Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$A$ is homeomorphic to $A\times A$ Is there any infinite topological space $A$ which is connected such that $A$ and $A\times A$ are homeomorphic?
| Yes, the easiest nice one is $\Bbb R^\mathbb{N}$ probably, in the product topology (product metric). If you want ugly spaces, the indiscrete topology on $\mathbb{N}$ also works.
In fact, for any connected space $C$, the space $C^\mathbb{N}$ will be an example.
| {
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Suppose that X is a cell complex with $\tilde{H}_{*}(X) = 0.$ Prove that the suspension $SX$ is contractible. The question is:
Suppose that X is a cell complex with $\tilde{H}_{*}(X) = 0.$ Prove that the suspension $SX$ is contractible.
I feel like this link contains a part (or maybe all) of the solution to this questi... | Sketch of solution :
1) the hypothesis implies that $X$ is connected (i.e. $0$-connected)
2)The given link then implies (or just by Van Kampen) $SX$ is $1$-connected
3) The suspension isomorphism, the Hurewicz theorem and the Whitehead theorem allow us to conclude
| {
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Normal subgroup of $S_3$? For the subgroup $N$ of $S_3$, $N = \{(1),(123),(132)\}$, I calculate that $(13)N = \{(13),(123),(23)\}$ and $N(13) = \{(13),(23),(12)\}$. Shouldn't this show that $N$ is not a normal subgroup, as opposed to what's printed here?
| Your computation of $(13)N$ is wrong. The $(123)$ should be $(12)$. One way to check this is that every permutation in $N$ is even, so the coset should consist only of odd permutations.
| {
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Normally distributed rain drops problem About 50% of raindrops land downtown, downtown is a perfectly circular space around the city centre. Assuming the coordinates of the raindrops are independent and distributed according to the standard normal about the city centre. What is the percentage of rain drops that land wi... | I believe you can use standard z-tables (e.g. https://en.wikipedia.org/wiki/Standard_normal_table) to work out the various percentages for the radius in question. The z-table will tell you the probability contained by the standard normal curve from 0 to whatever z-value you choose. Now, if you make the equivalence of... | {
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$X$ is a Hausdorff space and $f:X \rightarrow X$ a continuous function. Prove that $\{x \in X \mid f(x)=x\}$ is closed. (Is my proof correct?) Suppose $X$ is a Hausdorff space and $f:X \rightarrow X$ a continuous function. Prove that the set $\{x \in X \mid f(x)=x\}$ is closed in $X$.
I've already proved this propositi... | As an alternative proof for the general case (with both $f$ and $g$, and yes, we can of course take $g=\textrm{id}_X$ to derive the first from the second, as identities are always continuous), we can use nets: if $(x_i)_{i \in I}$ is a net in $X$ converging to some $x \in X$ and all $x_i, i \in I$ are in $C:=\{x\mid f... | {
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Boundedness of a linear operator on Hilbert space How can I prove that a linear operator $A$ on a Hilbert space $H$ that satisfies
$$ \langle x,Ay\rangle= \langle y,Ax\rangle$$
for all $x,y\in H$ is bounded (i.e., $\|Ax\|\leq c\|x\| $ for some constant $c>0$)?
| Let $B_{x}:y\rightarrow\left<y,Ax\right>$, then $B_{x}$ is a linear operator (beware that one shouldn't take $\left<Ax,y\right>$ because then it is conjugate linear, not linear), and $|B_{x}(y)|=\left|\left<y,Ax\right>\right|=\left|\left<x,Ay\right>\right|\leq\|x\|\|Ay\|$.
For fixed $y$, for all $x$ with $\|x\|\leq 1$,... | {
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What is a field of odd characteristic? I have some problem finding info about what a field of odd characteristic is?
An example:
Let $K$ be a field of odd characteristic. In [5], Bernstein and Lange
introduce Edwards curves defined by $x^2 + y^2 = c^2(1 + dx^2y^2)$
where $c, d ∈ K$ with $cd(1 − dc^4) \neq 0$. In ... | It is the smallest integer $p>0$ satisfying
$$0_K=\underbrace{1_K + 1_K \dots + 1_K}_{p \text{ times.}}$$
which is odd (specifically, it is a prime greater than 2.) There is a chance the writer was commenting on how weird the field is, but I would not think so (this is a joke.)
| {
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Prove or disprove that in an 8-element subsets of $\{1,2…,30\}$ there must exist two $4$-element subsets that sum to the same number. How can I show that for any set of $8$ distinct positive integers not exceeding $30$, there must exist two distinct $4$-elements subsets that same up to the same number?
I tried using pi... | The statement is false.
Take for example a subset with 7 odd numbers and 1 even number.
Then we divide this subset into two 4-element subsets. One of them will have 4 odd numbers whose sum will be an even number while the other will have 1 even number and 3 odd numbers which will add up to an odd number.
Example: 1,3,5... | {
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Non-emptiness of the relative interior in the infinite-dimensional case Let $E$ be a finite-dimensional space and let $C \subseteq E$ be a nonempty convex set. Then, the relative interior of $C$, which we denote by $\mbox{ri}(C)$, is nonempty.
If space $E$ is infinite-dimensional, does the result above still hold?
$$... | If I understood your notation right then a subset $C=\prod_{n=1}^{\infty} [-2^{-n};2^{-n}]$ of the space $\ell_2$ has empty relative interior, because $\operatorname{aff} C$ contains $x+e_n$ for each $x\in C$ and each $n$, where $e_n$ is the standard unit vector of $\ell_2$, but $x+2^{2-n} e_n\not\in C$.
| {
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How to find Mixed Nash Equilibrium and Correlated Equilibrium How do I find the mixed Nash equilibrium and Correlated Equilibrium of the following question? It looks impossible to find without any real numbers.
Given that $ M>>1>>\epsilon $
\begin{pmatrix}
(M,M)& (1+\epsilon,1+\epsilon)&(2\epsilon,2\epsilon)&(\epsilon... | Let $A$ be the payoff matrix you defined. Let $x = [x_1, x_2, x_3, x_4]^{\top}$. Let $y = Ax.$
The candidate to be a MNE is the vector $x$ such that:
$$y_1 = y_2 = y_3 = y_4.$$
By setting $x_4 = 1 -x_1-x_2-x_3$ and solving the previous system, one gets:
$$\begin{cases}
x_1 = \frac{\epsilon}{2\epsilon-M+1}\\
x_2 = \frac... | {
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Why does a sequence of unhappy numbers always loop back to itself? Given a positive integer $19$, it is said to be happy, because $1^2 + 9^2 = 82$, $8^2+2^2 = 68$, $6^2 + 8^2 = 100$, $1^2 + 0^2 + 0^2 = 1$. At each step we simply sum the square of all its digits, and if at some step the sum is equal to $1$, then we say ... | This is untrue. $2$ is unhappy and does not loop back to itself: $$2\to4\to16\to37\to58\to89\to145\to42\to20\to4$$
| {
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$\inf f(A) \leq f( \inf A)$ if $f$ is continuous Prove that $\inf f(A) \leq f( \inf A)$ if $f: [-\infty, + \infty] \to \mathbb{R}$ is continuous and $A \neq \emptyset$ is a subset of $\mathbb{R}$.
Attempt;
Put $a:= \inf A$. Choose a sequence in $A$ such that $a_n \to a$. Then
$$ \inf f(A)\leq\lim_{n \to \infty} \underb... | You have $\inf_x f(x) \le f(y)$ for all $y$ by definition. Ley $y = \inf A$ to finish.
| {
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Prove parallelogram has four triangles with same area using vectors I need to prove that a parallelogram has four triangles with same area using vectors only. Thought to prove it using area of triangle and parallelogram but with no success. May you help me please?
| Let $a$ and $b$ be the vectors of two nonparallel sides of you parallelogram (as in your linked picture). Now the diagonals of the parallelogram are given by $a+b$ and $a-b$. The four triangles to think about are-up to translation-the triangles formed by the pairs $(a, \tfrac{1}{2}(a+b))$, $(b, \tfrac{1}{2}(a-b))$, $... | {
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limit equaling zero at infinity I have a problem where I found the percentage of cells as time approaches infinity to be $\frac{100d}{c+d}$. All parameters are positive constants. The question asks are there circumstances when this quantity can be zero? I think if $d$ is small and $c$ is large, the percentage is very c... | Yes, it is true that if $c,d > 0$, then $\frac{100d}{c+d} > 0$ (so it can't equal $0$).
| {
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Given directrix, eccentricity, and focus get center of ellipse Given
Directrix: $x=2$
Focus: $(0,0)$
Eccentricity: $0.8$
Find the semi major axis $a$.
