Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show that T - I is a projection. Came across a question where:
I could solve (a) and (b), but have no clue what (c) means.
Please advice.
| By definition:
A projection is a linear transformation $P$ from a vector space to itself such that $P^2 = P$.
You have found that
$$
T=\begin{bmatrix}
1&0&0\\
1&1&1\\
1&0&2
\end{bmatrix}
$$
now you have to prove that:
$$
P=(T-I)=
\begin{bmatrix}
1&0&0\\
1&1&1\\
1&0&2
\end{bmatrix}-
\begin{bmatrix}
1&0&0\\
0&1&0\\
0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Misunderstanding Löwenheim–Skolem The Löwenheim–Skolem theorem shows that we can find a countable elementary submodel of $V$ that satisfies $ZFC$. [assuming, Con$(ZFC$)]. Call this set $U$. Then by the definition of elementary submodel, $V$ and $U$ must believe the same formulae. Let $\kappa$ be a cardinal in $U$ that ... | Good question! This is a subtle point. The error is when you write:
$(*)$ Then as $U$ countable, $\kappa$ must be countable (as seen from $V$).
This is not the case! Presumably, the reason for believing $(*)$ is (something like) "$\kappa$ in $U$, so $\kappa\subseteq U$," but this assumes that $U$ is transitive. (A se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Minimal Polynomial and Jordan Basis Claim: Assume $A:V\rightarrow V$ is an endomorphism with $\dim V=d$. The minimal polynomial of this linear transformation is $m(t)=(t-\lambda)^d$. Choose $v$ such that $(A-\lambda)^{d-1} v\neq 0$. Then, $V$ has basis $B=\{v, (A-\lambda)v, \cdots, (A-\lambda)^{d-1} v\}$ such that the ... | Let $v$ with $(A-\lambda )^{d-1}v\ne 0$. Denote
$$
v_i := (A-\lambda)^i v, \quad i=0\dots d-1.
$$
Then it holds for $i=0\dots d-2$
$$
A v_i = (A-\lambda) v_i + \lambda v_i = \lambda v_i + v_{i+1}.
$$
Hence the $i$-th row of the matrix representation contains $\lambda$ in the $i$-th columns and $1$ in column $i+1$. Pret... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Showing the equality $P(x,m) = \sum_a \{\frac{x}{2a} \} - \sum _b \{\frac{x}{2x}\}$ I am unsure how to start/proceed with the following problem:
Let $p_1,\cdots , p_m$ be the first $m$ odd primes and let $P(x,m)$ be the number of odd integers $\leq x$ and not divisible by any of these primes. Let $\{ u \} = [u + \frac... | Simply double counting and principle of inclusion and exclusion is enough,
Firstly, from the right side of the equality $\sum_a \{\frac{x}{2a} \} - \sum _b \{\frac{x}{2b}\}$ .......................................................(1)
$a$ has an even number of distinct prime factors$\implies \mu (a)=1$
$b$ has an odd num... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rows/Columns in Cantor's Diagonal Argument Imagine filling a grid diagonally, like in Cantor's diagonal argument:
\begin{array}{ |c|c|c|c|c|c|}
\hline
1 & 3 & 6 & 10 & 15 & \dots \\
2 & 5 & 9 & 14 & 20 & \dots \\
4 & 8 & 13 & 19 & 26 & \dots \\
7 & 12 & 18 & 25 & 33 & \dots \\
11 & 17 & 24 & 32 & 41 & \dots ... | We have $C(i,\,j)=C(1,\,i+j-1)-i+1$, which you can write as a quadratic using $C(1,\,j)=j(j+1)/2$. Now you should be able to solve $C(i,\,j)=n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Standard result for the gradient of a multidimensional Gaussian If $\vec x$ is a vector of dimension $n$ and $A$ is a symmetric matrix of dimension $n\times n$. I would like to know what is the standard result for computing the following expression?
$$\frac{\partial}{\partial {\vec x}}\exp(-{\vec x}^T\cdot A \cdot {\ve... | You have a composition of:
the diagonal (linear)
$$\Delta:\Bbb R^n\longrightarrow\Bbb R^n\times\Bbb R^n$$
$$x\longmapsto(x,x)$$
a symmetric bilinear form
$$B: \Bbb R^n\times\Bbb R^n\longrightarrow\Bbb R$$
$$(x,y)\longmapsto B(x,y) = -x^TAy$$
and the exponential
$$\exp: \Bbb R\longrightarrow\Bbb R.$$
By the chain rule,
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2643007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Bernoulli Uniform Bayes Estimator
My answer comes out as $(p|X)$~$BETA(x+1,-x+2)$, indicating that $p_{Bayes}=\frac{x+1}{3}$, but apparently the correct answer is $p_{Bayes}=\frac{\sum x_i+1}{n+2}$.
I don't understand where this answer comes from, however; can somebody here explain it? Thanks.
| You are calculating the posterior distribution incorrectly. I'll use $\theta$ instead of $p$ to avoid confusing the notation, so that $\theta \sim U(0,1)$ and $p(\theta) = 1$:
\begin{align*}
p(\theta | X) &= p(X | \theta) p(\theta)\\
&= \prod_{i=1}^n p(X_i|\theta) p(\theta)\\
&= p(\theta) \prod_{i=1}^n \theta^{X_i} (1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2643324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two unknowns in Arithmetic Progression I have a problem in my maths book which says
Find the arithmetic sequence in which $T_8 = 11$ and $T_{10}$ is the additive inverse of $T_{17}$
I don't have a first term of common difference to solve it, so I managed to make two equations to find the first term and common differ... | It's simple .
a+7d=11
And second eq is 2a+ 25d.
Multiply the first eq by cofficient of a from second eq . i.e.,
2a+14d=22 and
2a+25d=0
By subtracting both equations we get
d=-2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2643486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Nullity and rank bounds for a nilpotent matrix
Let $A=\mathbb R^{11}\to \mathbb R^{11}$ be a linear transformation such that $A^5=0$ and $A^4\neq 0$. Which of the following is true?
a) $\operatorname{null}A\le7$
b) $2\le\operatorname{null}A$
c) $2\le\operatorname{rk}A\le9$
I don't know how i think about $A$ fr... | Only the answer a) is true. In fact, more precisely, the following inequalities are true :
$4\leq \text{Rank}(A)\leq 9 $ and $3\leq\text{Nullity}(A)\leq 7$
To prove it you need to know the Jordan decomposition for a nilpotent linear transformation:
A nilpotent linear transformation of degree $u$ (i.e. $A^u=0$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2643609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove that the set $\{(x,y)\in \mathbb{R}^2 : x+y\geq0\}$ is closed using sequences. I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?
| Let $a = (x,y)$ is an accumulation point of the set. Then there exists a sequence $(a_n) \in \mathbb{R}^{2}$ converging to it. Assume that $a = (x,y)$ isn't in the set. Then we have that $x+y = N < 0$. Now take a ball with radius $\frac{|N|}{2}$ around $a$. Obviously that the intersection with the set is empty, but thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2643778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can you use the chain rule in vector calculus to compute the gradient of a matrix? From the definition of Jacobian I previously determined that the gradient of $x^TA$ with respect to $x$ for $x \in \mathbb{R}^m, A \in \mathbb{R}^{mxm}$ is equal to $A^T$
However, I want to now determine the gradient of
$x^TAx.$ From my... | Consider a scalar function $(\phi)$ of two vectors $(x,y)$
$$\eqalign{
\phi &= x^TAy = y^TA^Tx \cr
}$$
Its differential is
$$\eqalign{
d\phi &= x^TA\,dy + y^TA^T\,dx \cr
}$$
Now consider what happens in the case that $(y=x),$ so there is now a single vector argument
$$\eqalign{
d\phi &= x^T(A+A^T)\,dx \cr\cr
}$$
Depe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Counterexample request: a surjective endomorphism of a finite module which is not injective It's a classical and useful result that over a commutative, unital ring $A$, a surjective endomorphism of a finite module $M$ is an isomorphism. The standard proof seems to require commutativity in that one needs determinants a... | I think your intuition is entirely wrong here: usually, graded-commutative rings behave basically the same as commutative rings (as long as you restrict to graded modules, graded homomorphisms, etc). So you should expect that the result does still hold for graded-commutative rings (again, assuming your modules and hom... | {
"language": "en",
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Suppose $V$ is finite-dimensional and $E$ is a subspace of $\mathscr L(V)$ Suppose $V$ is finite-dimensional and $E$ is a subspace of $\mathscr L(V)$ such that $ST\in E$ and $TS \in E$ for all $S \in \mathscr L(V)$ and all $T\in E$. Prove that $E = \{0\}$ or $E=\mathscr L(V)$.
