Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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If $V=W_1+\cdots+W_k$ and $\dim(V) =\sum_{i=1}^{k}\dim(W_i)$, prove that $V = W_1 \oplus\cdots\oplus W_k$
Let $V$ be a vector space and let $W_1,\ldots,W_k$ be subspaces of $V$ such that $V = W_1 +\cdots+ W_k$ and $\dim(V) = \dim(W_1) +\cdots+ \dim(W_k)$. Prove that $V = W_1 \oplus\cdots\oplus W_k $.
My attempt:
$k=... | In fact
$$\begin{cases}
W_1 \cap W_2 = \{ 0 \} \\
(W_1+ W_2)\cap W_3 = \{0\}
\end{cases}$$
are sufficient conditions. The second one implies $(W_1+W_2) + W_3= (W_1+W_2) \oplus W_3$ and the first one that $W_1 + W_2 = W_1\oplus W_2$. Therefore
$$V= W_1+W_2+W_3 = W_1\oplus W_2 \oplus W_3.$$
However for the general case, ... | {
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Sketch the open balls for the Railway Metric Hi I'm having trouble picturing the open balls for the Railway metric
(,)={2(,) if ,,0 are collinear
2(,0)+2(0,) otherwise
I need to sketch the open balls of Bd(0,1), Bd((1,0),1) and Bd((1/2,0),1)
I have seen pictures of them like a "lollipop" but don't know how ... | The name "Railway Metric" comes from the following image. Suppose there are a number of railway lines which start at the origin $O$ and go radially outwards in straight lines. From a station $A$ on line $OA$ you can travel directly to any other point on the line $OA$ towards $O$ or away from $O$, but to reach a station... | {
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Why are the p-adic Integers $\mathbb{Z}_p$ called integers? I looked online but I can't find any reasoning as to why the p-adic integers are called integers. Are there certain properties that $\mathbb{Z}_p$ shares with $\mathbb{Z}$, and is there sufficeint overlap to really associate the name integers with $\mathbb{Z}_... | $\mathbb{Q}$ is the quotient field of the ring of (rational) integers $\mathbb{Z}$
And $\mathbb{Q}_p$ is the quotient field of the ring of p-adic integers $\mathbb{Z}_p$
| {
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Please help with indetifying subformulas in the propositional logic. In my textbook of the propositional logic I should indentify all subformulas of the following formula: ¬(a∧b)⟺(c⇒a).
According to the textbook the formula (a∧b)⟺(c⇒a) is not a subformula of the formula ¬(a∧b)⟺(c⇒a). Why?
They also argue that ¬(a∧b)⟺(c... | If the statement was ¬[(a∧b)⟺(c⇒a)] (i.e. if the statement was the negation of a biconditional), then (a∧b)⟺(c⇒a) would indeed be a subformula. But, the given formula ¬(a∧b)⟺(c⇒a) is a biconditional between ¬(a∧b) and c⇒a. So, those are both subformulas, but (a∧b)⟺(c⇒a) is not
Keep in mind that a subformula is not the ... | {
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Why doesn't this converge to $\pi$? $\lim_{n\to\infty}\frac12\sqrt{2-2\cos( \frac{2 \pi}{n} )} \times n$ Suppose a regular polygon with $n$ side has radius (line from center to point that connect sides *I don't know how to call it) of length $r$.
From cosine law, the side would has length of $$\sqrt{2r^2-2r^2\cos( \fr... | As I commented, I can't seem to replicate your graph. However, there is this:
Note that, by a Half-Angle Identity, we have $1-\cos\theta = 2\sin^2(\theta/2)$, so your expression reduces to $$\lim_{n\to\infty}n\sin\frac{\pi}{n}$$ which we can cleverly rewrite as $$\pi\cdot\lim_{n\to\infty}\frac{\sin(\pi/n)}{\pi/n}=\pi\c... | {
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Rigorous global optimization The work by Thomas Hales (see enter link description here) before the formal proof solves a number of global optimization problems that need to be solved exactly.
The strategy relies on following strategy:
*
*Use of interval arithmetic in order to be sure of where the values are located.... | See these papers:
*
*What can interval analysis do for global optimization? by Ratschek and Voller.
*Interval methods for global optimization by Wolfe.
and this book:
*
*Global Optimization Using Interval Analysis by Hansen and Walster.
See also a list by Neumaier.
| {
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Prove that $\mathcal{G}$ is $\sigma$-algebra Let $X, Y$ be arbitrary spaces, $f: X \to Y$ arbitrary function. Prove that if $\mathcal{F}$ is $\sigma$-algebra of X then family
$$
\mathcal{G} = \{G \subset Y : f^{-1}(G) \in \mathcal{F}\}
$$
is a $\sigma$-algebra of Y.
First, I have proved that it is closed under compleme... | The preimage of the empty set is also empty set!
In fact, suppose $f(A)=\emptyset$ and $A\neq \emptyset$. There exists $x\in A$, therefore $y\in f(A)=\emptyset$, a contradiction.
| {
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How should I notate a set of probability distributions? So I'm writing up a computer science paper. I want to notate that I'm estimating joint degree distributions for a bunch of graphs I am generating. I have seen joint degree distributions notated as $e_{ij}$ in other papers (where $i$ and $j$ are the degrees). I was... | As explained by @kccu, using $e_{i,j}$ would be a rather poor choice. This often denotes the edges between vertices $i$ and $j$.
I would use a capital "blackboard bold" P : $\mathbb{P}$, and hence $\mathbb{P}_{i,j}$
| {
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How can i find the domain of $f(x)$= $x^{1/x}$ on the negative numbers? I have been thinking of which negative values of $x$ yield a real number when plugged on $f(x)$ = $x^{1/x}$,
It is clear to see that it does not work for negative even numbers
$(-2)^{-1/2}$ = $(\frac{1}{\sqrt{-2}})$ is not real,
but for negative o... | I have already discussed some related problem here:
Is $(-1)^{2.16}$ a real number?
Basically for negative reals $x$, the value of $x^y$ can be extended via rational exponentiation to all $y\in\mathbb Q_{odd}$, the rational numbers represented by an irreducible fraction whose denominator is an odd natural.
This is made... | {
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Finding a generator for the intersection of two subgroups
In $\mathbb{Z}_{24}$ find a generator for $\langle 21\rangle\cap \langle 10\rangle$.
My attempt:
$\langle 21\rangle = \{21, 18, 15, 12, 9, 6, 3 ,0 \}$ from adding multiples of $21\pmod{24}$
$\langle 10\rangle = \{10,20,6, 16, 2, 12, 22, 8, 18, 4 , 14, 0\}$ fro... | So we have:
$$\langle 21\rangle=\{21,18,15,12,9,6,3,0\}$$
and
$$\langle 10\rangle=\{10,20,6,16,2,12,22,8,18,4,14,0\}$$
and therefore
$$ \langle 21\rangle \cap \langle 10\rangle = \{ 18,12,6,0\}$$
Since this is a pretty small set, you can check each member if they do or do not generate it. Then, you will find that $6$ ... | {
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How to prove that $\sum_{n=1}^{\infty}\frac{x}{n}\left(1+\frac{1}{n}-x\right)^n$ converges uniformly.
Prove that the following functional series converges uniformly for any $x$ from $E$.
$$
\sum_{n=1}^{\infty}u_n(x),\ \ \ u_n(x)=\frac{x}{n}\left(1+\frac{1}{n}-x\right)^n,\ \ x\in E=[0;1]
$$
I tried to use Weierstras... | $u_n(x)$ can also be estimated using the inequality between the geometric and the arithmetic mean:
$$
0 \le n^2 u_n(x) = (nx) \left(1+\frac{1}{n}-x\right)^n
\le \left( \frac{nx + n(1+\frac 1n - x)}{n+1} \right)^{n+1} \le 1 \, .
$$
| {
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$Ax = y$, possible to isolate matrix $A$? I have the matrix equation of real coefficients
$Ax = y$
where $y$ and $x$ are of different size. Is it possible to do a simple operation which will turn $A$ into a matrix function of both $y$ and $x$? I guess I can multiply on the right side with $x^{\top}$ and divide by the n... | Does this help?!
Let $ A\begin{bmatrix} 2 \\ 3 \end{bmatrix} =\begin{bmatrix} 10 \\ -2 \\ 5 \end{bmatrix}$, then we can find $A$ by assuming
$$A=\begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}$$ we get 3 equations
$$2a+3b=10,~ 2c+3d=-2,~ 2e+3f=5$$ as there are 6 unknowns one will get many solutions and hence ma... | {
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Show that $(1+x)^{(1+x)}>e^x$ How can I prove that $(1+x)^{(1+x)}>e^x$ for all $x>0$?
The problem arose as I tried to prove the well-known & intuitive econometric principle that the more often you compound your interest, the more interest you ultimately get (in maths, that $\frac{d}{dn}((1+\frac{1}{n})^n)>0$ for $n>0$)... | Your general case is easy with this corollary of the Mean value theorem:
Let $f,g$ be two differentiable functions defined on an interval $I$ and $x_0\in I$.
