Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that $\sum \frac{a_n}{a_n+3}$ diverges Suppose $a_n>0$ for each $n\in \mathbb{N}$ and $\sum_{n=0}^{\infty} a_n $ diverges. How would one go about showing that $\sum_{n=0}^{\infty} \frac{a_n}{a_n+3}$ diverges?
| Let $b_n=\dfrac{a_n}{a_n+3}$. If the $a_n$ are unbounded, then $b_n$ does not approach $0$, and therefore $\sum b_n$ diverges.
If the $a_n$ are bounded by $B$, then $b_n\ge \dfrac{1}{B+3} a_n$, and $\sum b_n$ diverges by comparison with $\sum a_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/262969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove every odd integer is the difference of two squares I know that I should use the definition of an odd integer ($2k+1$), but that's about it.
Thanks in advance!
| Eric and orlandpm already showed how this works for consecutive squares, so this is just to show how you can arrive at that conclusion just using the equations.
So let the difference of two squares be $A^2-B^2$ and odd numbers be, as you mentioned, $2k+1$. This gives you $A^2-B^2=2k+1$.
Now you can add $B^2$ to both si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 9,
"answer_id": 3
} |
Construction of an increasing function from a general function Supposing $f: [0,\infty) \to [0,\infty)$. The goal is to make an increasing function from $f$ using the following rule:-
If $t_1 \leq t_2$ and $f(t_1) > f(t_2)$ then change the value of $f(t_1)$ to $f(t_2)$.
After this change, we have $f(t_1) = f(t_2)$.
Le... | Consider the function $f(x)=1/x$ if $x>0$, with also $f(0)=0$. Then for example $f(1)$ will, for any $n>1$, get changed to $n$ on considering that $f(1/n)=n>f(1)=1$. Once this is done there will still be plenty of other $m>1$ for which $f(1/m)=m$ where $m>n$, so that $f(1)$ will have to be changed again from its presen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How do we prove that $f(x)$ has no integer roots, if $f(x)$ is a polynomial with integer coefficients and $f( 2)= 3$ and $f(7) = -5$? I've been thinking and trying to solve this problem for quite sometime ( like a month or so ), but haven't achieved any success so far, so I finally decided to post it here.
Here is my p... | Let's define a new polynomial by $g(x)=f(x+2)$. Then we are told $g(0)=3, g(5)=-5$ and $g$ will have integer roots if and only if $f$ does. We can see that the constant term of $g$ is $3$. Because the coefficients are integers, when we evaluate $g(5)$, we get terms that are multiples of $5$ plus the constant term $3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given real numbers: define integers? I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following:
Integer numbers are just special cases (a subset) of real numbers. Imagine a world where you know only real numbers. How are integers defined using mathema... | How about the values (Image) of
$$\lfloor x\rfloor:=x-\frac{1}{2}+\frac{1}{\pi}\sum_{k=1}^\infty\frac{\sin(2\pi k x)}{k}$$
But this is nonsense; we sum over the positive integers, and such, we can just define the integers as
$$x_0:=0\\x_{k+1}=x_k+1\\
x_{-k}=-x_k$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/263284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "68",
"answer_count": 9,
"answer_id": 6
} |
Circle geometry: nonparallel tangent and secant problem If secant and the tangent of a circle intersect at a point outside the circle then prove that the area of the rectangle formed by the two line segments corresponding to the secant is equal to the area of the square formed by the line segment corresponding to the ... | Others have answered this, but here is a source of further information:
http://en.wikipedia.org/wiki/Power_of_a_point
Here's a problem in which the result is relied on:
http://en.wikipedia.org/wiki/Regiomontanus%27_angle_maximization_problem#Solution_by_elementary_geometry
The result goes all the way back (23 centuries... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Game theory: Nash equilibrium in asymetric payoff matrix I have a utility function describing the desirability of an outcome state. I weigh the expected utility with the probability of the outcome state occuring. I find the expected utility of an action, a, with $EU(a) = \sum\limits_{s'} P(Result(a) = s' | s)U(s'))$ wh... | Set of concepts aimed at decision making in situations of competition and conflict (as well as of cooperation and interdependence) under specified rules. Game theory employs games of strategy (such as chess) but not of chance (such as rolling a dice). A strategic game represents a situation where two or more participan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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For what values of $a$ does this improper integral converge? $$\text{Let}\;\; I=\int_{0}^{+\infty}{x^{\large\frac{4a}{3}}}\arctan\left(\frac{\sqrt{x}}{1+x^a}\right)\,\mathrm{d}x.$$
I need to find all $a$ such that $I$ converges.
| Hint 1: Near $x=0$, $\arctan(x)\sim x$ whereas near $x=+\infty$, $\arctan(x)\sim\pi/2$.
Hint 2: Near $x=0$, consider $a\ge0$ and $a\lt0$. Near $x=+\infty$, consider $a\ge\frac12$ and $a\lt\frac12$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/263547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integers that satisfy $a^3= b^2 + 4$ Well, here's my question:
Are there any integers, $a$ and $b$ that satisfy the equation $b^2$$+4$=$a^3$, such that $a$ and $b$ are coprime?
I've already found the case where $b=11$ and $a =5$, but other than that?
And if there do exist other cases, how would I find them? And if not ... | $a=5, b=11$ is one satisfying it. I don't think this is the only pair.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/263622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
I realize that any of the class of functions $f:x\to (n\cdot x)$ gives a bijection between $\mathbb{N}$ and the subset of $\mathbb{N}$ whose members equal multiples o... | As great answers have been given already, I'd merely like to add an easy way to show that the set of finite subsets of $\mathbb{N}$ is countable:
Observe that
$$\operatorname{Fin}(\mathbb{N}) = \bigcup_{n\in\mathbb{N}}\left\{ A\subseteq\mathbb{N}: \max(A) = n \right\},$$
which is a countable union of finite sets as for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 7,
"answer_id": 3
} |
Fixed point in a continuous map
Possible Duplicate:
Periodic orbits
Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$.
Does $f$ necessarily have a fixed point?
| Here's a somewhat simpler (in my opinion) argument. It's essentially the answer in Amr's link given in the first comment to the question, but simplified a bit to treat just the present question, not a generalization. Start with any $a\in\mathbb R$. If we're very lucky, $f(a)=a$ and we're done. If we're not that luc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A Few Questions Concerning Vectors In my textbook, they provide a theorem to calculate the angle between two vectors:
$\cos\theta = \Large\frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\|\vec{v}\|}$
My questions are, why does the angle have to be $0 \le \theta \le \pi$; and why do the vectors have to be in standard position?
... | Given two points $x$ and $y$ on the unit sphere $S^{n-1}\subset{\mathbb R}^n$ the spherical distance between them is the length of the shortest arc on $S^{n-1}$ connecting $x$ and $y$. The shortest arc obviously lies in the plane spanned by $x$ and $y$, and drawing a figure of this plane one sees that the length $\phi$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof that $\sqrt{5}$ is irrational In my textbook the following proof is given for the fact that $\sqrt{5}$ is irrational:
$ x = \frac{p}{q}$ and $x^2 = 5$. We choose $p$ and $q$ so that the have no common factors, so we know that $p$ and $q$ aren't both divisible by $5$.
$$\left(\dfrac{p}{q}\right)^2 = 5\\ \text{ so ... | Exactly what MSEoris said, you can always reduce a fraction to such a point that they have no common factors, if $\frac{p}{q}$ had a common factor n, then $nk_0 = p$ $nk_1 = q $ then
$\frac{p}{q} = \frac{nk_0}{nk_1} = \frac{k_0}{k_1}$, now if $k_0, k_1$ have a common factor do the same and you will eventually get a fra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 5
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Convergence in $L^{p}$ spaces Set $$f_n= n1_{[0,1/n]}$$
For $0<p\le\infty $ , one has that $\{f_n\}_n$ is in $L^p(\mathbb R)$. For which values of $p$ is $\{f_n\}_n$ a Cauchy sequence in $L^p$? Justify your answer.
