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H: Classifying groups of order 21.
We have two sylow subgroups of orders 7 and 3. Let $n_3$ and $n_7$ denote the number of sylow subgroups for 3 and 7, respectively.
$n_7 \equiv 1 \mod 7$ and $n_7 | 3 \implies n_7 = 1$
$n_3 \equiv 1 \mod 3$ and $n_3 | 7 \implies n_3 = 1, 7$
Let $P_3 \cong \Bbb{Z}_3 $ and $P_7 \cong \B... |
H: Absolute value and roots
I've been trying to solve this problem and I always get 1, but the answer is $1 - 2x$.
If $x<\frac12$ then what is $\left|x-\sqrt{(x-1)^2}\right|$ ?
AI: Note that $\sqrt {t^2}=|t|$ for all real numbers $t$.
By the condition $x<\frac12$, it is clear that $x-1$ is negative, hence $\sqrt{(x-1)... |
H: why we want to use grassmannian space?
I wonder what's the special about grassmannian space? Why we want to use this space?
On wikipedia, it says:
"By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed coll... |
H: Unique solution to non linear system of equations with boolean coefficients
Say we have a system of $m$ equations of the form:
$$a_{11} x_1 + a_{12} x_2 + ... + a_{1n} x_n = p_1$$
$$...$$
$$a_{m1} x_1 + a_{m2} x_2 + ... + a_{mn} x_n = p_m$$
Where the $p_i,x_j \in \mathbb{R}$, but the $a_{ij}$'s are boolean values, ... |
H: Is it possible to divide a circle into $7$ equal "pizza slices" (using geometrical methods)?
Or is it possible to divide a circle into n equal "pizza slices" (I don't know how to call these parts, but I think you'll know what I mean), where n hasn't got a common divider with $360$? Or are the $360$ degrees just "ar... |
H: Preimage of invariant subspace
Suppose we have a linear map $A \colon V \to V$ on a finite- dimensional vector space, and $W \leq V$ it's invariant subspace. Then we have obviously $\operatorname{Ker} A + W \subseteq A^{-1}(W)$.
Is it then necessary $\operatorname{Ker} A + W = A^{-1}(W)$ ?
I can prove it in case $A... |
H: Projective Resolution of $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$
The question I have is on how to find a Projective Resolution of $\mathbb{Q}$ over $\mathbb{Z}$.
I have found that $\mathbb{Q}$ is flat, but it is not projective. I mention this as I wonder if there is some sort of trick I can use knowing that it ... |
H: Is it true player II must have a winning strategy, if the winning set is a closed but not open set?
Suppose, in a Gale-Stewart game, player I and player II choose from $\omega$ in a alternating fashion. If the outcome is in the winning set $W$, then player I wins. Otherwise player II wins. If $W$ is a closed but no... |
H: Why does one counterexample disprove a conjecture?
Can't a conjecture be correct about most solutions except maybe a family of solutions? For example, a few centuries ago it was widely believed that $2^{2^n}+1$ is a prime number for any $n$ . For $n=0$ we get $3$ , for $n=1$ we get $5$ , for $n=2$ we get $17$ , for... |
H: Decomposition an operator in terms of symmetric and anti-symmetric components
In linear algebra, we can write any operator as the sum of a symmetric and skew-symmetric parts:
$$A=A^{\mathrm{sym}}+A^{\mathrm{skew}}$$
where
$$A^{\mathrm{skew}}=\frac{1}{2}(A-A^T)$$
and
$$A^{\mathrm{sym}}=\frac{1}{2}(A+A^T)$$.
Can the ... |
H: What is meant by "constant" in the optional stopping theorem?
One of the three condition of the optional stopping theorem is that "There exists a constant $c$ such that $|X_{t\wedge \tau}| \leq c$ a.s. for all $t\in \mathbb N_0$".
In the article of Wikipedia on the optional stopping theorem, the second item in the ... |
H: Inequality for compact operator between Banach spaces
I've been pondering about the following Lemma for a while now, but can't think of a proof. In fact, I can't even think of a way to prove it.
Let $E$, $F$ and $G$ be Banach spaces, $T \in \mathcal{K}(E,F)$ (e.g. a compact operator) and $S \in \mathcal{L}(F,G)$ in... |
H: What justifies assuming that a level surface contains a differentiable curve?
My textbook's proof that the Lagrange multiplier method is valid begins:
Let $X(t)$ be a differentiable curve on the surface $S$ passing through $P$
Where $S$ is the level surface defining the constraint, and $P$ is an extremum of the f... |
H: If 5 coins are flipped what is the probability of getting only one head?