If $f(x_0)\ge g(x_0)$ and $f'(x)>g'(x)$ for all $x>x_0,\:x\in I$, then $f(x)>g(x)$ for all $x>x_0,\:x\in I$.
Indeed it is enough to compare the logs: set $f_a(... | {
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how does a surjective function not contradict the definition of a function. I was reading up on functions and different types of functions.
From wikipedia :
A function is a process or a relation that associates each element x
of a set X, the domain of the function, to a single element y of
another set Y
Surjectiv... | If you read those definitions carefully you will see that the "uniqueness" is in $Y$, not in $X$. For example consider the function from people in the US to states, which assigns to each person the state they live in. Each person lives in just one state (for mathematical and census purposes), but there are many people ... | {
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Lines AD, BE, and CF are concurrent Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that OD + DH = OE + EH = OF + FH and the lines $AD$, $BE$, and $CF$ are concurrent.
I am trying to solve the... | Hint: Let $H_a$, $H_b$, and $H_c$ denote the reflections of $H$ about $BC$, $CA$, and $AB$, respectively. Show that $H_a$, $H_b$, and $H_c$ are on the circumcircle of the triangle $ABC$. Now, $OD+DH=OD+DH_a$ for example. Pick $D$, $E$, and $F$ cleverly. The last hint is: with these good choices of $D$, $E$, and $F... | {
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Find all the values of the parameter 'a' for which the given inequality is satisfied for all real values of x. Find all the values of the parameter 'a' for which the inequality is satisfied for all real values of x.
$$a\cdot 9^x+4\cdot \left(a-1\right)\cdot 3^x+\left(a-1\right)>0$$
My attempt is as follows:-
$$a\cdot \... | There is an easier way :-):
Let $t = 3^x$, noting that we need $t>0$. The equation becomes
$at^2+4(a-1)t +(a-1) > 0$ for all $t>0$.
First note that if the above is true for all $t >0$ then we must have (taking
the limit as $t \downarrow 0$) that $(a-1) \ge 0$ (note $\ge$ not $>$).
In
particular, $a \ge 1$ must hold.
Th... | {
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Proving outer measure property I am self-studying analysis by Sheldon Axler. This is the one of exercise problem in his book. He uses $|\cdot|$ to indicate the outer measure.
Prove that if $A\subset \mathbb{R}$ and $t>0$, then $|A|=|A\cap(-t, t)|+|A\cap(\mathbb{R}\setminus(-t, t))|$.
$|A|\leq|A\cap(-t, t)|+|A\cap(\m... | For the first part note that exists a measurable set $G \supset A$ which is $G_{\delta}$ such that $|A|=|G|$
Indeed by the definition of the outer measure(with coverings of open intervals) we can find open sets $G_n \supset A$ such that $|G_n| \leq |A|+\frac{1}{n}$
Sending $n \to +\infty$ we have that $|G|=|A|$
Take ... | {
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Square Chessboard Problem Show that there is a $6$ x $4$ board whose squares are all black or white, where no rectangle has the four vertex squares of the same color. Also show that on each $7$
x $4$ board whose squares are all black or white, there is always a rectangle whose four vertex squares are the same color. ... | For the first part, the following construction seems to work:
$$\begin{matrix} 0&0&1&1&1&0\\1&1&0&0&1&0\\1&0&1&0&0&1\\0&1&0&1&0&1 \end{matrix}$$
Here, $1$ and $0$ stand for black and white respectively. For the $7\times 4$ case, let's assume such a configuration exists and derive a contradiction. Consider the possible ... | {
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Sets well ordered under different operations Proving $(a)\quad R^+ \cup \{0\}, <$
$(b)\quad [0,1], >$
$(c)\quad \text{The set of integers divisible by 5}, <$
$(d)\quad \{\{0,1,...,n\}|n ∈ N\},⊆$
I believe that:
(a) Is not well-ordered because of the fact that rational numbers would be in a continuous flow and would be... | Nice, they're correct. For (a)-(c) you've essentially already proved them through contradiction. Try to write them out in more detail.
Here's a sketch of (a): Take the rationals in $(0,1)$. If it had a minimum $q$, then take the midpoint of $0$ and $q$, contradiction.
| {
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At most one homomorphism between two groups Let $S$ be a set, and $F$, $G$ be groups.
Let $f: S \rightarrow F$ and $g: S \rightarrow G$ be functions.
I want to prove the following:
If $f(S)$ generates $F$, then there exists at most one homomorphism
$\psi:F \rightarrow G$ such that the following diagram commutes.
I k... | So $f$ and $g$ are just given maps between sets and $f$ is injective. If you want the diagram to commute, this defines where $\psi$ maps the generators of $F$. What you have to show is if you have a mapping of the group generators, there is a unique way to extend it to a group homomorphism.
So let $\sigma \in F$ be arb... | {
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A compact subspace of metric space with no isolated points. Suppose $X$ is a metric space which is connected and has no isolated points. Then I want to show that it will have some non-empty compact subspace $Y$ such that $Y$ has no isolated points. I don't know if my assertion is correct. If not, please give counterexa... | Counterexample. Let $X$ be a Bernstein subset of $\mathbb R^2$, i.e., a subset of $\mathbb R^2$ such that both $X$ and $\mathbb R^2\setminus X$ meet every uncountable closed subset of $X$. (Such sets exist, assuming the axiom of choice.) Then $X$ is a connected metric space with no isolated points, and every nonempty c... | {
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Maximising x(G) Let $G$ be a graph on $n$ vertices. To each vertex, we assign zero or more colours such that
*
*Any two vertices sharing an edge must have a common colour.
*Any two vertices not sharing an edge do not share any common colours.
Let $x(G)$ be the least number of colours needed to fulfil the conditio... | I would argue that the best you can do is using a maximal triangle-free graph (per Turan's construction, this is a complete bipartite graph whose partitions are as balanced as possible). It at least serves as a lower bound: take $n$ even (for simplicity, I'll handle odd in a bit), and construct the complete bipartite g... | {
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power of 2 involving floor function divides cyclic product if $S_n={a_1,a_2,a_3,...,a_{2n}}$} where $a_1,a_2,a_3,...,a_{2n}$ are all distinct integers.Denote by $T$ the product
$$T=\prod_{i,j\epsilon S_n,i<j}{(a_i-a_j})$$ Prove that $2^{(n^2-n+2[\frac{n}{3}])}\times \text{lcm}(1,3,5,...,(2n-1)) $ divides $T$.(where [] ... | HINT:
You're not quite correct for the powers of $2$. There are $2n$ numbers total, so there can in fact be exactly $n$ odds and $n$ evens (i.e. neither has to be $\ge n+1$.)
Instead: Let there be $k$ odds, and $2n-k$ evens. This gives you ${k \choose 2}$ factors of $2$ from the (odd - odd) terms and ${2n-k \choose 2... | {
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What's the quotient space of torus $S^1\times S^1$ under equivalence relation $(z,w)\sim (w,z)$ Consider the quotient space of torus $S^1\times S^1$ under the equivalence relation $(z,w)\sim (w,z)$.
I'm trying to visulaize it but find it pretty hard to do. Any hint?
| Identify the torus $S^1\times S^1$ with the square $[0,1]^2$ modulo the identification of $(0,w)$ with $(1,w)$ and $(z,0)$ with $(z,1).$ Then you can identify the quotient space by your equivalence relation with $\{(z,w)\in[0,1]^2 : z\ge w\}.$ Now let
\begin{align}
u & = z+w-1 \\
v & = -z+w+1 \\[12pt]
\text{so that } z... | {
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Having trouble finding examples of sets of functions that do NOT form a vector space In this question someone asks about showing that the set of all functions of the form $y(t) = c_1\cos\omega t + c_2\sin\omega t$ is a vector space. But doesn't literally any set of functions of the form $y(t) = c_1f(t) + c_2g(t) + \ldo... | Take your favourite differential equation, as long as it's not a homogeneous linear one. Then its solution set isn't a vector space under the usual definitions for functions of scaling and addition. Surely that's not a contrived example.
For the sake of physical examples, note that the linearity of quantum mechanics an... | {
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Roots over $\mathbb{C}$ equation $x^{4} - 4x^{3} + 2x^{2} + 4x + 4=0 $. I need roots over $\mathbb{C}$ equation $$x^{4} - 4x^{3} + 2x^{2} + 4x + 4 = 0$$
From Fundamental theorem of algebra we have statement that the equation have 4 roots over complex.
But I prepare special reduction:
$$ \color{red}{ x^{4} - 4x^{3} +... | Solving algebraically is perhaps not too messy if one is happy to have roots in terms of the solution to a cubic:-
Using the substitution $t=x-1$ we have $$ t^{4}- 4t^{2}-4t+3=0$$ and therefore $$ (t^{2}- 2)^2=4t+1.$$
For any $z$, $$ (t^{2}- 2+2z^2)^2=4z^2t^2+4t+4z^4-8z^2+1.$$
Let $z$ be a solution (found by Cardan's m... | {
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Cards in the deck
From a common deck ($52$ cards) three cards are drawn successively and without replacement. How many extractions in which the first card is hearts, the second is a king, and the third is not a lady?