I have started the proof, but I get lost ... | Here's one way to show that $I \in E$:
Let $T$ be a non-zero element of $E$. Let $v_1$ be a vector such that $T(v_1) \neq 0$. Extend $v_1$ into a basis $\{v_1,v_2,\dots,v_n\}$.
Select $S$ so that $ST$ is a the linear map satisfying
$$
ST(v_k) = \begin{cases}
v_1 & k=1\\
0 & k \neq 1
\end{cases}
$$
Call this map $T_1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that the union of finitely many compact sets is compact Show that the union of finitely many compact sets is compact.
Note: I do not have the topological definition of finite subcovers at my disposal. At least it wasn't mentioned. All I have with regards to sets being compact is that they are closed and bounded by... | Assume $K_j, j=1\cdots n$ are compact sets and
Note that$$\overline{A\cup B}=\overline{A}\cup\overline{ B} $$
and hence, for finite union and since each $K_j$ is closed we have,
$$\overline{\bigcup_{j=1}^{n}K_j}=\bigcup_{j=1}^{n}\overline{K_j } =\bigcup_{j=1}^{n} K_j $$ On the other hand, all each $K_j$ is bounded... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate a sum which almost looks telescoping but not quite:$\sum_{k=2}^n \frac{1}{k(k+2)}$ Suppose I need to evaluate the following sum:
$$\sum_{k=2}^n \frac{1}{k(k+2)}$$
With partial fraction decomposition, I can get it into the following form:
$$\sum_{k=2}^n \left[\frac{1}{2k}-\frac{1}{2(k+2)}\right]$$
This almost l... | Hint :
$1/2(\sum_{k=2}^{n}\dfrac{1}{k} -\sum_{k=2}^{n}\dfrac{1}{k+2} ):$.
$(1/2)\sum_{k=2}^{n}\dfrac{1}{k}=$
$(1/2)(\dfrac{1}{2} + \dfrac{1}{3} +.......\dfrac{1}{n})$ ;
$(1/2)\sum_{k=2}^{n}\dfrac{1}{k+2}=$
$(1/2)(\dfrac{1}{4}+...\dfrac{1}{n} +\dfrac{1}{n+1} + \dfrac{1}{n+2} ).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Thinking of sequence where $f_n'$ does not converge to $f'$ everyone,
we see in Rudin this example.
I was trying to think of another example that satisfies this property, but I could not. I could think of sequences of functions that were similar in form with a manipulation of $\sin(nx)$ on the top but could not think... | Consider the sequence of functions $f_n(x) = \frac{x}{1+nx^2}$. We have that $f'_n(x) = \frac{1-nx^2}{(1+nx^2)^2}$. Now obviously we have that $f(x) = 0$ and $f'(x) = 0$. On the other side we have that
$$\lim_{n \to \infty} f'_n(x) = \begin{cases}
0, & x \not = 0 \\
1, & x = 0
\end{cases}
$$
So the two "derivatives" do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Real Fundamental- System/Matrix of a Differential equation We consider:
$$y''' - 2y'' + 2y' - y = 0$$
The real solution to this equation is:
$$y(x) = c_3e^{x} + c_2e^{x/2}sin\left(\frac{\sqrt{3}x}{2}\right) + c_1e^{x/2}cos\left(\frac{\sqrt{3}x}{2}\right)$$
How do we now represent it as a fundamental- system/matrix ? ... | Write your DEQ as a System of First Order Equations, find eigenvalues / eigenvectors and proceed in the usual way.
We have $$y''' - 2y'' + 2y' - y = 0$$
To write it as a system of first order equations we let $x_1 = y$, so
$$\begin{align} x_1 ' &= y' = x_2 \\ x_2' &= y'' = x_3 \\ x_3' &= y''' = 2y'' - 2 y' + y = 2 x_3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $a_n=\sqrt[n]{f\left(\frac{1}{n} \right)^n+f\left(\frac{2}{n} \right)^n+\dots+f\left(\frac{n}{n} \right)^n}$ is convergent
$f$ takes positive values and is uniformly continuous. Prove that $$a_n=\sqrt[n]{f\left(\frac{1}{n} \right)^n+f\left(\frac{2}{n} \right)^n+\dots+f\left(\frac{n}{n} \right)^n}$$
is con... | Let $M=\max f|_{[0,1]}$. Then for every $\epsilon>0$, we find an interval of positive length $r$ such that $f(x)>M-\frac\epsilon2$ in that interval. Note that at least $nr-1$ of the points fall into that interval.
We conclude
$$ nM^n\ge f(\tfrac1n)^n+\ldots +f(\tfrac nn)^n\ge (nr-1)(M-\tfrac\epsilon2)^n$$
and so
$$ \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644649",
"timestamp": "2023-03-29T00:00:00",
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Which of these is not a subset of the powerset {0, 1}? The powerset of {0,1} is {{},{0},{1},{0,1}}. The answer to this problem says that {{0}}, {}, {{}}, are subsets of the powerset, but {0} is not a subset of the powerset.
However, this doesn't make any sense to me. Obviously {0} and {} are the only ones in the power... | {0} is an element from the power set, so {{0}} (a box containing the element {0}) is a subset of the powerset, but {0} itself is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the map $\mathbb{Z}_n^*\to \mathbb{Z}_m^*$ is surjective Let $m,n \in \mathbb{Z}$ such that $m|n$. Show that the map $$f: \mathbb{Z}_n^*\to \mathbb{Z}_m^*$$ $$f({a \pmod n}) = (a \pmod m)$$ is surjective. I am not able to figure out any simple way to tackle this... Any hints?
| Got it. Will prove by induction. We just have to show that the result holds when $n=mp$. Let $b \in \mathbb{Z}_m^*$.
If $b \not\equiv 0 \pmod p$ then $(b,m)=1 \land (b,p)=1 \implies (b,n)=1 \implies b \in \mathbb{Z}_n^*$. Now we have $f(b)=b$.
If $b \equiv 0 \pmod p$ then $m \not\equiv 0 \pmod p \implies b+m \not\equ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Number of set, say $A$, of subset such that $\sum |A_i|=|\xi|$ and that $A_{ij}=A_{\ell k}\iff \ell=i\land k=j$ My little brother asked me a question that I cannot answer and I would love to get some help with it.
Background
My brother tried to understand what ${n\choose k}$ means and came to conclusion that it gives ... | $$\binom{n}{k}B(k)$$
where $B(k)$ is the Bell number of $k$. That, is we count the number of ways to choose $k$ items, times the number the number of ways to partition a set of $k$ items.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644997",
"timestamp": "2023-03-29T00:00:00",
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Colored topological spaces? Let's say that for a topological space, we "color" it. By that, I mean we have some set of colors $C$, and we associate to each point in the space a color, and require continuous maps to preserve colors.
For example, we can color the faces of a polyhedron "white" and the edges "black". Two p... | Yes indeed. The 8 faces of the regular octahedron may be colored alternately black and white, yielding the overall symmetry of the regular tetrahedron, which is a subgroup of the octahedral symmetry. By contrast a chequerboard appears at first sight to yield a symmetry subgroup of the square tiling, however it turns ou... | {
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How to verify the uniqueness of a standard matrix? I think I have answered this question sufficiently, I just want to know if I am correct, or if I missed anything.
Here is the question:
Verify the uniqueness of A in Theorem 10. Let T : ℝn ⟶ ℝm be a linear transformation such that T($\overrightarrow{x}$) = B$\overrigh... | The step you are concerned about is indeed incorrect. Basically, your argument is that $(A-B)x=0$ for any vector $x$, and the zero matrix also satisfies $0_{m\times n}x=0$ for any vector $x$, therefore $A-B=0_{m\times n}$. But this logic is wrong: how do you know there can't be two different matrices which, when mult... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\lim\limits_{x\to0}\sin1/x$ What is
$$\lim\limits_{x\rightarrow0}{\left( \sin{\frac{1}{x}}\right)} $$?
Wolfram says "-1 to 1", but I don't know what that means.
In fact, I thought this limit didn't exist, so what does "-1 to 1" mean in this context?
| $$\Box \ \nexists \lim_{x\to 0}\bigg(\sin\frac1x\bigg).$$ Proof: Let $u = \dfrac{1}{x}$ then $$\lim_{x\to 0}\frac{1}{x} = \infty$$ since $$\lim_{x\to\infty}\frac{1}{x} = 0.$$ Therefore, we get $$\lim_{x\to 0}\bigg(\sin\frac 1x\bigg) = \lim_{u\to\infty}(\sin u)$$ but this cannot exist because sine is a periodic fluctuat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Mathematical methods for solving pair-wise cheapest path. Two trains need to go two different routes. The first train must start at $A_1$ and stop at $B_1$ and the second train $A_2$ and $B_2$. They start at the same time. Let us assume no two trains can be on the same track at the same time for risk of collision. How ... | There is probably a better solution (in terms of computational complexity), but here is a nice theoretical reduction:
Construct a pair graph $G^2$, where vertices are pairs of nodes of $G$, and edges are $(v_1, v_2) \to (u_1, u_2)$ if $v_i \to u_i$ are edges in $G$ and there is no collision. The cost of such edge is ob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do you write log to the base e? I was given a question, find $f'(1)$ of $f(x) = \ln \sqrt{2-x}$.