How would I do this question? I know if the question said: at least one head then I would do:
${5\choose0}=1$
$2^5=32-1 = 31 $
AI: We assume that the coin is fair and is flipped fairly.
There are $2^5$ equally likely strings of length $5$ made... |
H: Non-vanishing vector field on $\mathbb{R}P^{2n+1}$
I'm trying to cook up a non-vanishing vector field on $\mathbb{R}P^{2n+1}$. I know that $S^{2n+1}$ admits one, namely $(x_1,\dots,x_{2n+2})\mapsto (-x_2,x_1,\dots,-x_{2n+2},x_{2n+1})$. Moreover, I know that $S^{2n+1}$ is a smooth double cover of $\mathbb{R}P^{2n+1}... |
H: Understanding a proof about the Broccard problem
I was reading the paper by Berndt and Galway,"The Brocard–Ramanujan diophantine equation $n!=m^2$. And I got stuck in the part when it says:
(1)$$n!+1=m^2$$
(2)$$\left ( \frac{n!+1}{p} \right)=1 \text{ or }0$$
Let us say that we have a “solution” if (2) holds for eac... |
H: To which (logical) language belongs $\{p\} \Rightarrow q$?
According to my book, the essential difference between a logical implication $\{p\} \Rightarrow q$ and the statement $p \to q$ is that $p \to q$ is part of the propositional language, and $\{p\} \Rightarrow q$ is not. My question is, to which language does... |
H: Proving the limit superior exists
Suppose $(x_n)$ is a bounded sequence. Let $y_n = \sup \{x_n, x_{n + 1}, x_{n+2}, \dots \}$. Prove that $(y_n)$ converges.
The book said that
Clearly $(y_n)$ is decreasing and bounded below by a lower bound of $(x_n)$
Can someone explain to me why is it clearly decreasing? I h... |
H: Mathematical induction solution I don't understand
$$T(k) = 2T(\frac{k}{2})+k^2$$
$$T(k)\leq 2(c(\frac{k}{2})^2\log(\frac{k}{2}))+k^2$$
$$T(k)\leq \frac{ck^2\log\frac{k}{2}} { 2} + k^2$$
$$T(k)\leq \frac{ck^2logk}{2} - \frac{ck^2}{2} + k^2$$
$$T(k)\leq ck^2logk$$
There's something I don't understand with the last t... |
H: Understanding a Gram-Schmidt example
Here's the thing: my textbook has an example of using the Gram Schmidt process with an integral. It is stated thus:
Let $V = P(R)$ with the inner product $\langle f(x), g(x) \rangle = \int^{1}_{-1}f(t)g(t)dt$.
Consider the subspace $P_2(R)$ with the standard ordered basis $\bet... |
H: $\lim_{x \to\infty}\left(\frac{\pi}{2}-\tan^{-1}x\right)^{\Large\frac{1}{x}}$ Why aren't these two limits equal when they should be?
I was trying to solve this problem, which says to find the limit:
$$\lim_{x\to\infty}\left(\frac{\pi}{2}-\tan^{-1}x\right)^{\Large\frac{1}{x}}$$
Since the OP said the answer was one, ... |
H: What does it exactly mean for a subspace to be dense?
My understanding of rationals being dense in real numbers:
I know when we say the rationals are dense in real is because between any two rationals we can find a irrational number. In other words we can approximate irrational numbers using rationals. I think a mo... |
H: three dimensional cross product
Why do two three dimensional vectors $x$ and $y$ such that $x\cdot y$ does not equal $x\times y$ do not not exist?
They do not exist right? Please help me kinda lost in this.
AI: The dot product is a scalar quantity. The cross product is a vector quantity. To compare a scalar and ... |
H: Expected value of a stochastic harmonic series
It doesn't seem straightforward to put this into mathematical notation, but I'll do my best to explain the setup. Consider a harmonic series of the following type. For the sake of argument, say we have a set of 5 elements which are the reciprocals of the first 5 natu... |
H: How find the $f(x+f(y))+f(y+f(z))+f(z+f(x))=0,$
:$f:R\longrightarrow R$ ,and is continuous
such that $$f(x+f(y))+f(y+f(z))+f(z+f(x))=0,$$
find all $f$
AI: There are two solutions: either $f(x) = c-x$ for some constant $c$, or $f(x) = 0$ for all $x$.
We claim that $f(x+f(y)) = f(x) - f(y)$.
