Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to describe a three dimensional range. Consider the function $\mathbf{f}:\mathbf{R}^2\rightarrow\mathbf{R}^3$ given by $\mathbf{f}(\mathbf{x})=A\mathbf{x}$, where $A=\begin{bmatrix}
2 & -1 \\
5 & 0 \\
-6 & 3
\end{bmatrix}$ and the vector $\mathbf{x}$ in $\mathbf{R}^2$ is written as... | Hint :
In such cases it is best to describe the range as the set {$(x,y,z) |$.. conditions on x,y and z}. In this particular case, can x,y, and z take all the values freely or are there any constraints ? . When you reduce the equation to RREF. you will probably see one constraint. Express it in terms of a free variab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Associated Primes of Square free monomial ideals Let $I$ be a square free monomial ideal in $R=k[x_1,\dots,x_d]$. If $I$ is the edge ideal of a graph $G$, then the associated primes of $I$ are known.
My question is:
If $I$ is generated by square free monomials, are the associated prime ideals always of the form $(I:a... | Yes, they are. The result holds for associated primes of graded modules (see Bruns and Herzog, Lemma 1.5.6(b)(ii)), and monomials are the homogeneous elements in some grading on $k[x_1,\dots,x_n]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Notation Concerning Improper Integral and the Absolute Value of an Integrand I have the following integral
$$\int_{- \infty}^\infty e^{-|x|} dx$$
Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negatives preimages, I believe the integral must be evaluated in the following manner
$$\lim \... | Notice, the given function $$f(x)=e^{-|x|}\implies f(-x)=e^{-|-x|}=e^{-|x|}=f(x)$$ hence $f(x)=e^{-|x|}$ is an even function
Now, using property of definite integral $\color{blue}{\int_{-a}^{a}f(x)dx=2\int_{0}^{\infty}f(x)dx}$, we get
$$\int_{-\infty}^{\infty}e^{-|x|}dx=2\int_{0}^{\infty}e^{-|x|}dx$$
Since, $|x|=x\ \fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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If $a | b$, prove that $\gcd(a,b)$=$|a|$.
If $a | b$, prove that $\gcd(a,b)$=$|a|$.
I tried to work backwards. If $\gcd(a,b)=|a|$, then I need to find integers $x$ and $y$ such that $|a|=xa+yb$. So if I set $x=1$ and $y=0$ (if $|a|=a$) or if I set $x=-1$ and $y=0$ (if $|a|=-a$), is this good enough?
| If $a|b,\ |a||b\implies |a|\le gcd(a,b)\le a\le |a|\implies |a|=gcd(a,b)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Inverse Laplace transform with branch cut For the purpose of my research on persistent random walks I need to compute the inverse Laplace transform of
$$ F(s)=\frac{\mathrm e^{-b\sqrt{s^2-1}}}{s^2-1}.$$
I looked up in tables of integral transforms such as Erdélyi et al., Gradsteyn & Ryzhik, Prudnikov et al. with no suc... | I have a copy of G.E.Roberts and H.Kaufman "Table of Laplace Transforms", 1966.
On page 252, Item 3.2.55 is a more complex example but might apply if $v = 0$.
Inverse of $(s + (s^2 - a^2)^{1/2})^v \exp(-b\sqrt{s^2-a^2})/\sqrt{s^2 - a^2}$ is given as:
$0$ for $0 < t < b$
and $a^v ((t-b)/(t+b))^{v/2} I_v(a\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Group of order $432$ is not simple How do we prove that a group $G$ of order $432=2^4\cdot 3^3$ is not simple?
Here are my attempts:
Let $n_3$ be the number of Sylow 3-subgroups.
Then, by Sylow's Third Theorem, $n_3\mid 16$ and $n_3\equiv 1 \pmod 3$.
Thus, $n_3=1, 4$ or 16.
Next, let $Q_1$ and $Q_2$ be two distinct 3-s... | Let $G$ be a simple group of order $432$. Note that $|G|$ does not divide $8!$, thus no proper subgroup of $G$ has index at most $8$. Sylow’s Theorem forces $n_3(G) = 16 \not\equiv 1 \bmod 9$, so that there exist $P_3, Q_3 \in \mathsf{Syl}_3(G)$ such that $|N_G(P_3 \cap Q_3)|$ is divisible by $3^3$ and another prime. T... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Integral definition of derivative After seeing the integral definitions of div, grad, and curl, I'm left to wonder if we can define the regular derivative of a function $f: \Bbb R \to \Bbb R$ as the limit of an integral. For reference, here's the integral definition of grad
$$\operatorname{grad} f := \lim_{V\to 0} \fr... | No, it is not. By Lebesgue's differentiation theorem, for any $f\in L^1(\mathbb{R})$ and for almost every $x\in\mathbb{R}$ the following limit exists and it equals $f(x)$:
$$ \lim_{r\to 0}\frac{1}{2r}\int_{|t-x|<r}f(t)\,dt.$$
If $f$ is a continuous function it is absolutely continuous over any compact interval, hence:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Do these metrics determine the same Topology. Let $C([0,1])$ denote the set of all continuous real-valued functions on the interval $[0,1]$.
Given elements $f,g ∈ C([0,1])$, define
$d_{\infty}(f,g) = \displaystyle{\text{sup}_{x\in [0,1]}}| f(x)−g(x)|$ and
$\displaystyle{d_1(f,g) = \int_0^1 | f(x)−g(x)|dx}.$
(a) Show t... | Hint: For the last part, consider the sequence of functions $f_n$ that are zero everywhere except for a triangular bump going up to $1$ and then back down to zero between the endpoints $1/2^{n+1}$ and $1/2^{n}$. Is this sequence Cauchy in terms of your metric $d_1$, with a limit?. Is the sequence Cauchy according to $d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluating $\int \frac{\sin x}{\sin 4x}\mathrm dx$ $$\int \frac{\sin x}{\sin 4x}\mathrm dx$$
I have tried to do this by expanding $\sin 4x$ but it was worthless.
| Using $\displaystyle \bullet\; \sin 2x = 2\sin x\cdot \cos x$ and $\; \bullet\; \cos 2x = 1-2\sin^2 x$
Let $$\displaystyle I = \int\frac{\sin x}{2\sin 2x\cdot \cos 2x}dx = \int\frac{\sin x}{4\sin x\cos x\cdot \cos 2x}dx = \frac{1}{4}\int\frac{1}{\cos x\cdot \cos 2x}dx$$
Now $$\displaystyle I = \frac{1}{4}\int\frac{\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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solve the inequations and provide the solutions in brackets I have 2 inequations which I solve, but when I compare the result with websites like wolframalpha I get different results.
$|2-x|+|2+x| \leq 10$
I continue with $2-x+2-x\leq10$ OR $2-x+2-x\geq-10$
and I get the results $x\geq3$ and $x\leq7 $
$|6-4x|\geq|x-2|... | You have two critical points (points that evaluate to zero in the absolute values). These are $x=2$ and $x=-2$. Now you have three cases which you have to look at $x\in(-\infty,-2]$, $x\in (-2,2]$ and $x\in (2,\infty)$ [Note: these intervals are important because the expressions in the absolute value functions change s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Interesting $\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i$ I found that for m $\in N $
$$\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i.$$
I found it after doing an exercise.
For example: $$5^{2}-4^{2}+3^{2}-2^{2}+1^{2} = 1 + 2 + 3 + 4 + 5 = 15.$$
For me is the first time I saw this formula.
I think it is... | The proof is not that hard. Assume WLOG that $m$ is odd. Then
$$\begin{align}\sum_{i=0}^{m-1} (-1)^i (m-i)^2 &= \sum_{i=1}^m (-1)^{i+1} i^2 \\ &= 1^2-2^2+3^3-4^2 +\cdots+m^2\\&= 1^2+2^2+3^3+4^2 +\cdots+m^2-2 (2^2+4^2+\cdots(m-1)^2)\\&=\frac16 m (m+1)(2 m+1) - 2 \cdot 2^2 \frac16 \left (\frac{m-1}{2} \right )\left (\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimal primes of $\Bbb C[w,x,y,z]/(w^2x,xy,yz^2)$ and their annihilators I am examining the minimal primes of $R=\Bbb C[w,x,y,z]/(w^2x,xy,yz^2)$, intersections between them, and the annihilators of each minimal prime.