So I wrote
$$1/2 \ln (2-x)^{(-1/2)(-1)} = -1/2 \ln (2-x)^{-1/2}$$
$$= -(1/2\ln)/\sqrt{2-x}$$
But when I sub in $x = 1$ I get a SYNTAX error, I realised log base e cannot be put in my calculator. I don't know how to pu... | Given,
$$ f(x) = \ln \sqrt{2-x} $$
Use chain rule to differentiate.
$$ f^{'}(x) = \frac{1}{\sqrt{2-x}}\frac{1}{2 \sqrt{2-x}} (-1) $$
So, at $x=1$,
$$f^{'}(1) = \frac{-1}{2} $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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I don't know to find subgroups. $G=\mathbb Z_3\times\mathbb Z_5$
I don't know to find a subgroup.
Give me an example a subgroup of $G$ how to find that ?
| Because $(3,5)=1$, $G$ is cuclic and $([1]_3,[1]_5)$ is a generator of $G$.
Now as I commented for every positive $n| |G|=15$ there exists a subgroup of $G$ the one that is generated by $([1]_3,[1]_5)^n$.So we have:
$1)$ for $n=1$:$<([1]_3,[1]_5)>=G$
$2)$ for $n=3$ :$<([0]_3,[3]_5)>=\{0\}\times\mathbb{Z_5}$
$3)$for $n=... | {
"language": "en",
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"source": "stackexchange",
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Example of a non-closed subspace such that the quotient is not a Banach space As I've learnt recently in my Functional Analysis course, it is well known that if $X$ is a normed Banach space and $Y$ is a closed subspace, then the quotient $X/Y$ is a Banach space (e.g. How to show that quotient space $X/Y$ is complete wh... | Let $X$ be a Banach space and let $\alpha\colon X\longrightarrow\mathbb R$ be a discontinuous linear form. Then $\ker\alpha$ is a dense subspace of $X$. And $X/\ker\alpha$ is not a Banach space simply because the norm$$\|x+\ker\alpha\|=\inf\{\|x+y\|\,|\,y\in\ker\alpha\}$$is not a norm. In fact, it follows from the dens... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Product between normal and hyponormal operators which commute is hyponormal Let $A\in \mathcal{L}(E)$ be a normal operator i.e $A^{*}A=AA^{*}$.
Let $B\in \mathcal{L}(E)$ be an hyponormal operator i.e. $B^*B\geq BB^*$. If $AB=BA$. Why $AB$ is hyonormal?
I try to apply the following theorem:
Fuglede's theorem: Let $T,... | Since $A$ is normal and $AB=BA,$ we get $$AB^*=B^*A.$$ Similarly, since $A^*$ is normal,
and $A^*B^*=B^*A^*,$ we get
$$A^*B=BA^*.$$
Now note that
$(AB)^*AB=B^*(A^*A)B=B^*(AA^*)B=(B^*A)(A^*B)=(AB^*)(BA^*)=A(B^*B)A^*.$
Hence
$$A(B^*B)A^*\geq A(BB^*)A^*=(AB)(AB)^*.$$
This completes the proof.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute $\iint_D(x^2-y^2)e^{2xy}dxdy$. Compute $$\iint_D(x^2-y^2)e^{2xy}dxdy,$$ where $D=\{(x,y):x^2+y^2\leq 1, \ -x\leq y\leq x, \ x\geq 0\}.$
The area is a circlesector disk with radius $1$ in the first and fourth quadrant. Going over to polar coordinates I get
$$\left\{
\begin{array}{rcr}
x & = & r\cos{\theta... | Call $\alpha = \sin(2\theta)$ for simplicity.
$$\int_0^1 r^3 e^{\alpha r^2}\ dr = \int_0^1 \frac{d}{d\alpha} r e^{\alpha r^2} = \frac{d}{d\alpha} \int_0^1 r e^{\alpha r^2}\ dr$$
The latter can easily be done by parts once to get
$$\int r e^{\alpha r}\ dr = \frac{e^{a r} (a r-1)}{a^2}$$
Hence with the extrema it becomes... | {
"language": "en",
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Given two bounded sets $A,B$ and $\sup A<\sup B$, is there an element in $B$ that works as an upper bound for $A$? Originally I worked this questions out by just saying that $m=\sup B$ and $m>\sup A$. Then drawing the conclusion that $m$ is an upper bound for $A$. I figured this was wrong because we're unsure that $\su... | $\sup B$ is the smallest upper bound for $B$. This means that a strictly smaller number $\sup A$ cannot be another upper bound of $B$. Thus, $\mathbb R$ being totally ordered, there is a $b\in B$ such that $b\gt\sup A$. However, this means that, for every $a\in A$ you have $b\gt\sup A\gt a$, i.e. $b$ is an upper bound ... | {
"language": "en",
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"source": "stackexchange",
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How to find the values that $x$ can take to be a real number?
$$3 \cdot \sqrt{x+4} + 5 \cdot \sqrt [8]{6-x}$$
- How to find the values that $x$ can take to be a real number?
I'm a bit confused. However, I want to show my thinkings:
$$ x + 4 > 0 \implies x > -4$$
and
$$6-x>0 \implies 6 >x$$
My Kindest Regards,
| You should set
$$ x + 4 \ge 0 \implies x \ge -4$$
and
$$6-x\ge0 \implies 6 \le x$$
thus
$$-4\le x\le6$$
| {
"language": "en",
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$(a_n)_{n \geq 1}=\mathbb{Q}_+$ and $\sqrt[n]{a_n}$ is convergent
Is there any sequence $(a_n)_{n \geq 1}$ such that it contains all positive rational numbers without repetition, and $\sqrt[n]{a_n}$ is convergent?
My first guess is that there is no such sequence. I tried to build $a_n$ just like the sequence in the p... | Actually the standard one
$$ (a_n) = \left( \frac 11, \frac 21, \frac 12, \frac 31, \frac 13, \frac 41, \frac 32, \frac 23, \frac 14, \cdots\right) $$
works. The observation is that for the members $a_n$ in the $i$-th layer:
$$ \frac i1, \frac{i-1}{2}, \cdots, \frac{2}{i-1}, \frac 1i,$$
we have
$$ i \ge a_n \ge i^{-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why isn't the complex logarithm $\log z$ holomorphic on $\mathbb C -\{0\}$? Why isn't the complex logarithm $\log z$ holomorphic on $\mathbb C -\{0\}$? Why can't you just say take the $\arg z$ to be in $[0,2\pi)$ and then you don't have to worry about it being a multivalued function.
| The logarithm of a complex number depends on the arg function. If you start following a circle around the origin starting at a real number $r$, the arg function starts growing from zero until it nears $2\pi$ when it is finishing a full turn. In consequence, the arg function cannot be continuous on any circle that surro... | {
"language": "en",
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"source": "stackexchange",
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Determinant modulo $m$ for a matrix Let $A$ be a matrix having entries in $\mathbb{Z}_n$, i.e. the ring of intergers modulo $n$. Suppose $$A^m \equiv 0 \pmod{n}$$ for some positive integer $m$, then can we say that $$(\det(A))^m \equiv 0 \pmod n,$$ i.e., $\det A$ is also a nipotent modulo $n$?
According to me it is rig... | Yes. You are right. You can think about the determinant as a polynomial in the entries of $A$. Now, if you assume $A^m=0$ over $\mathbb{Z}_n$ then obviously a polynomial in its entries will be divisible by $n$ as well. From the equality $\det(A)^m=\det(A^m)$ you conclude what you want
| {
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"source": "stackexchange",
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How to decompose a complex number into a sum of two unitary modulus complex numbers? Is it possible to decompose any complex number $z = x + iy\in \mathbb{C}$ with $0\leq|z|\leq2$ into a sum of two unitary modulus exponentials ? i.e. $ z = e^{i\phi_1} + e^{i\phi_2}$ ?
I tried to decompose the problem $x + iy = \cos(\ph... | No. it is not, because$$\left|e^{i\phi_1}+e^{i\phi_2}\right|\leqslant1+1=2.$$
| {
"language": "en",
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"source": "stackexchange",
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Do the terms of an infinite series constitute a countable set? Given an infinite series (e.g. trigonometric expansion, exponential, whatever) $\sum_{\infty}T_{n}$, were one to consider the terms of this series as the members of a set $S$, it is obvious that the set would be an infinite one (given that the terms come f... | If I understand correctly your question, you construct from a formal series $\sum_{i \in I} T_i$ a set $S = \{T_i | i \in I\}$.