This was a Comp question I was not able to answer. I don't mind getting every details of the proof.
Wha... | Note that, we have $$\Vert f_{2n} -f_n\Vert_p^p = n^p \left(\dfrac1n - \dfrac1{2n}\right) + (2n-n)^p \dfrac1{2n} = \dfrac{n^{p-1}}2 + \dfrac{n^{p-1}}2 \geq 1 \,\,\,\,\,\,\, \forall p \geq 1$$
For $p<1$, and $m>n$ we have
$$\Vert f_m - f_n\Vert_p^p = n^p \left(\dfrac1n - \dfrac1m\right) + (m-n)^p \dfrac1m < n^p \dfrac1n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Sums of two probability density functions If the weighted sum of 2 probability density functions is also a probability density function, then what is the relationship between the random variables of these 3 probability density functions.
| I think you might mean, "What happens if I'm not sure which of two distributions a random variable will be drawn from?" That is one situation where you need to take a pointwise weighted sum of two PDFs, where the weights have to add to 1.
Suppose you have three coins in your pocket, two fair coins and one which lands ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Covariance of Brownian Bridge? I am confused by this question. We all know that Brownian Bridge can also be expressed as:
$$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$
Where the Brownian motion will end at b at $t = 1$ almost surely. Hence I can write it as:
$$Y_t = bt + I(t)$$
where $I(t)$ is a stochast... | I think the given representation of the Brownian Bridge is not correct. It should read
$$Y_t = a \cdot (1-t) + b \cdot t + (1-t) \cdot \underbrace{\int_0^t \frac{1}{1-s} \, dB_s}_{=:I_t} \tag{1}$$
instead. Moreover, the covariance is defined as $\mathbb{E}((Y_t-\mathbb{E}Y_t) \cdot (Y_s-\mathbb{E}Y_s))$, so you forgot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Examples of non-isomorphic abelian groups which are part of exact sequences Suppose $A_1$, $A_2$, $A_3$ and $B_1$, $B_2$, and $B_3$ are two
short exact sequences of abelian groups.
I am looking for two such short sequences where $A_1$ and $B_1$ is isomorphic
and $A_2$ and $B_2$ are isomorphic but $A_3$ and $B_3$ are ... | For the first pair take
$$0 \longrightarrow \mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0$$
and
$$0 \longrightarrow \mathbb{Z} \stackrel{3}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z} / 3\mathbb{Z} \longrightarrow 0.$$
For sequences with non-i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integrating $\int_0^{\infty} u^n e^{-u} du $ I have to work out the integral of
$$
I(n):=\int_0^{\infty} u^n e^{-u} du
$$
Somehow, the answer goes to
$$
I(n) = nI(n - 1)$$
and then using the Gamma function, this gives $I(n) = n!$
What I do is this:
$$
I(n) = \int_0^{\infty} u^n e^{-u} du
$$
Integrating by parts gives
... | You have
$$
I(n) = \lim_{u\to +\infty}u^ne^{-u}-0^ne^{-0}+nI(n-1)
$$
But $$\lim_{u\to +\infty}u^ne^{-u}=\lim_{u\to +\infty}\frac{u^n}{e^{u}}=...=0$$
and so
$$
I(n) =0-0+nI(n-1)=nI(n-1)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
The way into set theory Given that I am going through Munkres's book on topology , I had to give a glance at the topics included in the first chapter like that of Axiom of choice, The maximum principle, the equivalence of the former and the later etc. Given all this I doubt that I know enough of set theory , or more pr... | I'd recommend "Naive Set Theory" by Halmos. It is a fun read, in a leisurely style, starts from the axioms and prove the Axiom of Choice.
Also, see this XKCD. http://xkcd.com/982/
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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How can I show the Coercivity of this function? Let $S$ be the set of real positive matrices, $\lambda>0$ and $f:S\rightarrow\mathbb{R}$ defined by $$f(X)=\langle X,X\rangle-\lambda\log\det(X) $$
where $\langle X,X\rangle=\operatorname{trace}(X^\top X)$. How can one show that $f$ is coercive?
| Let $\mu = \max \{\det X : \langle X,X\rangle=1, X\ge 0\}$. The homogeneity of determinant implies that $\log \det X\le \log \mu+\frac{n}{2}\log \langle X,X\rangle$ for all $X\ge 0$. Therefore, $$f(X)\ge \langle X,X\rangle -\lambda \log \mu - \frac{\lambda n}{2}\log \langle X,X\rangle $$ which is $\ge \frac12 \langle X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Fun but serious mathematics books to gift advanced undergraduates. I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable.
In particular, I am looking for reada... | Modern Graph theory by Bela Bollobas counts as fun if they're interested in doing exercises which can be approached by clever intuitive arguments; it's packed full of them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "263",
"answer_count": 40,
"answer_id": 7
} |
Fun but serious mathematics books to gift advanced undergraduates. I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable.
In particular, I am looking for reada... | Dissections: Plane & Fancy by Frederickson, and the second side of the same coin: The Banach--Tarski Paradox by Tomkowicz and Wagon.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "263",
"answer_count": 40,
"answer_id": 39
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solution for equation For $a^2+b^2=c^2$ such that $a, b, c \in \mathbb{Z}$
Do we know whether the solution is finite or infinite for $a, b, c \in \mathbb{Z}$?
We know $a=3, b=4, c=5$ is one of the solutions.
| Assuming $m,n$ be any two positive integers such that $m < n$, we have:
$$a = n^2 - m^2,\;\; b = 2mn,\;\;c = n^2 + m^2$$
And then $a^2+b^2=c^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Sets of second category-topology A set is of first category if it is the union of nowhere dense sets and otherwise it is of second category.
How can we prove that irrational numbers are of second category and the rationals are of of first category?
| $\mathbb Q = \bigcup_{q \in \mathbb Q} \{ q \}$ hence the rationals are a countable union of nowhere dense sets.
Assume the irrationals are also a countable union of nowhere dense sets: $I = \bigcup_{n \in \mathbb N} U_n$. Then $\mathbb R = \bigcup_{q \in \mathbb Q} \{ q \} \cup \bigcup_{n \in \mathbb N} U_n$ is also a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Mind-blowing mathematics experiments We've all heard of some mind-blowing phenomena involving the sciences, such as the double-slit experiment. I was wondering if there are similair experiments or phenomena which seem very counter-intuitive but can be explained using mathematics? I mean things such as the Monty Hall pr... | If you let $a_1=a_2=a$, and $a_{n+1}=20a_n-19a_{n-1}$ for $n=2,3,\dots$, then it's obvious that you just get the sequence $a,a,a,\dots$. But if you try this on a calculator with, say, $a=\pi$, you find that after a few iterations you start getting very far away from $\pi$. It's a good experiment/demonstration on accumu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
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Why is 'abuse of notation' tolerated? I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange. It seems to all be forgiven with a wave of the hand. Why do we tolerate it at all?
I understand if later on in one's studies if things are assumed ... | When one writes/talks mathematics, in 99.99% of the cases the intended recipient of what one writes is a human, and humans are amazing machines: they are capable of using context, guessing, and all sorts of other information when decoding what we write/say. It is generally immensely more efficient to take advantage of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "91",
"answer_count": 10,
"answer_id": 1
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Exactness of Colimits Let $\mathcal A$ be a cocomplete abelian category, let $X$ be an object of $\mathcal A$ and let $I$ be a set. Let $\{ X_i \xrightarrow{f_i} X\}_{i \in I}$ be a set of subobjects. This means we get an exact sequence
$$
0 \longrightarrow X_i \xrightarrow{f_i} X \xrightarrow{q_i}X/X_i \longrightarrow... | I think you may have misquoted the question, because if $I$ is (as you wrote) merely a set, then a colimit over it is just a direct sum.