Let ($a, b, c$) be the possible $3$-tuples of withdrawals. For $a$ there are $4$ hearts, for $b$ the... | The number of possibilities are
If the first card is QH: 1 x 4 x 47
If the first card is KH: 1 x 3 x 46
If the first card is any other Heart: 11 x 4 x 46
The total is therefore 2350.
| {
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} |
Term for "functions that have a closed-form expression in terms of base functions $B$"? Suppose we have a set of "basic" functions $B=\{+,-,\cdot,/,\exp,\log,\sin \}$, and we want to define:
The set of functions $F_B$ which can be defined as $f(x)=\textit{application of elements of }B$.
Is there a term for this? I... | I would call them $B$-based or $B$-generated functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Are i.i.d. random variables with infinite expectation uncorrelated? If I have a sequence of i.i.d. random variables $(X_i)_{i\in \mathbb{N}}$, with $\mathbb{E}[X_i^+] = \infty$ and $\mathbb{E}[X_i^-] < \infty$, where $X_i^+ = \max(X_i, 0)$ and $X_i^- = \max(-X_i, 0)$ can I say that they are uncorrelated?
Because I hav... | Covariance of $X$ and $Y$ is defined only when $X$ and $Y$ have finite variance.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to find invertible matrix $P$ Such that $B=P^{-1}AP$
Let $A,B$ be the two $3 \times 3$ matrices
$$A=\begin{bmatrix}
3 & 2 &-5\\2 &6&-10\\1 &2 & -3 \end{bmatrix}$$
$$B=
\begin{bmatrix}
6 &20 &-34\\6 &32&-51\\4 &20 &-32 \end{bmatrix}
$$
Suppose that there is a Non Singular matrix $P$ that $P^{-1}AP=B$. Find $P$
My ... | When we don't know explicitly the eigenvalues, there are two methods
*
*We solve the equation $PB=AP$; the space of solutions has dimension $dim(C(A))$ where $C(A)$ is the commutant of $A$ (here $5$). The general solution is in the form $P=a_1P_1+\cdots+a_5P_5$ where the $(P_i)$ are known matrices. We randomly choo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Help with $\lim\limits_{x \to 0} \frac{x^2 \sin(2x)}{\log (1+(\sin3x)^3)}$ I'm preparing for my first exam in university (just recently enrolled in computer science) and I'm having difficulties working out this limit. I either currently lack the proper reasoning process to get it done or they haven't yet explained us a... | We have that
$$ \frac{x^2 \sin(2x)}{\log (1+(\sin3x)^3)}=\frac{(\sin(3x))^3}{\log (1+(\sin3x)^3)}\cdot \frac{(3x)^3 }{(\sin3x)^3}\cdot \frac{ \sin(2x)}{2x}\cdot \frac2{27}$$
then refer to standard limits as $u \to 0$
*
*$\frac{\log (1+u)}{u}\to 1$
*$\frac{\sin u}{u}\to 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Find the range of values of $k$ for which $kx^2 + 8x + k <6$ for all real values of $k$
Find the range of values of $k$ for which $kx^2 + 8x + k <6 $ for all real values of $k$.
I'm unsure if the discriminant must be greater than zero or less than zero.
My working steps: \begin{align}b^2 - 4ac = (8)^2 - 4(-2)(17-k) &... | We have
$$kx^2+8x+k<6 \iff kx^2+8x+k-6<0$$
and this is always true when $k<0$ and
$$b^2-4ac=64-4k(k-6)<0 \implies k^2-6k-16>0$$
that is $k<-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Typical Olympiad Inequality? If $\sum_i^na_i=n$ with $a_i>0$, then $\sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n$
Let $\sum_i^na_i=n$, $a_i>0$. Then prove that $$ \sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n $$
I have tried AM-GM, Cauchy-Schwarz, Rearrangement etc. but nothing seems to work. T... | A Hint for an Alternative Solution.
We want to show that $$\left(\frac{x^3+1}{x^2+1}\right)^4\geq 2x-1$$ for every $x\in\mathbb{R}$. By the AM-GM Inequality,
$$\left(\frac{x^3+1}{x^2+1}\right)^4+1\geq 2\,\left(\frac{x^3+1}{x^2+1}\right)^2\,.$$
Hence, it suffices to verify that
$$\left(\frac{x^3+1}{x^2+1}\right)^2\geq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Determine the radius of Convergence of this telescoping series Can anyone point it out how to find the radius of convergence of this series?
$\sum_{n= 1}^{\infty}\dfrac{1}{(x+n) \cdot (x+n - 1)}$
I tried the Ratio Test, but the limit goes to 1.
Any help is appreciated, thank you :)
| This can be rewritten as:
$$\sum_{n=1}^{\infty} \left(\frac{1}{x+n}-\frac{1}{x+n+1}\right)=\frac{1}{x+1}$$
Since it's a telescopic series and not a power one, this is always convergent (for every real $x \neq -1$).
As @Doug M noticed, $x$ cannot be a negative integer since the sum is not bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3391126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Evaluate $\sum_{n=1}^\infty 2^{-\frac{n}{2}}$
Find $$\sum_{n=1}^\infty 2^{-\frac{n}{2}}$$
I know that the final numerical value of that is $1+\sqrt2$ but not sure how to get that. Any identities, formula or hints would be helpful.
I tried re-expressing it as $\frac{1}{\sqrt2}+\frac{1}{2}+\frac{1}{2\sqrt2}+\frac{1}{4... | You have $$\sum_{n=1}^\infty 2^{-\frac n2}=\sum_{n=1}^\infty (2^{-\frac 12})^n = \frac{2^{-\frac{1}2}}{1-2^{-\frac{1}2}} = 1+\sqrt 2$$
by the same formula you gave.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3391264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Find the number of real solutions of the equation $2^x+x^2=1$ My attempt is as follows:-
$$2^x+x^2=1$$
$$\left(1+x\cdot log(2)+\frac{x^2\cdot (log(2))^2}{2!}+\frac{x^3\cdot (log(2))^3}{3!}+\dots\right)+x^2=1$$
$$x\cdot log(2)\left(1+x^2+\frac{x\cdot log(2)}{2!}+\frac{x^2\cdot (log(2))^2}{3!}\right)=0$$
$$x=0, \left(1+x... | Consider the function $$f(x)=2^x+x^2-1.$$
Its first and second derivatives are $$f'(x)=\ln2\cdot2^x+2x$$
and $$f''(x)=(\ln2)^2\cdot2^x+2.$$
Observe:
*
*Second derivative is strictly positive
*First derivative has one root at $x_{min}=\dfrac1{\ln2}W\bigg(\dfrac{(\ln 2)^2}2\bigg)\approx-0.28454$
*$f(x_{min})\approx-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $\int \frac{{dx}^2}{dy}$ valid? It appears to me that
$\int \frac{{dx}^2}{dy}$ can be rewritten as
$\int \frac{dx}{dy}\cdot{}dx$ which in turn can be rewritten as
$\int f^{-1}{^\prime}(y)\cdot{}dx$. (it is assumed that $y=f(x)$ although y may not be a function of x; multiple solutions for y may exist for a given val... | Your interpretation of $dx^2$ as $(dx)(dx)$ may get confused with $ d(x^2)$.
Otherwise your thought process is straightforward and worths further investigation.
You example makes perfect sense.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3391507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Uniqueness of fractional linear transformations for ad-bc=1 Show that any fractional linear transformation can be represented in the form $$f(z) = \frac{az+b}{cz+d}$$
where $ad-bc = 1$. Is this representation unique?
This is just the definition of a fractional linear transformation. I am having trouble proving that $a... | If you replace the coefficieints $a,b,c,d$ by $ta,tb,tc,td$ where $t \neq 0$ you get the same function $f$. When you do this $ad-bc$ becomes $t(ad-bc)$. So there is no question of proving that if $f$ has the given form then $ad-bc$ must be $1$. What is true is that you can always take $t=\frac 1 {ad-bc}$ so that in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3391621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there known undecidable statements which are Gödel sentences, but were not always known to be such? As I understand it, Gödel and Rosser proved constructively that for any consistent, effectively generated system within which basic arithmetic can be carried out, an infinite amount of statements which can be proven ... | Statements that are undecidable in Peano arithmetic, but can be stated in it and are provable in something larger such as ZF or second-order arithmetic, include $\varepsilon_0$ induction, Goodstein's theorem and the Paris–Harrington theorem. I'm not sure, though, if they can be formatted as Gödel sentences. See also @D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3391785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Maximum distance between samples with equal value in a sequence of i.i.d. discrete samples Let $\left(X_i\right)_{i=1}^n$ be a sequence of i.i.d. samples with discrete outcome space $S=\{s_1,...,s_k\}, k<\infty$, $s_i \in \mathbb{R}$ with respective probabilities $p_1,...,p_k$. Define the maximum distance between two ... | Let $X_{j}$
indicate the event that there does not exist $k\in\left[j+1,\dots,j+\epsilon n\right]$
such that $s_{j}=s_{k}$.