Then the cardinal of $S$ is less than or equal to the cardinal of $I$, almost by definition.
In particular, if $I = \mathbb{N}$, or if $I = \mathbb{Z}$, then $S$ is countable.
PS : note tha... | {
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Will LQR act like MPC in reality? MPC is a predictive controller. Which means that MPC will analyse the best input values $u$ to get the shortest way from setpoint to reference point in trajectories $x$.
MPC is very well used in the industry. But my question is:
As I heard, LQR with saturation limits on $u$ is equal to... | No, an LQR controller (or trivially saturated LQR controller) will not give the same control signal as an MPC controller. You can (and typically want to) tune he MPC controller though such that it coincides with the LQR feedback once the system enters the state where the LQR feedback is and remains unsaturated.
If LQR... | {
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Show $\text{Hom}_R(M,N)$ is an abelian group Define a commutative ring $R$ where $M$ and $N$ are left $R$-modules, and denote by $\text{Hom}_R(M,N)=\{f:M\to N|f(rm)=r\cdot f(m)\text{ }\forall r\in R,m\in M\}$. Then, we want to prove that $\text{Hom}_R(M,N)$ is an $R$-module.
To do this, we need to show first that $\tex... | Hint:
$\DeclareMathOperator\Hom{Hom}\Hom_R(M,N)$ is a subset of the abelian group $\;N^M$ (set of all maps from $M$ to $N$), so all you to prove is it's a subgroup, i.e. it's not empty, the sum of two linear maps is linear, and the opposite of a linear map is linear.Proving in a similar way that the product of a line... | {
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Does it make mathematical sense to do an absolute convergence test if the original series diverges? Reason I ask I know a series can converge but then when you apply the absolute convergence test it may diverge. I understand this part. One concludes absolute convergence is a stronger condition!
But what happens if ... | Note that if $\sum a_n$ diverges, with $a_n$ negative, also $\sum |a_n|=-\sum a_n$ diverges.
| {
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"source": "stackexchange",
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Is $f(z)= |z|$ continuous on the complex plane? So I understand that the absolute value of $z=a+b\mathbf i=\sqrt{a^2+b^2}$, I just don't know if its enough to just say that this is continuous so $f$ is continuous or if I have to go through an epsilon-delta proof. A brief explanation of the structure of the proof would ... | By triangle inequality,
$$||z_1|-|z_2|| \leq |z_1-z_2|$$
As $z_1-z_2\to 0$, $f(z_1)-f(z_2) \to 0$.
Hence yes, it is continuous.
| {
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Change of coordinates matrix, proper subspace I don’t fully understand this exercise and it’s really frustrating.
It says something like this:
Consider the space $P_2[R]$ with basis:
$B_1 = \{x^2 + x + 1, x, x - 1\}$
$B_2 = \{x^2 - x + 1, x, 2\}$
If $S \in P_2$ is a “proper” subspace and $L_1$ and $L_2$ bases of $S$.... | a)Does the change or coordinates matrix from L₁ to B₁ exist?What would be their size?
It can't exist because $L_1$ has $1$ or $2$ elements whereas $B_1$ has $3$ elements. The matrix of the change of basis should be invertible.
b)Does the change or coordinates matrix from L₁ to L₁ exist?What would be their size?
Of cour... | {
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Probability of $3+3$ cards, out of $6$ cards drawn from a solitaire
A solitaire consists of $52$ cards. We take out $6$ out of them (wihout repetition). Find the probability there are $3+3$ cards of the same type (for example, $3$ "1" and $3$ "5").
Attempt.
First approach. There are $\binom{13}{2}$ ways to choose $2... | In order to fully clear your confusion, let us tackle a simpler problem first.
We are dealing with drawing w/o replacement, (hypergeometric distribution)
If asked to find the Pr of drawing $2$ red and $3$ blue balls from a pool of $5$ red and $4$ blue balls,
Using the multiplication rule, $P(RRBBB)$ in that particular ... | {
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Why $K^0 = \{0\}$? I am reading "Linear Algebra" by Takeshi SAITO.
Why $n \geq 0$ instead of $n \geq 1$?
Why $K^0 = \{0\}$?
Is $K^0 = \{0\}$ a definition or not?
He wrote as follows in his book:
Let $K$ be a field, and $n \geq 0$ be a natural number.
$$K^n = \left\{\begin{pmatrix}
a_{1} \\
a_{2... | It can be thought of as a "useful" definition. Any subspace of $K^n$ is isomorphic to $K^m$ for some $m\leq n$. If you don't define $K^0=\{0\},$ then this isn't true for the $0$-subspace.
Another approach is to define $K^n$ as the set of functions from a set of $n$ elements to $K$. When $n=0$, the set of functions from... | {
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How to find the indefinite integral? $$\int\frac{x^2}{\sqrt{2x-x^2}}dx$$
This is the farthest I've got:
$$=\int\frac{x^2}{\sqrt{1-(x-1)^2}}dx$$
| As $0<x<2,$
$$\dfrac{x^2}{\sqrt{2x-x^2}}=\dfrac{x^{3/2}}{\sqrt{2-x}}$$
set $x=2\sin^2t,x^{3/2}=\text{?}$
$dx=\text{?}$ and $\sqrt{2-x}=+\sqrt2\cos t$
| {
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Any suggestions on using induction to prove this inequality? I am solving an exercise and can't advance in the following induction:
$$n\log n - n + 1 \leq \log n!.$$
If necessary, i put the complete question.
*
*Update
Calculate
$$\lim_{n\to \infty}\frac{n!e^{n}}{n^{n}}$$
following the steps bellow:
A. Show... | Starting from the fundamental $(1+\frac{1}{n})^n \leq e$ for all $n>0$, we get the inequality $$\tag{*} en^n \geq (n+1)^n$$
I'd prefer to work with exponentials over logs, so note that your inequality is equivalent to $$\tag{H} n! \geq e\left(\frac{n}{e}\right)^n $$
For the inductive step, we assume $n! \geq e\left(\fr... | {
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First Order Logic - unsatisfiable set of formulas I know what is an unsatisfiable set of formulas in first order logic and I'm studying how to prove the unsatisfiability.
What I don't understand, I'm sorry a think as an engineer, is what get in practice when I prove that set of formulas is unsatisfiable. Can you intuit... | Intuitively, proving that a set of formulas is unsatisfiable gives you something like the formal version of the engineering maxim: "Cheap. Fast. Reliable. Pick two."
Is this what you were asking?
| {
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Number of non-integer solutions in an diophantine equation of order 2
Consider the equation $x^2 + y^2 = 2015$ where $x\geq 0$ and $y\geq 0$. hoe many solutions $(x, y)$ exist such that both $x$ and $y$ are non-negative integers?
*
*Greater than two
*Exactly two
*Exactly one
*None
I tried all the combinations o... | An even number that's a square is always a multiple of $4$. An odd number that's a square is always one larger than a multiple of $4$. So the sum of two perfect squares is always either
*
*A multiple of four (if they are both even)
*One larger than a multiple of four (if one of them is odd)
*Two larger than a mul... | {
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Help understanding the quaternion group of order $8$ From Wikipedia:
In group theory, the quaternion group $Q_8$ (sometimes just denoted by $Q$) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication.
There are many representation of $Q_8$, in one d... | $$\begin{array}{cccc}
1 = \left(\!\!\begin{array}{rr}1 & 0\\0&1\end{array}\!\!\right),&
x = \left(\!\!\begin{array}{rr}0 & 1\\-1 & 0\end{array}\!\!\right),&
x^2 = \left(\!\!\begin{array}{rr}-1 & 0\\0 & -1\end{array}\!\!\right),&
x^3 = \left(\!\!\begin{array}{rr}0 & -1\\1 & 0\end{array}\!\!\right),\\
\\
y = \left(\... | {
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Number of even terms in a polynomial related to an exponent For a polynomial $(x_1+x_2+...+x_N)^{2k}$, I am trying to show that the number of fully even terms is $≤a_kN^k$, where $a_k$ only depends on $k$ and is constant for a constant $k$. When I say "fully even terms" I mean terms where only even exponents appear.
Fo... | The terms you want to count are the terms of the form $x_1^{2k_1}\cdots x_N^{2k_n}$ where $k_i \ge 0, i=1,...,N,$ and $\sum{k_i} = k.$ This is exactly the number of terms in $(x_1+x_2+ \cdots x_N)^k$ or $N^k.$ We get a one-to-one correspondence by dividing/multiplying all the exponents by $2.$
My comment about parti... | {
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Exponential equation - logarithmisation is the transformation of this equation: $$9^x + 6^x = 2× 4^x$$
into this: $$\log_2 (9^x) + \log_2 (6^x)=\log_2 (2×4^x)$$ correct? I want to know because I really want to solve this equation.
| It's $f(x)=0$, where $$f(x)=\left(\frac{3}{2}\right)^{2x}+\left(\frac{3}{2}\right)^{x}-2.$$
We see that $f$ increases, which says that our equation has one root maximum.