Anyway, let me point out why "the colimit functor preserves colimits (and in particular cokernels)" is relevant. Exactness of a sequence of the form $A\to B\to C\to0$ is equivalent ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Extra-Challenging olympiad inequality question We have the set $\{X_1,X_2,X_3,\dotsc,X_n\}$. Given that $X_1+X_2+X_3+\dotsb +X_n = n$, prove that:
$$\frac{X_1}{X_2} + \frac{X_2}{X_3} + \dotsb + \frac{X_{n-1}}{X_n} + \frac{X_n}{X_1} \leq \frac{4}{X_1X_2X_3\dotsm X_n} + n - 4$$
EDIT: yes, ${X_k>0}$ , forgot to mention :)... | Let
$$
\begin{eqnarray}
L(x_1,\ldots, x_n) &=& \frac{x_1}{x_2} + \frac{x_2}{x_3} + \ldots + \frac{x_n}{x_1} \\
R(x_1,\ldots, x_n) &=& \frac{4}{x_1 x_2 \ldots x_n} + n - 4 \\
f(x_1,\ldots, x_n) &=& R(x_1,\ldots, x_n) - L(x_1,\ldots, x_n)
\end{eqnarray}
$$
The goal is to prove that $f(x_1,\ldots, x_n) \ge 0$ for all $n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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Limit of a function whose values depend on $x$ being odd or even I couldn't find an answer through google or here, so i hope this isn't a duplicate.
Let $f(x)$ be given by:
$$
f(x) = \begin{cases}
x & : x=2n\\
1/x & : x=2n+1
\end{cases}
$$
Find $\lim_{x \to \infty} f(x).$
The limit is different depen... | Your first statement following the word "attempt" has the correct intuition: "this limit doesn't exist because we have different values for" ... $\lim_{x\to \infty} f(x) $, which depends on x "which could be either odd or even." (So I'm assuming we are taking $x$ to be an integer, since the property of being "odd" or "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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$f$ continuous in $[a,b]$ and differentiable in $(a,b)$ without lateral derivatives at $a$ and $b$ Does anyone know an example of a real function $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ such that the lateral derivatives
$$ \lim_{h \to a^{+}} \frac{f(x+h)- f(x)}{h} \quad \text{and} \quad \lim_{h \to b^{-... | $$f(x) = \sqrt{x-a} + \sqrt{b-x} \,\,\,\,\,\, \forall x \in [a,b]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Integral $\int_{0}^{1}\ln x \, dx$ I have a question about the integral of $\ln x$.
When I try to calculate the integral of $\ln x$ from 0 to 1, I always get the following result.
*
*$\int_0^1 \ln x = x(\ln x -1) |_0^1 = 1(\ln 1 -1) - 0 (\ln 0 -1)$
Is the second part of the calculation indeterminate or 0?
What am I ... | Looking sideways at the graph of $\log(x)$ you can also see that $$\int_0^1\log(x)dx = -\int_0^\infty e^{-x}dx = -1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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Prove that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\geq\frac{1}{2}(a+b+c)$ for positive $a,b,c$ Prove the following inequality: for
$a,b,c>0$
$$\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\geq\frac{1}{2}(a+b+c)$$
What I tried is using substitution:
$p=a+b+c$
$q=ab+bc+ca$
$r=abc$
But I cannot reduce $a^2(b+c)... | Hint: $ \sum \frac{a^2 - b^2}{a+b} = \sum (a-b) = 0$.
(How is this used?)
Hint: $\sum \frac{a^2 + b^2}{a+b} \geq \sum \frac{a+b}{2} = a+b+c$ by AM-GM.
Hence, $\sum \frac{ a^2}{ a+b} \geq \frac{1}{2}(a+b+c)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding power series representation How we can show a presentation of a power series and indicate its radius of convergence?
For example how we can find a power series representation of the following function?
$$f(x) = \frac{x^3}{(1 + 3x^2)^2}$$
| 1) Write down the long familiar power series representation of $\dfrac{1}{1-t}$.
2) Differentiate term by term to get the power series representation of $\dfrac{1}{(1-t)^2}$.
3) Substitute $-3x^2$ everywhere that you see $t$ in the result of 2).
4) Multiply term by term by $x^3$.
For the radius of convergence, once yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Uncountable closed set of irrational numbers Could you construct an actual example of a uncountable set of irrational numbers that is closed (in the topological sense)?
I can find countable examples that are closed, like $\{ \sqrt{2} + \sqrt{2}/n \}_{n=1}^\infty \cup \{ \sqrt2 \}$ , but how does one construct an uncoun... | Explicit example: translation of a Cantor-like set.
Consider the Cantor set $$C := \Big\{ \sum \limits_{n=1}^{+\infty} \frac{\varepsilon_n}{4^n}\ \mid\ (\varepsilon_n)_n \in \{0,1\}^{\mathbb{N}}\Big\}.$$
It is uncountable and closed. Consider now the number $$x := \sum \limits_{n=1}^{+\infty} \frac{2}{4^{n^2}}.$$
The c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 7,
"answer_id": 6
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Proving a Geometric Progression Formula, Related to Geometric Distribution I am trying to prove a geometric progression formula that is related to the formula for the second moment of the geometric distribution. Specifically, I am wondering where I am going wrong, so I can perhaps learn a new technique.
It is known, an... | You have $$\sum_{k=0}^{\infty} ka^k = \dfrac{a}{(1-a)^2}$$
Differentiating with respect to $a$ gives us
$$\sum_{k=0}^{\infty} k^2 a^{k-1} = \dfrac{(1-a)^2 - a \times 2 \times (a-1)}{(1-a)^4} = \dfrac{1-a + 2a}{(1-a)^3} = \dfrac{a-1+2}{(1-a)^3}\\ = \dfrac2{(1-a)^3} - \dfrac1{(1-a)^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/265142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Helly's selection theorem (For sequence of monotonic functions) Let $\{f_n\}$ be a sequence of monotonically increasing functions on $\mathbb{R}$.
Let $\{f_n\}$ be uniformly bounded on $\mathbb{R}$.
Then, there exists a subsequence $\{f_{n_k}\}$ pointwise convergent to some $f$.
Now, assume $f$ is continuous on $\mathb... | It's absolutely my fault that i didn't even read (c) in the link. I extend the theorem in the link and my argument below is going to prove;
"If $K$ is a compact subset of $\mathbb{R}$ and $\{f_n\}$ is a sequence of monotonic functions on $K$ such that $f_n\rightarrow f$ pointwise on $K$, then $f_n\rightarrow f$ uniform... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What exactly is steady-state solution? In solving differential equation, one encounters steady-state solutions. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t \rightarrow \infty$. But the steady-state solution is given as $f(t)$, and this means that th... | Example from dynamics: You can picture for yourself a cantilever beam which is loaded by a force at its tip say: $F(t) = \sin(t)$. At $t=0$ the force is applied, then you get the transient state, after some time the system will become in equilibrium: the steady-state. In this state no changes are applied to the system.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 6,
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Absoluteness of $ \text{Con}(\mathsf{ZFC}) $ for Transitive Models of $ \mathsf{ZFC} $. Is $ \text{Con}(\mathsf{ZFC}) $ absolute for transitive models of $ \mathsf{ZFC} $? It appears that $ \text{Con}(\mathsf{ZFC}) $ is a statement only about logical syntax. Taking any $ \in $-sentence $ \varphi $, we can write $ \text... | Yes, $\text{Con}(\mathsf{ZFC})$ is an arithmetic statement ($\Pi^0_1$ in particular, because it says a computer program that looks for an inconsistency will never halt) so it is absolute to transitive models, and your proof is correct.
By the way, there are a couple of ways you can strengthen it. First, arithmetic sta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Math question please Rolle theorem? I have to prove that the equation $$x^5 +3x- 6$$ can't have more than one real root..so the function is continuous, has a derivative (both in $R$) . In $R$ there must be an interval where $f'(c)=0$, and if I prove this,than the equation has at least one real root. So $5x^4+3 =0$ ..th... | So you want to prove $5x^4+3=0$ has only one root at $0$? That's not true as $5x^4+3>0$. This establishes the proof
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/265429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Combinatorics alphabet If say I want to arrange the letters of the alphabet a,b,c,d,e,f such that e and f cannot be next to each other.