I can write the cartesian equation $x^2+y^2=e^2(2-x)^2$ and work the center by manipulating it. However I've been looking for a formula for the semi major axis $a$, in t... | Let the focus and directrix be $F=(f,0)$ and $x=d$; let $D:=(d,0)$ be the foot of the perpendicular from the focus to the directrix. The points on any conic satisfy
$$\text{eccentricity}=\frac{\text{distance from focus}}{\text{distance from directrix}} \tag{1}$$
In particular, an endpoint $P:=(p,0)$ of the major axis ... | {
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Find Dirichlet series of $2^n$ How can I find the Dirichlet series of $2^n$?
The Dirichlet series of a sequence $\{a_n\}_{n=1}^\infty$ is defined as $f(s) = \sum_{i = 1}^\infty \frac{a_n}{n^s}$.
If $\{a_n\}_{n=1}^\infty$ is multiplicative, then we have the following formula for Dirichlet series: $\sum_{i = 1}^\infty \f... | $\forall s \in \mathbb{R}$, $\lim_{n \to \infty} \frac{2^n}{n^s} = \infty$
Thus the Dirichlet series $\sum_{n = 1}^\infty \frac{2^n}{n^s}$ diverges $\forall s \in \mathbb{R}$, according to Cauchy convergence test.
| {
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Why can't we multiply matrices entrywise? Why can't we multiply corresponding elements like addition is done?
Is there a specific reason why it won't be significant?
By definition, we have to multiply a row by columns.
Why such a definition other than multiplying corresponding elements?
Please ignore my ignorance. I h... | You can do such element-wise multiplication of matrices, but it obviously represents a different kind operation. The 'standard' way of multiplying matrices has important applications in linear algebra and, as such, in many areas of science and engineering. The element-wise multiplication has other (typically less used)... | {
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Baby Rudin exercise 8.26 Solution Manual:
https://minds.wisconsin.edu/bitstream/handle/1793/67009/rudin%20ch%208.pdf?sequence=4&isAllowed=y
In the second part of the exercise, we are asked to prove exercises 24 and 25 without the assumption of differentiability of $\gamma(t)$. However, isn't the definition of $Ind(\ga... | Read the set up to the exercise a little more carefully. We formulate the winding number of a continuous curve by first approximating the continuous curve with a trigonometric polynomial (which is continuously differentiable) and then taking the winding number of that.
| {
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$y''+y'+y=\sin^2x$: particular solution? The problem I am trying to solve is finding the particular solution of the equation:
$$y''+y'+y=\sin^2x$$
I don't know what format the particular solution has. Once I know that, I can probably solve the problem with little difficulty.
I haven't seen any examples in my textbo... | $$y''+y'+y=\sin^2x$$
This particular solution works fine too:
$$y_p=Ae^{2ix}+Be^{-2ix}+C$$
| {
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About perfectly normal spaces. In some books the terms: regular and $T_3$; normal and $T_4$; completely normal and $T_5$; perfectly normal and $T_6$ are synonyms, but in some books, the difference is that regular, normal, completely regular and perfectly regular spaces are not $T_1$ and $T_3, T_4, T_5, T_6$ spaces are ... | There are no examples for (2) and (3), for essentially the same reason that (as proved in Henno's answer) there are no examples for (1). A perfectly normal space must be $ \mathrm R _ 0 $ (which is a non-$ \mathrm T _ 0 $ version of $ \mathrm T _ 1 $), and Henno's answer is just the $ \mathrm T _ 0 $ version of the pr... | {
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A function $f: X \to Y$ is continuous if and only if $ ^{−1} (C) $ is closed in $X $ for every closed set $C$ in $Y$. Since a mapping $f$ of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $ ^{−1} (V)$ is open in $X$ for every open set $V$ in $Y$ and since a set is closed if and only if i... | Yes, the key is that the preimage behaves nicely with all the set operations.
| {
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Evaluate the limit $\lim_{n\to\infty} n(\sqrt{n}-\sqrt{n+1})$ Evaluate the limit $\lim_{n\to\infty} n(\sqrt{n}-\sqrt{n+1})$
I am trying to evaluate this limit. First I multiplied by the conjugate to obtain: $a_n=\dfrac{-n}{\sqrt{n}+\sqrt{n+1}}$
I was able to show the limit by taking $f(x)=a_x$ and then applying l'Hôpit... | Or we can look at $n(\sqrt{n+1}-\sqrt{n})=\dfrac{n}{\sqrt{n+1}+\sqrt{n}}\geq\dfrac{n}{2\sqrt{n+1}}=\dfrac{\sqrt{n}}{2\sqrt{1+\dfrac{1}{n}}}\rightarrow\infty$ as $n\rightarrow\infty$, so the limit is $-\infty$.
| {
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Translation of a set Let $A\subset \mathbb R$ be Lebesgue measurable set. Is it true that if $\ \forall r\in(0,1)$
$$A\cap (A+r)\neq \emptyset$$
then $\lambda(A)>0$?
$$$$
I think that is linked to the Vitali set, but I did't manage to prove it.
| No.
Take for instance the Cantor set on $[0,1]$
We have that $m(C)=0$ and $C-C=[-1,1]$
Thus $(0,1) \subseteq C-C\Longrightarrow C \cap(C+r) \neq \emptyset,\forall r \in (0,1)$
| {
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Three squares of a chess board being randomly chosen at random, what is the chance that two are of one color and one of other? I applied this concept;
$\Rightarrow$ there are total of 64 squares out of which 32 are white and others are black.
Now I considered two cases that, (1) the two squares of same colour are w... | The number of ways to choose two black and one white square is ${32 \choose 2}32$ with the first factor from choosing the two black squares and the second from choosing the white square. Two white and one black is the same by symmetry. There are ${64 \choose 3}$ ways to choose three squares, so the chance you want is... | {
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Prove a sequence is convergent and find its limit I tried to prove that $a_{1}=s,\ a_{n+1}=s+a_{n}^{2}$ is a monotonically increasing series, but I didn't know how to prove that it is bounded from above.
about the limit, I tried to compare between the limit of $a_n$ and the limit of $a_{n+1}$ but I received: $L= S + L... | If the limit $L$ does exist, $L = s + L^2$ is a quadratic equation in $L$. That has at most two real roots. If there are no real roots, there is no possibility of convergence. If there are, the next step might be to look at a cobweb plot
of the function $f(x) = s + x^2$.
| {
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Suppose $f \geq 0$, if continuous on [a,b] and $\int_{a}^{b} f(x)dx=0$. Prove that $f(x)=0$ for all $x\in [a,b]$. My attempt:
Let P the partition of $[a,b]$. Let $x_{i}^{*} \in [x_{i-1},x_{i}]$ and f is non negative in $[x_{i-1},x_{i}]$.
Since f is continuous on $[a,b]$, then $f$ is R-integrable with $\int_{a}^{b} f(x... | I would have used the Mean value theorem to solve this.
Take a generic point $c \in (a ,b).$ You know that:
$$\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx = 0.$$
Since $f$ is continuous in $[a, b]$ (with $b > a$), then there exist $d \in (a, c)$
and $e \in (c, b)$ such that:
$$\int_{a}^{c} f(x) dx... | {
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$u$-Subtitution with a definite integral that has a constant gives different answer I have the following integral:
$$\int_0^\pi (2+\cos^2(t)\sin(t))\,\mathrm dt$$
Choosing $u = \cos(t)$, I would get the following result:
$$2u - \frac{u^3}{3}$$
which is:
$$2\cos(t) - \frac{\cos^3(t)}{3}\Bigg|_0^\pi.$$ However, if I solv... | The latter is correct. You really only needed u-substitution for the trig part of that integral. You've incorrectly applied the substitution for $\mathrm dt$ when you used the substitution for the integral of the constant.
$$\begin{align}
\int_{0}^{\pi}2\,\mathrm dt = -\int_{u(0)}^{u(\pi)}\frac{2}{\sin(t)}\,\mathrm du ... | {
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Axiom of Choice --Example Problem Q: Suppose that for any set $X$ and any function $f:X\rightarrow X$ there exists $g:X\rightarrow X$ such that $fgf=f$. Prove that any set has a choice function.
My attempt:
Let $A=\left \{a,b,c... \right \}\subset X$ be an arbitrary non-empty set.
Choose $f:X\rightarrow X$ defined by
$... | The issue of $X$ having at least $2$ element is not a big deal.
However, you seemed to aim to prove the axiom of choice for a single set $A$, instead of a given family of nonempty sets. (Note that your definition of $f$ and $g$ depends on $A$.)
So, your proof should rather start with assuming a family $(A_i)_{i\in I}$ ... | {
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Proof that if a sequence of random variables converges weakly to a constant, then it converges to it in probability Is my proof correct? A sequence of random variables {$\xi_n$}$\xrightarrow{w}c$ means by definition that $F_{\xi_n}(t)\rightarrow F_c(t)$ for every $t$ such that $F_c(t)$ is continuous.