But, $0$ is a root and we are done!
Your reasoning is wrong because $\log(a+b)$ is not always equal to $\log{a}+\log{b}.$
| {
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How to solve the equation $x^2+2=4\sqrt{x^3+1}$? From the Leningrad Mathematical Olympiad, 1975:
Solve $x^2+2=4\sqrt{x^3+1}$.
In answer sheet only written $x=4+2\sqrt{3}\pm \sqrt{34+20\sqrt{3}}$.
How to solve this?
| HINT.-The given answer is root of the quadratic equation
$$x^2-2(4+2\sqrt3)x+c=0$$ where $c$ is certain constant. It follows
$$x=4+2\sqrt3\pm\sqrt{{(4+2\sqrt3)^2-c}}=4+2\sqrt3\pm\sqrt{34+20\sqrt3}$$
You get so the value of $c$ from which you can verify the solution.
| {
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What is the limit of $3^{1/n}$ when n approaches infinity Graphically, I see that $\lim_{n->\infty}3^{1/n}$ approaches $1$. However, how to show $\lim_{n->\infty}3^{1/n} = 1$ step by step?
| Bernoulli's Inequality, which, for integer exponents, can be proven using a simple inductive argument, says
$$
\left(1+\frac2n\right)^n\ge3\ge1
$$
Taking roots, we get
$$
1\le3^{1/n}\le1+\frac2n
$$
Then, the Squeeze Theorem ensures that
$$
\lim_{n\to\infty}3^{1/n}=1
$$
| {
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The order of quantifiers I am reviewing for an exam, and I have come across a question that does not contain a solution, so I wanted to verify my answer.
Question 1: If $\exists y \forall x P(x, y)$ is true, then $\forall x \exists y P(x, y)$ is also true.
To me that appears true, because if there exists at least one p... | Correct. To demonstrate that something does not follow, it is often helpful to provide a concrete counterexample, e.g. you could assume that $P(x,y)$ stands for $x$ has $y$ as a parent. So then $\forall x \exists y P(x,y)$ becomes the claim that everyone has a parent (true), but $\exists y \forall x P(x,y)$ becomes t... | {
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Limiting Probabilities I have the following question in which I need some help.
A mark of chain on states {0,1,2,3,4,5} has the transition probability matrix
\begin{bmatrix}
1/3 & 2/3 & 0 & 0 & 0 & 0 \\
2/3 & 1/3 & 0 & 0 & 0 & 0 \\
0 & 0 & 1/4 & 3/4 & 0 & 0 \\
0 & 0 & 1/5 & 4/5 & 0 & 0 \\
1/4 & 0 & 1/4... | Since the last two states are transient, you know that the last two columns of $P^\infty$ will be zero. Also, the nonzero entries of the last two rows will be the corresponding absorption probabilities. Now you need to fill in the two $2\times2$ blocks corresponding to the two absorbing classes. You should be able to w... | {
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How do we prove that the irrational numbers have no upper bound From Calculus to Apostol I know that real numbers do not have upper bound, I also know that irrational numbers belong to real numbers. Would the mathematical proof be different?
I quote the theorems to determine that the real numbers are not upper bounded.... | Let $n$ be an integer value, then $n+\frac{1}{\sqrt2}$ is irrational.
Since the set of integer is not bounded from above, the set of irrational number is not bounded from above since $n+\frac{1}{\sqrt2}> n$.
Remark: there is nothing special about the number $\frac1{\sqrt2}$, it can be replaced by any positive irration... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2649780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Correlation for random graph (Erdos-Renyi) Consider $n$ vertices labeled $1, 2, . . . , n$ and suppose that between each of the $n$ pairs of distinct $2$ vertices an edge is, independently, present with probability $p$. The degree of vertex $i$, designated as $D_i$, is the number of edges that have vertex $i$ as one o... | You employ linearity of expectation.
$$
\mathbb E[D_i] = \mathbb E\left[\sum_{k \ne i} X_{ik}\right] = \sum_{k \ne i} \mathbb E[X_{ik}] = \sum_{k \ne i} p = (n-1)p.
$$
You can do the same thing with $\mathbb E[D_i D_j]$ and $\mathbb E[D_i^2]$, too; the expression inside the expected value will be the product of two s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2649920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Is an absorbing state necessarily recurrent?
Definition:
For a state i, state i is an absorbing state IFF the probability that state i returns to state i, $p_{ii}$, is 1 and $p_{ij}=0$
Definition:
A state i is recurrent/ persistent if the probability of state i returning to state i k-times is $p^{k}_{ii}=1$
From here... | You are correct: an absorbing state must be recurrent.
To be precise with definitions: given a state space $X$ and a Markov chain with transition matrix $P$ defined on $X$.
A state $x \in X$ is absorbing if $P_{xx} = 1$; neccessarily this implies that $P_{xy} = 0, \, y \neq x$.
Given $x \in X$, the first return time is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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What is the best method to solve the limit $\lim_{x\to \infty}\biggl(1+\sin\frac{2}{x^2}\biggr)^{x^2}$? By the looks of it, I would say the following is a Neperian limit: $$\lim_{x\to \infty}\biggl(1+\sin\frac{2}{x^2}\biggr)^{x^2}$$
but I could not find a way to algebraically bring it in the form: $$\lim_{x\to \infty}\... | You can use this:
If $a \to 1$ and $b \to \infty$, then $\lim a^b = \exp(\lim (a-1)b)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Solving the Summation Cases Let $n$ be a positive integer. Prove that
$\displaystyle \sum_{k=1}^{n} \dfrac{(-1)^{k-1}} {k} \binom{n} {k} = 1 +\dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n}$
I and my friend discussed this two days ago. In this case, we prove that
it goes to $\displaystyle \sum_{k=1}^{n} \dfrac{1}{k... | There's a beautiful proof of this identity from the integral representation:
$$H_n = \int_0^1 \frac{1-x^n}{1-x}dx$$
This is easy to confirm because:
$$\frac{1-x^n}{1-x} = 1 + x + \dots + x^{n-1}$$
Then,
\begin{align*}
H_n &= \int_0^1 \frac{1-x^n}{1-x}dx\\
&= \int_0^1 \frac{1-(1-u)^n}{u}du\\
&= \int_0^1 \frac{1-\sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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A self adjoint matrix of operators Let $F$ be a complex Hilbert space. We recall that an operator $A\in\mathcal{B}(F)$ is said to be hyponormal if $A^*A\geq AA^*$ (i.e. $\langle (A^*A-AA^*)z,z \rangle\geq 0$ for all $z\in F$).
A pair $S=(S_1,S_2)\in\mathcal{B}(F)^2$ is called hyponormal if
$$S'=\begin{pmatrix}[S_1^*, S... | Any positive operator $T$ is selfadjoint:
$$
\langle T^*x,x\rangle=\langle x,Tx\rangle=\langle Tx,x\rangle,
$$
where the last equality is due to $T$ being positive. Now, by polarization, $\langle T^*x,y\rangle=\langle Tx,y\rangle$ for all $x,y$, so $T^*=T$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inequality Proof $\frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2}\geq \frac{3\sqrt{3}}{2}$ Let $a,b,c\in \mathbb{R}^+$, and $a^2+b^2+c^2=1$, show that:
$$ \frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2}\geq \frac{3\sqrt{3}}{2}$$
| The function $f(x)=\frac{1}{x(1-x^2)}$ takes its minimum at $x=\frac{1}{\sqrt 3}$ on $(0,1)$.
Thus
$$\begin{eqnarray*}\frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2} & = & a^2 f(a) + b^2 f(b) + c^2 f(c)\\
& \ge & (a^2+b^2+c^2)f\left(\frac{1}{\sqrt 3}\right)= \frac{3\sqrt 3}{2}.\end{eqnarray*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many bit strings?
How many bit strings of length $8$ have either exactly two $1$-bit among the first $4$ bits or exactly two $1$-bit among the last $4$ bits?
My solution:
A bit only contains $0$ and $1$, so $2$ different numbers, i.e., $0$ and $1$. For the first part we have $2^6=64$ ways. Similar for the other ... | Let $A$ be the set of bit strings with exactly two $1$-bit among the first $4$ bits, and $B$ be the set of bit strings with exactly two $1$-bit among the last $4$ bits.
\begin{align}
\#A &= \binom{4}{2} 2^4 = 6\cdot2^4 \\
\#B &= 2^4 \binom{4}{2} = 6\cdot2^4 \\
\#A\cap B &= \binom{4}{2}^2 = 6^2 \\
\#A\cup B &= \#A + \#B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Why do we use $\frac{\pi}{180}$ to convert from degrees to radians? If I want to convert from degrees to radians, I can use the function that takes degree value as an input, multiplies it with $\frac{\pi}{180}$ and returns the radian value: $\operatorname{DtoR}(d)=d \times \frac{\pi}{180}$.