I would think the answer was $6\times4\times4\times3\times2$ as there are first 6 letters then 4 as e cannot be next to f.
Thanks.
| The $6$ numbers without any restriction can be arranged in $6!$ ways.
If we put $e,f$ together, we can arrange the $6$ numbers in $2!(5!)$ ways, as $e,f$ can arranged in $2!$ ways.
So, the required number of combinations is $6!-2(5!)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Average limit superior Let $\mathcal{l}_\mathbb{R}^\infty$ be the space of bounded sequences in $\mathbb{R}$. We define a map $p: \mathcal{l}_\mathbb{R}^\infty\to\mathbb{R}$ by
$$p(\underline x)=\limsup_{n\to\infty} \frac{1}{n}\sum_{k=1}^n x_k.$$
My notes claim that
$$\liminf_{n\to\infty} x_n\le p(\underline x)\le \lim... | Let $A = \liminf_{n \to \infty} x_n$ and $B = \limsup_{n \to \infty} x_n$. For any $\epsilon > 0$, there is $N$ such that for all $k > N$, $A - \epsilon \le x_k \le B + \epsilon$. Let $S_n = \displaystyle \sum_{k=1}^n x_k$. Then for $n > N$,
$$ S_N + (n-N) (A - \epsilon) \le S_n \le S_N + (n-N) (B + \epsilon) $$
and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cohen–Macaulayness of $R=k[x_1, \dots,x_n]/\mathfrak p$ I'm looking for some help for the following question:
Let $k$ be a field and $R=k[x_1, \dots ,x_n]$.
Show that $R/\mathfrak p$ is Cohen–Macaulay if $\mathfrak p$ is a prime ideal
with $\operatorname{height} \mathfrak p \in\lbrace 0,1,n-1,n \rbrace$.
My proof... | Hint. $R$ integral domain, $\operatorname{ht}(p)=0\Rightarrow p=(0)$. If $\operatorname{ht}(p)=1$, then $p$ is principal. If $\operatorname{ht}(p)=n−1,n$, then $\dim R/p=1,0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/265596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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an equality related to the point measure For any positive measure $\rho$ on $[-\pi, \pi]$, prove the following equality:
$$\lim_{N\to\infty}\int_{-\pi}^{\pi}\frac{\sum_{n=1}^Ne^{in\theta}}{N}d\rho(\theta)=\rho(\{0\}).$$
Remark:
It is easy to check that for any fixed positive number $0<\delta<\pi$, then $$|\int_{\delt... | Try to show that $\int_{-\delta}^\delta e^{in\theta}d\rho(\theta)\rightarrow \rho(\{0\})$, a kind of generalized Riemann Lebesgue Lemma. Then your result will follow by the fact that you are taking a Cesaro average of a sequence that converges. I believe you need some kind of sigma finite condition on your $\rho$ for t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the tangent function (like in trig) and tangent lines the same? So, a 45 degree angle in the unit circle has a tan value of 1. Does that mean the slope of a tangent line from that point is also 1? Or is something different entirely?
| The $\tan$ function can be described four different ways that I can describe and each adds to a fuller understanding of the tan function.
*
*First, the basics: the value of $\tan$ is equal to the value of $\sin$ over $\cos$.
$$\\tan(45^\circ)=\frac{\sin(45^\circ)}{\cos(45^\circ)}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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Who are the most inspiring communicators of math for a general audience? I have a podcast series (http://wildaboutmath.com/category/podcast/ and on Itunes https://itunes.apple.com/us/podcast/sol-ledermans-podcast/id588254197) where I interview people who have a passion for math and who have inspired others to take an i... | I recommend Art Benjamin. He's a dynamic speaker, has given lots of math talks to general audiences (mostly on tricks for doing quick mental math calculations, I think), and is an expert on combinatorial proof techniques (e.g. he's coauthor of Proofs That Really Count). Benjamin is a math professor at Harvey Mudd Col... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 13,
"answer_id": 3
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Question With Regards To Evaluating A Definite Integral When Evaluating the below definite integral $$\int_{0}^{\pi}(2\sin\theta + \cos3\theta)\,d\theta$$
I get this.$$\left [-2\cos\theta + \frac{\sin3\theta}{3} \right ]_{0}^{\pi} $$
In the above expression i see that $-2$ is a constant which was taken outside the in... | $$\int_{0}^{\pi}(2\sin\theta + \cos3\theta)d\theta=\left [-2\cos\theta + \frac{\sin3\theta}{3} \right ]_{0}^{\pi} $$ as you noted so $-2$ as you see in @Nameless's answer is just for cosine function. Not for all terms.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Problem from "Differential topology" by Guillemin I am strugling one of the problems of "Differential Topology" by Guillemin:
Suppose that $Z$ is an $l$-dimensional submanifold of $X$ and that $z\in Z$. Show that there exsists a local coordinate system $\left \{ x_{1},...,x_{k} \right \}$ defined in a neighbourhood $... | I found answers from Henry T. Horton at the question Why the matrix of $dG_0$ is $I_l$. and Augument, and injectivity. very helpful for solving this question.
To repeat your question, which is found in Guillemin & Pallock's Differential Topology on Page 18, problem 2:
Suppose that $Z$ is an $l$-dimensional submanifold... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Proving the stabilizer is a subgroup of the group to prove the Orbit-Stabiliser theorem I have to prove the OS theorem. The OS theorem states that for some group $G$, acting on some set $X$, we get
$$
|G| = |\mathrm{Orb}(x)| \cdot |G_x| $$
To prove this, I said that this can be written as
$$ |\mathrm{Orb}(x)| = \frac{|... | You’re on the right track. By definition $G_x=\{g\in G:g\cdot x=x\}$. Suppose that $g,h\in G_x$; then $$(gh)\cdot x=g\cdot(h\cdot x)=g\cdot x=x\;,$$ so $gh\in G_x$, and $G_x$ is closed under the group operation. Moreover, $$g^{-1}\cdot x=g^{-1}\cdot(g\cdot x)=(g^{-1}g)\cdot x=1_G\cdot x=x\;,$$ so $g^{-1}\in G_x$, and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Given a ratio of the height of two similar triangles and the area of the larger triangle, calculate the area of the smaller triangle Please help, I've been working on this problem for ages and I can't seem to get the answer.
The heights of two similar triangles are in the ratio 2:5. If the area of the larger triangle i... | It may help you to view your ratio as a fraction in this case. Right now your ratio is for one-dimensional measurements, like height, so if you were to calculate the height of the large triangle based on the height of the small triangle being (for example) 3, you would write:
$3 \times \frac52 =$ height of the large ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simple integral help How do I integrate $$\int_{0}^1 x \bigg\lceil \frac{1}{x} \bigg\rceil \left\{ \frac{1}{x} \right\}\, dx$$
Where $\lceil x \rceil $ is the ceiling function, and $\left\{x\right\}$ is the fractional part function
| Split the integral up into segments $S_m=[1/m,1/(m+1)]$ with $[0,1]= \cup_{m=1}^\infty S_m$. In the segment $m$, we have $\lceil 1/x \rceil=m+1$ and $\{1/x\} = 1/x- \lfloor 1/x\rfloor = 1/x - m$ (apart from values of $x$ on the boundary which do not contribute to the integral).
This yields
$$\begin{align}\int_0^1 x \bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Algebraic manipulation of normal, $\chi^2$ and Gamma probability distributions If $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$, then
$$
\frac{n - 1}{\sigma^2}S^2 \sim \chi^2_{n - 1}
$$
where $S^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i^2- \bar{x})^2$, and there's a direct relationship between the $\chi^2_p$ and Gamma($\alpha, \... | Suppose $X\sim \operatorname{Gamma}(\alpha,\beta)$, so that the density is $cx^{\alpha-1} e^{-x/\beta}$ on $x>0$, and $\beta$ is the scale parameter. Let $Y=kX$. The density function of $Y$ is
$$
\frac{d}{dx} \Pr(Y\le x) = \frac{d}{dx}\Pr(kX\le x) = \frac{d}{dx} \Pr\left(X\le\frac x k\right) = \frac{d}{dx}\int_0^{x/k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definition of $L^0$ space From Wikipedia:
The vector space of (equivalence classes of) measurable functions on $(S, Σ, μ)$ is denoted $L^0(S, Σ, μ)$.