So, i have $F_{\x... | Well, implication (?) is not true on its own in general - it would fail if many of the $\xi_n - c$ were equal to $\varepsilon$ with positive probability - but the conclusion is in fact true in this case. A quick fix is just to use $\varepsilon/2$ instead of $\varepsilon$.
Notation comment: people usually expect $\va... | {
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Integrating $\frac1{a+b\cos(x)}$ using $e^{ix}$ In calculus class our teacher demonstrated us the evaluation of the definite integral $\int_0^\pi \dfrac{1}{a+b\cos(x)}dx=\dfrac{\pi}{\sqrt{a^2-b^2}}$, for $a\ne 0, b \ne 0, |\dfrac ba \lt 1|$, but there's a part I could not grasp.
Our teacher started out by setting up an... | In the given method, the fractions of the form $\dfrac1{1+z}$ are developed using Taylor, then integrated term-wise. Doing that, you retrieve the series for $\log(1+x)$, which are applied to $\alpha$ and $\beta$.
As the roots are complex, the computation of the logarithms is a little involved. In the end, only an imagi... | {
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"url": "https://math.stackexchange.com/questions/3469375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Example of application of Krasner's Lemma I'm learning about valued fields at the moment, and I stumbled upon these notes
http://www-personal.umich.edu/~wuyifan/ExpositoryArticles/NumberTheory/LocalFields/Krasner%27s_Lemma.pdf
As proposition 2.1 it states that when $\eta$ is a primitive $p$-th ($p$ odd) root of unit th... | It is easier to check first the elementary proof.
Let $K = \Bbb{Q}_p(\zeta_p)$ and $O_K$ its ring of integers with residue field $O_K/(\pi)$.
*
*From $$\zeta_p^p - 1 \equiv (\zeta_p-1)^p \equiv 0\bmod \pi \implies v_p(\zeta_p-1)> 0$$ $$\implies v_p(\sum_{l=0}^{a-1}\zeta_p^l) = v_p(a)=0\implies v_p(\zeta_p-1)=v_p(\z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is my proof of the irrationality of $\sqrt{3}$ legitimate? I read the proof of the irrationality of $\sqrt{3}$ in my textbook (Richard Hammack's Book of Proof), and I was wondering if my proof is legitimate as well.
Prop: $\sqrt{3}$ is irrational.
Suppose by way of contradiction that $\sqrt{3}$ is rational. Hence, $\s... | There are at least two flaws:
*
*you say $m,n$ are both not even, which in fact means neither $m$ nor $n$ are even.
*you say "so the fraction is reduced", but is $\frac{15}{25}$ reduced ?
Allowing only irreducible fractions, $$\sqrt3=\frac pq\iff p^2=3q^2.$$
So $p^2$ is a multiple of $3$, and so must $p$ be. Then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Chess match where first to win game wins match. Probability of winning is p and q, of draw is 1-p-q. Find mean, PMF, and variance of match duration. Question
Fischer and Spassky play a sudden-death chess match whereby the first player to win a game wins the match. Each game is won by Fischer with probability $p$, by Sp... | The expressions are wrong. P(Fisher wins)=$p\sum_{n=0}^\infty (1-p-q)^n=p\frac{1}{1-(1-p-q)}=\frac{p}{p+q}$
Let $X$ be the duration of the match . $E(X)=(p+q)\sum _{k=1}^\infty k(1-p-q)^{k-1}=\frac{1}{p+q}$, $E(X^2)=(p+q)\sum_{k=1}^\infty k^2(1-p-q)^{k-1}=\frac{2-p-q}{(p+q)^2}$ The variance is $\frac{1-p-q}{(p+q)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3470122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$S^2$ without $n$ points is homeomorphic to $S^2$ without $m$ points if and only if $n = m$
Consider the unit sphere $S^2$ with the subspace topology of $\mathbb{R}^3$. Now let $n,m$ be positive integers. Prove that $S^2$ with $n$ different points removed from it is homeomorphic to $S^2$ with $m$ different points remo... | Not sure why algebraic topology seems to be eschewed here, but for the harder direction I would guess that that $H_1(X)\cong\Bbb Z^{n-1}$, whereas $H_1(Y)\cong\Bbb Z^{m-1}$.
For this one can use that $X$ deformation retracts onto a "rose with $n-1$ petals".
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculus implicit differentiation question I stumbled upon this calculus implicit differential question: find
$\cfrac{du}{dy} $ of the function $ u = \sin(y^2+u)$.
The answer is $ \cfrac{2y\cos(y^2+u)}{1−\cos(y^2+u)} $. I understand how to get the answer for the numerator, but how do we get the denominator part?
And c... | $$u = \sin(y^2+u) \implies\frac{du}{dy} = \cos(y^2+u)\times\left(2y+\frac{du}{dy}\right)$$
$$\implies \frac{du}{dy} (1-\cos(y^2+u)) = 2y\cos(y^2+u) \implies \frac{du}{dy}=\frac{2y\cos(y^2+u)}{1-\cos(y^2+u)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3470444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determining the Fundamental Matrix Using Generalized Eigenvectors
Determine $\mathit{e}^{At}$ by using generalized eigenvector method to find a fundamental matrix of $x'=Ax$ with $A=\begin{bmatrix}
5 &-4 &0 \\
1&0 &2 \\
0& 2 &5
\end{bmatrix}$.
I just want to know whether my solution is okay?
I found the eigenv... | $e^{At} = P e^{Jt} P^{-1}$
and
$e^{Jt} = \begin{bmatrix} e^{J_1 t} && 0 \\ 0 && e^{J_2 t} \end{bmatrix}$
Firstly, as $J_1 = 0$, $e^{J_1t} = 1$ .
Now, if we open taylor series of exponential around $\lambda_2$
$e^{xt} = \sum \frac{e^{\lambda_2t}}{n!}(xt-\lambda_2t)^n $
$e^{J_2 t} = \sum \frac{e^{\lambda_2 t}}{n!}(J_2t-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3470586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An exercise in differential topology Problem: Given a smooth submanifold $M\subset\mathbb{R}^k$, show that the tangent bundle space $$TM=\{(x,v)\in M\times\mathbb{R}^k:v\in TM_x\}$$ is also a smooth manifold. Show that any smooth map $f:M\rightarrow N$ gives rise to a smooth map $$df:TM\rightarrow TN$$ where $$d(\text{... | One way to define the tangent space to a submanifold $M\subset \Bbb R^k$ at some point $x$ is to consider the set of all derivatives at $t=0$ of all smooth curves $f:\Bbb R\to M$ such that $f(0)=x$. As such, the tangent space $T_xM$ is indeed defined as a subspace of $\Bbb R^k$ (whose dimension is the dimension of $M$)... | {
"language": "en",
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"source": "stackexchange",
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Expected runtime analysis for sums of four squares (Rabin and Shallit) I've been reading a paper of Rabin and Shallit ("Randomized Algorithms in Number Theory"), which gives a brief sketch of an ERH-conditional algorithm to compute a representation of a positive integer $n$ as a sum of four squares, originally presente... | As discussed in the comments, apparently the number used need to be a prime. We only need to find a square root of $-1$ modulo that number, and the prime density from ERH is used only to show that we have a certain probability of doing so.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3470873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of Sum of Orthogonal Let $\{x_n\}$ be an orthonormal basis of a Hilbert space $H.$
Can anyone help me how to show that $\sum_{n=0}^{\infty}|(x_n,x)|^2$ is convergent and $\|x\|^2=\sum_{n=0}^{\infty}|(x_n,x)|^2?$
This is to show that $(x_n,x)\rightarrow 0.$
I used Bessel's Inequality to show that $\|x\|^2\g... | The usual definition of a basis for a Hilbert space denoted $\{x_n\}$ you mean that the set is orthonormal and that
$$x = \sum_{n=1}^\infty \langle x,x_n\rangle x_n$$
meaning that
$$\lim_{N\rightarrow \infty} \left\|x-\sum_{n=1}^N\langle x,x_n\rangle x_n\right\| = 0\quad \text{Convergence in the Hilbert space norm}$$
N... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is a locally compact Hausdorff vectorspace with countable base a topological vector space? my question is related to the very first part of Chapter 3 of Resnicks book "Extreme Values, Regular Variation and Point Processes.
He is claiming that one should think of a locally compact Hausdorff space with countable base as... | This is obviously false, because the topology could have nothing to do with the vector space structure. For instance, with $K=\mathbb{R}$ and $X=\mathbb{R}$ with its usual vector space structure, we can pick a bijection between $X$ and $[0,1]$ and thus give $X$ a topology which is homeomorphic to the usual topology on... | {
"language": "en",
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Without calculating the square roots, determine which of the numbers:$a=\sqrt{7}+\sqrt{10}\;\;,\;\; b=\sqrt{3}+\sqrt{19}$ is greater.