And if I want to go from rad... | The radian is defined as the plane angle subtended by any circular arc divided by its radius.
When the circular arc is actually congruent to the circle, the length is $2\pi r=2\pi=$ $\tau$ (for a unit circle). The angle subtended by this arc is $360^\text{o}$, and therefore $1\:\text{radian}=\frac{360}{\tau}=\frac{180}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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Dynamic Programming: Dividing a chocolate bar
We have a chocolate bar of $F\times C$ squares, some of the "chocolate squares" contains almonds. We can only split the chocolate bar by cutting it horizontally or vertically, obtaining two chocolate bars, one of $k\times C$ squares and the other of $F-k\times C$ squares (... | If $i_1 < i_2+1$ or $j_1<j_2+1$
\begin{align}
&T(i_1, i_2, j_1, j_2)\\
&=
\min\big\{\underbrace{\min_{i_1\leq i<i_2}\left\{T(i_1,i, j_1, j_2) + T(i,i_2, j_1, j_2)\right\}}_{\text{Horizontal splits}}, \underbrace{\min_{j_1\leq j<j_2}\left\{T(i_1,i_2, j_1, j) + T(i_1,i_2, j, j_2)\right\}}_{\text{Vertical splits}}\big\} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the reduction-formula for $\int\sec^n(x)dx$ only valid for $n\ge3$? I have a question regarding the following reduction formula:
$$I_n=\int\sec^n(x)dx=\frac{1}{n-1}(\sec^{n-2}x\tan x)+\frac{n-2}{n-1}I_{n-2}+C$$
My calculus book states that it is only valid for $n\ge3$. Why is this the case? How does one intuit s... | Consider the beginning of the derivation:
$$\int \sec^n(x) dx = \int \sec(x)^{n-2} \sec(x)^2 dx = \sec(x)^{n-2} \tan(x) - \int (n-2) \sec(x)^{n-2} \tan(x)^2 dx.$$
In the case $n=1$, this is true but it is not useful, because you get the same multiple of $I_1$ on the right side as you already had, so you can't isolate $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Equivalence of categories $\Delta$ and $\Delta_{\text{big}}$, and the generators of the algebra $\mathbb{Z}[\Delta]$ I have been given that $\Delta_{\text{big}}$ is the category of all finite ordered sets with order preserving maps as the morphisms and $\Delta \subset \Delta_{\text{big}}$ be its full small subcategory ... | You can use the folowing characterization of equivalence of categories: a functor $F:C\rightarrow D$ is an equivalence of categories if and only if it is fully faithful and essentially surjective.
The canonical embedding $\Delta\rightarrow \Delta_{big}$ is fully faithful and essentially surjective. This implies that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove a floor function is onto/surjective I have function $u(x) = \lfloor x \rfloor$ mapped from $\mathbb{R}$ to $\mathbb{Z}$ which I need to prove is onto.
I know that standard way of proving a function is onto requires that for every $Y$ in the co-domain there should exist an $x$ in the domain such that $u(x) = y$
I ... | Note that for an integer $z$, $$ \lfloor z \rfloor=z$$
Thus the function $$u(z) = \lfloor z \rfloor$$ is onto.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Math gambling question. Why is the net loss $0$? Suppose you are playing a gambling game. There is a $50/50$ chance of losing $1$ dollar, and winning $1$ dollar. Your starting money is $1$ dollar, and you keep playing until you either lose all your money, or you finish $1000$ rounds.
Using a computer, with a sample siz... | On each round of the game, one's expected loss is zero, whether or not
one has been eliminated. If one is active, one's expected loss on a round
is $\frac12(1+(-1))=0$. If one is eliminated, one always loses zero.
By linearity of expectation, one's expected loss in $10^4$ rounds
is $10^4\times 0$.
This problem is essen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partial of Modulo operator ? (with non-integers) I am trying to derive gradient for a special neural network, but got stuck on the Modulo Arithmetic.
With usual funcitons such as $f(a,x) = a/x$ the partials would be $\frac{1}{x}$ and $-\frac{1}{x^2}$, but am struggling to find such a rule on the internet for Modulo op... | A reasonable definition for your module-extended-to-reals is a piecewise-defined function:
$$\mod(A,x) = A -nx$$
for $A/x\le n<A/x +1$, $n\in\Bbb Z$.
Partial derivatives will exist in the interior of the chunks.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of ways of choosing seven children from a classroom of 32 (15 boys, 17 girls) with at least 1 boy I know that the correct solution can be calculated as:
$$ \binom {32} {7} - \binom {17}{7}$$
But why is the following solution incorrect? (I am interested in why the following reasoning is incorrect, I realize that ... | I think the best explanation is to take a group with lets say 4 boys and 3 girls.
With your method, any of those boys could have been the first one to be chosen, and the rest was picked as the group of six kids from the remaining 31 children. Meaning, in your expression $15 {31 \choose 6}$ every group with four boys i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 5,
"answer_id": 2
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determine distribution function from density function The variable $\xi$ has the following density function:
$$f(x)=\begin{cases}x/50&0<x<10\\
0&else\end{cases}$$
How do I determine its distribution function?
| You get the CDF by integrating $f(x)$:
$$F(x) = \int_0^xf(s)\;ds = \int_0^x\frac{s}{50}\;ds=\frac{x^2}{100},\qquad 0<x<10$$
You can write
$$F(x) = \begin{cases}
0 & x\leq 0\\
\frac{x^2}{100} & 0<x<10 \\
1 & x \geq 10
\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$ imply $\exists d \in F_n(d^{pq}=a)$? Suppose $F_n$ is a free group of rank $n$, and $a$ is an element of $F_n$, such that $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$, where $p$ and $q$ are coprime integers. Does there always $\exists d \in F_n... | Consider the subgroup $\langle b,c \rangle$ of $F$. As a subgroup of a free group, it is itself free, but $a$ is in its centre, and the only free group with nontrivial centre is the infinite cyclic group.
So $\langle b,c \rangle = \langle g \rangle$ for some $g \in F$, and $a = g^k$ for some $k \in {\mathbb Z}$. Since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If and only if condition for Simpson's paradox Suppose that female and male students apply to schools A and B. Given that $p>q$ and $r>s$ where $p$ is the ratio of female students accepted to A, $q$ is the ratio of males accepted to A, $r$ is the ratio of females accepted to B and $s$ is the ratio of males accepted to ... | You may have difficulty proving the if direction, as it not always true. Indeed in a sense it is usually not true
Suppose there are $40$ students in total and the number of applicants of each gender to each college is $10$, and consider $p=\frac{8}{10}$, $q=\frac{6}{10}$, $r=\frac{4}{10}$ and $s=\frac{2}{10}$, implyin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the function $u_{j}(x,y)$ and $v_{j}(x,y) $ so that $w_{j}(x,y)+iv_{j}(x,y)$ and $u_{j}(x,y)+iw_{j}(x,y)$ My function $w_{j}(x,y)=x+3y$ is harmonic
$w_{xx}+w_{yy}=0+0$
In order to find $v_{1}(x,y)$ I use Cauchy-Riemann
$v_{y}=w_{x}$ and $v_{x}=-w_{y}$
So for $v_{y}$ and $v_{x}$
$v_{1}(x,y)=\int1dy=x+A(X)$
$v_{1}... | Given the harmonic function
$w = x + 3y, \tag 0$
we have, by the Cauchy-Riemann equations,
$v_x = -w_y = -3, \tag 1$
and
$v_y = w_x = 1; \tag 2$
thus
$\nabla v = \begin{pmatrix} v_x \\ v_y \end{pmatrix} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}; \tag 3$
it follows that, given any two points $(x_0, y_0), (x, y) \in \Bbb R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this sum integer? $b! \pi+1/(b+1)+1/(b+2)(b+1)+...$, where $b \neq 0$. Is $S$ an integer ?
$S= \: b! \:\pi+\frac{1}{b+1}+\frac{1}{(b+2)(b+1)} + \frac{1}{(b+3)(b+2)(b+1)} + ...$
$b \neq 0$.