This doesn't seem connected to the definition of $L^p(S, Σ, μ), \forall p \in (0, \infty)$ as being the set of measurable functions $f$ such that $\int_S |f|^p d\mu <\... | Note that when we restrict ourselves to the probability measures, then this terminology makes sense: $L^p$ is the space of those (equivalence classes of) measurable functions $f$ satisfying
$$\int |f|^p<\infty.$$
Therefore $L^0$ should be the space of those (equivalence classes of) measurable functions $f$ satisfying ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
"answer_id": 3
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Summing elements of a sequence Let the sequence $a_n$ be defined as $a_n = 2^n$ where $n = 0, 1, 2, \ldots $
That is, the sequence is $1, 2, 4, 8, \ldots$
Now assume I am told that a certain number is obtained by taking some of the numbers in the above sequence and adding them together (e.g. $a_4 + a_{19} + a_5$), I m... | Fundamental reason: The division algorithm.
For any number $a\geq 0$, we know there exists a unique $q_0\geq 0$ and $0\leq r_0<2$ such that
$$a=2q_0+r_0.$$
Similarly, we know there exists a unique $q_1\geq 0$ and $0\leq r_1<2$ such that
$$q_0=2q_1+r_1.$$
We can define the sequences $q_0,q_1,\ldots$ and $r_0,r_1,\ldots$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$a_{n}$ converges and $\frac{a_{n}}{n+1}$ too? I have a sequence $a_{n}$ which converges to $a$, then I have another sequence which is based on $a_{n}$: $b_{n}:=\frac{a_{n}}{n+1}$, now I have to show that $b_{n}$ also converges to $a$.
My steps:
$$\frac{a_{n}}{n+1}=\frac{1}{n+1}\cdot a_{n}=0\cdot a=0$$ But this is wro... | For $a_n=1$, clearly $a_n \to 1$ and $b_n \to 0$. So the result you are trying to prove is false.
In fact, because product is continuous, $\lim \frac{a_n}{n+1} = (\lim a_n) (\lim \frac{1}{n+1})=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/266345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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L'Hospital's Rule Question. show that if $x $ is an element of $\mathbb R$ then $$\lim_{n\to\infty} \left(1 + \frac xn\right)^n = e^x $$
(HINT: Take logs and use L'Hospital's Rule)
i'm not too sure how to go about answer this or putting it in the form $\frac{f'(x)}{g'(x)}$ in order to apply L'Hospitals Rule.
so far i'... | $$\lim_{n\to\infty} (1 + \frac xn)^n =\lim_{n\to\infty} e^{n\ln(1 + \frac xn)} $$
The limit
$$\lim_{n\to\infty} n\ln(1 + \frac xn)=\lim_{n\to\infty} \frac{\ln(1 + \frac xn)}{\frac1n}=\lim_{n\to\infty} \frac{\frac{1}{1 + \frac xn}\frac{-x}{n^2}}{-\frac1{n^2}}=\lim_{n\to\infty} \frac{x}{1 + \frac xn}=x$$
By continuity o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to prove everything in mathematics by theorem provers such as Coq? Coq has been used to provide formal proofs to the Four Colour theorem, the Feit–Thompson theorem, and I'm sure many more. I was wondering - is there anything that can't be proved in theorem provers such as Coq?
A little extra question is ... | It is reasonable to believe that everything that has been (or can be) formally proved can bew proved in such an explicitly formal way that a "stupid" proof verification system can give its thumbs up. In fact, while typical everyday proofs may have some informal handwaving parts in them, these should always be able to b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Test for convergence the series $\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}$ Test for convergence the series
$$\sum_{n=1}^{\infty}\frac{1}{n^{(n+1)/n}}$$
I'd like to make up a collection with solutions for this series, and any new
solution will be rewarded with upvotes. Here is what I have at the moment
Method 1
We kn... | Let $a_n = 1/n^{(n+1)/n}$.
Then
$$\begin{eqnarray*}
\frac{a_n}{a_{n+1}} &\sim& 1+\frac{1}{n} - \frac{\log n}{n^2}
\qquad (n\to\infty).
\end{eqnarray*}$$
The series diverges by Bertrand's test.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/266547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Computing left derived functors from acyclic complexes (not resolutions!) I am reading a paper where the following trick is used:
To compute the left derived functors $L_{i}FM$ of a right-exact functor $F$ on an object $M$ in a certain abelian category, the authors construct a complex (not a resolution!) of acyclic obj... | Compare with a projective resolution $P_\bullet\to M\to 0$. By projectivity, we obtain (from the identiy $M\to M$) a complex morphism $P_\bullet\to A_\bullet$, which induces $F(P_\bullet)\to F(A_\bullet)$. With a bit of diagram chasing you shold find that $H_\bullet(F(P_\bullet))$ is the same as $H_\bullet(F(A_\bullet)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Quotient Group G/G = {identity}? I know this is a basic question, but I'm trying to convince myself of Wikipedia's statement. "The quotient group $G / G$ is isomorphic to the trivial group."
I write the definition for left multiplication because left cosets = right cosets. $ G/G = \{g \in G : gG\} $ But how is this iso... | If G is a group and N is normal in G, then G/N is the quotient group. G/N as a group consists of cosets of the normal subgroup N in G and these cosets themselves satisfy the group properties because of normality of N. Now G is clearly normal in G. Hence G/G consists of the coset that is all of G. Thus this group has on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 8,
"answer_id": 5
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Want to show $\sum_{n=2}^{\infty} \frac{1}{2^{n}*n}$ converges Want to show $\sum_{n=2}^{\infty} \frac{1}{2^{n}*n}$ converges. I am trying to show this by showing that the partial sums are bounded. I have tried doing this by induction but am not seeing how to pass the inductive assumption part. Do I need to instead loo... | Since you mentioned induction:
Let $s_m = \sum_{n=2}^{m} \frac{1}{2^{n}*n}$. Then $s_m \leq 1-\frac{1}{m}$.
$P(2)$ is obvious, while $P(m) \Rightarrow P(m+1)$ reduces to
$$1-\frac{1}{m}+\frac{1}{2^{m+1}(m+1)} \leq 1- \frac{1}{m+1}$$
which is equivalent to:
$$m \leq 2^{m+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/266788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 8,
"answer_id": 4
} |
difficulty understanding branch of the logarithm Here is one past qual question,
Prove that the function $\log(z+ \sqrt{z^2-1})$ can be defined to be analytic on the domain $\mathbb{C} \setminus (-\infty,1]$
(Hint: start by defining an appropriate branch of $ \sqrt{z^2-1}$ on $\mathbb{C}\setminus (-\infty,1]$ )
It j... | Alternatively, you can just take the standard branch for $\sqrt{z}$ excluding $(-\infty,0]$ and then compute $\sqrt{z-1}\sqrt{z+1}$ which is defined for $z+1,z-1\notin(-\infty,0]$, that is, for $z\notin(-\infty,1]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/266857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
how to find the nth number in the sequence? consider the sequence of numbers below,
2 5 10 18 31 52 . . .
the sequence goes on like this.
My Question is,
How to find the nth term in the sequence?
thanks.
| The sequence can be expressed in many ways.