We have:
$$p:=\Pr\left[X_{j}=1\right]= \sum_{i=1}^{k}p_i\left(1-p_{i}\right)^{\epsilon n}$$
Let $X=\sum_{j=1}^{n}X_{j}$, then
$$\mathbb{E}\left[X\right]=np$$
which approaches zero whenever ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3391884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\mathbb{E}\big(e^{X}\big|$ $e^{Y}\big)\overset{?}{=}\mathbb{E}\big(e^{X}\big|$ $Y\big)$ I have a quick question. Is the following reasoning correct? If not, why?
I know that $X|Y\in N(\rho Y, 1-\rho^2)$.
I want to deduce an expression for $\mathbb{E}\big(e^{X}\big|$ $e^{Y}\big)$. My idea was as following:
$\mathbb{... | Yes, provided that $e^{X}$ is integrable. It is because $\sigma(Y)=\sigma(e^{Y})$.
Proof: Clearly $e^{Y}$ is $\sigma(Y)$-measurable, so $\sigma(e^{Y})\subseteq\sigma(Y)$.
On the other hand, $Y=\ln\left(e^{Y}\right)$ which is $\sigma(e^{Y})$-measurable,
so $\sigma(Y)\subseteq\sigma(e^{Y})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3392203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to show this function $h:\Bbb{R}^n\setminus{\{0\}} \to S^{n-1} \times \Bbb{R}$ is continuous? Is the function $h:\Bbb{R}^n\setminus{\{0\}} \to S^{n-1} \times \Bbb{R}$ defined by $x=(x_1, \dots, x_n) \mapsto \left(\frac{x_1}{\|x\|}, \dots, \frac{x_n}{\|x\|}, \log\|x\|\right)$ continuous?
I tried to do it using open ... | To prove that a map $x : f(x)=(f_1(x), \dots, ,f_p(x))$ is continuous where $x \in \mathbb R^n$, it is sufficient to prove that each $x \mapsto f_i(x)$ is continuous.
$x \mapsto \Vert x \Vert$ is continuous as it is the square root of a polynomial map which is continuous.
$x \mapsto \frac{x_i}{\Vert x \Vert}$ is contin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3392371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve with eigenfunction expansion $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} + e^{-t} + e^{-2t} \cos \frac{3\pi x}{L}$
Q: Solve the following non-homogeneous problem:
\begin{align*}
\frac{\partial u}{\partial t} &= k \frac{\partial^2 u}{\partial x^2} + e^{-t} + e^{-2t} \cos \frac{3\pi x}... | Letting
$$u \sim \sum_{n \ge 0} A_{n}(t) \cos \left( \frac{n \pi x}{L} \right)$$
then substituting into the PDE (and noting that $1 \equiv \cos(0 \pi x/L)$) yields
$$\sum_{n \ge 0} A_{n}'(t) \cos \left( \frac{n \pi x}{L} \right) = -k \sum_{n \ge 0} \left( \frac{n \pi}{L} \right)^{2} A_{n}(t) \cos \left( \frac{n \pi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3392558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Finding the maximum possible variation of a permutation of size N Given a permutation of size N, you can create a distance matrix that shows the distance between each element to each other element, giving a complete description of the permutation.
ie 1,2,3,4 creates
\begin{array}{|c|c|c|c|c|}
\hline
& \textbf{1} & \te... | If I understand correctly, this can be formulated as a maximization version of the quadratic assignment problem. I get the following values for $N \le 10$. Please confirm whether these match what you obtained.
N D(N)
1 0
2 0
3 2
4 8
5 16
6 28
7 44
8 68
9 96
10 134
I obtained these v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3392662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to show that this sequence is monotonic increasing?
Let $k>1$ and define a sequence $\left\{a_{n}\right\}$ by $a_{1}=1$ and $$a_{n+1}=\frac{k\left(1+a_{n}\right) }{\left(k+a_{n}\right)}$$
(a) Show that $\left\{a_{n}\right\}$ is monotonic increasing.
Assume $a_n \geq a_{n-1}$. Then,
$$a_{n+1} = \frac{k(1+a_n)... | Turn the question of whether $(a_n)$ is monotone increasing into an inequality purely in terms of a single term $a_n$. In particular,
$$a_n \le a_{n+1} = \frac{k(1 + a_n)}{k + a_n}.$$
Simplifying, making the temporary assumption that $k + a_n > 0$,
$$a_n(k + a_n) \le k(1 + a_n) \iff a_n^2 - k \le 0 \iff a_n \in [-\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3392871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given a function $f$ infinitely differentiable at a point $c$ does there exist a neighborhood of $c$ in which $f$ is infinitely differentiable? Suppose $f$ is a real valued function defined on a subset of the reals and $f$ is infinitely differentiable at $c$.Then is it possible that there does not exist any neighborhoo... | This is similar to the idea in Kavi Rama Murthy's answer. First, note that for $k = 0, 1, 2, \dots$, you can construct a function $h_k : [-1, 1] \to \mathbb{R}$ which is $C^k$ but is not $(k+1)$ times differentiable at any point: just take repeated antiderivatives (integrals) of a continuous nowhere differentiable func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3393040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Equivalence between Latin squares I have two Latin squares of order 6. Is there any way to check whether they are isomorphic? I mean any program or online tool?
$
L_1=
\left[ {\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6\\
2 & 4 & 5 & 1 & 6 & 3 \\
3 & 1 & 2 & 6 & 4 & 5\\
4 & 5 & 6 & 2 & 3 & 1\\
5 & 6 & ... | The standard way of doing this is described in McKay, Meynert, and Myrvold, 2006 (link). Essentially this method is: convert the two Latin squares to graphs, and compare the canonical labels of the graphs computed e.g. using Nauty. Exactly which graphs to convert to depends on the equivalence type; "isotopism", "isom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3393183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If a real matrix has a complex eigenvalue on the unit circle, when is it a root of unity? Let $A$ be a real matrix with integer entries, and suppose $z$ is a complex eigenvalue of $A$ with $|z|=1$. As shown in this answer, $z$ need not be a root of unity (i.e. there need not exist an $m$ with $z^m = 1$). Under which co... | all 1 and 0 is not good enough. This first characteristic polynomial is the one given by Jose Carlos Santos in your earlier question If an eigenvalue of an integer matrix lies on the unit circle, must it be a root of unity?
$$
\left(
\begin{array}{cccc}
1&1&1&0 \\
1&1&0&1 \\
0&1&0&0 \\
0&0&1&0 \\
\end{array}
\right)
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3393377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to rotate a line in the complex plane? How do I rotate the the line $arg(z) = 0$ by $\frac{\pi}{4}$ radians counter-clockwise about the origin in the complex plane.
The general transformation is $z\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}$ however how do I algebraically find the image of the rotation of the line?
Thank... | If I understand both the OP's desire and gimusi's response correctly, then an alternative approach is to algebraically prove that for complex z and w, arg(zw) = arg(z) + arg(w).
Thus, when |w| = 1, rotating z by arg(w) is equivalent to multiplying z by w. Therefore, rotating z by $\;\pi/4\;$ is equivalent to multiplyi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3393622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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New to proofs, need help with how to approach a beads-and-wires proof puzzle I am working on a proof puzzle about beads and wires. We are given 4 axioms about the objects you can create with beads and wires.
*
*Axiom 1. You must have exactly 3 beads.
*Axiom 2. There is exactly one wire between each pair of beads.
... | Sketch:
Label the beads $B_1,B_2,B_3$ and suppose that $B_1$ was on all the wires, we will derive a contradiction.
Let $W_{ij}$ be the unique wire containing both $B_i,B_j$, for $i\neq j$.
If $W_{12},W_{23}$ were the same wire then all three beads would be on that one wire, which would contradict Axiom $3$. Thus th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3393765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differential equation $x'(t) e^{-x'(t)^2} = c$ with Lambert W function. Let $x(t)$ be a smooth function, find a solution of $x'(t) e^{-x'(t)^2} =c$.
I first saw this DE in a question of
Frederic Chopin (Integral involving piecewise continuous function), when I started working on the problem I consulted Wolfram Alpha.
W... | Square the equation and multiply with $-2$,
$$
(-2x'^2)e^{-2x'^2}=-2c^2.
$$
Then apply Lambert-W as the inverse of the function $ve^v=u$,
$$
-2x'^2=W(-2c^2).
$$
Now select one of the square roots so that the sign of $x'$ is the sign of $c$ and integrate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3393867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Defective Chips, Why not use nCk? Below is an example that I understand the solution to, but not why the problem should be approached using percentages versus n chose k.
Example 1 from Khan Academy
A manufacturer of processing chips knows that 2%, percent of its chips are defective in some way.
