 or ... or 8.(i.e., there's 8 different kinds of marbles)
How many samples are required to estimate the frequency of occurrence of each kind (... | This is quite straight forward in socio-economic statistics. An example in health care is given here. There also a general calculator.
Hope this helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/414780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the area: $(\frac xa+\frac yb)^2 = \frac xa-\frac yb , y=0 , b>a$ Find the area: $$\left(\frac xa+\frac yb\right)^2 = \frac xa-\frac yb,$$ $ y=0 , b>a$
I work in spherical coordinates.
$x=a\cdot r\cdot \cos(\phi)\;\;,$
$y=b\cdot r\cdot \cos(\phi)$
Then I get the equation and don't know to do with, cause "a" and "... | Your shape for $a=1$, $b=2$ is as below
It is much easier if you would use line parametrization rather than polar coordinates. Let $y=m\,x$ then
$$\bigg(\frac xa+\frac{mx}b\bigg)^2=\frac xa-\frac {mx}b\Rightarrow x(m)=\frac{1/a-m/b}{\big(1/a+m/b\big)^2}$$
and
$$y(m)=m\,x(m)=m\frac{1/a-m/b}{\big(1/a+m/b\big)^2}$$
To fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
when Fourier transform function in $\mathbb C$? The Fourier transform of a function $f\in\mathscr L^1(\mathbb R)$ is
$$\widehat f\colon\mathbb R\rightarrow\mathbb C, x\mapsto\int_{-\infty}^\infty f(t)\exp(-ixt)\,\textrm{d}t$$
When is this indeed a function in $\mathbb C$? Most of calculations you get functions in $\mat... | There is a basic fact about Fourier transform on the Schwartz space: Let $f\in \mathcal{S}(\mathbb{R})$, then $\widehat{f'} = it\widehat{f}$. Thus, if $\widehat{f}$ is real-valued, then $\widehat{f'}$ is complex-valued.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/414992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
distributing z different objects among k people almost evenly We have z objects (all different), and we want to distribute them among k people ( k < = z ) so that the distribution is almost even.
i.e. the difference between the number of articles given to the person with maximum articles, and the one with minimum artic... | Each person will get either $\lfloor \frac{z}{k}\rfloor$ or $\lceil \frac{z}{k}\rceil$ objects. These are the floor and ceiling functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/415076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Is it possible to derive all the other boolean functions by taking other primitives different of $NAND$? I was reading the TECS book (The Elements of Computing Systems), in the book, we start to build the other logical gates with a single primitive logical gate, the $NAND$ gate. With it, we could easily make the $NOT$ ... | As you noted, it's impossible for the constants to generate all of the boolean functions; the gates which are functions of a single input also can't do it, for equally-obvious reasons (it's impossible to generate the boolean function $f(x,y)=y$ from either $f_1(x,y)=x$ or $f_2(x,y)=\bar{x}$).
OR and AND can't do it eit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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