Without calculating the square roots, determine which of the numbers:
$$a=\sqrt{7}+\sqrt{10}\;\;,\;\; b=\sqrt{3}+\sqrt{19}$$
is greater.
My work (I was wondering if there are other way... | There are indeed other ways to do this. Your solution is great, but if you were just curious about another method, here is one:
$$
\begin{align}
\sqrt{7} + \sqrt{10} \quad &? \quad \sqrt{3} + \sqrt{19} \\
\sqrt{10} - \sqrt{3} \quad &? \quad \sqrt{19} - \sqrt{7}
\end{align}
$$
Note that instead of comparing $a$ and $b$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding a Recurrence Relation and Solving I am trying to find a recurrence relation for the following question:
For integer n ≥ 1, let h(n) be the number of length $n$ words consisting of A’s and B’s, that contain either at least an “AA”, or at least an “ABB”. Find a recurrence relation satisfied by $h(n)$ (with neces... | This is an odd problem, because it seems easier to find a closed expression for $h(n)$ than a recurrence relation!
After all, how many words of length $n$ don't meet the criteria? Such a word can start with any number of B's. But once it hits the first A, it needs to alternate between A and B to avoid having either a... | {
"language": "en",
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Definite integral help I'm working on a physics problem and I got to the integral:
$$\int_0^\infty (a+b+x^2)^{-\frac{3}2} dx = \frac{1}{(a+b)}$$
I am just trying to understand how this is achieved. Because the indefinite integral yields
$$x*(a+b)^{-1}*(a+b+x^2)^{-\frac{1}2}$$
Evaluating this from 0 to $\infty$, to me, ... | The indefinite integral of $(C+x^2)^{-\frac{3}{2}}$ actually equals $\frac{x}{C\sqrt{C+x^2}}$. If we set $C=a+b$, the answer will coincide with yours.
I guess that in your approach you missed $x$ coming from deriving $x^2$ inside brackets.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show not Lebesgue integrable using step functions I have shown that conditionally $\int_0^\infty \int_0^\infty f(x,y) dx dy = \int_0^\infty \int_0^\infty f(x,y) dx dy = \frac{\pi}{2}$ where $f(x,y) = e^{-xy} \sin(x).$ This part is relatively easy because
$$\int_0^\infty e^{-xy} \sin(x) dy = \frac{\sin(x)}{x} $$
Now I ... | It is readily apparent that $f$ is not Lebesgue integrable since $F(x) = \int_0^\infty e^{-xy} |\sin x| \, dy = \frac{|\sin x|}{x}$ is not integrable over $[0,\infty)$. If $f$ were integrable, the iterated integral must be finite by Tonelli's theorem.
Alternatively, using your suggested approach, take $A_{jk} = \left[... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the minimal polynomial for $\cos(\frac{2\pi}{5})$ and $\sin(\frac{2\pi}{5})$ Let $\omega$ be the primitive 5th root of $1$, then $\cos(\frac{2\pi}{5}) = \frac{w+w^{-1}}{2}$ and $\sin(\frac{2\pi}{5}) = \frac{w-w^{-1}}{2i}$. How to find the minimal polynomial of $\frac{w+w^{-1}}{2}$ then? (without using the Chebysh... | Hint: square it, then make use of the fact that the 5 fifth roots of unity sum to 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3471903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Using the Poisson approximation to estimate the number of trials required to guarantee at least one success Suppose that on average, out of $N$ trials, $q$ succeed. $q$ is much smaller than $N$. For a concrete example, suppose $N = 100$ and $q = 2$.
Let $n$ be the number of trials run in a particular experiment. How la... | You have miscalculated the probability in the Poisson case. Indeed, since the Poisson probability is given by
$$
\mathbb P\bigl(\textrm{Poisson}(\lambda)=k\bigr)=\frac{\lambda^ke^{-\lambda}}{k!},
$$
we have that the probability of no successes is $$\frac{\lambda^0e^{-\lambda}}{0!}=e^{-\lambda},$$
not $\lambda$ as you h... | {
"language": "en",
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Show that $2(\sin y + 1)(\cos y + 1) = (\sin y + \cos y + 1)^2$ The question states:
Show that: $$2(\sin y + 1)(\cos y + 1) = (\sin y + \cos y + 1)^2$$
This is what I have done
$2(\sin y + 1)(\cos y + 1) = 2(\sin y + \cos y + 1)^2$
L. H. S. = R. H. S.
From L. H. S.
$2(\sin y +1)(\cos y + 1) = 2(\sin y... | Expanding the RHS,
$$\color{blue}{(\sin y + \cos y + 1 )^2} = \sin^2 y + \cos^2 y + 1 +2\cos y \sin y + 2\cos y + 2\sin y$$
$$= 1+ 1 +2\cos y \sin y + 2\cos y + 2\sin y=2(1+\cos y \sin y + \cos y + \sin y) = 2 \left[(1+\cos y) + (\sin y + \cos y \sin y ) \right] = 2 \left[(1+\cos y) + \sin y(1 + \cos y) \right]=\color... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3472158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Multivariate Lagrange inversion with zero powers (also asked in MO)
The multivariate Lagrange inversion formula, which I found in a couple of papers (such as this and this), is as follows. If $f_i=t_ig_i(f)$, $1\le i\le k$, then
$$[\boldsymbol{t^n}]h(\boldsymbol{f(t)})=\frac{1}{n_1n_2\cdots n_k}[\boldsymbol{x^{n-1}}]\s... | Let's see how the $n_i$ come as reciprocal factors into the formula. They come from the factorial denominators of the Taylor series terms.
Look at the one-variable Lagrange-Bürmann formula:
$$[t^n]h(f(t))=\frac{1}{n}[x^{n-1}](h'(x)g(x)^n).$$
It can be proved that
$$(h(f(t)))^{(n)}|_0=(h'(x)g(x)^n)^{(n-1)}|_0$$
Now we t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3472280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Weak topologies in normed spaces Let $(X,\|\cdot\|)$ be a normed space over complex or real field and $\tau_X$ is the topology generated by the norm $\|\cdot\|$, i.e. just norm topology. 1) Is there a locally convex topology $\tau$ such that $\tau$ is weaker than $\tau_X$ and stronger than $\sigma(X,X^*)$, where $X^*$ ... | Let's first consider the case that $X$ is finite-dimensional. Since then there is only one Hausdorff vector space topology on $X$, and $\sigma(X,X^{\ast})$ is Hausdorff, the answer to the first question is "no" if we understand "weaker" and "stronger" in the strict sense [in the non-strict sense the answer is trivially... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about why this hyperplane section divisor is singular at a certain point I have a small question about an excerpt from a passage about intersection numbers in "Basic Algebraic Geometry I" by Igor Shafarevich.
Let $ X \subset \mathbb{P}^{3} $ be a nonsingular surface of degree $ m $ and $ L \subset X $ a line.... | $E$ is singular at $y\in L\cap C$ because $y$ is on multiple irreducible components: every point which is on multiple irreducible components is singular, as the local ring of such a point has at least two minimal primes (corresponding to the distinct irreducible components it's on) while a regular local ring is a domai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3472548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the expected area of a cyclic quadrilateral inscribed in a unit circle? Choose four points randomly on the circumference of a circle with radius $1$. Connect them to form a quadrilateral. What is the expected area of this quadrilateral?
I have attempted to simulate to find an answer but not sure how to approac... | The four central angles have the same distribution, and the expected area is $4$ times the expected area of one of the four triangles spanned by the central angles.
The probability density of the central angle is proportional to the volume it leaves to the remaining two points: $f_\alpha(\alpha)\propto(2\pi-\alpha)^2$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving $\sinh x = kx$ Can we solve the equation $\sinh x = kx$ for $x$ in terms of elementary functions? I've tried reexpressing the hyperbolic sine as exponentials and converting the equation into a quadratic in $e^x$, but this doesn't seem to make the problem any easier. I've considered expanding $\sinh x$ as a Tayl... | As mentioned in the comments, the only solution for $k\le1$ is $x=0$ and for $k>1$ there are 3 solutions: $x=0,\pm x_\star$. Although there is no closed form in terms of special functions such as the Lambert W function known, it is not hard to numerically compute $x_\star$, the positive nonzero solution. For example, w... | {
"language": "en",
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If $f(x\cdot y)$ = $f(x). f(y)$ $\forall$ $x,y$ and $f(x)$ is continuous at $x = 1$. Prove following If $f(x\cdot y)$ = $f(x). f(y)$ $\forall$ $x,y$ and $f(x)$ is continuous at $x = 1$. Prove that $f(x)$ is continuous for all $x$ except at $x = 0$. Given $f(1)\ne0$.
$$f(1)=f(1)\cdot f(1)$$
$$f(1)(f(1)-1)=0$$
$$f(1)=0 \... | Obviously $f(x+y)=f\left(x\cdot\left(1+\frac yx\right)\right)=f(x)\cdot f\left(1+\frac yx\right)$ where $x\not = 0$.