Suppose an inspector r... | Let $X$ be the number of defective chips among the four.
\begin{align}
\Pr(X\ge1) = {} & \Pr(X=1) + \Pr(X=2) + \Pr(X=3) + \Pr(X=4) \\[12pt]
= {} & {}_4C_1 \,\,0.98^1(1-0.98)^3 + {}_4C_2\,\,0.98^2(1-0.98)^2 \\[2pt]
& {} + {}_4C_3 \,\,0.98^3(1-0.98)^1 + {}_4C_4 \,\,0.98^4 \\[12pt]
= {} & 4\cdot0.98^1(1-0.98)^3 + 6\cdot0.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\sqrt{3-\sqrt{3}} \in L = \mathbb{Q}(\sqrt{3+\sqrt{3}})$? Is $\sqrt{3-\sqrt{3}} \in L = \mathbb{Q}(\sqrt{3+\sqrt{3}})$?
I know that $\frac{1}{\sqrt{3+\sqrt{3}}} = \frac{\sqrt{3-\sqrt{3}}}{\sqrt6}$. So I just need to know whether $\sqrt6 \in L$. Since $\sqrt 3 = (\sqrt{3+\sqrt{3}})^2 - 3$, I only need to know if $\s... | Use the tracial method, for example. The trace map on a number field $Q(\alpha)$ assigns to each $\beta \in \mathbb Q(\alpha)$ the quantity $\sum_{\sigma} \sigma(\beta)$ where $\sigma(\beta)$ are all the conjugates of $\beta$.
Suppose that $\sqrt 2 \in L$. Then $\sqrt 3 \in L$ implies that $\mathbb Q(\sqrt{3+\sqrt 3}) ... | {
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"url": "https://math.stackexchange.com/questions/3394344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solve $x\equiv 1\bmod2, x \equiv 1\bmod5$ and $x \equiv 0\bmod3$ $$x\equiv 1\mod2\\ x \equiv 1\mod5\\x \equiv 0\mod3$$
Somehow I got the wrong solution
Here's how I got them
$b_i$ | $N_i$ | inverse| Product
2 | 20 | 4 |160
2 | 12 | 3 | 72
0 | 15 | 3 | 0
Sum of products is 232 then you modulus by 2*3*5=30 which... | Why modulo $60$? The least common multiple is $2\cdot3\cdot5=30$.
Indeed, $x=5a+1$; from $5a+1\equiv1\pmod{2}$ we deduce $a=2b$; then
$$
10b+1\equiv0\pmod{3}
$$
is the same as $b\equiv2\pmod{3}$, so $b=3c+2$. Recapitulating,
$$
x=5a+1=10b+1=30c+21
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Path connected subset $\mathbb{R}^2$ We all know that $\mathbb{R}^2$ is path connected. Consider the subset $S=\{(x,y):\mid \mid (x,y)\mid \mid \geq c, \text{for fixed $c$ in $\mathbb{R}$}\}.$ It can be geometrically visualize that $S$ is path connected.
But I need a precise path between any two points of $S$. I think ... | Suppose that the points are given by $x=r_1 e^{i\theta_1},y=r_2 e^{i\theta_2}$. Then a path can be given by
$$\gamma(t)=((1-t)r_1+tr_2)e^{i((1-t)\theta_1+t\theta_2)}$$
for $t\in[0,1]$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tensor deviator calculation rules In the field of continuum/solid mechanics, there are often deviatoric tensors defined, like for the derivation (comma in einstein notation) of a displacement
$$\mathrm{dev}(u_{i,j})=u_{i,j}-\frac{1}{3}\,u_{k,k}\,\delta_{ij},$$ where $\delta_{ij}$ is the Kronecker delta.
If one performs... | $\require{cancel}$
First note that
$$
\delta_{ii} = n \tag{1}
$$
where $n = 3$ is the dimension of your tensors. With this in mind
\begin{eqnarray}
\sigma_{ij}~{\rm dev}(u_{ij}) &=& \sigma_{ij} \left(u_{ij} - \frac{1}{n}u_{kk}\delta_{ij} \right) \\
&=& \sigma_{ij}u_{ij} - \frac{1}{n} \sigma_{ij}u_{kk} \delta_{ij} \\
&=... | {
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A $3$-$4$-$5$ right $\triangle ABC$ ($AC=5$) fits in a square such that $A$ is also vertex of the square. Find the side of square.
A right angle triangle ABC of sides $3$, $4$, and $5$ ($AC=5$) is fit in a square such that $A$ is also vertex of square . Find the side of square.
Some error is occurring while uploading... | If $A$ is a vertex of the square; and $B$ in on a side of the square not adjacent to $A$; and $C$ is on the other side not adjacent to $A$.
And if the side of the square in $s$ we can, using cartesian coordinates, assume $A$ is at $(0,0)$ and $B$ is at $(s,y)$ and $C$ is at $(x, s)$.
Then we have $s^2 + y^2 = 4^2$ and ... | {
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Differential of a Lipschitz map is Borelian Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz map. We say that $f$ is differentiable in $x \in \mathbb R^n$ in the Frechet sense if exists a linear operator $A: \mathbb R^n \to \mathbb R$ such that
$$ \lim_{h \to 0} \frac{f(x+h) - f(x) - A\cdot h}{\|h\|} = 0 \quad (I) $$
... | Partial answer.
For $n=1$
$$g_n(x):= \frac{f(x+\frac{1}{n})-f(x)}{\frac{1}{n}}$$
So $f'(x)=\lim_ng_n(x)$
Since $g_n$ are Borel measurable,then $\limsup_ng_n,\liminf_ng_n$ are Borel measurable.
So $D_f=\{x:\limsup_ng_n(x)=\liminf_ng_n(x)\}$ is Borel measurable.
Also $f':D_f \to \Bbb{R}$ is Borel measurable as a pointw... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Poisson distribution, defining lambda I'm currently working on the following exercise:
The number of hits, X, per baseball game, has a Poisson distribution.
If the probability of a no-hit game is $\frac13$ , what is the probability of
having 2 or more hits in specified game?
If I understood correctly the lambda r... | Note you have two cases when lambda is equal to two or equal to three. Start from there, you have two cases I guess
| {
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How did Rudin come up with this exact expression? To show that for every rational $p>0$ such that $p^2<2$ one can get a rational $q$ with $p^2<q^2<2$ and $p<q$ Rudin writes,
We now examine this situation a little more closely. Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of al... | It's always seemed easier to me, as a matter of intuition, to find $n\in \mathbb N$ such that $(p+1/n)^2<2.$ This is the same as making $(2p)/n+1/n^2<2-p^2.$ And because $1/n^2\le 1/n,$ it suffices to find $n$ such that
$$(2p)/n+1/n= (2p+1)/n<2-p^2.$$ Now you're just a manipulation away from applying the Archimedian pr... | {
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The proof of $(n+1)!(n+2)!$ divides $(2n+2)!$ for any positive integer $n$
Does $(n+1)!(n+2)!$ divide $(2n+2)!$ for any positive integer $n$?
I tried to prove this when I was trying to prove the fact that ${P_n}^4$ divides $P_{2n}$ where $n$ is a positive integer, where $P_{n}$ means the multiplication of all $k!$ fr... | Although the expression $\frac{(2n+2)!}{(n+1)!(n+2)!}$ is not quite a binomial coefficient, it can be expressed as the difference of two binomial coefficients (and so it is in fact an integer):
$$ \frac{(2n+2)!}{(n+1)!(n+2)!} = \frac{1}{n+2} \binom{2n+2}{n+1} = \binom{2n+2}{n+1} - \binom{2n+2}{n+2} $$
This restates tha... | {
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if $A \subseteq B$, then $A \cap C \subseteq B\cap C$ So I tried doing this problem myself, and the answer that I got seems right, yet at the same time I feel like the way I did it is kind of.... wonky? It seems weird basically, and I was hoping someone can help me validate my answer
proof:
Suppose $x \in A \subseteq... | The "$x$" you are referring to in lines 1 and 2 are a different "$x$" than you have in the rest of the proof. And don't care about the $x\in A$ specifically but just that it leads to a general conclusion that we will use for the later $x$.
If I were to edit your proof but leave your thought process and pacing completl... | {
"language": "en",
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Tangent space of $SL(2)$ at $A$ Consider $SL(2)=\{A \in \mathbb R^{2 \times 2}|\det A=1\}$.
I want to determine the tangent space of $SL(2)$ at $A \in SL(2)$. Let's call it $T_A$.
$A=\begin{pmatrix}a_1 && a_2 \\ a_3 && a_4\end{pmatrix}$.
$SL(2)=\{A \in \mathbb R^{2 \times 2}|F(A)=0\}$,
where $F:\mathbb R^{2 \times 2}\t... | That's one way to do this. That said, there are a couple possible improvements.
First one : $F$ is a function from $\mathbb{R}^{2 \times 2} \to \mathbb{R}$. The derivative a real function of multiple real variables is not a vector, but a covector (or a row-vector).