Also, from here Is this sum rational or not? $1/(q+1)+1/(q+2)(q+1)...$ where $q$ is an integer
$\frac{1}{b+1}+\frac{1}{(b+2)(... | Hint:
$$S_b=\frac{1}{b+1}+\frac{1}{(b+2)(b+1)} + \frac{1}{(b+3)(b+2)(b+1)} + ...+\: b! \:\pi=\dfrac{1}{b+1}(1+\frac{1}{b+2}+\frac{1}{(b+3)(b+2)} + \frac{1}{(b+4)(b+3)(b+2)} + ...+\: (b+1)! \:\pi)=\dfrac{S_{b+1}+1}{b+1}\to\\S_b=\dfrac{S_{b+1}+1}{b+1}$$can you finish now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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I have a bag with 3 coins in it. One of them is a fair coin, but the others are biased trick coins. When flipped, the three coins come up heads with probability 0.5, 0.3, 0.6 respectively. Suppose that I pick one of these three coins entirely at random and flip it three times.
1. What is P(HTT)? (i.e., it comes up head... | First:
$$\frac13\cdot \frac18+\frac13\cdot 0.3\cdot 0.7^2+\frac13\cdot 0.6\cdot 0.4^2=0.122667.$$
Second:
$$\frac{\frac13\cdot \frac18}{0.122667}=\frac{0.041667}{0.122667}=0.3397.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Color $27$ unit cube so that by rearranging, they could form a blue $3\times3$ cube, a green one, and a red one? I searched but there's not much useful information. My instinct is that it is not possible, but I don't know how to show it.
To make it clear, there are $27$ unit cubes, that is, $6\times27$ sides to be colo... | For the $3 \times 3 \times 3$ case it is possible:
Haskell code:
{-# LANGUAGE FlexibleContexts #-}
import Diagrams.Prelude
import Diagrams.Backend.Cairo.CmdLine (defaultMain)
v x = [x,x,x]
e x = [x,x]
f x = [x]
cubes =
[ v red ++ v green
, v green ++ v blue
, v blue ++ v red
]
++
concatMap (replicate 6)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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How many times do a couple of bells ring close enough that they are heard as one? This is a problem from the book: Recreations in the theory of numbers by Albert Beiler.
The problem is this: You have 2 bells $B_1,B_2$.
$B_1$ rings every $\frac{4}{3}$ seconds while $B_2$ rings every $\frac{7}{4}$seconds.
How many stro... | In $28''$ bell $B_1$ rings $21$ times, and its sound covers $21''$ of the $28''$. The probability that a random sound of bell $B_2$ (a Dirac $\delta$) is not heard then is ${3\over4}$. During these $28''$ bell $B_2$ emits $16$ Dirac-sounds, only $4$ of which will then actually be heard (elementary number theory takes ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Algorithm complexity using iteration method I want to find the complexity of
$$T(n) = T\left(\frac{n}{2}\right) + n \left(\sin\left(n-\frac{π}{2}\right) +2\right)$$ by iteration method.
Assume $T(1) = 1$.
\begin{align*}
T(n) &= T\left(\frac{n}{2}\right) + n \left(\sin\left(n-\frac{π}{2}\right) +2\right)\\
&= T\left... | The term to be simplified is a combination of $\sum 2^k\cos 2^k$ and $\sum 2^k\sin 2^k$ or collectively $\sum 2^ke^{i2^k}$.
As far as I know, there is no closed form expression for such a "super-geometric" summation.
Anyway, using that $|e^{it}|\le1$, the absolute value of the sum is bounded by $2n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Intuition behind "Exclusive or" I am trying to understand the intuition behind Exclusive or.
Why it is called exclusive or?
Wikipedia says: it gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true
Wait! why "or" is ambiguous when both operands are true? Or says nothing about... | "Or" is ambiguous in daily life. Some times when people say "or", the option of both is not considered a valid option ("Will you take the red pill or the blue pill?"). Some times it is ("Did you go to some fancy school, or are you just smart?"). You have to tell from the context, which means that the word itself has bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Reduced row echelon with imaginary numbers Working on the following problem:
Let $v = \begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}
w = \begin{bmatrix} -3 \\ i \\ 8 \\ \end{bmatrix}
y = \begin{bmatrix} h \\ -5i \\ -3 \\ \end{bmatrix}$
For what values of $h$ is the vector $y$ in the plane generated by the vectors $v$ and $... | \begin{align}
\begin{bmatrix}
1 & -3 & h \\
0 & i & -5i \\
-2 & 8 & -3
\end{bmatrix}\xrightarrow{R_3\leftarrow R_3+2R_1}
\begin{bmatrix}
1 & -3 & h \\
0 & i & -5i \\
0 & 2 & -3+2h
\end{bmatrix}\xrightarrow{R_3\leftarrow R_3+2iR_2}
\begin{bmatrix}
1 & -3 & h \\
0 & i & -5i \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
If $f$ is continuous on an interval and $g(x) = \sin f(x)$, then does there exist $k \in \mathbb{Z}$ such that $f(x) = (-1)^k\arcsin g(x) + k\pi$?
If $f$ is continuous on an interval $I$ and $g(x) = \sin f(x)$, then does
there exist $k \in \mathbb{Z}$ such that $f(x) = (-1)^k\arcsin g(x) + k\pi$?
Here, $\arcsin$ is... | The reasoning is fine ; however $x \mapsto k_x$ need not be continuous. It is continuous at each point $x$ such that $\arcsin(g(x)) \neq \pm \frac{\pi}{2}$. But in the remaining cases, you can say nothing (you can swith to $k+1$ or $k-1$ without discontinuity). See below for a counterexample.
Short answer : this is fal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simple word problem causes dispute A company purchases 200 VCR units at a price of \$220 each. For each order of 200, there is a fee of \$25 added as well.
If the company sells each VCR unit at a price marked up 30 percent, what is the profit per unit?
-First dispute was that no one buys VCRs anymore. Agreed! Let's lo... | This dispute is if the order fee applies per order or per VCR, and it seems from the wording that the per order fee must apply per order, not per VCR, so the second approach is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Euler Totient Function - show that if $q$ is prime and divides $m$, then $\phi(qm) = q\phi(m)$,
Show that if $q$ is prime and divides $m$, then $\phi(qm) = q\phi(m)$, while if $q$ doesn't divide $m$, then $\phi(qm) = (q-1)\phi(m)$ where $\phi$ is the Euler totient function, i.e. $\phi(m) = n \prod_{i=1}^n \left(1 - \f... | Note that if $q \mid m$ then we have that $q$ appears in the prime factorization of $m$. So we get that:
$$\phi(qm) = qm\left(1 - \frac 1q\right)\left( 1 - \frac 1{p_1}\right) \cdots \left(1 - \frac 1{p_n}\right) = q\phi(m)$$
If $q \not \mid m$ it doesn't appear in the prime factorization then we have that $\phi(m) = m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to calculate ad(X)? I have looked everywhere, and maybe its really simple, and I am just being stupid, but I really don't know how to calculate ad(X). I understand that ad_x(y)=[x,y], but i just want to calculate ad(x)? I also know that Ad(g)(X) = g^(-1)Xg. "g inverse multiplied by X multiplied by g", but the deter... | If you already know structure constants in the base you're using, then the coefficients of the adjoint action are easily computed as
$$(ad(e_i))^k_j=c_{ij}^k,$$
since
$$ad(e_i)(e_j)=[e_i, e_j]=\sum{c_{ij}^k}e_k.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Invariant subspace of all dimensions. Let $V$ be a finite dimensional vector space over $ \Bbb C,$ and suppose that $\dim(V) = n$. Prove that if
$T \in L(V)$ then for each $k$ with $0 \leq k \leq n$, $T$ has an invariant subspace of dimension $k.$
First of all it seems clear to me that we wish to use induction its just... | Hints for another approach without triangular form:
The characteristic polynomial of $\;T\;$ has all its roots in $\;\Bbb C\;$, say $\;a_1,...,a_k\;$ . Induction on $\;n\;$ : for $\;n=1\;$ the claim is trivial as the trivial space and the whole space are $\;T\,-$ invariant.
Assume for $\;\dim V<n\;$ and we prove for $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to find a Lyapunov function? $$\dot{x} = x - x^2 - xy$$
$$\dot{y} = y - y^2 - xy$$
Is there a general method to find a Lyapunov function? Why do I feel like finding a Lyapunov function is like shooting in the dark and luck-dependent? I tried to guess some function with $x$ and $y$, but it doesn't work. I also tried... | First an observation on the direct solution of the system. Add both equations to get
$$
\frac{d}{dt}(x+y)=(x+y)-(x+y)^2
$$
which is a logistic equation for $u=x+y$ with a stable point at $x+y=1$ and an unstable at $x+y=0$,
$$
\dot u=u-u^2\text{ or }\frac{du^{-1}}{dt}=1-u^{-1}\implies 1-u^{-1}=(1-u_0^{-1})e^{-t}.
$$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Laplace-Beltrami operator in $\mathbb{R}^m$ The Laplace-Beltrami operator, $\Delta\colon \Omega^{k}(M)\longrightarrow \Omega^{k}(M)$ is defined as $\Delta=d\delta + \delta d$ where $d$ is the usual exterior derivative and $\delta$ is the codifferential: $$\delta=(-1)^{m(k+1)+1}*d*$$ where $*$ is the Hodge operator, $\a... | Note the ambiguity: when you write $\delta$, it's really $\delta :\Omega^k \to \Omega^{k-1}$. In the case for $\Delta = \delta d$, $\delta$ is adding on one forms and thus $k=1$. So
$$ \delta = (-1)^{m(k+1)+1} * d* = (-1)^{2m+1} *d*= - *d*.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding value of Quadratic If the quadratic equations $3x^2+ax+1=0$ and $2x^2+bx+1=0$ have a common root, then the value of $5ab-2a^2-3b^2$ has to be find.