If a minimal prime does not contain x, then it must contain both w and y. If it contains x, then it ... | $$(w^2x,xy,yz^2)=(\underline{w}^2,xy,yz^2)\cap(\underline{x},xy,yz^2)=(w^2,\underline{x},yz^2)\cap(w^2,\underline{y},yz^2)\cap(x,yz^2)=(w^2,x,y)\cap(w^2,x,z^2)\cap(w^2,y)\cap(x,y)\cap(x,z^2)=(w^2,y)\cap(x,z^2)\cap(x,y)$$ is a reduced primary decomposition, so the associated (minimal) prime ideals are $(y,w),(x,z),(x,y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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An example of a representation which is simultaneously of real and quaternionic type Can anyone provide an example of a (complex) representation which is simultaneously of real (with a structure map $j^2 = 1$)and quaternionic (with a structure map $j^2 = -1$) type? As this cannot be true for irreducible representations... | Take any rep $V$ over $\Bbb R$ and consider $\Bbb H\otimes_{\Bbb R}V$ as a complex vector space.
Say $V$ is a representation of $G$ over $\Bbb R$. Assume we've picked a basis. Then $G$ acts by real matrices, and if we extend scalars (by tensoring with a larger ring of scalars) $G$ still acts by those real matrices.
Si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Varying Order of Covariant Differentiation Does the order of covariant differentiation matter?
Will E$_i$$_j$,$_k$$_l$$_t$ =E$_i$$_j$,$_l$$_t$$_k$ ?
Does it matter if the tensor E$_i$$_j$ is continuously differentiable?
| In general no, for example on a reimannian manifold where the christoffel symbols and its derivatives do not vanish there is a loss in commutativity:
$$\nabla_a \nabla_bT^c \ne \nabla_b \nabla_aT^c$$
infact:
$$\nabla_a \nabla_bT^c - \nabla_b \nabla_aT^c = R^{c}_{dab}T^d$$
Where $R$ is called the reimann christoffel ten... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Normal distribution with P(x=a) and P(x≥a)
Given: height of $1000$ students normally distributed with $\mu=174.5\,\mathrm{cm}$, $\sigma=6.9\,\mathrm{cm}$
Find: a. $P(x<160\,\mathrm{cm})$ b. $P(x=175\,\mathrm{cm})$ c. $P(x\geq188\,\mathrm{cm})$
For a., I used the formula $z=x-\mu/\sigma$, which gives $-2.10$. Looking ... | One of the properties of a continuous probability distribution like the normal distribution is that the probability of any individual outcome is zero. This is true even if it is possible for that outcome to happen.
For example, it is possible that the height of a student is exactly 175 cm, but if you model height with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that for each number $n \in \mathbb N$ is the sum of numbers $n, n + 1, n + 2, ..., 3n - 2$ equal to the second power of a natural number. I've got a homework from maths:
Prove that for each number $n \in \mathbb N$ is the sum of numbers
$n, n + 1, n + 2, ..., 3n - 2$ equal to the second power of a natural
... | By the trick that we know from Gauss we get that $$\sum_{i = 1}^s i = \frac{s(s+1)}{2}$$
If we fill in $s = 3n - 2$ and substract the sum up to $s = n-1$ we see that $$S_n = \sum_{i = n}^{3n - 2} i = \sum_{i = 1}^{3n - 2} i - \sum_{i = 1}^{n-1} i$$
so $$S_n = \frac{(3n - 2)(3n - 1)}{2} - \frac{(n-1)(n)}{2}$$ which giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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A metric space is complete when every closed and bounded subset of it is compact
$X$ is a metric space such that every closed and bounded subset of $X$ is compact . Then $X$ is complete.
Let $(x_{n})_{n}$ be a Cauchy sequence of $X$. Now every Cauchy sequence in a metric space is bounded. If $(x_{n})$ is convergent ... | Here is a modified version of the proof that I believe fills all the holes:
Consider a Cauchy sequence $(x_{k} : k)$ in $X$.
Let the limit points of this sequence be denoted $L$. I claim that $L$ is nonempty and in fact contains exactly one point. Then clearly the only point in $L$ is the limit, so it converges.
First ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Intuitive explanation of binomial coefficient formula Regarding the formula for binomial coefficients:
$$\binom{n}{k}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}$$
the professor described the formula as first choosing the $k$ objects from a group of $n$, where order matters, and then dividing by $k!$ to adjust for overcounting... | Suppose you had 5 books on a shelf and wanted to pick 3. If order mattered (permutations), there would be $5 \cdot 4 \cdot 3$ ways of doing so. If order did not matter (combinations), you would discount the extra permutations. That is, "ABC" would be the single representative of all of its $3!$ permutations. The total ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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choosing N object from set of M>N maximizing overall ratio value/weight I have a set of M objects each with a certain value and weight. From this set I want to take out N objects ($N<M$ of course) and maximize the ratio:
total value of the N objects / total weight of the N objects
It's a kind of an optimal selection pr... | Supponse $M = 2$ and you have one item with very large weight $w$ and very large value $v$, such that the ratio $v/w$ is maximal by a large amount for this item. Then if you are forced to take another item, usually your best bet will be to choose the item with a very small weight, so that the ratio is relatively unchan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $\left( {{\partial ^2 f}\over{\partial t\,\partial s}}\right)\bigg|_{t = s = 0} = ab - ba?$ This is a followup to my previous question here.
Let $a, b \in M_n(\mathbb{R})$. Consider the function$$f: \mathbb{R}^2 \to M_n(\mathbb{R}), \text{ }(t, s) \mapsto e^{t \cdot a} e^{s \cdot b} e^{-t \cdot a} e^{-s \cdot b}.$$I... | First, note that we do have the usual product rule $(fg)' = f'g + fg'$
for taking (partial) derivatives when products don't commute. I won't go through the details here, but the exact same proof that works for functions $\mathbb{R} \to \mathbb{R}$ goes through. Extending to multi-term products, we get:
\begin{align*}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Function to describe teardrop shape If I fill a plastic ziploc-shaped bag with water, the cross section profile should be sort of teardrop shaped (assuming we ignore the edge effects of the bag being sealed on the sides as well as the top and bottom). The bag should "sag"/get wider until to get the center of gravity as... | This involves physics of equilibrium to form an ODE which decides its shape.
Assumptions are: Bag is full, does not stretch, too long compared to height, force per unit length $N$ a.k.a. surface tension is proportional to height of water (of density $\gamma$) gravity load column above it, the hydraulic pressure is pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Properties of $f(x)=\ln(1+x^2)+x+2$ vs $g(x)=\cosh x+\sinh x$ This is from an MCQ contest.
Consider the two functions: $f(x)=\ln(1+x^2)+x+2$ et $g(x)=ch(x)+sh(x)$.
The real number $c$ such that: $(f^{-1})'(2)=g(c)$
*
*$1]$ $c=-1$
*$2]$ $c=0$
*$3]$ $c=1$
*$4]$ None of the above statements is correct
... | You were on the right track. Note that
$$\frac{df^{-1}(x)}{dx}=\left.\frac{y^2+1}{(y+1)^2}\right|_{y=f^{-1}(x)}$$
Now, when $f^{-1}(2)=y$, $\log (1+y^2)+y=0\implies y=0$. So, we have $f^{-1}(2)=0$ and this means that
$$\left.\frac{df^{-1}(x)}{dx}\right|_{x=2}=\left.\frac{y^2+1}{(y+1)^2}\right|_{y=f^{-1}(2)=0}=1$$
S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Solving $\lim _{x\to 1}\left(\frac{1-\sqrt[3]{4-3x}}{x-1}\right)$ $$\lim _{x\to 1}\left(\frac{1-\sqrt[3]{4-3x}}{x-1}\right)$$
So
$$\frac{1-\sqrt[3]{4-3x}}{x-1} \cdot \frac{1+\sqrt[3]{4-3x}}{1+\sqrt[3]{4-3x}}$$
Then
$$\frac{1-(4-3x)}{(x-1)(1+\sqrt[3]{4-3x})}$$
That's
$$\frac{3\cdot \color{red}{(x-1)}}{\color{red}{(x-1)}... | I think you overlooked this multiplication $(1+(4-3x)^{1/3})(1-(4-3x)^{1/3})$ which equals $1-(4-3x)^{2/3}$ not $1-(4-3x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
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A high-level reason that $u \cdot (v \times w) = (u \times v) \cdot w$? I can do the algebra to show that for $u, v, w \in \mathbb{R}^3$, this identity is true:
$$u \cdot (v \times w) = (u \times v) \cdot w$$
But is there a more high-level reason? I didn't expect the cross and dot product to be connected in this surpri... | Yes. The volume of the parallelepipid is equal to the scalar triple product. Consider the geometric definitions of the cross-product and inner-product to see this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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How to prove that n! have a higher order of growth than n^(log n)? I am aware that n^n have a higher order of growth than n!, but how about n^(log n)? Is there a way to get an alternative form of n^(log n) such that when taking the
lim n to infinity [alternative form(n^(log n))] / (n!)
it would equate to 0?
| Using a multiplicative variant of Gauss's trick we have:
$$
(n!)^2 = (1 \cdot n) (2 \cdot (n-1)) (3 \cdot (n-2)) \cdots
((n-2) \cdot 3) ((n-1) \cdot 2) (n \cdot 1)
\ge n^n
$$
So
$$
\dfrac{n^{\log n}}{n!}
\le
\dfrac{n^{\log n}}{n^{n/2}}
\le
\dfrac{n^{n/4}}{n^{n/2}}
=
\dfrac{1}{n^{n/4}}
\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this a proper algebraic proof that $Re(z)\leq\left|z\right|$? Given that $z=(x+iy)\in \mathbb{C}$, I'm supposed to show that $Re(z)\leq\left|z\right|$.