So, we have $$\displaystyle\lim_{h\to 0} f(x\pm h)=f(x)\cdot f\left(1\pm\frac hx\right)$$
And since we are naturally observing the points where $x\not = 0$, $\frac hx\to 0$. And, $f(x)$ is continuous at... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\int f' g d \lambda = - \int f g' d \lambda$ I know that this result is quite elementary, but nevertheless, I don't think that the result is trivial. So, let $f,g \in C^1 (\mathbb R)$ with $f, g, f', g' \in L^2 (\mathbb R)$.
Why is it true that $\displaystyle\int_{\mathbb R} f'g\,d\lambda = - \int_{\mathbb R} fg'\,d\l... | You are right that this is usually treated "in the literature" (i.e., in textbooks and papers) as trivial, although it is not quite that.
First, note (by Cauchy Schwarz) that $F := f \cdot g \in L^1$ is continuously differentiable with $F'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \in L^1$. It is well-known (see for ins... | {
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To find the Jordan Canonical Form Consider a matrix A (5x5) with all entries = 1. Here the entries are considered as elements of $F_5$ ,the finite field of order 5.
What is the Jordan canonical form?
I have found out that $A^2=0$ and thus the minimal polynomial is $x^2$.
So I know there are (two 2x2 blocks and one 1x1... | Hint: If $A$ is an $n \times n$ matrix, then $n - \operatorname{rk}(A)$ is the total number of Jordan blocks that $A$ has associated with $\lambda = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3473852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why my expression for acceleration doesn't work? So i have an object that moves in a straight line with initial velocity $v_0$ and starting position $x_0$. I can give it constant acceleration $a$ over a fixed time interval $t$. Now what i need is that when the time interval ends this object should stop exactly at a poi... | I will use $t_0$ rather than $t$, since this is also a fixed quantity.
What you are doing doesn't work for arbitrary $t_0$, $x_0$, $x_1$, and $v_0$.
Since your only unknown is supposed to be $a$, from the first equation you get
$$a = -\frac{v_0}{t_0}$$
From the second equation you get
$$a = \frac{2(x_1-x_0-v_0t_0)}{t_... | {
"language": "en",
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Does there exist a division ring without unity? In abstract algebra I have only ever seen division introduced via multiplicative inverses, namely starting from a ring with unity $R$ and then adding the condition that each element $x$ has an inverse element $x^{-1}$ such that $xx^{-1}=x^{-1}x=1$. But I can also imagine ... | Let $R$ be a ring without unit. Suppose
for each ordered pair $(a,b)\in R$ with $b\neq 0$, there exists a unique $c\in R$ such that $a=bc=cb$.
We claim that $R$ is a ring with unit.
Let $b \in R$, $b \ne 0$. Then there is a unique $e_b$ such that $b = be_b = e_bb$.
We must show that $e_a = e_b$ for all nonzer... | {
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Evaluating the limit $\lim_{x\to0}\frac{1}{x^3}\int_{0}^{x}\sin(\sin(t^2))dt$ $$\lim_{x\to0}\frac{1}{x^3}\int_{0}^{x}\sin(\sin(t^2))dt$$
This is a compound question from me.
*
*I don't know how to begin evaluating this limit. My guess would be that I would have to find the value of this Riemann's integral and then p... | Without L'Hopital:
\begin{align*}
\dfrac{1}{x^{3}}\int_{0}^{x}\sin(\sin t^{2})dt=\dfrac{1}{x^{3}}\left(x\sin(\sin x^{2})-\int_{0}^{x}t\cos(\cos t^{2})2tdt\right).
\end{align*}
Note that
\begin{align*}
\dfrac{1}{x^{3}}(x\sin(\sin x^{2}))=\dfrac{\sin(\sin x^{2})}{\sin x^{2}}\dfrac{\sin x^{2}}{x^{2}}\rightarrow 1.
\end{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3474189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
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Expected value of Distinct Elements from maximum sequence of permutation of 1 to n Let $a_1,a_2 ,... a_n$ be a permutation of 1 to $n$. We define sequence $b = b_1 , b_2 ,... ,b_n$ as $b_i = max ~~{a_1,a_2,...a_i}$. Find Expected Value of $X$: distinct numbers in $b_i$.
For example if the permutation is $1, 3, 2$ then ... | Let $I_i$ be the indicator variable that $a_i = b_i$, namely that $a_i$ is the maximum of the first $i$ values of the permutation.
Hint: Show that $X = \sum I_i$.
Hint: Show that $ E[I_i] = \frac{1}{i}$.
Hence, $ E[X] = \sum_{i=1}^n \frac{1}{i}$.
This agrees with the $n=3, 4$ cases that you calculated.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3474373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving Zeta Relations Without Direct Evaluation Is it possible to derive the following $\zeta$ relations without actually finding the values themselves?
\begin{align*}
2 \zeta(2)^2 &= 5\zeta(4)\\
4\zeta(2)\zeta(4) &= 7\zeta(6)\\
3\zeta(2)\zeta(6) &= 5\zeta(8)
\end{align*}
These small integer relations make it look lik... | $\mathbf{\text{Hint:}}$(after seeing that you are much interested for even zeta values)
The pattern you look towards is seen from the fact that $$\zeta(2n)=\frac{(-1)^{n+1}(2\pi)^{2n}B_{2n}}{2(2n)!}$$
Where $B_n$ are the Bernoulli numbers and $n\in\Bbb{N}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3474518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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How many positive integers have less than $90000$ have the sum of their digits equal to $17$? How many positive integers have less than $90000$ have the sum of their digits equal to $17$?
I tried to write the number as $ABCDE$ and use some math with that (so we need $A + B + C + D + E = 17$), and I tried to use stars a... | Stars and bars sounds like a good idea. You want to put $17$ balls in $5$ buckets, with no more than $9$ balls in any one bucket. First do it without the $9$-ball restriction. Now you have to subtract the number of ways that have $10$ or more balls in a bucket. Since there are only $17$ balls, there can't be more t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3474620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $x$ be a real number such that $|x|<1.$ Which of the following is false? Let $x$ be a real number such that $|x|<1.$ Which of the following is false?
*
*If $x\in \mathbb Q$, then $\sum_{m\ge 0}x^m \in \mathbb Q$
*If $\sum_{m\ge 0}x^m \in \mathbb Q$ then $x\in \mathbb Q$
*If $x\notin \mathbb Q$ then $\sum_{m\ge... | Everything you have done is correct except that you cannot take $x=1+\frac 1 {\sqrt 2}$ since $|x|<1$. Take $x=1-\frac 1 {\sqrt 2}$ instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3474758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Triangle inscribed in a circle,2 points fixed and 1 moving. The track of centroid makes a circle but how do I prove it without cartesian coordinate? Triangle ABC and circle O. A and B are fixed, but C is moving on the circle.
So I have triangle ABC and circle O. A and B are fixed on the circle, but C is moving around t... | This is not too hard to see using ordinary geometry.
Let $A$, $B$ be fixed points on circle with center $O$, and $C$ any other point on the circle.
In triangle $ABC$, bisecting $CB$, $AB$ at $E$, $F$, and joining $AE$, $CF$, then $G$ is the centroid of $\triangle ABC$.
In fixed triangle $AOB$, bisect $AO$ at $D$, and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Assuming matrix $B$ is symmetric, can I prove that $A$ is symmetric $A,B$ are square matrices and $A(I+B)=I$, $B$ is symmetric, can I prove that $A$ is symmetric as well?
| Given $B$ is symmetric.
$A(I+B) = I =(I+B)^T \times A^T = (I+B) A^T$
$$A(I+B)A^T = A = IA^T =A^T$$ so $A$ is symmetric!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3475002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine if this statement about Big O notation is true or not. $f(n) = n^2 + n^{0.5}$
$g(n) = [g(n-1)]^2 + [g(n-2)]^2$ for $n \geq 3$, where $g(1) = 1$ and $g(2) = 2$
The statement: $2^{2^{f(n)} }= Ω(g(n))$
The $\lim_{n \rightarrow \infty} \frac{2^{2^{f(n)} }}{g(n)}$ can't be computed easily since $g(n)$ has a recurr... | The $g_n$ are given in sequence $A000283$ in $OEIS$ (have look here). I you look at the formula, in year 2003, Benoit Cloitre proposed
$$g_n=\left\lfloor A^{2^{n}}\right\rfloor$$ where
$$A=1.23539273778543688962233101322844082434745718691367945473360\cdots$$ is "almost" $[\log(5) ]^{e^\gamma}-\log(3)=1.235392625$
Now... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that the series $\sum_{x∈X}(f(x) + g(x))$ is absolutely convergent, and $ \sum_{x∈X}(f(x) + g(x)) = \sum_{x∈X}f(x) + \sum_{x∈X}g(x)$
Let $X$ be an arbitrary set (possibly uncountable), and let $f:X → R$ and $g: X → R$ be functions such that the series $\sum_{x∈X} f(x)$ and $\sum_{x∈X} g(x)$ are both absolutely ... | For any finite subset $F$ of $X$, we have
\begin{align*}
\sum_{x\in F}|f(x)+g(x)|\leq\sum_{x\in F}|f(x)|+\sum_{x\in F}|g(x)|\leq\sum_{x\in X}|f(x)|+\sum_{x\in X}|g(x)|<\infty,
\end{align*}
so
\begin{align*}
\sup_{F\subseteq X, F~\text{finite}}\sum_{x\in F}|f(x)+g(x)|<\infty.