I tried by eliminating the terms but ended with $(2a-b)x=1$. Can you please suggest how to proceed further?
| The common root, $r$, is also root of
\begin{align*}
3x^2+ax+1-(2x^2+bx+1)&=0\\
x^2+(a-b)x&=0\\
x(x+a-b)&=0
\end{align*}
The root cannot be $zero$, so we get $r=b-a$ as the common root.
Also, the common root is a root of
\begin{align*}
2(3x^2+ax+1)-3(2x^2+bx+1)&=0\\
(2a-3b)x-1&=0\\
x&=\frac1{2a-3b}\implies r=\frac1{2a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Compute $\iiint_R 6z \ dV.$
Integrate the function $$f(x,y,z)=6z$$ over the tetrahedral
$$R=\{(x,y,z):x\geq0, \ y\geq 0, \ z\geq 0, \ 5x+y+z \leq 5\}.$$
This tetrahedral can obtain hegiths in $z$-axis from 0 to 5 in the first octant. Drawing this out, I get that the bounds are
\begin{array}{lcl}
0 \leq x \leq 1 \\... | Yes, it is correct.\begin{align}
&\int_0^1 \int_0^{5-5x} \int_0^{-5x-y+5} 6z \,\,\, dz dy dx \\
&=\int_0^1 \int_0^{5-5x} 3(-5x-y+5)^2 \,\,dy dx \\
&=\int_0^1 \int_0^{5-5x} 3(5x+y-5)^2 \,\,dy dx \\
&=\int_0^1 -(5x+0-5)^3 \, dx \\
&=-5^3 \int_0^1(x-1)^3 \, dx \\
&= -5^3 \frac{(x-1)^4\mid_0^1}{4}\\
&=\frac{125}{4}
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Show that $\frac1x+\log x-\log(x+1)\ge0$ for $x\ge1$ without differential calculus I can't seem to solve this exercise. I want to show that: $\frac1x+\log x-\log(x+1)\ge0$ for $x\ge1$.
I've tried looking for bounds on $\log x-\log(x+1)$, but there's nothing promising I can derive. I'm thinking of multiplying out $x$, y... | $$\frac1x+\log x-\log(x+1)\ge0$$
$$\iff\frac1x\ge-\log x+\log(x+1)=\log\frac{x+1}x=\log\left(1+\frac1x\right)$$
Define $y=\frac1x$, so that $0<y\le1$:
$$\iff y\ge\log(1+y)$$
Now define $z=y+1$ so that $y=z-1$ and $1<z\le2$:
$$\iff z-1\ge\log z$$
which is an inequality you have.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving differential equation using a pair of integrals as helper I found an interesting problem, but I could only solve half of it:
Given $$ I_{1} = \int \frac{e^{-x} + \sin{x}}{e^{-x} + \sin{x} + \cos{x}} dx \\ I_{2} = \int \frac{\cos{x}}{e^{-x} + \sin{x} + \cos{x}} dx $$
and $f:(0,\frac{\pi}{2})\rightarrow \... | for your second integral: substitute $$t=e^x\sin(x)+e^x\cos(x)+1$$ then we get
$$dt=2e^x\cos(x)dx$$ and the integral will be $$\frac{1}{2}\int \frac{1}{t}dt$$
and at first you can write $$\frac{\cos(x)}{e^{-x}+\sin(x)+\cos(x)}=\frac{e^x\cos(x)}{1+e^x\sin(x)+e^x\cos(x)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does the presence of non harmonic real parts of complex functions imply that they are not holomorphic? If $f $ is a complex function such that $f(x,y)=u+iv$ and $u$ is not harmonic can we say that $f$ is not holomorphic and therefore not analytic?
| Yes, you can (at least in a domain). For every holomorphic function in a domain, its real and imaginary parts are harmonic functions. See https://en.wikipedia.org/wiki/Holomorphic_function#Properties
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Counter-example: If $J$ is prime then $f^{-1} (J) $ is prime. $f$ need not be unital. Let $R$ and $S$ be commutative with 1 and $ $ $ f:R\rightarrow S$ is a ring homomorphism which need not be surjective or unital i.e. $f(1_R)=1_S$
I know that for surjective or unital ring homomorphisms the statement - "If $J$ is prime... | Yes: the zero map would then be a morphism of rings, and the preimage of every prime ideal is then $R$, which is not a prime ideal (by definition)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Prove Borel-Cantelli's lemma
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let
$(A_n)_{n\geq1}$ be a sequence of events in $\mathcal{F}$. Prove that
$\mathbb {P}(\limsup_{n \geq 1} A_n) = 0$ if $\sum_{i=1}^\infty
\mathbb{P}(A_i)$ converges.
My attempt:
$$0 \leq\mathbb {P}(\limsup_{n \geq 1} A_... | The last equality is an elementary result from calculus:
Set $a: = \sum_{i=1}^\infty \Bbb P (A_i)$. Then
$$
\lim_{n \to \infty} \sum_{i=n}^\infty \Bbb P (A_i) =
\lim_{n \to \infty} \sum_{i=1}^\infty \Bbb P (A_i) - \sum_{i=1}^{n-1} \Bbb P (A_i) =a -\lim_{n \to \infty} \sum_{i=1}^{n-1} \Bbb P (A_i) = a-a =0.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why Can't Vector Projections Between Two Vectors Be the Same The question is as follows:
Give two reasons why the projection of u onto v is not the same as the projection of v onto u.
I was thinking that the directions of vectors u and v are not the same so that's one way that the projections might differ. Furthermor... | The projectoin can be the same, if they are equal. When they are not equal, you have two cases:
*
*they are colinear: then the two projections are the identity on both, and so different.
*they are not colinear: the projection onto $u$ is colinear with $u$, while the projection onto $v$ is colinear with $v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why is $\alpha = x^2dx^1 - x^1dx^2$ not a differential of a function? Why is $\alpha = x^2dx^1 - x^1dx^2$ not a differential of a function?
I know that a $df$ can be expressed:
$$ df= \frac{\partial f}{\partial x^i}dx^i$$
And I assumed that $\alpha$ is the differential of a function and get a necessary condition $f=0$ ... | Hint
We have
$$
\frac{\partial f}{\partial x^1}=x^2
$$
and
$$
\frac{\partial f}{\partial x^2}=-x^1
$$
so:
$$
\frac{\partial^2 f}{\partial x^1\partial x^2}\ne \frac{\partial^2 f}{\partial x^2\partial x^1}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$y''+\epsilon y'=\epsilon$, where $y(0)=1$, $y'(0)=0$ I am trying to solve $y''+\epsilon y'=\epsilon$, where $y(0)=1$, $y'(0)=0$ using perturbation theory.
Using the substitution $y=y_{0}+y_{1}\epsilon$ I got the series $y=1+\epsilon(1+\frac{x^{2}}{2})+O(\epsilon ^{2})$.
However wolframalpha tells me the exact solution... | The exaxt solution has exponental
$$y''+\epsilon y'=\epsilon$$
Just integrate
$$y'+\epsilon y=\epsilon x+ K_1$$
$$e^{\epsilon x}y'+e^{\epsilon x}\epsilon y=e^{\epsilon x}(\epsilon x+ K_1)$$
$$(e^{\epsilon x}y)=\int e^{\epsilon x}(\epsilon x+ K_1)dx$$
$$y=e^{-\epsilon x}K_2 + \frac {K_1}{\epsilon}+e^{-\epsilon x}\int ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How to prove that $AB$ is invertible if and only if $A$ is invertible? Let $A$ be a matrix and $B$ an invertible matrix. Show that $AB$ is invertible if and only if $A$ is invertible.
I know how to do this using determinants, but how else could you prove this?
| $A$ is invertible $\implies$ $AB$ is invertible: This is because $(AB)^{-1}=B^{-1}A^{-1}$.
$A$ is invertible $\impliedby$ $AB$ is invertible: use the proven implication ($\implies$) above, applied to the matrices $AB$ and $ABB^{-1}=A$, with the fact that, since $B$ is invertible, $B^{-1}$ is also invertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 0
} |
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