This is my attempt:
Note that $\sqrt{Re(z)} = \sqrt{x} \leq \sqrt{x^2} \leq \sqrt{x^2 + y^2}$.
Since $\sqrt{x^2} = x$, this shows that $x = Re(z) \leq \sqrt{x^2 + y... | I think it is easier to write the flow as
$$|z|=\sqrt{x^2+y^2}\ge \sqrt{x^2}=|x|=\left|\text{Re}(z)\right|\ge \text{Re}(z)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Proving (for integers) that primality implies irreducibility Let $Z = \{n \in \mathbb{Z} \; | \; |n| > 1\}$. Let $p\in Z$.
$p$ is irreducible if for some $a,b \in \mathbb{Z}$:
$$p = ab \implies |a| = 1 \vee |b| = 1$$
$p$ is prime if for $a,b \in \mathbb{Z}$:
$$p \mid ab \implies p\mid a \vee p\mid b$$
I would like to... | You started with $p=ab$, and then weakened this hypothesis to $p|ab$. In fact, with the original, stronger, hypothesis, you have $u=1$, so $b$ is a unit since it divides $1$.
| {
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Addition in $R_S$ is well defined Let $R$ be a commutative ring with $1 \neq 0$ and suppose S is a multiplicatively closed subset of $R \backslash {\{\, 0 \,\} }$
containing no zero divisors. We have the relation ∼ defined on $R × S$ via $(a, b) ∼ (c, d) ⇔ ad = bc$.
Let $R_S$ denote the set of equivalence classes $\fra... | There is a standard method: Pick $(a^\prime, b^\prime)$ with $(a,b) \sim (a^\prime, b^\prime)$ (hence $ab^\prime = a^\prime b$) and show $\frac{ad+bc}{bd} = \frac{a^\prime d+b^\prime c}{b^\prime d}$. This is really straight forward.
Since the addition is symmetric, there is no need to also allow an other representative... | {
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Identity is the only real matrix which is orthogonal, symmetric and positive definite
Show that identity is the only real matrix which is orthogonal, symmetric and positive definite
All I could get using above information was that $A^2=I$, hence it is its own inverse.
Using the fact that $A$ is positive-definite, I g... | Hint: $A$ is symetric and positive-definite. Use diagonalization.
| {
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Category Kernel Question (Unique homomorphism $\alpha_0:\ker\phi\to L$) There is a a question with two parts (prove or disprove), one of which I know how to do, but the other part I am stuck.
Let $\phi:M'\to M$ be a homomorphism of abelian groups. Suppose that $\alpha:L\to M'$ is a homomorphism of abelian groups such t... | Part (i) is false: Take $M=L=0$ and $M'=\Bbb{Z}/2\Bbb{Z}$ so that there are unique homomorphisms
$$\phi:\ M'\ \longrightarrow\ M\qquad\text{ and }\qquad\alpha:\ L\ \longrightarrow\ M',$$
and they clearly satisfy $\phi\circ\alpha=0$. Then $\ker\phi=M'$ so the inclusion map $\mu:\ \ker\phi\ \longrightarrow\ M'$ is the i... | {
"language": "en",
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Minimum value of the expression given below. Is $a,b,c$ are three different positive integers such that :
$$
ab+bc+ca\geq 107
$$
Then what is the minimum value of $a^3+b^3+c^3-3abc$.
I expanded this expression to $(a+b+c)((a+b+c)^2-3(ab+bc+ca))$ and tried to find the minimum value of $(a+b+c)$ by AM-GM inequalities but... | Yes thats the correct answer. We can do it as $3(ab+bc+ca)>321$
and for the value to be non negative (a+b+c)^2 should be just greater than 321 so we can suppose it to be 18 as 18^2=324 and hence (a+b+c)>18
for minimum value it should 18.
and also abc should be greatest for minimum value
so we struck to values 5,6,7 .
... | {
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How does the Torsion of two vector fields act on their corresponding flows? Let $X$ and $Y$ be vector fields defined on an open neighborhhod of a smooth manifold $M$ endowed with an (arbitrary) affine connection $\nabla$ (i'm not assuming anything apart from it being a connection on $TM$).
I'm trying to understand the... | So from Samelson Lie Bracket and Curvature, its shown that
$$ (\nabla_Xs)^v =[X^h,s^v]$$ where $ X \in \Gamma(TM)$ and $s \in \Gamma(E)$ and $\nabla$ is the covariant derivative on $E$. The $s^v$ is the vertical extension and is defined as below
Like wise the $X^h$ is the horizontal lift of the vector field and is de... | {
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A question on Gershgorin disk Suppose that the $n$ Gershgorin discs of $A \in {\mathbb{C}^{n \times n}}$ are
mutually disjoint.
If $A $ has real main diagonal entries and
its characteristic polynomial has only real coefficients
Why is every eigenvalue of $A$ real?
| Hint: if your polynomial has only real coefficients, then if $a+bi$ is a root, then so is $a-bi$. If $a+bi$ is in a disk centered at a real number, is $a-bi$ also in that disk? Can you have two values in the same disk when the disks are disjoint?
| {
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Is there always a shortest path using vertices of low eccentricity? We say a graph $G$ is self-centered if $\text{rad}(G)=\text{diam}(G)$, so if its radius equals its diameter. In other words, the eccentricity of every vertex is equal.
Consider the following claim: let $G$ be a graph that is not self-centered, and let ... | This is not true.
Let $G$ be a wheel graph with at least 8 spokes and subdivide each spoke.
You can easily see that the center vertex is the only central vertex of $G$
and the vertices on the rim are the only vertices with maximum eccentricity (4).
However, if you take two vertices on the rim that are at distance two, ... | {
"language": "en",
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Show that every subsequence converging to the same limit implies limit of sequence exists. I need to prove that if $\exists$ subsequences $a_{n_{k_{1}}}$ and $a_{n_{k_{2}}}$ (of $a_{n}$) that converge to different limits, then the sequence $a_{n}$ does not converge. I'm not sure how to do this. If I suppose $a_{n_{k_{1... | Suppose $a_n$ does converge to some value $\alpha$.
This means that $\forall \epsilon>0 \exists N>0$ s.t $\forall n >N$: $|a_n - \alpha| < \epsilon$.
Let $a_{n_k}$ be a subsequence of our original $a_n$. Since both sequences are infinite, and $N$ is a finite number, we can infer that there is some $\beta$ such that $\f... | {
"language": "en",
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When is the additive identity not the zero vector? My teacher cryptically mentioned today that the zero vector is not always the additive identity. When asked for clarification I was told "we'll get there".
He did confirm it is always 0 in matrices filled with real numbers, but I can't think of or find any matrix, whet... | That depends on what you mean by the zero vector.
If you want, you can consider $\mathbb{R}_+$ (the set of positive real numbers) as a vector space, where you define $x\oplus y = xy$ and $\lambda x = x^\lambda$. Then the "additive identity" is actually $1$ (but should probably be called the zero vector in this strange... | {
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What happens if we remove the non-negativity constraints in a linear programming problem? As we know, a standard way to represent linear programs is
$$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & A x \leq b\\ & x \geq 0\end{array}$$
with the associated dual
$$\begin{array}{ll} \text{minimize} & b^T y... | If you remove the non-negativity constraint on $x$ then the constraints of the dual program become $A^T y = c$. Similarly, if you drop the non-negativity constraint on $y$ your primal constraints become $A x = b$.