As Matt N. and M. Strochyk mentioned:
$$ a_{n+2}= a_{n}+a_{n+1}+3,$$
$$ a_1 = 2 \quad (n\in \mathbb{N})$$
Or as this one for example:
$$ a_{n+1}= a_{n}+\frac{(n-1)n(2n-1)}{12}-\frac{(n-1)n}{4}+2n+1,$$
$$ a_1 = 2 \quad (n\in \mathbb{N})$$
It's interesting that the term:
$$ b_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 6
} |
Show $\lim\limits_{n\to\infty} \sqrt[n]{n^e+e^n}=e$ Why is $\lim\limits_{n\to\infty} \sqrt[n]{n^e+e^n}$ = $e$? I couldn't get this result.
| Taking logs, you must show that
$$\lim_{n \rightarrow \infty} {\ln(n^e + e^n) \over n} = 1$$
Applying L'hopital's rule, this is equivalent to showing
$$\lim_{n \rightarrow \infty}{en^{e-1} + e^n \over n^e + e^n} = 1$$
Which is the same as
$$\lim_{n \rightarrow \infty}{e{n^{e-1}\over e^n} + 1 \over {n^e \over e^n}+ 1} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 8,
"answer_id": 7
} |
A non-linear maximisation We know that $x+y=3$ where x and y are positive real numbers. How can one find the maximum value of $x^2y$? Is it $4,3\sqrt{2}, 9/4$ or $2$?
| By AM-GM
$$\sqrt[3]{2x^2y} \leq \frac{x+x+2y}{3}=2 $$
with equality if and only if $x=x=2y$.
Second solution
This one is more complicated, and artificial (since I needed to know the max)$.
$$x^2y=3x^2-x^3=-4+3x^2-x^3+4=4- (x-2)^2(x+1)\leq 4$$
since $(x-2)^2(x+1) \geq 0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Convergence of the series Im trying to resolve the next exercise:
$$\sum_{n=1}^\infty\ e^{an}n^2 \text{ , }a\in R $$
I dont know in which ranges I should separe the a value for resolving the limit and finding out the convergence.
| Write it as
$\sum_{n=1}^\infty\ r^n n^2$
where $r = e^a$ satisfies $0 < r$.
If $r \ge 1$ (i.e., $a \ge 0$), the sum clearly diverges.
If $r < 1$ (i.e., $a < 0$), you can get an explicit formula
for $\sum_{n=1}^m\ r^n n^2$
which will show that the sum converges.
Therefore the sum converges for
$a < 0$ and diverges for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
2 heads or more in 3 coin toss formula what is the formula to calculate the probabilities of getting 2 heads or more in 3 coin toss ?
i've seen a lot of solution but almost all of them were using method of listing all of possible combination like HHT,HTH,etc
what i am trying to ask is the formula and/or method to calcu... | The simplest is by symmetry. The chance of at least two heads equals the chance of at least two tails, and if you add them you get exactly $1$ because one or the other has to happen. Thus the chance is $\frac 12$. This approach is not always available.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/267186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Infinite sum of floor functions I need to compute this (convergent) sum
$$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$
But I have no idea how to get rid of the floor thing. I thought about some variable substitution, but it didn't take me anywhere.
| We'll let $M=2^k$ throughout.
Note that $$f(j)=j-M\left\lfloor\frac{j}{M}\right\rfloor$$
is just the modulus operator - it is equal to the smallest positive $n$ such that $j\equiv n\pmod {M}$
So that means $f(0)=0, f(1)=1,...f(M-1)=M-1,$ and $f(j+M)=f(j)$.
This means that we can write:
$$F(z)=\sum_{j=0}^{\infty} f(j)z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
On the metric subspace $S = [0,1]$ of the Euclidean space $\mathbb R^1 $, every interval of the form $A = [0,a)$ or $(a, 1]$ where $0<a<1$ is open set in S. These sets are not open in $\mathbb R^... | Hint: To show $(x,1]$ not open in $\mathbb R$ simply show that there is no open neighborhood of $1$ included in the half-closed interval.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/267293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Isosceles triangle Let $ \triangle ABC $ be an $C$-isosceles and $ P\in (AB) $ be a point so that $ m\left(\widehat{PCB}\right)=\phi $. Express $AP$ in terms of $C$, $c$ and $\tan\phi$.
Edited problem statement(same as above but in different words):
Let $ \triangle ABC $ be a isosceles triangle with right angle at $C$.... | Edited for revised question
Dropping the perpendicular from $C$ onto $AB$ will help. Call the point $E$.
Also drop the perpendicular from $P$ onto $BC$, and call the point $F$. Then drop the perpendicular from $F$ onto $AB$, and call the point $G$.
This gives a lot of similar and congruent triangles.
$$\tan \phi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Mathematical competitions for adults. I live in Mexico city. However, I am not so interested in whether these exist in my vicinity but want to learn if they exist, if they do then It might be easier to make them more popular in other places.
Are there mathematical competitions for adults? I have been into a couple of m... | Actually I know only one. I was looking for the same thing and found your question.
"Championnat International des Jeux Mathématiques et Logiques"
http://www.animath.fr/spip.php?article595
Questions and answers must be in french. But there is no requirement for participation. Any age and nationalities are welcome.
Qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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$A\unlhd G$ , $B\unlhd G$ and $C\unlhd G$ then $A(B∩C)$ is a normal subgroup of $G$ If $A$ normal to $G$ , $B$ normal to $G$ and $C$ normal to $G$ then how can I show that$$A(B∩C)\unlhd G$$
how can i solve this problem? Thanks!
| You know that if $B,C$ be subgroups of a group so does their intersection. Moreover if one of subgroups $A$ and $B\cap C$ are normal in $G$, so we have a theorem saying $A(B\cap C)\leq G$ also. Now show that the normality of $A(B\cap C)$ in $G$. In fact, show that: $$\forall x\in A(B\cap C), g\in G$$ we have $g^{-1}xg\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Closed set in $\ell^1$
Show that the set $$ B = \left\lbrace(x_n) \in \ell^1 : \sum_{n\geq 1} n|x_n|\leq 1\right\rbrace$$
is compact in $\ell^1$.
Hint: You can use without proof the diagonalization process to conclude that every bounded sequence $(x_n)\in \ell^\infty$ has a subsequence $(x_{n_k})$ that converges i... | We can use and show the following:
Let $K\subset \ell^1$. This set has a compact closure for the $\ell^1$ norm if and only if the following conditions are satisfied:
*
*$\sup_{x\in K}\lVert x\rVert_{\ell^1}$ is finite, and
*for all $\varepsilon>0$, we can find $N$ such that for all $x\in K$, $\sum_{k\geqslant N}|x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Boundedness of an integral operator Let $K_n \in L^1([0,1]), n \geq 1$ and define a linear map $T$ from $L^\infty([0,1]) $to sequences by
$$ Tf = (x_n), \;\; x_n =\int_0^1 K_n(x)f(x)dx$$
Show that $T$ is a bounded linear operator from $L^\infty([0,1]) $to $\ell^\infty$ iff
$$\sup_{n\geq 1} \int_0^1|K_n(x)| dx \lt \i... | Yes, you can choose $f$ as you want. If $T$ is bounded then
$$
\exists C>0\qquad\left\vert \int_0^1K_n(x)f(x)\,dx\right\vert\leq C \Vert f\Vert_{L^\infty}\qquad \forall f\in L^\infty\quad \forall n\in\mathbb{N}.
$$
Fix $m\in\mathbb{N}$, if we take $f=\text{sign}(K_m)\in L^\infty$ then
$$
\int_0^1 \vert K_m(x)\vert\,dx\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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convergence of weighted average It is well known that for any sequence $\{x_n\}$ of real or complex numbers which converges to a limit $x$, the sequence of averages of the first $n$ terms is also convergent to $x$. That is, the sequence $\{a_n\}$ defined by
$$a_n = \frac{x_1+x_2+\ldots + x_n}{n}$$
converges to $x$. ... | Weighted averages belong to the class of matrix summation methods.
Define
$$W:=\left(\begin{matrix}W_{1,1},W_{1,2},\ldots\\W_{2,1},W_{2,2},\ldots\\\vdots\\\end{matrix}\right)$$
Represent the sequence $\{x_n\}$ by the infinite vector $X:=\left(\begin{matrix}x_1\\x_2\\\vdots\end{matrix}\right)$, and $\{b_n\}$ by the v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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What is the value of the given limit?