\end{align*}
This proves the absolute conve... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Percentage of people who have good credit ratings given that their ratings will improve
Suppose that 75% of all people with credit records improve their credit ratings within three years. Suppose that 18% of the population at large have poor credit records, and of those only 30% will improve their credit ratings withi... | Try visualizing the situation for 1.000 people.
750 people will improve their ratings.
180 people have poor ratings - meaning the remaining 820 have good ratings.
30% of the 180 people will improve - that is 54 people.
Since a total of 750 people improved that means that from the
820 people with good ratings we need 7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3475476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $[-\frac{\pi}{2}, \frac{\pi}{2} ]$ the set of values for $f(x) = \arctan \sqrt{x^2-1} + \arcsin \frac{1}{x}$? I am given the function:
$$f:D \rightarrow \mathbb{R} \hspace{3cm} f(x) = \arctan \sqrt{x^2-1} + \arcsin \frac{1}{x}$$
where $D$ is the maximum domain of the function. I am told that the set of values of... | Hint
Let $\arctan\sqrt{x^2-1}=y,\dfrac\pi2>y\ge0,x=\pm\sec y$
If $x>0,x=\sec y$
$\arcsin\dfrac1x=\arcsin(\cos y)=\dfrac\pi2-\arccos(\cos y)=\dfrac\pi2-y$
If $x<0,x=-\sec y$
$\arcsin(-\cos y)=-\arcsin(\cos y)=?$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $m$ and $n$ are integers, show that $\left|\sqrt{3}-\frac{m}{n}\right| \ge \frac{1}{5n^{2}}$ If $m$ and $n$ are integers, show that $\biggl|\sqrt{3}-\dfrac{m}{n}\biggr| \ge \dfrac{1}{5n^{2}}$.
Since $\biggl|\sqrt{3}-\dfrac{m}{n}\biggr|$ is equivalent to $\biggl|\dfrac{ \sqrt{3}n-m}{n}\biggr|$
So I performed the fol... | You're asking to prove, for integers $m$ and $n$ (with the assumption $n \neq 0$), that
$$\left|\sqrt{3}-\frac{m}{n}\right| \le \frac{1}{5n^2} \tag{1}\label{eq1A}$$
Note if $m = 0$, \eqref{eq1A} obviously holds. Otherwise, as this other answer states, WLOG, we may assume both $m$ and $n$ are positive since if they have... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Vector space structure on extensions of vector bundles Let $F, G$ be vector bundles on a scheme over the field of complex numbers. I know that the set $V:=Ext^1(F, G)$ has the structure of an additive group given by the Baer sum of extensions. But $V$ also has the vector space structure. Namely, if $a\in\mathbb{C}^*$ a... | In fact, $\operatorname{Ext}^1(F,G)$ admits two vector spaces structures. One which comes from $F$ and another one which comes from $G$. But you can show that these two structures are identical here.
The construction of the first one is as follow : let $a\in\mathbb{C}$ and consider the multiplication by $a$ map $a:F\to... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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What is a non-trivial covering space? I've come across this term many times but its meaning seems to be always assumed. Sometimes it looks like it means the covering space is connected or path-connected sometimes just that it is not equal to the space $X$ being covered. So what is exact definition? thanks
| To say that $X$ is a nontrivial covering space of $Y$ means that there exists a covering map $f : X \to Y$ such that $f$ is not a homeomorphism, equivalently $f$ is not one-to-one, equivalently the degree of $f$ is $\ge 2$ (recall that the degree is the cardinality of any fiber $f^{-1}(y)$, which is well-defined indepe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there an infinite set of finite strings such that no element is a subsequence of another? Of course, this is meant to be over a finite alphabet. My intuition is that this doesn't exist over any such alphabet, so that's what I'd want to know how to prove.
I'm also interested in questions like "can such a set be compu... | This answer is translated (with small modifications) from here.
$\Sigma$ is a finite alphabet.
$\Sigma^\ast$ is the set of finite strings over $\Sigma$ (Kleene star).
$x\preceq y$ means that $x$ is a subsequence of $y$.
We'll prove that there is no infinite set $S \subseteq \Sigma^\ast$ such that no element of it is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3476176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Find $m$ if $f(x)=x^m\sin\frac{1}{x}$ is continuous and is not differentiable If $f(x)=\begin{cases}
x^m\sin\dfrac{1}{x}, & x\ne 0 \\
0, & x=0
\end{cases}$.
Find $m$ if $f(x)$ is continuous and is not differentiable
My attempt is as follows:-
Let's find the condition of continuity
$$\lim_{x\to0^{+}}x^m\sin\dfrac{1}... | I have to admit I cannot really understand what you are trying to do, and how you come with your conditions for $p$ and $q$.
The first condition is that you need
$$
\lim_{x\to0} x^m\sin\frac1x=0.
$$
Since the sine is bounded, when $m>0$ we have $\left|x^m\sin\frac1x\right|\leq|x|^m$, so the limit is $0$ and the func... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\mathbb Q(\sqrt m,\sqrt n)=\mathbb Q(\sqrt {m}+\sqrt {n})$. Show that $\mathbb Q(\sqrt m,\sqrt n)=\mathbb Q(\sqrt {m}+\sqrt {n})$
My attempt: It is obvious that $\mathbb Q(\sqrt {m}+\sqrt {n}) \subset \mathbb Q(\sqrt m,\sqrt n) $ .
Is this proof is correct?
| It is good, but you can shorten it to just deduce that
$$
2(m-n)\sqrt{n}\in\mathbb{Q}(\sqrt{m}+\sqrt{n})
$$
If $m=n$ the statement $\mathbb{Q}(\sqrt{m}+\sqrt{n})=\mathbb{Q}(\sqrt{m},\sqrt{n})$ is obvious, so we can assume $m\ne n$. Thus $\sqrt{n}\in\mathbb{Q}(\sqrt{m}+\sqrt{n})$ and so also
$$
\sqrt{m}=(\sqrt{m}+\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3476455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that surrounded by zero exist smooth function $y(x)$
Let the function $G$ be such that $G(x)=\sum_{ij}G_{ij}(x)x_ix_j$ for some $G_{ij}\in C^{\infty}$ that vanish at zero. Prove that in a neighborhood of zero there exists a smooth function $y(x)$ such that:
$$Q(x+y(x))=Q(x)+G(x), y(0)=0, dy(0)=0$$ where $Q(x... | This is a way of deriving the OP's desired result, using a deus ex machina argument that I happened to know.
Given $G$ and a non-degenerate quadratic form $Q$ as stated, the
function $f(x)=Q(x)+G(x)$ has a non-degenerate critical point at $x=0$, with $f(0)=0$, $df(0)=0$, and $d^2f(0)=2Q$. By the Morse Lemma there exis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3476568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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if $\{f_n\}\to f$ in $L^{p_2}(E)$ then $\{f_n\}\to f$ in $L^{p_1}(E)$. Assume that $E$ has finite measure and $1 \le p_1 < p_2 \le \infty$. Show that if $\{f_n\}\to f$ in $L^{p_2}(E)$ then $\{f_n\}\to f$ in $L^{p_1}(E)$.
my attempt: let $p=\frac{p_2}{p_1}$ and $1=\frac{1}{p}+\frac{1}{q}$, and pick $n$ such that $\|f_... | Good. But you should also take care of the case that $p_{2}=\infty$, but this is easy:
\begin{align*}
\|f_{n}-f\|_{L^{p_{1}}}^{p_{1}}=\int_{E}|f_{n}-f|^{p_{1}}\leq\|f_{n}-f\|_{L^{\infty}}^{p_{1}}\int_{E}=\|f_{n}-f\|_{L^{\infty}}^{p_{1}}\mu(E).
\end{align*}
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the interval of convergence for $\sum\limits_{n=1}^{\infty}\frac{x^n}{n4^n}$. I am working on this problem, but I am not exactly sure about my answer. Can you help me how to do the steps to find the interval of convergence?