For these primal dual pairs of LPs strong duality still holds. That means that if your primal LP has a bou... | {
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Taylor expansion for $\arcsin^2{x}$ I stumbled upon this particular expansion that was included in this post.
$$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$
This caught my eye because I remember trying to derive a Taylor series for $\arcsin^{2}(x)$ a while ... | You can find a derivation for the Taylor series of $\frac{\arcsin(x)}{\sqrt{1-x^2}}$ in this nice answer. Since $$\frac{d\arcsin^2(x)}{dx} = \frac{2\arcsin(x)}{\sqrt{1-x^2}}$$
the Taylor series for $\arcsin^2(x)$ follows by integration.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Equivalence relations
Having trouble proving this is an equivalence relation.
Is it suffice to say that let $x y z$ be any string in $\Sigma^*$,
$(xz \in L \iff yz \in L) \rightarrow (yz \in L \iff xz \in L)$ shows that $xRy \rightarrow yRx$?
| For any $x\in \Sigma^*$, define $L(x)$ to be the language $\{z\in \Sigma^*\mid xz\in L\}$. Then the definition of $R$ can be rephrased as
$$xRy \iff L(x)=L(y).$$
This is clearly reflexive, symmetric, and transitive, but assuming this is an elementary course you probably want to at least write out transitivity, making ... | {
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Convergence of $\sum_{k=0}^{\infty}{x^k\over (k+1)!}$ How can I prove the convergence of $$\sum_{k=0}^{\infty}{x^k\over (k+1)!}$$
and what is the limit function? I think that I need to use the fact that $\sum_{k=0}^{\infty}{x^k\over k!}$ is a convergent series.
Any comments, suggestions or hints would be really appre... | Hint: notice that $\sum\limits_{k=0}^{\infty}\frac{x^k}{(k+1)!}=\frac{1}{x}\sum\limits_{k=0}^{\infty}\frac{x^{k+1}}{(k+1)!}=\frac{1}{x}\sum\limits_{k=1}^{\infty}\frac{x^k}{k!}$.
| {
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Show if $N$ is normal subgroup of $G$ and $H$ is a subgroup of $G$, then $N \cap H$ is normal subgroup of $H$. Show if $N$ is normal subgroup of $G$ and $H$ is a subgroup of $G$, then $N \cap H$ is normal subgroup of $H$.
attempt: Then recall $N \cap H$ is normal if and only if $h(N \cap H) h^{-1} \subset N \cap H$.
... | To show $N ∩ H $ is normal in $H$ we must show that the conjugate of $N ∩ H$ by an element
of $h$ is $N ∩ H$. So let $h ∈ H$. Then $h(N ∩ H)h
^{−1} = hNh^{−1} ∩ hHh^{−1}$ but
$hNh^{−1} = N$ since $N$ is normal in $G$, and $hHh^{−1} = H$ since $H$ is a subgroup of $G$,
so $g(N ∩ H)g^
{−1} = N ∩ H$, as desired.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Boundedness of an operator in $L^p$ space Let's define the following operator
$$\mathcal{J}_\epsilon:=(I-\epsilon\Delta)^{-1},$$
where $\epsilon>0$ and $\Delta$ is the Laplacian. We know that $$\Vert\mathcal{J}_\epsilon f\Vert_{L^p(\Omega)}\leq\Vert f\Vert_{L^p(\Omega)},$$ for every $1<p<\infty$, with $\Omega$ an open... | You may have already had this proven, but I thought I would include it for completeness.
The integral of $e^{-2\pi ix\cdot\xi}$ over a sphere of radius $r$ is
$$
\begin{align}
2\pi r\int_{-r}^re^{-2\pi it|\xi|}\,\mathrm{d}t
&=2\pi r\int_{-r}^r\cos\left(-2\pi t|\xi|\right)\,\mathrm{d}t\\
&=\frac{2r}{|\xi|}\sin\left(2\p... | {
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Root system for Lie algebras, Does $t_{\alpha+\beta}=t_\alpha+t_\beta$? For a root system, where $\alpha,\beta \in \Delta$ and $\alpha+\beta\in \Delta$
Does $t_{\alpha+\beta}=t_\alpha+t_\beta$?
Where $t_\alpha$ I know is denoted in different ways depending on author, one author calls these root vectors. I.e. for $\alph... | You don't really give a definition for $t_\alpha$, and I have not seen this notation before. But I assume that $h_\alpha$ and $e_\alpha$ have their standard meaning, and take the relation $t_\alpha = \frac{\langle \alpha, \alpha \rangle}{2}h_\alpha$ you give as definition, where I further assume that $\langle \cdot ,\c... | {
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For which $k$ with $0Because PARI/GP is not very fast in primilaty testing, I did not check the
pairs $k \times 3571\# \pm 1$ in ascending order, but I begun with $k=200,000$
and got the twin prime pair
$$210637\times 3571\# \pm 1$$
*
*Is there a $k$ with $0<k<210637$, such that $k\times 3571 \# \pm 1$ is a twin pri... | I don't know about PARI/GP, but NewPGen can sieve pretty fast.
Running the interval $0<k<200000$ for one minute eliminated about 170000 candidates.
https://primes.utm.edu/programs/NewPGen/
@Edit: I could have perhaps explained the programm a bit. It only sieves, for testing you need another program. Pfgw performs proba... | {
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How do I solve $\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x^2}}{\sqrt{1+x}-1}$ indeterminate limit without the L'hospital rule? I've been trying to solve this limit without L'Hospital's rule because I don't know how to use derivates yet. So I tried rationalizing the denominator and numerator but it didn't work.
$\lim_{x ... | Set $x=\cos2y$
$$\lim_{x \to 0}\dfrac{\sqrt{1+x}-\sqrt{1-x^2}}{\sqrt{1+x}-1} =\lim_{y\to\pi/4}\dfrac{\sqrt2\cos y-\sin2y}{\sqrt2\cos y-1}$$
$$=-\sqrt2\lim_{y\to\pi/4}\cos y\cdot\lim_{y\to\pi/4}\dfrac{\sin y-\sin\dfrac\pi4}{\cos y-\sin\dfrac\pi4}$$
Method$\#1:$
$$\lim_{y\to\pi/4}\dfrac{\sin y-\sin\dfrac\pi4}{\cos y-\sin... | {
"language": "en",
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Is it possible to describe the Collatz function in one formula? This is related to Collatz sequence, which is that
$$C(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$
Is it possible to describe the Collatz function in one formula? (without modular conditions)... | You can use floor/ceiling maps. If you let $f(n) = 2*(n/2 - floor (n/2 ))$ then this function is the indicator of whether $n$ is odd or even, (i.e. $f(n) = 1$ if $n$ is odd and $f(n) = 0$ if $n$ is even), and so then you can define $C(n) = f(n)*(3n + 1) + (1 - f(n))*(n/2)$.
However, note that looking at the floor of $n... | {
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equality of complex analysis property I need to prove the following that
$e^{\bar{z}} = \bar{e^{z}}$
with $e^z := \Sigma_{k = 0}^{k = \infty} \frac{z^k}{k!}$
The problem that I am having is first we know $e^z$ defined that way always converge, but what I don't understand is we have to see that $\bar{z^k}$ put into th... | First, we have
$$\overline{\lim_{N\to \infty}S_N}=\lim_{N\to \infty}\overline{S_N}$$
Second, we have
$$\overline{S_N}=\overline{\sum_{k=0}^Nf_k(z)}=\sum_{k=0}^N\overline{f_k(z)}$$
Third, we have
$$\overline{f_k(z)}=\overline{\left(\frac{z^k}{k!}\right)}=\frac{\overline{z^k}}{k!}$$
Finally, we have
$$\overline{z^k}=\bar... | {
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Justifying differentiation of infinite product Let $(z_n)_{n\in\mathbb{Z}}$ be a sequence of complex numbers s.t. the product $P(z):=\prod_{n=1}^{\infty}{\left(1-\frac{z}{z_{-n}}\right)\left(1-\frac{z}{z_{n}}\right)}$ is absolutely convergent for every $z\in\mathbb{C}$, and hence defines an entire function with zeros a... | Yes, this can be done. You can find this on ProofWiki: Derivative of Infinite Product of Analytic Functions
You don't even need the absolute convergence of the product $f=\prod f_n$, locally uniform convergence (=uniform convergence on compact sets, modulo some details, depending on the author, but many authors do not ... | {
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Proving $\sum_{k=0}^n {n \choose k}{m-n \choose n-k} = {m \choose n}$ I have tried using the principle of mathematical induction. The base case is simple, but I am having trouble with the induction step. Any help would be great, thanks !
| Although a MI method may be used, a Combinatorial argument is much more simple. We assume that we want to select n distinct objects of a total of m objects. We can always select them directly and get RHS.Alternatively, we can separate the m objects into two groups, one of n and the other of m-n.
Now assume we take 0 fr... | {
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Is my algebraic space a scheme? Consider $\mathcal{M}_{1,1}$ over $\bar{\mathbb{Q}}$.