Put $x = y = z = 0$,... |
H: Conditions for Fubini's theorem
To preface this post, I have to admit that I have extremely little measure theory knowledge and I get lost when trying to read about Fubini's theorem for this reason. In the theorem statement for Fubini's theorem, it says that
$$\int_{A\times B}|f(x,y)|d(x,y) < \infty.$$
I get that i... |
H: When can you treat a limit like an equation?
Lately, I've been very confused about the weird properties of limits. For example, I was very surprised to find out that $\lim_{n \to \infty} (3^n+4^n)^{\large \frac 1n}=4$ , because if you treat this as an equation, you can raise both sides to the $n$ power, subtract, a... |
H: Solving a system of equations, why aren't the solutions preserved?
I have the equations $$6x^2+8xy+4y^2=3$$
$\qquad$ $\qquad$ $\qquad$$\qquad$$\qquad$$\qquad$$\qquad$$\qquad$$\qquad$and $$2x^2+5xy+3y^2=2$$
This question can be found here, and the answer written by "response" went like this:
Multiply the second by 8... |
H: Proof that a Hilbert space is not locally compact.
The proof in my textbook is: Construct $B(x,\epsilon)$. If $x=\langle x_{1},x_{2},\dots\rangle$, defined a sequence $x_{n}=\langle x_{1},x_{2},\dots,x_{n}+\epsilon,\dots\rangle$. Clearly, $d(x_{i},x_{j})=\sqrt{2}\epsilon$ when $i\neq j$. Since the sequence $\langle... |
H: How can we prove that this triangle is Equilateral Triangle?
This is a problem which was sent to me by a friend , but i couldn't solve it , in particular , i don't have ideas for that .
I hope you can help by hints or any thing .
Here is the problem in the image .
AI: This is known as Morley's trisector theorem. |
H: Isomorphisms on categories
This time I have a conceptual question. Reading Categories for the Working Mathematician, I noticed the author gives two different notions between isomorphisms:
1) An isomorphism $T:B\rightarrow C$ of categories is a functor $T$ from $B$ to $C$, which is bijection both on objects and arro... |
H: Logical systems that are complete but not sound
I was wondering, are there any commonly used logics(with both notions of deductions and of semantics) that are complete but not sound? I'm looking for an example that has actually proven useful to logicians. Thank you for your wisdom!
Sincerely,
VIen
AI: If a logical ... |
H: Finding an orthonormal basis using Gram Schmidt process
OK, here's a question with polynomials. We want to find an orthonormal basis using Gram Schmift.
Assuming that we are in a vector space V, $R^2[X]$ where {$f = \lambda_0+\lambda_1X+\lambda_2X^2$}. Since X is a polynomial it gets treated differently than if it... |
H: A generalization of Waring's problem
Let $f(x)$ be a polynomial with integer coefficients such that $$\lim_{x\to +\infty}f(x)=+\infty.$$
Is it true that there always exist two integers $K$ and $R$ (depend on $f(x)$), such that every positive integer $n$ can be expressed as $$n=r+\sum_{i=1}^kf(x_i),$$
where $|r|\leq... |
H: All distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$
This question is from a past exam.
Find all distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$
Attempt/Thoughts?
Since $\mathbb{Z}_4$ is cyclic we are looking for cyclic subgroups of the g... |
H: Why is $V_{4}$ the semi direct product of $Z_{2}$× $Z_{2}$
I'm trying to understand what is a semi direct product , so by the definition semi-direct product of G , I'd need two groups , $N$ and $H$ , where :
$H∩N$ = {e}
$H \cdot N$ = $G$
If $H=N=Z_{2}$ , then : $Z_{2}∩Z_{2}≠{e}$ .
Which contradicts $H∩N$ = {e}... |
H: For any set $A\subseteq\mathbb{R}^n$, we have $ \overline{A^{\circ}} = \overline{\overline{A^{\circ}}^{\,\circ}}$
I have to prove that for any set $A\subseteq\mathbb{R}^n$,
$$ \large\overline{A^{\circ}} = \overline{\overline{A^{\circ}}^{\,\circ}} $$
This is what I got so far: for any set $A$ I'm using these definit... |
H: Determinant with polynomials
Question
In $\Bbb R _{n-2}[X]$, let $f_1(x),\dots, f_n(x) \in \Bbb R _{n-2}[X]$ and let $a_1,\dots, a_n \in \Bbb R$
$$A=\begin{bmatrix} f_1(a_1) &f_1(a_2) & \dots & f_1(a_n) \\ \vdots & \vdots & & \vdots \\f_n(a_1) &f_n(a_2) & \dots & f_n(a_n)\end{bmatrix}$$
What is $\det(A)$?