My answer is $L = \left| \frac{x}{4} \right| < 1$.
| A quick fomula for power series:
If $a_n=n^{b}\left(\frac{1}{a}\right)^n$ where $a,b\in\mathbb{R}\wedge a\neq0$ we have
$$\sum _{n=1}^{\infty }\:n^{b}\left(\frac{1}{a}\right)^n\left(x-c\right)^n\text{ which converges on }\left\{\begin{array}{l}
(c-a,c+a),a\in\mathbb{R}\wedge b\ge0
\\ [c-a,c+a),a>0\wedge -1\le b<0
\\ (c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3476813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to quickly yet convincingly claim that edge contractions preserve outerplanarity Let $G$ be a simple outerplanar graph with $n$ vertices. Let the vertex $v \in V(G)$ have degree 2 and be a member of a bounded face formed by a chordless cycle $C$ of more than 3 vertices.
Given the above conditions, I want to prove ... | The easiest way to justify the claim is the fact that a graph is outerplanar if and only if it does not contain the graphs $K_4$ or $K_{2, 3}$ as a minor. A graph $H$ is a minor if a graph $G$ if $H$ can be obtained from $G$ by a sequence of edge contractions, edge deletions, and vertex deletions.
In case you're not al... | {
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"timestamp": "2023-03-29T00:00:00",
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How to show that $f(x,y)$ is continuously differentiable on $\mathbb{R}^2$? I have been given the function $f(x,y)=\begin{cases} \frac{x^3y-xy^3}{x^2+y^2}, \quad \quad (x,y)\neq 0 ; \\ 0 \quad \quad \quad \quad \quad (x,y)=0.\end{cases}$.
Is it enough to compute all partial derivates and then showing that they are co... | Yes it should be enough to compute all partial derivates and then showing that they are continuous. But notice the following:
Your $f$ is two times partial differentiable, but
$\partial_2\partial_1f(0,0)=-1\neq 1=\partial_1\partial_2f(0,0)$
hence (according to Schwarz's theorem) your partial differentials $\partial_1\p... | {
"language": "en",
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Find $f(10)$ for the following conditions Let $f(x)$ be a real valued function not identically zero satisfies the equation,
$f(x + y^n) = f(x) + (f(y))^n$ for all real $x$ & $y$ and $f'(0)\ge 0$ where $n>1$ is an odd natural number. Find $f(10)$
Putting $x=0,y=0$
$$f(0)=f(0)+f(0)$$
$$f(0)=0\tag{1}$$
Putting $x=0,y=1$
... | To complete your thoughts just note that by the mean value theorem $$f(10) = f(10) - f(0) = 10 \cdot f^\prime (t)$$ for some $t \in (0, 10)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3477490",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Find all polynomials $P(x)$ with odd degree such that $P(x^2 - 2) = P^2(x)-2$ The problem says :
Find all polynomials $P(x)$ with odd degree such that
$$P(x^2 - 2) = P^2(x)-2$$
I tried a lot if ways (using high school mathematics) but the only solution I have so far is $P(x) = x$. Can anyone solve this problem usin... | This extended comment has the sole purpose of showing a simple Mathematica code to print the $P(x)$ polynomials up to the tenth degree that satisfy the relation $P(x^2-2) - (P(x))^2 + 2 = 0$.
P[x_] = Sum[ToExpression[StringJoin["a", ToString[n]]] x^n, {n, 0, 10}];
coeff = CoefficientList[P[x^2 - 2] - P[x]^2 + 2, x];
ze... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3477678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Riemann integrability of a piecewise function over $[1, 7]$ I have the following function
$f(x) = \begin{cases} 2 & \text{if } 1 \leq x \leq 2, \\ 3 & \text{if } 2 < x \leq 4,\\ 1 & \text{if } 4 < x \leq 7. \end{cases}$
How do I determine if the function is Riemann integrable within $[1, 7]$?
I think I have to check ... | hint
Let $\epsilon>0$ given and consider the partition $\sigma$ definef by
$$\Bigl(1,2-\frac{\epsilon}{7},2+\frac{\epsilon}{7} ,4-\frac{\epsilon}{7}, 4+\frac{\epsilon}{7},7\Bigr)$$
then
$$U(f,\sigma)-L(\sigma,f)=$$
$$(3-2)\frac{2\epsilon}{7}+(3-1)\frac{2\epsilon}{7}=\frac{6\epsilon}{7}<\epsilon$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3477792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show $\langle T(x),y\rangle=\langle x,S(y)\rangle$ for all $x, y$ implies, S is the adjoint operator? Suppose that $H$ is a Hilbert space and $T$ and $S$ are two functions from $H$ to $H$.
If
$$
\langle T(x),y\rangle=\langle x,S(y)\rangle
$$
for all $x, y \in H$, show that $T$ and $S$ are continuous linear opera... | Hint: Closed Graph Theorem easily gives continuity of $T$ and $S$. By definition of $T^{*}$ we get $\langle x , T^{*}y \rangle =\langle x , Sy \rangle $ for all $x$ and $y$. Put $x=T^{*} y-Sy$ to see that $\|T^{*} y-Sy\|^{2}=0$ which gives $T^{*} y=Sy$ for all $y$.
Details for continuity: let $x_n \to x$ and $Tx_n \to ... | {
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} |
Calculating the line integral for the circle and the square
Consider the region $S$ bounded between the square with corners at the points (4,4),(-4,4),(-4,-4) and (4,-4) (oriented counterclockwise), and the circle of radius 1 centered at (-1,0) (oriented clockwise) and
$$
F(x,y)=\left(\frac{-y}{(x+1)^2+y^2}, \frac{... | It looks like you are making a mistake when calculating the integral for the circle. You have
$$
x=-1+\cos t,\ \ \ y=-\sin t
$$
(where the minus sign accounts for the clockwise direction of the curve). Then $(x+1)^2+y^2=\cos^2t+\sin^2t=1$, and
$$
\int_{\text{circle}}F\cdot dr=\int_0^{2\pi} \left(\sin t,\cos t \right)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the logical reason to use a proof by contradiction Given an arbitrary set $A$ of real numbers. I want to decide if the set $A$ is infinite.
My question is: What is the logical reason to use a proof by contradiction.
I can think for example that there is no known method to prove this directly. But I am not conv... | The reason most people do proof by contradiction isn't really logical.
When you do a direct proof, you have $n$ assumptions and $1$ predetermined statement to prove.
When you do proof by contradiction, you have $n+1$ assumptions and you succeed when you prove any false or contradictory statement.
Some people find the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 4
} |
Prove if $u,v,w$ linearly independent then ${\{u+v+w,v-w,2w}\}$ linearly independent
Prove if $u,v,w$ linearly independent then ${\{u+v+w,v-w,2w}\}$
linearly independent
what I did is :
I need to Prove that for $x,y,z \in F$
$x(u+v+w) + y(v-w) + z(2w) = 0 $
implies that $x=y=z=0$
$x(u+v+w) + y(v-w) + z(2w) = xu... | The systematic way is to consider matrices:
$$
\begin{pmatrix} u' \\ v' \\ w' \end{pmatrix}
=
\begin{pmatrix} 1 & 1 & \hphantom{-}1 \\ 0 & 1 & -1 \\ 0 & 0 & \hphantom{-}2 \end{pmatrix}
\begin{pmatrix} u \\ v \\ w \end{pmatrix}
$$
The matrix is triangular with nonzero diagonal entries. Therefore, it is invertible.
Thus,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
show the function $\cos^2(x)$ and $\sin^2(x)$ belong to Trig2(R) and find their coordinates for the basis In the next question it can be used without proof that the familie
$(1,\cos(x),\sin(x),\cos(2x),\sin(2x))$ is lineart independent and thus a basis for trig2(R)
b) show that the function $\cos^2(x)$ and $\sin^2(x... | $ \cos (2x)= \cos^2 x- \sin^2x = \cos^2 x-(1- \cos^2 x)= 2 \cos^2 x-1,$ hence
$$ \cos^2 x= \frac{1}{2}( \cos(2x)-1).$$
Can you proceed ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding a min on $\sqrt{f(x)}$ is equal to min on $f(x)$ During a discussion about RMS, one said that finding the min of a function or of it square root is the same because square root is monotonic increasing.