I have an algebraic stack $\mathcal{M}$ finite etale over $\mathcal{M}_{1,1}$
I can prove that it is an algebraic space (essentially because all its "hidden fundamental groups" are trivial - ie, all geometric points of $\mathcal{M}$ ... | So the answer is yes, and follows from two facts:
*
*For an abelian variety $A/S$, and $\sigma\in Aut_S(X)$, if there is a point $s\in S$ such that $\sigma|_{A_s} = id$, then $\sigma = id$. (This is called "rigidity", and can be found in Mumford's Geometric Invariant Theory)
*For a stack affine over $\mathcal{M}_{1... | {
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Writing specific word as product of commutators in free groups Let $F$ be a free group on $\{a,b,c,\cdots\}$. Then a word
$$a^mb^nc^k\cdots \hskip1cm \mbox{(finite expression)}$$
lies in commutator subgroup $[F,F]$ if and only if sum of powers of $a$ is zero, sum of powers of $b$ is zero, and so on. This follows by co... | In Applications of topological graph theory to group theory, Gorenstein and Turner give such an algorithm. Furthermore, the number of commutators in the product they find is minimal.
I give some details on my blog, and in particular I show how the argument gives $$x^2yx^{-1}y^{-2}x^{-1}y=[x^2yx^{-1},y^{-1}x^{-1}].$$
| {
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"source": "stackexchange",
"question_score": "3",
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No finitely generated abelian group is divisible My question might be ridiculously easy. But I want prove
No finitely generated abelian group is divisible.
Let $G$ be a finitely generated abelian group. By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^... | Proof with no structure: let $G$ be a f.g. divisible abelian group. Assuming by contradiction that $G\neq 0$, it has a maximal (proper) subgroup $H$; then $G/H$ is a simple abelian group, hence cyclic of prime order, hence is not divisible, contradiction since being divisible passes to quotients.
The same shows that a ... | {
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Prove that $(a+1)(a+2)(a+3)\cdots(a+n)$ is divisible by $n!$ so I have this math problem, I have to prove that $$(a+1)(a+2)(a+3)\cdots(a+n)\text{ is divisible by }n!$$ I'm not sure how to start this problem... I completely lost. Here's what I know: $(a+1)(a+2)(a+3)\cdots(a+n)$ is like a factorial in that we are multipl... | Let
$$ f(a,n) = (a+1)(a+2)\cdot\ldots\cdot(a+n).\tag{1}$$
We have:
$$ f(a+1,n)-f(a,n) = n\cdot f(a+1,n-1) \tag{2} $$
hence the claim follows by applying a double induction.
| {
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Does a cofiber sequence of CW complexes induce a cofiber sequence of skeleta? Suppose $X\to Y \to Z$ is a cofiber sequence of CW complexes. We can replace the maps with homotopic cellular maps $X_n\to Y_n \to Z_n$ taking the $n$-skeleton of $X$ to the $n$-skeleton of $Y$ and similarly for the second map.
Is the result... | No. For example, $S^1 \to \bullet \to S^2$ is a cofiber sequence, but it doesn't remain a cofiber sequence after taking $1$-skeleta.
| {
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"url": "https://math.stackexchange.com/questions/1451041",
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least-squares problem with matrix Consider the samples of vectors $(x_0,x_1,...,x_k)$ where $x\in \mathbb{R}^m$ and $(y_0,y_1,...,y_k)$ where $y\in \mathbb{R}^n$.
I need to find the matrix $A\in \mathbb{R}^{n\times m}$ of the following least-squares problem :
$\min_A S= \sum\limits_{i=0}^k (y_i-Ax_i)^T(y_i-Ax_i)$
To ... | You can rewrite it as
$$\sum\limits_{i=0}^k (y_i-Ax_i)^T(y_i-Ax_i)=\sum\limits_{i=0}^k ||y_i-Ax_i||_2^2=||Y-AX||_F^2$$
where X and Y are the matrices of the samples by column, and the $||\cdot||_F$ is the Frobenius norm. In this way you can transpose everything
$$\dots=||Y^T-X^TA^T||_F^2$$
and you can treat it as a lea... | {
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"timestamp": "2023-03-29T00:00:00",
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Why is the gradient of implicit surface the normal vector (i.e. parallel to the normal line)? If we have an implicit surface $g(x, y, z)=...$, the gradient is a normal vector. But why? Do I have to start with tangent vectors and then show the gradient is perpendicular? Not sure how to approach this.
| Fix a point $p$ on the surface and take a curve $\gamma=\gamma(t)$ lying on the surface and passing through $p$, say $\gamma(0)=p$. That implies
$$
g(\gamma(t))=0
$$
for all $t$. Differentiate this relation termwise.
This argument shows that $\nabla g(p)$ is orthogonal to the velocity vector $\dot{\gamma}(0)$ for all ... | {
"language": "en",
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Why is $0\to\ker\varepsilon\to P_0\xrightarrow\varepsilon N\to 0$ a projective resolution? Let $R$ be a commutative ring with $1$.
My professor defined $\operatorname{Tor}_i^R(M,N)$ as follows: Tensor a projective resolution $\dots\to P_1\to P_0\to M\to 0$ with $N$ and set
$$\operatorname{Tor}_i^R(M,N):=\ker(P_i\otimes... | In general $\ker \varepsilon$ is not projective, otherwise every module would have a projective resolution of length one.
Consider the following counter example: Let $R=K[x,y]$ where $K$ is a field and let $M=R/(x,y)$. Let $\epsilon: R \to M$ be the map that sends an element to its residue class. The kernel is the ide... | {
"language": "en",
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How to find the range of a function without a graph? Okay so I know how to find the domain of the function $f(x) = \frac{3x^2}{x^2-1}$, which is $x\neq 1$ and $x\neq -1$, but I'm totally confused on how to find the range without using a graph.
| Write the function as
$$f(x)=\frac{3x^2-3+3}{x^2-1}=3+\frac3{x^2-1}$$
As $f$ is even, we may suppose $x\ge 0,\enspace x\ne q 1$.
Now the range of $x^2-1$ is $[-1,+\infty)$ and for $f(x)$, the value $x^2=1$ is excluded, hence the range of $\;\dfrac3{x^2-1}$ is $(-\infty,-3] $ for $x\in [0,1)$ and $(0,+\infty)$ for $x>1... | {
"language": "en",
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Proving that the Empty Set $\emptyset$ is Unique Verification I know that this question has been asked before, but I am presenting my proof for verification. I do not want anyone to give me a proof or suggest a better one; I only want to know if the following proof is legitimate.
Let $A$ and $B$ be empty sets. Let $U$ ... | Your proof is not legitimate in ZF (in which there is no "universe" in which $A$ and $B$ abide). (I would say more but you explicitly asked me not to.)
| {
"language": "en",
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Is $t^3+t+1$ irreducible over $\mathbb{Q}(\sqrt{-31})$? To prove this, I substituted $a + b \sqrt{-31}$ into the polynomial, where $a,b$ are rational. But it is very complicated. Is there another way to check irreduciblity of $t^3+t+1$ over $\mathbb{Q}(\sqrt{-31})$?
| A cubic polynomial $f$ that is irreducible over $\mathbb{Q}$ is irreducible over any quadratic extension of $\mathbb{Q}$.
To see this, let $\alpha$ be a root of $f$, and note that $\mathbb{Q}(\alpha)$ has degree $3$ over $\mathbb{Q}$, so it cannot be contained in an extension of degree $2$.
| {
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Let $H$ be a group. Let $a, b$ be fixed positive integers and $H=\{ax+by\mid x,y\in \Bbb Z\}.$ Show that $d\mathbb Z =H$ where $d=\gcd(a,b)$.
Let $H$ be a group. Let $a, b$ be fixed positive integers and $$H=\{ax+by\mid x,y\in \Bbb Z\}.$$ Show that $d\mathbb Z =H$ where $d=\gcd(a,b)$.
Given $d=\gcd(a,b)$ then $d|a, ~... | Citing you
By Euclidian algorithm, there exist $u, v$ such that $ua+vb=\gcd(a,b)=d$
so you're done since given $dz\in d\mathbb Z$,
$$dz=z(ua+vb)=a(zu)+b(zv)$$
which is a linear combination of $a$ and $b$, thus $dz\in H$. This proves $d \Bbb Z \subset H$.
| {
"language": "en",
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Show that the separable differential equation M(x) + N(y)y' = 0 is exact. Show that the separable equation
$M(x) + N(y)\frac{dy}{dx} = 0$
is exact.