Here, you are, in order :
*
*working with the gradi... | {
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"source": "stackexchange",
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Probability: random elevator stops I have a elevator probability problem but with a twist (instead of people exiting the elevator, it is the elevator stopping on each level). Require some help understanding and completing the probability questions.
Problem:
4 people go into the elevator of a 5-storey shophouse. Assume ... | You did not specify the events $X_a$ and $X_b$, but from the context I assume it is meant that
$$ X_a = 1\{\text{elevator stops at floor } a\}. $$
You have to find $\mathbb{E}X_aX_b$, which is equal to the probability that the elevator stops at both floor $a$ and $b$. In the question it is only stated that $a,b \in \{1... | {
"language": "en",
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"source": "stackexchange",
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Determinant of a circulant matrix I am interested in the calculation of the determinant of a $N\times N$ matrix with the following shape (for $N=5$)
\begin{equation}
A = \begin{pmatrix}
a & b & 0 & 0 & b \\
b & a & b & 0 & 0 \\
0 & b & a & b & 0 \\
0 & 0 & b & a & b \\
b & 0 & 0 & b & a \\
\end{pmatrix}
\end{equation}
... | Following the formula from the link in the comment, we have the following. Let $\omega$ denote an $N$th root of unity, then
$$
\det(A) = \prod_{j=0}^{N-1} [a + b(\omega^j + \omega^{-j})] =
\prod_{j=0}^{N-1} [a + 2b\cos(2\pi j/N)].
$$
Hopefully this is sufficient for your purposes.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove inequality $|y \ln{y} - x \ln{x}| < 2 |\ln{\frac{1}{|y-x|}}|$ when $x,y \in (0,1]$, $x \neq y$. EDIT: Counter-example found. Statement is FALSE.
However, I think the argument still has value. It is true if you restrict the domain to $[0.223,0.716]$. Maybe $[\frac{3}{10},\frac{7}{10}]$ so it’s less “obvious”?
Prov... | It's wrong. Try $x=1$ and $y=0.05.$
In this case $RHS-LHS=-0.047...<0.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Quick question on an approximation from physics I've seen written in a physics problem sheet that for a small $\epsilon>0$, the following approximation is considered
$$
(1+2\epsilon)^{-\frac12}\approx 1-\epsilon.
$$
Any reason why this is the case? Is this any special approximation in physics?
| This and many other cases are covered by Newton's Generalized Binomial Theorem. Here, we have $$(1+2\epsilon)^{-\frac12} = \displaystyle\sum_{i=0}^\infty \binom {-\frac 12}i (2\epsilon)^i$$
And of course, $\epsilon^i~\forall~i>0$ is very small in comparison to the rest of the sum so we omit those terms. So, our sum app... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Parametrize $x^8-x^3-x=y^8-y^3-y$ I was playing around earlier with the function $f(x)=x^3-x$. I decided that its inverse would be way too messy for me to try to calculate explicitly, but I became interested in the distinct real values $x,y$ for which $f(x)=f(y)$. I noticed that by assuming $y=kx$ I could simplify this... | For polynomials of order $n$, I guess the trick you used is this:
$$
f(x)=p(x)=\prod_1^n (x-p_i)
$$
and for the requested equation $f(y)=f(x)$, this reduces to:
$$
\prod_{i=1}^n {(y-p_i) \over (x-p_i)}=1
$$
If we introduce $n-1$ auxiliar variables, $k_1,...,k_{n-2},k$:
$$
y=k_i(x-p_i)+p_i, i:1...n-1\\
y={1 \over \prod_... | {
"language": "en",
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What is the sum of $k^2(n - k)$ for $k = 1$ to $k = n$? What is the value of $\sum\limits_{k=1}^n k^2(n - k)$?
The problem I had was:
for a square grid of size $n \times n$ how many squares have their corners on the intersecting points of the grid. (There are $n \times n$ points, the square's length is $n - 1$).
For e... | \begin{align*}
\sum_{k=1}^n k^2(n-k)&=n\sum_{k=1}^n k^2-\sum_{k=1}^n k^3 \\[5pt]
&=n\left(\frac{n}{6}(n+1)(2n+1)\right)-\frac{n^2}{4}(n+1)^2\\[5pt]
&= \frac{n^2}{12}(n^2-1),
\end{align*}
by linearity of summation and the formulae for $\sum k^2$ and $\sum k^3$.
| {
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Indefinite integral $\int \frac{1}{2-\cos(x)}\,dx$ has discontinuities. How to fix? Using the standard tangent half-angle substitution, we get $$\int \frac{1}{2-\cos(x)}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}(\sqrt{3}\tan\frac{x}{2}) + C$$
The resulting antiderivatives are piecewise continuous functions with discontinuities... | Your problem is the $+C$ term. Take for example $\frac{1}{x}$. Its most general antiderivative is usually given as
$$\int \frac{1}{x}dx = \ln|x| + C$$
But this is not correct. Notice that the piecewise function
$$f(x)=\begin{cases} \ln(-x) +2 & x < 0\\ \ln(x) -1 & x> 0\\ \end{cases}$$
is also an antiderivative of $\fr... | {
"language": "en",
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if $f \geq 0$ Lebesgue measurable the set $A := \{(x,y)\in \mathbb{R}^2 \mid 0 < y < f(x)\}$ is measurable - proof verification Let $f : \mathbb{R} \to \mathbb{R}^+$ be a positive, Lebesgue measurable function.
Denote the set $A := \{(x,y)\in \mathbb{R}^2 \mid 0 < y < f(x)\}$ .
Show that $A$ is Lebesgue measurable.
Is... | You can write
$$\{ 0 < y < f(x)\} = \{y>0\}\cap\{f(x)-y>0\} = \{y>0\}\cap P^{-1}(0,\infty)$$
where $$P:\mathbb R^2 \to \mathbb R , \quad P(x,y) = f(x) - y.$$
$P$ is measurable, so the result follows.
Closer to your attempt (along the lines of Jake's comments)- For $q\in\mathbb Q_+$, let $$A_q = B_q \times (0,q), \quad... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does $\lim_{s\to \infty}F(s)=0$ for all Laplace transforms? Let $f(t)$ be a piece-wise continous function of exponential order $\alpha$.
Then $F(s)$ exists.
I must prove then that $\lim_{s\to\infty} F(s)=0$ but i have no idea on how to do it.
I tried to prove it by the $\varepsilon,\delta$ definition of limits, using t... | If
$$|f(t)|\lt Me^{\alpha t}$$
then:
$$F(s)=\int_{0}^{\infty} |f(t)|e^{-st} \, dt < \int_0^\infty Me^{-st+\alpha t} \, dt$$
thus:
$$\lim_{s\to \infty}F(s)=\lim_{s\to \infty} \int_0^\infty |f(t)|e^{-st} \, dt<\lim_{s\to \infty} \int_0^\infty M e^{-st+\alpha t} \, dt=0$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is a function with a countable set of discontinuities, Riemann Stieltjes integrable? We all know that a function $f:[a,b]\to \mathbb{R}$ such that it has only finitely many discontinuities and $\alpha$ is a monotonic increasing function Then $f$ is Riemann Stieltjes integrable.
My question is that can we replace this... | The first statement is incorrect. If $f$ and $\alpha$ are both discontinuous from the right or both discontinuous from the left at even a single point, then the Riemann-Stieltjes integral does not exist. A proof of this is given below.
On the other hand, as long as there are no such shared discontinuities, then the int... | {
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Solve heat equation given conditions
I don't quite understand what I am supposed to do with the last given condition $u(x,0)$.
I solved for all the different cases of $λ$ and got that
$$u(x,t) = e^{-3n^2π^2t}(Bsin(nπx)) $$
where $B≠0$
From here I get confused. In our answer key it simply says:
In order to satisfy t... | According to my calculation,
$u(x,t) = \sum^{\infty}_{n=1}a_n cos(n\pi x)e^{-3(n\pi)^2t}$
Using the given initial condition, we can know that
When $n = 3 \rightarrow a_3 = 2$
and $n=5 \rightarrow a_5 = 4$. and $a_n = 0$ otherwise.
Therefore, $u(x,t) = 2\cos(3\pi x)e^{-3(9\pi^2)t} \ + 4\cos(5\pi x)e^{-3(25 \pi^2)t}$... | {
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"timestamp": "2023-03-29T00:00:00",
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Combinatorics question about six letter sequences with repetition The question I'm trying to answer is as follows:
"How many six-letter “words” (sequences of letters with repetition) are there in which the first and last letter are vowels? In which vowels appear only (if at all) as the first and last letter?"
For the f... | For the first case I agree with your solution, we have
*
*$5^2\cdot 26^4$
For the second case we have
*
*$26^2\cdot 21^4$
there is not double counting because repetition are allowed.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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True or False: If the product of n elements of a group is the identity element, it remains so no matter in what order the terms are multiplied. I am working my way through Charles Pinter's book: A Book of Abstract Algebra. From recommendations on this site, I found a page/web address on Wisconsin University's Math Depa... | Since this is a true or false question, it is not that the question is phrased incorrectly, but rather that the answer is that it is false.