Possible Duplicate:
How can I prove Infinitesimal Limit
Let $$\lim_{x\to 0}f(x)=0$$ and $$\lim_{x\to 0}\frac{f(2x)-f(x)}{x}=0$$
Then what is the value of $$\lim_{x\to 0}\frac{f(x)}{x}$$
| If $\displaystyle\lim_{x\to0}\frac{f(x)}{x}=L$ then
$$
\lim_{x\to0}\frac{f(2x)}{x} = 2\lim_{x\to0}\frac{f(2x)}{2x} = 2\lim_{u\to0}\frac{f(u)}{u} = 2L.
$$
Then
$$
\lim_{x\to0}\frac{f(2x)-f(x)}{x} = \lim_{x\to0}\frac{f(2x)}{x} - \lim_{x\to0}\frac{f(x)}{x} =\cdots
$$
etc.
Later note: What is written above holds in cases... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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The control of norm in quotient algebra Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and a constant $c>0$ such that for any $v \in B_1$,$${\left\| {Tv} \right\|_{{B_1}}... | To start from a very simple case: If $B_1$ is a Hilbert space and $S$ is finite-dimensional such that we have
$$ \|Tv\| \le c\|v\| + \|Sv\| \quad \forall v$$
then we can find a finite-dimensional $A$ satisfying
$$\tag{1} \|Av\| \le \|Sv\|$$
and
$$\tag{2} \|(T-A)v\| \le c \|v\|$$
for every $v \in B_1$.
Proof: Write $B_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find all entire functions $f$ such that for all $z\in \mathbb{C}$, $|f(z)|\ge \frac{1}{|z|+1}$ Find all entire functions $f$ such that for all $z\in \mathbb{C}$, $|f(z)|\ge \frac{1}{|z|+1}$
This is one of the past qualifying exams that I was working on and I think that I have to find the function that involved with $f$... | Suppose $f$ is not constant. As an entire non-constant function it must have some sort of singularity at infinity. It cannot be a pole (because then it would have a zero somewhere, cf the winding-number proof of the FTA), so it must be an essential singularity. Then $zf(z)$ also has an essential singularity at infinity... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
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move a point up and down along a sphere I have a problem where i have a sphere and 1 point that can be anywhere on that sphere's surface. The Sphere is at the center point (0,0,0).
I now need to get 2 new points, 1 just a little below the and another little above this in reference to the Y axis. If needed or simpler to... | The conversion between Cartesian and spherical coordinates is
$$ (x,y,z) = (R \sin{\theta} \cos {\phi},R \sin{\theta} \sin {\phi}, R \cos{\theta})$$
where $R$ is the radius of the earth/sphere, $\theta = $ $+90^{\circ}$- latitude, and $\phi=$ longitude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/268064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Proof by induction on $\{1,\ldots,m\}$ instead of $\mathbb{N}$ I often see proofs, that claim to be by induction, but where the variable we induct on doesn't take value is $\mathbb{N}$ but only in some set $\{1,\ldots,m\}$.
Imagine for example that we have to prove an equality that encompasses $n$ variables on each sid... | If the statement in question really does not "work" if $n>m$, then necessarily the induction step $n\to n+1$ at least somewhere uses that $n<m$. You may view this as actually proving by induction
$$\tag1\forall n\in \mathbb N\colon (n>m\lor \phi(m,n))$$
That is, you first show
$$\tag2\phi(m,1)$$
(which of course impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Differing properties due to differing topologies on the same set I have been working on this problem from Principles of Topology by Croom:
"Let $X$ be a set with three different topologies $S$, $T$, $U$ for which $S$ is weaker than $T$, $T$ is weaker than $U$, and $(X,T)$ is compact and Hausdorff. Show that $(X,S)$ is... | If $X$ is finite, the only compact Hausdorff topology on $X$ is the discrete topology, so $T=\wp(X)$. In this case there is no strictly finer topology $U$. If $X$ is infinite, the discrete topology on $X$ is not compact, so if $T$ is a compact Hausdorff topology on $X$, there is always a strictly finer topology.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/268218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Sampling from a $2$d normal with a given covariance matrix How would one sample from the $2$-dimensional normal distribution with mean $0$ and covariance matrix $$\begin{bmatrix} a & b\\b & c \end{bmatrix}$$ given the ability to sample from the standard ($1$-dimensional) normal distribution?
This seems like it should b... | Say you have a random variable $X\sim N(0,E)$ where $E$ is the identity matrix. Let $A$ be a matrix. Then $Y:=AX\sim N(0,AA^T)$. Hence you need to find a matrix $A$ with $AA^T = \left[\matrix{a & b \\ b & c}\right]$. There is no unique solution to this problem. One popular method is the Cholesky decomposition, where yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
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A Question on $p$-groups. Suppose $G$ is a group with order $p^{n}$ ($p$ is a prime).
Do we know when we can find the subgroups of $G$ of order $p^{2}, p^{3}, \cdots, p^{n-1}$?
| (Approach modified in light of Don Antonio's comment below question). Another way to proceed, which may not be so common in textbooks, and which produces a normal subgroup of each possible index, is as follows. Suppose we have a non-trivial normal subgroup $Q$ of $P$ (possibly $Q = P$). We will produce a normal subgrou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A complex equation I want to solve the following equation:
$g(s)f(s)=0$
where $f$ and $g$ are defined in the complex plane with real values and they are not analytic.
My question is:
If I assume that $f(s)≠0$, can I deduce that $g(s)=0$ without any further complications?
I am a little confused about this case: if $... | First, note that your question is really just about individual complex numbers, not about complex-valued functions.
Now, as you note, if $(x + i y)(u + i v) = 0$ then this implies that $x u - v y = x v + u y = 0$. However, the only way these equations can hold is if either $x + i y = 0$ or $u + i v = 0$.
Multiplying... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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how many number like $119$ How many 3-digits number has this property like $119$:
$119$ divided by $2$ the remainder is $1$
119 divided by $3$ the remainderis $2 $
$119$ divided by $4$ the remainder is $3$
$119$ divided by $5$ the remainder is $4$
$119$ divided by $6$ the remainder is $5$
| You seek numbers which, when divided by $k$ (for $k=2,3,4,5,6$) gives a remainder of $k-1$. Thus the numbers you seek are precisely those which are one less than a multiple of $k$ for each of these values of $k$. To find all such numbers, consider the lowest common multiple of $2$, $3$, $4$, $5$ and $6$, and count how ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
} |
Question on determining the splitting field It is not hard to check that the three roots of $x^3-2=0$ is $\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}$, hence the splitting field for $x^3-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}]$. However, since $\sqrt[3]{2... | Note that $(\alpha\zeta_n^k)^n = \alpha^n\zeta_n^{nk}=\alpha^n=a$, $0\le k<n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/268676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Are there real-life relations which are symmetric and reflexive but not transitive? Inspired by Halmos (Naive Set Theory) . . .
For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have the other two.
One can construct each of... | Actors $x$ and $y$ have appear in the same movie at least once.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/268726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "155",
"answer_count": 15,
"answer_id": 3
} |
Trouble with form of remainder of $\frac{1}{1+x}$ While asking a question here I got the following equality
$\displaystyle\frac{1}{1+t}=\sum\limits_{k=0}^{n}(-1)^k t^{k}+\frac{(-1)^{n+1}t^{n+1}}{1+t}$
I'm trying to prove this with Taylor's theorem, I got that $f^{(n)}(x)=\displaystyle\frac{(-1)^nn!}{(1+x)^{n+1}}$ so th... | Taylor will not help you here.
Hint: $\sum_{k=0}^n r^k=\frac{1-r^{n+1}}{1-r}$ for all positive integers $n$, when $r\neq 1$.
Take $r=$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/268768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What does it mean $\int_a^b f(G(x)) dG(x)$? - An exercise question on measure theory I am reading Folland's book and definitions are as follows (p. 108).
Let $G$ be a continuous increasing function on $[a,b]$ and let $G(a) = c, G(b) = d$.