Thought
W... |
H: Strong convergence of operators
I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about:
A sequence of operators $T_n\in L(X, Y)$ (here, $X$ and $Y$ are Banach Spaces) converges strongly to an operator $T\in L... |
H: Proving that not defined value is equal to something
My younger brother (9th Grader) got the following maths problem-
Given: $$2^a = 3^b = 6^c$$
Prove:
$$c=\frac{a * b}{a+b}$$
From my elementary knowledge of mathematics it seems like a=b=c=0.Also, (ab)/(a+b) is not defined and not defined can be equal to 0. Which m... |
H: Proving that exists only one basis which is dual to a given basis
Question
Let $V$ be a finite dimensional vector space over $\Bbb F$ and $V^*$ it's dual space. Let $f_1 ... f_n$ be a basis for $V^*$. Prove that $\exists ! e_1 ... e_n$ - basis for $V$ s.t. $f_1 ... f_n $ is its dual basis.
Thought:
Someone show... |
H: Euler angles to rotation matrix. Rotation direction
So we have a 2D rotation matrix for counterclockwise (positive) angle "$a$":
$\begin{pmatrix}
\cos(a) & -\sin(a) \\
\sin(a) & \cos(a)
\end{pmatrix}$.
For clockwise (negative) angle:
$\begin{pmatrix}
\cos(a) & \sin(a) \\
-\sin(a) & \cos(a)
\end{pmatrix}$.
When conv... |
H: Find the 3rd degree of polynomial having trigonometry as a root
I am struggling to solve the problem that asking to find the 3rd degree of polynomial having all coefficients as integers, which satisfying
$f(\cos \frac{\pi}{7})=0$.
I know that I should use the euler's equation, but I don't know how to apply it to ... |
H: Computing determinant of X-matrix using permutations and blocks
Calculate the following determinant $$\det\begin{bmatrix} a_1 & 0 & ... &0&b_1 \\ 0&a_2 & ... &b_2&0 \\ &&... \\ 0 & b_{2n-1} & ... &a_{2n-1}&0\\ b_{2n} & 0 & ... &0&a_{2n}\end{bmatrix}$$
We thought about using the idea of a determinant of blocks. U... |
H: How do I show that $\int_0^\infty \frac{\sin(ax) \sin(bx)}{x^{2}} \, \mathrm dx = \pi \min(a,b)/2$
Recently I found a claim saying that
$$
\int_0^\infty \left( \frac{\sin ax}{x}\right)\left( \frac{\sin bx}{x}\right) \mathrm{d}x= \pi \min(a,b)/2
$$
from what I can see this seems to be true. I already know that
$\... |
H: Must certain rings be isomorphic to $\mathbb{Z}[\sqrt{a}]$ for some $a$
Consider the group $(\mathbb{Z}\times\mathbb{Z},+)$, where $(a,b)+(c,d)=(a+c,b+d)$. Let $\times$ be any binary operation on $\mathbb{Z}\times\mathbb{Z}$ such that $(\mathbb{Z}\times\mathbb{Z},+,\times)$ is a ring. Must there exist a non-square ... |
H: For what $a>0$ does this integral converge? $\int_{1}^{\infty} {\frac{x^a(\sin x+2)}{x^{2a}\ln(a)}}$
For what $a>0$ does this integral converge?
$$\int_{1}^{\infty} {\dfrac{x^a(\sin x+2)}{x^{2a}\ln(a)}}$$
So I said:
$\left|\dfrac{x^a(\sin x+2)}{x^{2a}\ln(a)} \right| \leq \left| \dfrac{3x^a}{x^{2a}\ln(a)} \righ... |
H: If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
Was revising for a Linear Algebra when I came across this question.
"Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$,
show that rank($A$) $\leq n/2$"
My attempt:
I was thinking of the theorem from Cayley-Hamilton but... |
H: system of differential linear equations $y'=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}y$
find the solution to the problem $y'=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}y, y(0)=\begin{pmatrix}4\\0\end{pmatrix}$
I know i have to find the eigenvalues and eigenvectors of the matrix $A=\begin{pmatrix}1 & 1\\0 & 1\end{pmat... |
H: Am I understanding this integration right?
This is the snippet of a problem from this PDF here. What I dont understand is why they retain the $Sin$ part for evaluation after integration when all that it is going to evaluate to is 0.