Your claim that it can only hold if the group is abelian is not true for all such $a, b, c$, which we can see in any group by $a=b=c=e$ and other less trivial examples. What you n... | {
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Existence of Right Angle in Hilbert Axioms Hilbert, in Foundations of Geometry briefly mentions that the existence of right angles is a corollary to the supplementary angle theorem. (i.e. If two angles are congruent, then their supplementary angles are congruent).
How does existence of right angles follow from this?
| Well, it's not an immediate conclusion from this fact. However, this fact is used in the proof. The proof goes as follows:
Take a line $L$ and a point $p$ not lying on $L$. Next take a point $a\in L$ and choose a ray $A$ with origin $a$ which is contained in $L$. Let $P:=\overrightarrow{ap}$ and let $M$ be a halfplane ... | {
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What does the semicolon (";") mean in "$a\in(2;3)$"? I'm working on a linear program and I have the following constraint:
I'm wondering what does the ";" mean? At first I thought it meant the variable $a$ can only be $2$ or $3$, but that's what $(2, 3)$ is for, right?
| The open interval of numbers between $a$ and $b$ is often denoted as $(a,b)$. However, in some countries where comma $(,)$ is used as decimal points, a semicolon $(;)$ may be used in place of a comma as a separator to avoid ambiguity: for example, the open interval from $0$ to $1$ would be written as $(0;1)$.
In the ex... | {
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Reccurence Relation for columns of Pascal's triangle I am given the recurrence equation $$F_k(n) = \sum_{j = 0}^{k - 1}(-1)^j\binom{k}{j + 1}F_k(n - j - 1)$$ and asked to solve it. Here k is a positive integer and $F_k(k - 1) = 1$ and $F_k(n) = 0$, $n < k -1$ I have a conjecture that the solution is $$F_k(n) = \frac{(... | The result can be derived using generating functions. The recurrence can be written as
$$F_k(n)=\sum_{j=1}^{k}(-1)^{j-1}\binom{k}{j}F_k(n-j)$$
(for a bit nicer look) and is valid for $n\geqslant k$. Now suppose $k$ is fixed, and let $F(x)=\sum\limits_{n=0}^{\infty}F_k(n)x^n$. Then, multiplying the recurrence by $x^n$ a... | {
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More infinite nested square roots Let us consider the following nested square root:
$$A=\sqrt{b_1+\sqrt{b_2+ \sqrt{b_3+\cdots}} }.$$
Consider the following three cases:
*
*$b_k = k^2 + k -1 \Rightarrow A = 2.$
*$b_k = (k+2)^4 - 4(k+2)^3+5(k+2)^2-4(k+2)+1 \Rightarrow A = 4.$
*$b_k = 4^{k-1} \Rightarrow A = 2.$
The ... | For the first case, we can start with $\:2 = \sqrt{1+\sqrt{9}} = \sqrt{(1^2+1-1)+\sqrt{(2+1)^2}}\:$. Then we repeatedly apply the identity
$$(k+1)^2 = (k^2+k-1) + (k+2) = (k^2+k-1) + \sqrt{(k+2)^2}$$
So we get
$$\begin{align}
2 = \sqrt{1+\sqrt{9}} &= \sqrt{(1^2+1-1)+\sqrt{(2+1)^2}}\\[1.5ex]
&= \sqrt{(1^2+1-1)+\sqrt{2^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3398472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is $\sqrt{x^2}$ even or odd? Is the function $x\mapsto \sqrt{x^2}$ even or odd?
Mentioning the square root does not have negative sign,
$\sqrt{x^2} = \pm x$
As it is clear LHS is even and RHS is odd for both sign, which one is true?
| It’s the same thing as the absolute value of X, which is even. Why? Because squaring and then square rooting any real quantity keeps it the same but makes positive (from when we squared it).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3398626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Example of smooth $f: \mathbb C \to \mathbb C$ holomorphic on annulus with non-vanishing line integral Is it possible to construct a smooth (when viewed as a map on $\mathbb R^2$) map $f: \mathbb C \to \mathbb C$ such that $f$ is holomorphic on an annulus $r < |z| < R$ however the contour integral $\int_{|z| = \epsilon... | It is very possible to take the meromorphic function $f(z) = \frac1z$, and then change it only on the unit disc to become smooth as an $\Bbb R^2\to \Bbb R^2$ function in the entire complex plane. It will then be analytic on, say, the annulus $1<|z|<2$ and still have non-vanishing contour integral on that annulus.
More ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3398802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
upper limit of a sequence of sets Let $\langle A_n:n\in\Bbb N\rangle$ be a sequence in $X$,$\lim\sup A_n=\bigcap_{n\in N}\bigcup_{k>n} A_k,$,can we conclude that $A_n\subset lim\sup A_n$ when n tends to $\infty$? If not,what,s the relationship between $A_n$ and $lim\sup A_n$ when $n$ is large enough.
| Put $A_{2n}:= \{0,n\}$ and $A_{2n+1}:=\{n\}$. Then $\limsup A_n = \{0\}$ but there is no relation between $A_n$ and $\limsup A_n$ in the sense of inclusion. In $\limsup A_n$ you get exactly those elements which are contained in infinitely many $A_n$'s so you have enough freedom to destroy any strict relation in this di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3398976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The value which is not possible for $\sum_{k=1}^n k^3$ is
a)25
b)36
c)225
d)441
The value becomes $$\left[\frac{(n)(n+1)}{2}\right]^2$$
Since all options are perfect squares, I don’t really know what to do with This.
The discrimaint ie. $b^2-4ac>0$ for all options if we solve the quadratic equation for each option.
T... | Let
$$\left[\frac{(n)(n+1)}{2}\right]^2=x$$
It is quadratic in $n$.
$$n^2+n-2\sqrt{x}=0$$
The discriminant $$1+4*1*2\sqrt{x}=1+8\sqrt{x}$$ should be a perfect square because $n$ must be whole number.
So, number need not only be a square number. It should satisfy the above condition as well.
$25$ doesn't satisfy it and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Write the expression in terms of $\sin$ only $\sin(4x)-\cos(4x)$ I am currently taking a Precalc II (Trig) course in college. There is a question in the book that I can't figure out how to complete it. The question follows:
Write the expression in terms of sine only:
$\sin(4x)-\cos(4x)$
So far I have $A\sin(x)+B\cos(x)... | Use that
$$\cos(4x)=1-2\sin^2(2x)=1-2(2\sin x \cos x)^2=1-8\sin^2 x(1-\sin^2x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find all possible real 2x2 matrices A I am trying to find all possible real $2×2$ matrices $A$ such that $A^{2019} = \begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}$
I am really stuck with this one, could someone provide some insight to help me solve it? Thanks!
| We may identify $$ \begin {bmatrix}a&b\\-b&a\end{bmatrix}$$ with the complex number $$a+bi$$
Thus the matrix $$ \begin {bmatrix}0&1\\-1&0\end{bmatrix}$$ will be identified with $i$
The problem is finding all $2019$ complex numbers $z$ such that $$z^{2019}=i=e^{i \pi /2 }$$
These are $$\cos (\theta_k ) + i \sin (\thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can a quadratic involving complex inputs be solved by the quadratic formula as well? If a quadratic equation $\gamma(z) = az^2 + bz + c = 0,$ where $a,b,c \in \mathbb{R},$ takes as input values from the complex plane, may one use the quadratic formula to solve the quadratic equation? Won't these solutions necessarily i... | For your particular problem, if $a, b, c,$ and $d$ are all real numbers, there are three possibilities:
*
*The expression under the square root sign, $(d-a)^2 + 4cb,$ is a positive real number. Then you have two solutions for $z,$ both of which have zero imaginary component.
*The expression under the square root si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
"paths" on graphs as topological paths? Disclaimer: have not studied topology yet, currently using Wikipedia to scrap by
It just seems to me that there should be some way to connect paths in graph theory to paths (I am interested only in undirected simple graphs if it ever matters) in topology. Fix a path in some graph... | You can embed any (finite) graph (simple, undirected) into $\Bbb R^3$ faithfully: vertices get mapped to points in $3$-space and every edge is a homeomorpic copy of $[0,1]$, where distinct edges only can meet at endpoints/vertices.
In such a representation of a graph $G$ as a subspace $\hat{G}$ of $\Bbb R^3$, there is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Factor $x^{35}+x^{19}+x^{17}-x^2+1$ I tried to factor $x^{35}+x^{19}+x^{17}-x^2+1$ and I can see that $\omega$ and $\omega^2$ are two conjugate roots of $x^{35}+x^{19}+x^{17}-x^2+1$. So I divide it by $x^2+x+1$ and the factorization comes to the following
$$(x^2+x+1)(x^{33}-x^{32}+x^{30}-x^{29}+x^{27}-x^{26}+x^{24}-x^{... | Can one find the factors by hand? Well, perhaps with a bit of guessing.