The homework sheet that the teacher gave us said it should be a one-liner. I'm not sure how to prove it that easy (as to put it in one line).
I have done this so far:
$M(x... | $$M(x) + N(y)\frac{dy}{dx} = 0 \tag 1$$ is exact because an integral of $(1)$ is $$\int M(x) \, dx+\int N(y) \, dy = \text{ constant } $$ and is a solution of $(1).$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$4ab-2-5b=0$ have no solution $a,b$ are natural numbers .
Show that $4ab-2-5b=0$ has no solution .
By contradiction : $b(4a-5)=2$
So $4a-5=${1,2} which gives $a\in\mathbb{Q}$ ( contradiction)
So the equation has no solution !
Is my proof correct ?
| The essence of your proof is correct, but you've written it down in an awful fashion. Contrary to popular belief, mathematics isn't just writing down some formulae and implications. Every proof must contain some explaining words.
In this case, a proper proof could look like this:
Assume there is an integer solution $a,... | {
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Different ways finding the derivative of $\sin$ and $\cos$. I am looking for different ways of differentiating $\sin$ and $\cos$, especially when using the geometric definition, but ways that use other defintions are also welcome.
Please include the definition you are using.
I think there are several ways to do this ... | To the first order, the tangent is a good approximation of the boundary of a shape.
So can you approximate $(\cos(\theta+d\theta),\sin(\theta+d\theta))$ in terms of $(\cos\theta,\sin\theta)$? In other words, what's:
$(\cos(\theta+d\theta),\sin(\theta+d\theta))-(\cos\theta,\sin\theta)$
Well, it's tangent to the circle, ... | {
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Based on a coordinate system centered in a sphere where is $M(x, y, z) = 6x - y^2 + xz + 60$ smallest? I am trying to work through a few examples in my workbook, and this one has me completely dumbfounded.
Suppose I have a sphere of radius 6 metres, based on a coordinate system centred in that sphere, at what point on... | On your sphere $x^2+y^2+z^2=6^2$, so that $y^2=\ldots$. Substitute this into $M$ to have a function of $x$ and $z$ only.
| {
"language": "en",
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The closure of $c_{00}$ is $c_{0}$ in $\ell^\infty$ let
$x=(x(1),x(2),\ldots,x(n),\ldots)\in c_0$
So for any $\varepsilon>0$ there exists a $n_0\in\mathbb{N}$ such that $|x(n)|<\varepsilon_2$ for all $n\geq n_0$.
Now for all $n\geq n_0$ ,$||x_n−x||_\infty=\sup{|x(m)|}_{m\geq n_0}\leq \varepsilon_2<\varepsilon$
i have a... | I assume what you want to show is that the closure with respect to the sup-norm, of the space of all sequences that are eventually zero ($c_{00}$) is the space of all sequence that tend to $0$ that is $c_0$.
To show this some sequence $x=(x(1), x(2), \dots) \in c_0$ is fixed and one needs to show that for every $\eps... | {
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Unit of a purely infinite, simple C*-algebra Suppose that we have a purely infinite, simple C*-algebra with unit $1$. Can we find two projections $p,q$ both equivalent to the identity such that $1=p+q$ and $pq=0$?
Well, there are two projections equivalent to $1$ such that $pq=0$ but what can we can we say about $p+q$?... | You cannot always get p + q = 1. We can see this using K-theory.
Suppose p,q are orthogonal projections, both Murray von-Neumann equivalent to 1, such that p + q = 1. Let [-] denote the class of a projection in K_0. Then we have
[1] = [p+q] = [p] + [q] = [1] + [1]
Thus [1] = 0. But the class of the unit is not always... | {
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What is the definition of a set? From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition.
My question is what the definition of a set is?
I have noticed that many other definitions st... | One of the reasons mathematics is so useful is that it is applicable to so many different fields. And the reason for that is that its logical structure starts with "undefined terms" that can then be given definitions appropriate to the field of application. "Set" is one of the most basic "undefined terms". Rather th... | {
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Limit at infinity for sequence $ n^2x(1-x^2)^n$ I'm supposed to prove that this sequence goes to zero as n goes to infinity.
$$\lim_{n\to \infty} {n^2x (1-x^2)^n}, \mathrm{where~} 0 \le x \le 1$$
I've been trying a few things (geometric formula, rewriting $(1-x^2)^n$ as $\sum_{k=0}^n \binom{n}{k} (-1)^k x^{2k} $) and m... | First let us observe that $x(1-x^2)^n$ is smaller than $(1-x^2)^n$ on the interval, since $|x|<1$.
The largest point of $(1-x^2)$ is at $x=0$ which is $1$ and the smallest is $0$ at $x=1$ (easy to check) and monotonically decreasing. So the function $(1-x^2)^n$ will be bounded by 1. Any $f(x_0) = (1-{x_0}^2)^n$ will de... | {
"language": "en",
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Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$ I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral
$$\int_0^\infty \sin (x^p)\, {\rm d}x$$
I am interested in a Laplace approach, double integral etc. For some weird reason I cannot get it to wo... | The integral evaluates to
$$
\int_0^{\infty}\sin x^a\ dx=\Gamma\left(1+\frac{1}{a}\right)\sin\frac{\pi}{2a},
$$
but the way I know uses complex analysis.
Added by request:
We will integrate the function $\exp(-x^a)$ around the circular wedge of radius $R$ and opening angle $\pi/(2a)$, for $a>1$. By the Residue Theorem... | {
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What are some things we can prove they must exist, but have no idea what they are? What are some things we can prove they must exist, but have no idea what they are?
Examples I can think of:
*
*Values of the Busy beaver function: It is a well-defined function, but not computable.
*It had been long known that there ... | I'm suprised that the following two haven't shown up:
*
*What is the smallest Riesel number?
*What is the smallest Sierpiński number?
In both cases we know they exist because they are smaller than or equal to 509,203 and 78,557 respectively.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1452844",
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Prove there is a line that cuts the area in half Suppose you are given two compact, convex sets A and B in the plane, prove there exists a line such that the area of A and B is simultaneously divided in half.
Can you help me with this proof? What I think I have to do is give a fuction that measures the area and then us... | Okay, this is convoluted so bear with me.
For any point x in the plane you can construct a line, L, from the origin of the plane through x. For each real number r (positive or negative) you can construct a line perpendicular to L that is r distance from the origin. Precisely one of these lines at a specific distance ... | {
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ZF without regularity allows P(A) member of A Last week, I was doing a bit of reading around on some axiom of regularity(/foundation)-related questions, and found an answer in one of them (which I cannot seem to locate, for the life of me, right now) which stated that, although it was a fairly technical result, one cou... | No, it most certainly does not imply that $\mathcal P(A)\in A$. Simply because $\mathcal P(\varnothing)\notin\varnothing$.
Moreover, you get an entire model of $\sf ZF$ inside any model of $\sf ZF-Reg$. So there will be plenty of sets which do not have $\mathcal P(A)$ as their element.
But it is consistent. The easiest... | {
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function $f(x)=x^{-(1/3)}$ , please check whether my solution is correct? Let $f(x)=x^{-(1/3)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from -1 to 1. Which of the following statements is/are TRUE?
*
*f is continuous in [-1, 1]
*f is not bounded in [-1, 1]
*A is nonzero a... | The following statements certainly have standard definitions: 1) "$f$ is continuous at $x$" ; 2) "$f$ is continuous". For the function $f(x) = x^{-1/3}$, it's certainly true that "$f$ is continuous" and that "$f$ is not continuous at $0$".
However, what about the statement, "$f$ is continuous on $[a,b]$"? Perhaps the... | {
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"timestamp": "2023-03-29T00:00:00",
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Closed form of: $\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $ $\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $
Does a closed form of the above integral exists?
$n$ is a positive integer
| I want to add a proof which uses complex analysis:
First observe that our integral can be split as follows:
$$
I_n=-\underbrace{\int_0^{\pi/2}dx\left(ix^{n+1}+\log(\frac{1}{2i})x^n\right)}_{A_n}+\underbrace{\int_0^{\pi/2}dxx^n\log(1-e^{-2ix})}_{B_n}
$$
The first part is trivial and yields $A_n=\log(\frac{1}{2i})\frac{x... | {
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"timestamp": "2023-03-29T00:00:00",
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Disjoint Set Proof Let A1, A2, ... be an arbitrary infinite sequence of events, and let B1, B2, ... be another infinite sequence of events defined as follows:
$B_1 = A_1, B_i = \frac {A_i}{\bigcup^{i-1}_{j=1} A_j}$ for i = 2, 3, ...