Is this make any sense?
| Finding a minimum of a function is equivalent to finding the point at which the derivative of the function is equal to zero on a given interval. If we can show that the sign of the derivative of any $f(x)$ is equal to the sign of the derivative of any $\sqrt{f(x)}$ then we can show that the min or minimums for $\sqrt{f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Set of finite signed measures with the weak*-topology is a topological vector space For a compact space $X$ I want to show that the space of finite signed measures on the Borel-$\sigma$-algebra on $X$, equipped with the weak*-topology, i.e. the corsest topology such that
$\mu\mapsto\int f~\mathrm{d}\mu~$ is continuous... | Perhaps you should do it by nets:
Assume that $(\mu_{\alpha},\nu_{\alpha})\rightarrow(\mu,\nu)$ weak$^{\ast}$, then $\displaystyle\int fd(\mu_{\alpha}+\nu_{\alpha})=\int fd\mu_{\alpha}+\int fd\nu_{\alpha}\rightarrow\int fd\mu+\int fd\nu=\int fd(\mu+\nu)$, so $\mu_{\alpha}+\nu_{\alpha}\rightarrow\mu+\nu$ weak$^{\ast}$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit as $y\to x$ of $(\sin y-\sin x)/(y-x)$ without L’Hospital $$\lim_{y\rightarrow x}\frac{\sin(y)-\sin(x)}{y-x}$$
Is there any cool trig identity I could use to solve this? We don't have L’Hospital yet, so I have to calculate it otherwise. I tried solving this using the series expansion of sine:
$$\cdots =\lim_{y\ri... | In the comments, someone pointed out that you can show that $\lim_{y\to x}\dfrac{\sin(y)-\sin(x)}{y-x}=\cos(x)$ if we know that:
$$\lim_{\theta\to0}\dfrac{\sin(\theta)}{\theta}=1,\quad(1)$$
$$\lim_{\theta\to0}\dfrac{\cos(\theta)-1}{\theta}=0.\quad(2)$$
Here is a link to that argument: Solving a limit given a limit
I wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Inverse of a multiplication operator Given an operator $M$: $L^2([0,1])\rightarrow L ^2([0,1])$:
$$M(f)(x) = x^2f(x) $$
I am trying to show if $(I + M)$ is invertible and what $|| (I+M)^{-1} ||$ is.
I am aware of the theorem which says if $||M|| < 1$ then $I-M$ and hence $I+M$ is invertible and allows computation of $... | Let $P=I+M$ and $Qf=\dfrac{1}{1+x^{2}}\cdot f(x)$, it is routine to check that $PQf=f$ and $QP f=f$, so $P$ is algebraic invertible.
We also note that
\begin{align*}
\|Qf\|_{L^{2}}^{2}=\int_{0}^{1}\dfrac{1}{(1+x^{2})^{2}}|f(x)|^{2}\leq\int_{0}^{1}|f(x)|^{2}dx=\|f\|_{L^{2}}^{2},
\end{align*}
so $\|Q\|\leq 1$.
Now we let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3478925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Decomposing $SO_3$
(Artin Algebra, 9.4.9)Let $H_i$ be the subgroup of $SO_3$ of rotations about the $x_i$-axis, $i=1,2,3$. Prove that every element of $SO_3$ can be written as a product $ABA'$, where $A$ and $A'$ are in $H_1$ and $B$ is in $H_2$. Prove that this representations is unique unless $B=I$.
I know that $A... | for $ A \in H_1$ it has the form
$$
\begin{matrix}
1 & 0 & 0 \\
0 & c_\theta & s_\theta \\
0 & -s_\theta & c_\theta \\
\end{matrix}
$$
For any $ G \in SO_3 $ :
$$
\begin{matrix}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33} \\
\end{matrix}
$$
Assume that G is not in $H_1$ otherwi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A polynomial $p(x,y)$ that is never $0$ if $y\neq 0$ I would like to find the best proof of the following fact:
If a polynomial $p\in k[x,y]$ (where $k$ is an algebraically closed field) is such that $p(a,b)\neq 0$ on the set $\{(a,b):b\neq 0\}$, then in fact $p\in k[y]$ (that is, $p$ does not depend on $x$). I do know... | As an amusing complement to anomaly's excellent answer, notice that the polynomial $p(x,y)=a_0(y)$ being nonzero for $y\neq0$ is necessarily of the form $f(x,y)=cy^n$ where $c\in k^*$ and $n\in \mathbb N$.
Notice also that the theorem is false for non algebraically closed fields, as witnessed by the polynomial $x^2+1\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Concluding critical points of function are not in the domain for $f(x,y) = x^{3} - x +y^2 - 2y$ My question has to do more with determining if the points are in the domain or not. So I am asked to find the extrema of the function $f(x,y) = x^{3} - x +y^2 - 2y$ over the closed trinagular region $(-1,0), (1,0), (0,2)$.
T... | What you have is fine.
We can say that the triangle is bounded by the lines
$y = 2x + 2, y = -2x + 2, x = 0$
or the region is below the line
$y = \begin {cases} 2x + 2 & x\le 0\\ -2x + 2 & x>0\end{cases}$
Plugging the points $x = \pm \sqrt {\frac 13} \approx \pm\frac {4}{7}$
In fact $\frac {4}{7}$ is sligthly less tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this proof of $\mathcal{L}\{\sin{at}\}=\frac{a}{s^2+a^2}$ valid? I found this proof of $\mathcal{L}\left\{\sin{at}\right\}=\frac{a}{s^2+a^2}$ on Proof Wiki:
This proof is way easier than others since it uses the linearity of the Laplace Transform. However, I am confused by the author's use of $\operatorname{Im}$. I... | Inspired by the above, the conclusion can be proved in another way.
We know that
$$
{\cal L}\left\{ {{e^{iat}}} \right\} = \frac{1}{{s - ia}}
$$
and
$$
{\cal L}\left\{ {{e^{ - iat}}} \right\} = \frac{1}{{s + ia}}.
$$
With the help of Euler's Formula, $\sin at$ can be written as
$$
\sin at = \frac{1}{{2i}}\left( {{e^{ia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are Banach norms Fréchet differentiable? Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff
$$\lim_{h \to 0} \frac{\|f(x + h) - f(x) - Ah\|_W}{\|h\|_V} = 0$$
We call a $f$... | No this is not always true. Take $(V, \Vert \cdot \Vert)=(\mathbb R^2, \sup (\vert x \vert, \vert y \vert))$. The norm is not Fréchet differentiable when $x = \pm y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Calculate sum of series $\sum_{n=1}^\infty \frac{x+a}{n(x+a) + n^2}$. I've been stuck with calculating the sum of series of the following problem. Can you help me?
$$\sum_{n=1}^\infty \frac{x+a}{n(x+a) + n^2}$$
for real numbers $a\geq 0$ and $x\geq 1$.
| After @URL's comment, using
$$S_p=\sum_{n=1}^p\frac{x+a}{n(x+a)+n^2}=\sum_{n=1}^p\frac1n-\sum_{n=1}^p\frac1{n+(x+a)}$$ and using generalized harmonic numbers, we have
$$S_p=H_{a+x}+H_p-H_{a+x+p}$$
Now, using the asymptotics
$$H_q=\gamma +\log \left({q}\right)+\frac{1}{2 q}-\frac{1}{12
q^2}+O\left(\frac{1}{q^3}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
let $d = \gcd(m,n), m, n > 0$ Bézout gives $d = mx + ny, x, y \in \mathbb{Z}$ prove that.. ... prove that it is always possible to choose $x < 0$.
I did $m = qn + r$ and $\gcd(m,n) = \gcd(n, \operatorname{rem}(m, n)) = \gcd(n, r)$
But I do not know where to go from here.
| If we suppose $x, y$ are given and $x>0$, we know the other solutions in integers of the equation $\;mX+nY=d\;$ are given by
$$X=x-kn,\quad Y=y+kn\qquad( k\in \mathbf Z),$$
so choose $k$ so large as to ensure that $x'=x-kn<0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Basic binary operation on set Hey please help me answer this question:
Given a set $A$ with at least 2 elements which on it the binary operation * is defined in that manner:
for every $a,b\in A, a*b=b$.
Check if the binary operation * is commutative, associative and idempotent.
| Say $A$ has at least two different elements, name them $5$ and $2$. Then $$5*2 = 2\ne 5 = 2*5$$
so it is not commutative.
Since we have also:
$$a*(b*c) = a*c = c$$ and $$(a*b)*c = b*c=c$$
we see it is associative. Also we have $b*b = b$ so ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Are there uncountably many disjoint uncountable real null sets? It is easy to think of a countable family of disjoint Cantor sets, and their union is of course a null set.
It is equally trivial to define an uncountable family of Cantor sets but, how can it be ensured that they are pairwise disjoint?
Would their union... | First do it for $C=\{0,1\}^{\mathbb{N}}$. For every $b=(b_n)_n\in \{0,1\}^{\mathbb{N}}$ let $C_b=\{(a_n)\ \mid\ a_{2n} = b_n \textrm{ for all } n\}$. Clearly $(C_b)_b$ form a partition of $C$ into compact subset of measure $0$.
Now $[0,1]$ is measure equivalent to $C$ under the map
$a\mapsto s(a) = \sum a_n/2^{n+1}$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
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