In the big factor
$$\begin{align}p_{33}(x)&=x^{33}-x^{32}\\&+x^{30}-x^{29}\\&+x^{27}-x^{26}\\&+x^{24}-x^{23}\\&+x^{21}-x^{20}\\&+x^{18}\\&-x^{16}+2x^{15}-x^{14}\\&-x^{13}+2x^{12}-x^{11}\\&-x^{10}+2x^{9}-x^{8}\\&-x^{7}+2x^6-x^5\\&-x^4+2x^3-x^2\\&-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3400035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is infiniteness of $K$ essential for the isomorphism between coordinate rings and polynomial functions? Let $S$ be an algebraic sets in n-dimensional affine space, $K$ an infinite field, an exercise in Aluffi’s Algebra:Chapter 0 (page 414, ex 2.12) asks a proof that there is an isomorphism of $K$- algebra between the p... | A very simple counterexample: on the prime field of characteristic $p$, the Frobenius map $\;\mathbf F_p\longrightarrow\mathbf F_p$, $x\longmapsto x^p$ is equal to the identity. Yet, in the polynomial ring $\mathbf F_p[X]$, $X^p\ne X$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3400185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Derivative of cumulative distribution function Lets say f is probability funktion to cdf F.
I want to compute $$lim_{ \Delta t \rightarrow 0+}\frac{ P( t<X<t +\Delta t )}{\Delta t} $$
I should get $f(t) $. How do I get to that?
| $$ \lim_{ \Delta t \rightarrow 0+}\frac{ P( t<X<t +\Delta t )}{\Delta t} = \lim_{ \Delta t \rightarrow 0+}\frac{ P(X < t + \Delta t) - P ( X \leq t )}{\Delta t} $$
$$ = \lim_{ \Delta t \rightarrow 0+}\frac{ F( t + \Delta t ) - F( t )}{\Delta t} = F'(t) = f(t) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3400325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find a parameter $m$ of a second order polynomial for which the polynomial is injective on a given interval. I am given the function:
$f : \mathbb{R} \rightarrow \mathbb{R}$
$ f(x)=x^2-mx+2$
$ m \in \mathbb{R}$
And I am asked to find $m \in \mathbb{R}$ for which the function is injective on the interval $[-1, 1]$. Here... | Tthe position of the max/min of the quadratic $f(x)=Ax^2+Bx+C$ is $x_0=-\frac{B}{2A}$. If The function has to be one to one (injective) fo $x \in [a,b]$ it has to either monotonically increase or decrease in $[a,b]$, then $x_0=\le a ~or~ x_0 \ge b.$
So here $$x_0=-m/2 \le -1 ~or~ x_0=-m/2 \ge 1 \implies m \in (-\infty,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3400466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Natural map from $\mathbb{X}$ to $\mathbb{X}/\mathbb{M}$, that is not closed I know that by considering projection $q : \mathbb{R}^2 \to \mathbb{R}$, $(x, y) \to x$, and the closed subset
$$G = \left\{(x, y) : y \ge \frac 1 x, x > 0\right\}$$
will prove that $q$ is not a closed map.
But I'm having some difficulty in... | The term 'projection' for the natural quotient map $X\to X/M$ is rather illustrative.
However, the projection $q:\Bbb R^2\to \Bbb R$ can also be viewed as a quotient map, namely take $X=\Bbb R^2 $ (as a vector space), and its subspace $M=\{(0,y):y\in\Bbb R\}$.
Then, identifying the quotient $\Bbb R^2 /M$ with $\Bbb R$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3400766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is $\le$ a binary operation? I am a little confused with $\le$. Is it a binary operation?
I know the definition of binary operation that is any function from $A×A \rightarrow A$ and $A$ non empty.
Kindly help me to understand the concept.
| A binary operation on a set $A$ is a function $A \times A \to A$. You need to be able to plug any elements of $A$ into the function and get an element of $A$ back. Addition on the naturals is a binary operation. You can take any two naturals, add them, and get a natural as the result.
A relation on $A$ is a set of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3400901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Sub-module of a Tensor Product of ring and module. Suppose $R$ is ring with unit and $M$ be unital $R$-module and $N$ be sub-module of $M$. Also let $S$ be a ring containing R as subring and $1_S=1_R$. I have two questions.
$1.$ Can we claim that $S\otimes_R N$ is sub-module of $S\otimes_R M$ ?
I think this should no... | Let $S=k[x,y]/(xy)$ and $R=k[x]/(xy)\subseteq S$. Now, $R$ is an integral domain, so we can set $M$ to be its field of fractions, and $N=R\subseteq M$. Then $S\otimes_R N=S$, but $$(y,1)=\left(y,\frac{x}{x}\right)=\left(xy,\frac{1}{x}\right)=\left(0,\frac{1}{x}\right)=(0,0)$$ in $S\otimes_R M$, and so the inclusion is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $(X,d)$ is a metric space, is $X^{m+1}\to\{1,...,m\}, (x,x_1,...,x_m)\mapsto\min(\operatorname{argmin}_{k\in\{1,...,m\}}d(x,x_k))$ measurable? Given a a metric space $(X,d)$ and $m\in\mathbb{N}$, equip $X\times X^m$ with the product Borel $\sigma$-algebra and $\{1,...,m\}$ with the $\sigma$-algebra $2^{\{1,...,m\}}$... | Yes. Denoting the map by $f$, we have
\begin{align*}
\{f(x,x_1,\cdots,x_m) = k\}
&= \left( \bigcap_{i < k} \{ d(x,x_i) > d(x, x_k) \} \right) \cap \left( \bigcap_{i > k} \{ d(x,x_i) \geq d(x, x_k) \} \right).
\end{align*}
Since each function $(x,x_1,\cdots,x_m) \mapsto d(x, x_i) - d(x, x_k)$ is continuous, $\{ d(x,x_i)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Doubt about open sets in the Product topology So we know that in the product topology a basis element is gonna be of the form $\prod U_\alpha$ where $ U_\alpha \neq X_\alpha $ for a finite number of $\alpha$. So my thing is that we know that a product of closed sets in the product topology is gonna be closed so why is... | Because $\prod_\alpha X_\alpha\setminus\prod_\alpha F_\alpha\neq\prod_\alpha(X_\alpha\setminus F_\alpha)$. In fact, $\prod_\alpha X_\alpha\setminus\prod_\alpha F_\alpha$ is the union of all the products $\prod_\alpha Y_\alpha$ where $Y_\alpha=X_\alpha\setminus F_\alpha$ for one specific $\alpha$, whereas for all other ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Investigate the continuity of a function I'm trying to determine whether the following function is continuous:
$y=\frac{1}{\sqrt{1+x}-\sqrt{1-x}}$. Imo it is continuous, because it is a composition of two continuous functions $y=\frac{1}{x}$ and $y=\sqrt{1+x}-\sqrt{1-x}$.
So my answer is, that the function is continuo... | You are correct but a more detailed answer would explain the domain as well.
The domain is $$[-\sqrt 2 /2,0)\cup (0, \sqrt 2/2]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving $\lim_{x\to 4} \left(\frac{\sqrt {2x-1}}{\sqrt {x-3}}\right) = \sqrt 7$ using $ \varepsilon - \delta$
Prove that $$\lim_{x\to 4} \left(\frac{\sqrt {2x-1}}{\sqrt {x-3}}\right) = \sqrt 7$$ using $\varepsilon - \delta$.
We find $\delta$ such that
$0<|x-4| <\delta$
$$\left|\frac{\sqrt {2x-1}}{\sqrt {x-3}}-\sqrt 7... | We want to show that $\lim\limits_{x\to 4} \dfrac{\sqrt{2x-1}}{\sqrt{x-3}}=\sqrt{7}$. By the $\epsilon-\delta$ limit definition, this means $\forall \epsilon >0,\exists \delta >0 \space(0<|x-4|<\delta \Rightarrow \left |\dfrac{\sqrt{2x-1}}{\sqrt{x-3}}-\sqrt{7} \right |<\epsilon)$. Simplifying, we obtain $\left |\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Show that there exist $\gamma\in[\alpha,\beta] $ such that $\int\limits_{E}f|g|=\gamma\int\limits_{E}|g|$
Suppose $f:E\rightarrow \mathbb{R}$ is measurable, $g:E\rightarrow R$
is Lebesgue integrable and there exists $\alpha,\beta\in\mathbb{R}$ such that
$\alpha\leq f(x)\leq\beta$ for almost every $x\in E$. Show th... | Just to clarify that if $\displaystyle\int|g|=0$, then $g=0$ a.e. and $\gamma\in[\alpha,\beta]$ can be taken as arbitrary.
If it were not, then $\gamma=\dfrac{\displaystyle\int f|g|}{\displaystyle\int|g|}$ would be a candidate. And we know that $(\cdot)\geq(\cdot\cdot)$ implies that $\displaystyle\int(\cdot)\geq\int(\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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