Prove that B1,B2,... is a disjoint collection of events:
I have no idea how to tackle th... | I'll sketch a proof. Use induction on $B_1,...,B_n$. $B_1$ by itself is obviously a family of disjoint sets so the base case is easy. Let $m=n+1$. If $B_1,...,B_n$ are disjoint, it shouldn't be too hard to show that $B_1,...,B_m$ are disjoint. Basically, $B_m$ is disjoint from the sets $A_1,...A_n$ which are supersets ... | {
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Solving limits without using L'Hôpital's rule $$\lim_{x \to \frac{\pi}{2}}\frac{b(1-\sin x) }{(\pi-2x)^2}$$
I had been solving questions like these using L'Hôpital's rule since weeks. But today, a day before the examination, I got to know that its usage has been 'banned', since we were never officially taught L'Hôpital... | Hint: Use trigonometry identities:
$$\sin(x)=\cos\left(\frac{\pi}{2}-x\right)\\\sin\frac{t}{2}=\sqrt{\frac{1}{2}(1-\cos t)}$$
Specifically, set $t=\frac{\pi}{2}-x$ then you want:
$$\lim_{t\to 0} \frac{b(1-\cos t)}{4t^2}$$
Then use the trig identities above, replacing $1-\cos t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1453522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Integral $\int_{0}^{1}\frac{\log^{2}(x^{2}-x+1)}{x}dx$ Here is an integral I derived while evaluating another. It appears to be rather tough, but some here may not be so challenged :)
Show that:
$$\int_{0}^{1}\frac{\log^{2}(x^{2}-x+1)}{x}dx=\frac{11}{9}\zeta(3)-\frac{\pi}{72\sqrt{3}}\left(5\psi_{1}\left(\frac13\righ... | Asset at our disposal: $$\sum\limits_{n=0}^{\infty} \frac{x^{2n+2}}{(n+1)(2n+1)\binom{2n}{n}} = 4(\arcsin (x/2))^2$$
Differentiation followed by the substitution $x \to \sqrt{x}$ gives:
$\displaystyle \sum\limits_{n=0}^{\infty} \frac{x^{n}}{(2n+1)\binom{2n}{n}} = \frac{2\arcsin (\sqrt{x}/2)}{\sqrt{x}\sqrt{1-(\sqrt{x}/... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integration involving negative roots yields two answers? I have this problem that is driving me insane.
I am asked to evaluate:
$$\int^1_{-1} \sqrt[3]{t} -2 \ dt$$
Now I evaluate this correctly to
$$[\frac{3^{4/3}t}{4}-2t]^1_{-1}$$
From here I work this problem has follows:
$$[\frac{3(1)^{4/3}}{4}-2(1)]-[\frac{3(-1)^{... | You are correct. The answer that Wolfram gives is not correct in the context.
Note that there are also two complex third roots of $-1$, they are $$\frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$$
Wolfram is using the first of these.
| {
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"question_score": "1",
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What is the sum of the cube of the roots of $ x^3 + x^2 - 2x + 1=0$? I know there are roots, because if we assume the equation as a function and give -3 and 1 as $x$:
$$
(-3)^3 + (-3)^2 - 2(-3) + 1 <0
$$
$$
1^3 + 1^2 - 2(1) + 1 > 0
$$
It must have a root between $[-3,1]$. However, the root is very hard and it appeared ... | using the information in the equation
$$
\sum_{i=1}^3 r_i^3 = -\sum_{i=1}^3 (r_i^2 - 2r_i + 1) \\
= -5 -\sum_{i=1}^3 r_i^2
$$
also, from the well-known expressions giving the elementary symmetric functions of the roots in terms of the coefficients,
$$
\sum_{i=1}^3 r_i^2 = (\sum_{i=1}^3 r_i)^2 - 2(r_1r_2+r_2r_3+r_3r_1) ... | {
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"source": "stackexchange",
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Are these two statements logically equivalent? XOR and If/then Consider the statement "Everyone in Florida plays either basketball or soccer, but no one plays both basketball and soccer." Let $B(x)$ be the propositional function "$x$ plays basketball", and $S(x)$ be "$x$ plays soccer".
I am expressing this statement us... | No. The second formula can be satisfied by a population of couch potatoes ie $\neg B(x) \land \neg S(x)$ is consistent with the second formula, but not the first.
| {
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Is $V=L$ a single first-order sentence? Is $V=L$, the axiom of constructibility, a single first-order sentence?
Since $V=L$ really stands for $(\forall x)(\exists \alpha)[\alpha \in \mathrm{On} \wedge x \in L_\alpha]$, so my question might as well be "is $x \in L_\alpha$ a single first-order formula?".
I was confused a... | Yes, it is a first-order statement.
Recall that the definition of $L_\alpha$ is an inductive definition. $L_0=\varnothing$, $L_{\alpha+1}=\operatorname{Def}(L_\alpha,\in)$ and for limit ordinals $L_\alpha=\bigcup_{\gamma<\alpha}L_\gamma$.
It might seem that the successor definition is not "internal", but in fact it is.... | {
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"answer_id": 0
} |
Obtaining a derivative using limit definition We have the following limit $$ \lim_{h \to 0} \dfrac{(x+h)^{1/4}-x^{1/4}}{h}$$
I want to find this limit (which I figure is just the derivative of $x^{1/4}$) using only elementary methods (algebra, mostly).
So you can rewrite the function as $\dfrac{4}{h}(\sqrt{x+h} - \sqrt... | Use substitution: set $y=x^{\tfrac14}, \enspace k=(x+h)^{\tfrac14}-x^{\tfrac14}$. Note $k\to 0\;$ as $h\to 0$. Let's rewrite the variation rate:
$$\frac{(x+h)^{1/4}-x^{1/4}}{h}=\frac k{(y+k)^4-y^4}=\frac k{4y^3k+6y^2k^2+4yk^3}=\frac 1{4y^3+6y^2k+4yk^2},$$
which tends to
$$\frac1{4y^3}=\frac1{4x^{3/4}}$$
as $h$ (or $k$)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1454238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Evaluating a complex line integral Suppose $f(z) = y $. I want to find $\int_{\gamma} y $ where $\gamma$ is the curve joining the line segments $0$ to $i$ and then to $i+2$.
Try:
Let $\gamma_1(t) = it $ there $ 0 \leq t \leq 1 $ and $\gamma_2 = t + i $ where $0 \leq t \leq 2 $. So,
$$ \int_{\gamma} y = \int_{\gamma_1} ... | We have $f(z)=y$ on $\gamma$.
On $\gamma_1$, $z=it$ and $dz=idt$. Thus, $y=t$ and
$$\int_{\gamma_1}f(z)\,dz=\int_0^1t(idt)=\frac{i}{2}$$
On $\gamma_2$, $z=t+i$ and $dz=dt$. Thus, $y=1$
$$\int_{\gamma_2}f(z)\,dz=\int_0^21(dt)=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1454323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find maximum of a polynomial? How would I find the value of p which the function f is the maximum:
$f(p)= 100 p(1-p)^{99}$ $p\geq0$
And in general, can setting the derivative equal to $0$ and solving for the variable lead to the values in function's domain that produces the maximum value ?
| Here is another way; clearly we seek $p \in (0, 1)$, so $p, 1-p$ are both positive.
$$\frac{99}{100}f(p) = (99p) \cdot \underbrace{(1-p)(1-p)\cdots(1-p)}_{99 \text{ times}} $$
Now the RHS is the product of $100$ terms, which sum to a constant, viz. $99$. Therefore its maximum is when all terms are equal, viz. $\dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1454457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\int_{\gamma} \frac{ |dz| }{z} $ where $\gamma $ is the unit circle I need to evaluate $\int_{\gamma} \frac{ |dz| }{z} $ where $\gamma $ is the unit circle.
Attempt: Let $\gamma(t) = e^{it} $ where $t \in [0,2 \pi]$. How can I get rid of the absolute values in the differential?
| The notation $\lvert dz\rvert$ denotes the arc-length measure. For a piecewise continuously differentiable curve $\alpha \colon [a,b] \to \mathbb{C}$ and a continuous function $f$ defined (at least) on the set $\alpha([a,b])$, we have by definition
$$\int_{\alpha} f(z)\,\lvert dz\rvert = \int_a^b f(\alpha(t))\cdot \lve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1454678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
"Ambiguous up to scale" , Explanation required. I am reading "Computer Vision: Models, Learning, and Inference" in which author writes at several points (like on page 428-429) that although matrix A seems to have 'n' degree of freedom but since it is ambiguous up to scale so it only has 'n-1' degree of freedom? Can any... | It might be illuminating to learn about projective spaces (just the idea.) I will explain. Define $\mathbb{RP}^1$ (so called projective line) as set of all straight lines in a plane going through origin. Any such line is uniquely determined by specifying one of its points $(x,y) \neq 0$. However, for any nonzero $\lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1454803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
what is a curve ? Is the concept of derivative limited to curves only? I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..
| A curve is the graph of an application \begin{align*}\gamma :A\subset \mathbb R&\longrightarrow \mathbb R^n\\ t&\mapsto (\alpha_1(t),...,\alpha_n(t))\end{align*}
where \begin{align*}
\alpha_i:A&\longrightarrow \mathbb R\\
t&\mapsto \alpha_i(t).