What is asked in the question is:
If $f$ is a Borel measurable and integrable ... | This is likely either Riemann-Stieltjes or Lebesgue-Stieltjes integration (most likely the latter, given the context).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/268821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$ Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally:
$$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$
If this statement is true, what is the proof ?
| This is true in $ZFC$ because of Zermelo's Well-Ordering Theorem; given two sets $A,B$, since they are well-orderable there exist alephs $\aleph_{\alpha},\aleph_{\beta}$ with $|A|=\aleph_{\alpha}$ and $|B|=\aleph_{\beta}$, since alephs are comparable, the cardinalities of $A$ and $B$ are comparable.
Furthermore, this i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
Check my proof of an algebraic statement about fractions I tried to prove the part c) of "Problem 42" from the book "Algebra" by Gelfand.
Fractions $\frac{a}{b}$ and $\frac{c}{d}$ are called neighbor fractions if their difference $\frac{cb-ad}{db}$ has numerator ±1, that is, $cb-ad = ±1$. Prove that:
b) If $\frac{a}{b... | First of all, the proof is correct and I congratulate you on the excellent effort. I will only offer a few small comments on the writing.
It's not clear until all the way down at (5) that you intend to do a proof by contradiction, and even then you never make it explicit. It's generally polite to state at the very begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How to calculate $\sum \limits_{x=0}^{n} \frac{n!}{(n-x)!\,n^x}\left(1-\frac{x(x-1)}{n(n-1)}\right)$ What are the asymptotics of the following sum as $n$ goes to infinity?
$$
S =\sum\limits_{x=0}^{n} \frac{n!}{(n-x)!\,n^x}\left(1-\frac{x(x-1)}{n(n-1)}\right)
$$
The sum comes from CDF related to sampling with replacemen... |
Let $N_n$ denote a Poisson random variable with parameter $n$, then
$$
S_n=\frac{n-2}{n-1}n!\left(\frac{\mathrm e}n\right)^n\mathbb P(N_n\leqslant n)+\frac2{n-1}.
$$
As a consequence, $\lim\limits_{n\to\infty}S_n/\sqrt{n}=\sqrt{\pi/2}$.
To show this (for a shorter proof, see the end of this post), first rewrite e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Diagonalizable unitarily Schur factorization Let $A$ be $n x n$ matrix.
What exactly is the difference between unitarily diagonalizable and diagonalizable
matrix $A$? Can that be that it is diagonalizable but not unitarily diagonalizable?
What are the conditions for Schur factorization to exist? For a (unitarily) diag... | Diagonalization means to decompose a square matrix $A$ into the form $PDP^{-1}$, where $P$ is invertible and $D$ is a diagonal matrix. If $P$ is chosen as a unitary matrix, the aforementioned decomposition is called a unitary diagonalization. It follows that every unitarily diagonalizable matrix is diagonalizable.
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compact subspaces of the Poset On page 172, James Munkres' textbook Topology(2ed), there is a theorem about compact subspaces of the real line:
Let $X$ be a simply-ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact.
My question is whether there is a genera... | Many topologies have been defined on partial orders and lattices of various types. One of the most important is the Scott topology. Let $\langle P,\preceq\rangle$ be a partial order. A set $A\subseteq P$ is an upper set if ${\uparrow\!\!x}\subseteq A$ whenever $x\in A$, where ${\uparrow\!\!x}=\{y\in P:x\preceq y\}$. A ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Difference between $u_t + \Delta u = f$ and $u_t - \Delta u = f$? What is the difference between these 2 equations? Instead of $\Delta$ change it to some general elliptic operator.
Do they have the same results? Which one is used for which?
| The relation boils down to time-reversal, replacing $t$ by $-t$. This makes a lot of difference in the equations that model diffusion. The diffusion processes observed in nature are normally not reversible (2nd law of thermodynamics). In parallel to that, the backward heat equation $u_t=-\Delta u$ exhibits peculiar and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
$\infty - \infty = 0$ ? I am given this sequence with square root. $a_n:=\sqrt{n+1000}-\sqrt{n}$. I have read that sequence converges to $0$, if $n \rightarrow \infty$. Then I said, well, it may be because $\sqrt{n}$ goes to $\infty$, and then $\infty - \infty = 0$. Am I right? If I am right, why am I right? I mean, ho... | $$\sqrt{n+100}-\sqrt n=\frac{100}{\sqrt{n+100}+\sqrt n}\xrightarrow [n\to\infty]{}0$$
But you're not right, since for example
$$\sqrt n-\sqrt\frac{n}{2}=\frac{\frac{n}{2}}{\sqrt n+\sqrt\frac{n}{2}}\xrightarrow [n\to\infty]{}\infty$$
In fact, "a difference $\,\infty-\infty\,$ in limits theory can be anything
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/269337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 7,
"answer_id": 2
} |
How to be sure that the $k$th largest singular value is at least 1 of a matrix containing a k-by-k identity In section 8.4 of the report of ID software, it says that the $k$th largest singular value of a $k \times n$ matrix $P$ is at least 1 if some subset of its columns makes up a $k\times k$ identity.
I tried to figu... | The middle part of the SVD (which containts the singular values) does not change if you permute columns, so you may put the $k$ columns mentioned first. So assume the matrix has the form $A=[\begin{smallmatrix}I&B\end{smallmatrix}]$ where $I$ is a $k\times k$ identity, and $B$ is $k\times(n-k)$. Compute $$AA^T=I+BB^T\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
non constant bounded holomorphic function on some open set this is an exercise I came across in Rudin's "Real and complex analysis" Chapter 16.
Suppose $\Omega$ is the complement set of $E$ in $\mathbb{C}$, where $E$ is a compact set with positive Lebesgue measure in the real line.
Does there exist a non-constant boun... | Reading Exercise 8 of Chapter 16, I imagine Rudin interrogating the reader.
Let $E\subset\mathbb R$ be a compact set of positive measure, let $\Omega=\mathbb C\setminus E$, and define $f(z)=\int_E \frac{dt}{t-z}$. Now answer me!
a) Is $f$ constant?
b) Can $f$ be extended to an entire function?
c) Does $zf(z)$ h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving that $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$, where $\alpha$ is the golden ratio I got stuck on this exercise. It is Theorem 1.15 on page 14 of Robbins' Beginning Number Theory, 2nd edition.
Theorem 1.15. $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$.
Proof: Exercise.
(image of r... | Using Binet's Fibonacci Number Formula,
$\alpha+\beta=1,\alpha\beta=-1$ and $\beta<0$
\begin{align}
F_n-\alpha^{n-1}=
&
\frac{\alpha^n-\beta^n}{\alpha-\beta}-\alpha^{n-1}
\\
=
&
\frac{\alpha^n-\beta^n-(\alpha-\beta)\alpha^{n-1}}{\alpha-\beta}
\\
=
&
\beta\frac{(\alpha^{n-1}-\beta^{n-1})}{\alpha-\beta}
\\
=
&
\beta\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Eigenvalues of the matrix $(I-P)$ Let $P$ be a strictly positive $n\times n$ stochastic matrix.
I hope to find out the stability of a system characterized by the matrix $(I-P)$. So I'm interested in knowing under what condition on the entries of $P$ do all the eigenvalues of the matrix $(I-P)$ lie (not necessarily str... | One sufficient condition (that is not necessary) is that all diagonal entries of $P$ are greater than or equal to $1/2$. If this is the case, by Gersgorin disc theorem, all eigenvalues of $I-P$ will lie inside the closed disc centered at $1/2$ with radius $1/2$, and hence lie inside the closed unit disc as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/269598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Linear independence of $\sin(x)$ and $\cos(x)$ In the vector space of $f:\mathbb R \to \mathbb R$, how do I prove that functions $\sin(x)$ and $\cos(x)$ are linearly independent. By def., two elements of a vector space are linearly independent if $0 = a\cos(x) + b\sin(x)$ implies that $a=b=0$, but how can I formalize t... | Although I'm not confident about this, maybe you can use power series for $\sin x$ and $\cos x$? I'm working on a similar exercise but mine has restricted both functions on the interval $[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/269668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 11,
"answer_id": 7
} |
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