\end{align*}
And yes, derivative are limited to curve. For a surface or any... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1454919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proofs for Taylors theorem and other forms Let $f \in C^k[a,b]$.Show that for $x,x_0 \in [a,b]$,
$$f(x)=\sum\limits_{j=0}^\mathbb{k-1}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+{1\over k!}{\int_{x_0}^x f^{(k)}(t)(x-t)^k \,dt}$$
and after this use this result for proving the following forms of Taylor's theorem:
$i)$ if we assum... | Set
$$
\varphi(x)=\sum_{j=0}^{k-1}\frac{f^{(j)}(x)}{j!}(z-x)^j
$$
then
$$
φ'(x)=\frac{f^{(k)}(x)}{(k-1)!}(z-x)^{k-1}
$$
and thus
$$
φ(z)-φ(x)=\int_x^z\frac{f^{(k)}(t)}{(k-1)!}(z-t)^{k-1}\,dt
$$
Now change the variables $x$ to $x_0$ and then $z$ to $x$ to obtain
$$
f(x)=\sum_{j=0}^{k-1}\frac{f^{(j)}(x_0)}{j!}(x-x_0)^j+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equation $x^{n+1}-2x+1=0$
Show without derivatives that the equation $x^{n+1}-2x+1=0,n>1,n\in\mathbb{R}$ has at least one solution $x_0\in (0, 1)$
I have thought that I need to use Bolzano for the equation as $f(x)$. So, we have: $$f(0)=1>0$$ $$f\left( \frac{1}{2}\right) =2^{-n-1}>0$$ $$f(1)=0$$
There is a problem he... | You want some $a \in(0,1)$ such that $f(a)<0$.
Writing $b =\frac{1}{a}$ this is equivalent to finding a $b >1$ such that
$$
1-2b^n+b^{n+1} <0
$$
or
$$2b^n > 1+ b^{n+1}$$
Now by Bernoulli we have
$$b^n >1+n(b-1)$$
So by choosing $n(b-1)>1$ you have $b^n>2$ and hence
$$\frac{1}{2}b^n>1 \,.$$
Also, as long as $b <\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 4
} |
Under what conditions does matrix multiplication commute? This is just a check on my reasoning, I guess.
So for two matrices $A, B$ to commute, the following must hold:
$$(AB)_{ij} = \sum_{k=1}^{n}a_{ik}b_{kj} = \sum_{k=1}^{n}b_{ik}a_{kj} = (BA)_{ij}$$
This can happen if for all $i, j, k$:
a. $a_{ik}=a_{kj}$ and $b_{ik... | Two matrices commute when they are simultaneously triangularisable, i.e., when there is some basis in which they are both triangular. Roughly speaking, it is when they have the same eigenvectors, probably with different eigenvalues. (But then there are degenerate cases, which make it all more complicated.)
This propert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Vocabulary question: singularity for an analytic map I have a question that is purely on vocabulary. My native language is not english, so I would like to know the usual convention for the following.
When people say "let $f: X \to Y$ be an analytic map", where $X,Y$ are, say, complex manifolds, do they allow for singul... | No, an analytic map has no singularities. "Holomorphic" is a synonym in English, in all cases of which I'm aware. "Meromorphic" functions may have poles, but no essential singularities. I don't know a word for a function which is analytic at most points, but with arbitrary singularities permitted.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Lebesgue measure of numbers whose decimal representation contains at least one digit equal $9$ Let $A$ be the set of numbers on $[0, 1]$ whose decimal representation contains at least one digit equal $9$. What is its Lebesgue measure $\lambda(A)$?
| We don't care about the countable set of $x\in[0,1]$ possessing two different decimal expansions. Denote by $A_k$ $(0\leq k\leq9)$ the set of numbers in $[0,1]$ having $k$ as first digit after the decimal point. The sets $A_k$ with $k<9$ can be written as $$A_k={k\over10}+{1\over 10}A\qquad(0\leq k<9)\ .$$ The scaling... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
If $a$ , $b$ have the same minimal polynomial there exists a field automorphism such that $\sigma(a)=b$ Two elements are said to be conjugates if they have the same minimal polynomial.
It is true that for every extension $L/K$, $a\in L$ and $b$ conjugate to $a$ in $L$ there exists an element $\sigma \in \text{Gal}(L/K)... | This is false. For instance, consider $K=\mathbb{Q}$, $L=\mathbb{Q}(\sqrt[4]{2})$, $a=\sqrt{2}$, and $b=-\sqrt{2}$. Then $a$ and $b$ are conjugate over $K$, but no automorphism of $L$ can send $a$ to $b$ since $a$ has a square root in $K$ but $b$ does not.
It is true if $L$ is normal over $K$. In that case, there is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why do the decision variables in a linear optimization problem have to be non-negative? Why do the decision variables in a linear optimization problem have to be non-negative? I mean it makes sense in certain scenarios (when you are talking about how many pairs of two shoes to make for maximum profit) as you can't make... | Yes, you are right. A variable can be negative. If at least one of the variable is negative (0 inclusive), then you can transform the problem to a problem with only non-negative variables. Therefore you still have the standard form.
Numerical example:
$\texttt{max} \ \ 2x_1-x_2$
$x_1-2x_2\leq 5$
$3x_1-x_2\leq 7$
$x_1\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Vector Proof for triple product How can I prove/disprove A x (B x C)=(A x B) x C +B x (A x C) ?
I know I could equate the right side to:
B(A dot C)-C(A dot B)
But I don't know where to go from there.
| This is the Jacobi identity for the vector cross product. Since you already have the identity:
$$
\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b})
$$
applying this to both sides should show you they are equivalent. The left-hand side is:
$$
LHS=\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why can there be an infinite difference between two functions as x grows large, but a ratio of 1? I learned in grade school that the closer $a$ and $b$ are to one another, the closer $\frac{a}{b}$ is going to be to $1$. For example, $\frac{3}{\pi}$ is pretty close to 1, and $\frac{10^{100}}{42}$ isn't even close to 1.
... | For the limit, $$\lim_{x \to \infty} \frac{x^2}{x^2+x}$$
The $x$ term becomes insignificant and $x^2$ is the dominant term as $x$ gets bigger.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 5,
"answer_id": 0
} |
How to find the set for which a function is harmonic LEt $f(z) = Im( z + \frac{1}{z} ) $. I need to find the set where $f$ is harmonic. IS there a way to find this without too much computations?
| Although @Almentoe has shown that $f$ is harmonic is the punctured plane that deleted the origin, here we analyze the problem directly - "brute force."
First, note that we can write for $z\ne 0$
$$f(z)=\text{Im}\left(z+\frac1z\right)=y-\frac{y}{x^2+y^2}$$
Then, for $(x,y)\ne (0,0)$, we have
$$\frac{\partial^2 f}{\par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Diagonalization of matrix and eigenvalues of $A+A^2+A^3$ Let $$A=
\begin{pmatrix}
6 & 4 & 1 \\
-6 & -1 & 3 \\
8 & 8 & 4 \\
\end{pmatrix}
$$
Find a non-sigular matrix P and a diagonal matrix D such that $A+A^2+A^3=PDP^{-1}$
No idea what to do
| You do not have to calculate $A+A^2+A^3$.
Suppose that you know how to diagonalize the matrix $A$, i.e., you can find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$.
(I will leave the computations to you. You can have a look at other posts tagged diagonalization or on Wikipedia. I guess you c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
When to use product or set notations in calculating Probability. Problem:
The probability that it will rain today is 0.5.The probability that it will rain tomorrow is 0.6.The probability that it will rain either today or tomorrow is 0.7.What is the probability that it will rain today and tomorrow?
*
*Why we just ... | The probability of a union of events is always the sum of probabilities of the events minus the probability of their intersection. This is how probability measures are required to work (among other things), so: $\Pr(A\cup B)=\Pr(A)+\Pr(B)−\Pr(A\cap B)$. This is equivalently:
$$\Pr(A\cap B)=\Pr(A)+\Pr(B)−\Pr(A\cup ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1456142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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