If I am not wrong, $Sin(n\pi) = 0$
All that should be left after integrating by... |
H: Smallest n digit number that can divide a n digit number
Is there any simple way to find the smallest n digit number that can divide n digit number.
For Example:
Lets take a two digit number xx. I want to find the smallest two digit(yy) number that can divide the number xx.
And if its three digit xxx then i need t... |
H: $Var(X-Y)$ strange outcome
THe time $Y$ between the arrival of two jobs has the probabilitiy density
$$
f(y) = \frac{1}{2}e^{-\dfrac{y}{2}}
$$
The duration, $Z \sim Unif[1,3]$ and is independent of Y
Assume that it is difficult to determine the probability densities of $Y$ and $Z$ and that one only has informat... |
H: Fixed point and non-fixed point function
For constructing another proof I need two functions explicitly and therefore
I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the closed interval $[a,b] \subset \mathbb{R}$ where it is defined) has always ... |
H: Integral of $\frac{2}{(x^2+2x+5)^{1/2}}$
Please, I am stuck here for hours. Can somebody help me find the following integral $$\frac{2}{(x^2+2x+5)^{1/2}}$$ Only thing I can do is turn this into $$\frac{2}{(x^2+2x+1+4)^{1/2}} = \frac{2}{((x+1)^2+2^2)^{1/2}}$$ Don't know if that helps. Thank you!
AI: Hint
$$\sqrt{x^2... |
H: Consider a set equipped with two idempotent functions that commute.
Consider a set $X$ equipped and two functions $f,g : X \rightarrow X$. Assume $f$ and $g$ commute with each other. Finally, call $x \in X$ a fixed point of $f$ iff $f(x)=x.$
Then we can show that if $x$ is a fixed point of $f$, then so too is $g(x)... |
H: Question on Cauchy filters
Let $(X,\mathcal{V})$ is a uniform space and $\xi$ is a Cauchy filter on $(X,\mathcal{V})$.
$o(\xi)$ is the family of all open subsets of $X$ containing at least one element of $\xi$.
what does "containing at least one element of $\xi$" mean? that is, for every elements $V$ of $o(\xi)... |
H: How to check if three coordinates form a line
Assume I have three coordinates from a world map in Longitude + Latitude. Is there a way to determine these three coordinates form a straight line?
What if I was using a system with bounds that defines the 2 corners (northeast - southwest) in Long/Lat?
The long & lat ar... |
H: I didn't understand the proof of the chain rule
From a lecture note about analysis:
For differentiable functions $f$ and $g$
$$
(g\circ f)'(x_0)=g'(f(x_0))f'(x_0).
$$
Proof: Consider the limit $$\lim_{x \to x_0}{\frac{g(f(x))-g(f(x_0))}{x-x_0}}.$$ Note that $$\lim_{x \to x_0}{\frac{g(f(x))-g(f(x_0))}{x-x_0}}=\lim_{... |
H: Modification of the triangle inequality
We know from the triangle inequality that $X+Y \geq Z$,
My question is under what conditions of $a,b,c$ (acute, obtuse or right angle) that $Z >X $ and $Z \geq Y $
AI: The sine rule tells you that
$$\frac{Z}{\sin c} = \frac{X}{\sin a}$$
So $Z > X$ iff $\sin c > \sin a$. S... |
H: Evaluation of $\lim_{N\to\infty}\sum_{k=1}^N \frac{1}{k+\alpha N}$
The series
$$S(\alpha,N)=\sum_{k=1}^N \frac{1}{k+\alpha N}$$
can be evaluated as:
$$S(\alpha,N)=\Psi(N+1+\alpha N)-\Psi(1+\alpha N)$$
The limit $$L(\alpha)=\lim_{N \to \infty}S(\alpha)$$
can be evaluated for different values of the parameter $\alph... |
H: Is the centralizer of a normal subgroup equal to the centralizer of the intersection of that normal subgroup with the commutator subgroup?
Let $G$ be a group, $N$ a normal subgroup of $G$, $C_G(X)$ the centralizer of $X$ in $G$ and $G'$ the commutator subgroup of $G$. Is it true that $C_G(N) = C_G(N \cap G')$?
If t... |
H: In how many ways can 5 identical balls be placed in a (3x3) grid such that each row contains atleast one ball?
I tried two methods here. First, the usual one of checking different arrangements with a 3,1,1 distribution and the other with a 2,2,1 distribution and calculating their respective combinations. Works out ... |
H: Isolated Singularities
Consider the following functions and determine which kind of singularities they have in $z_0$. If it is a removeable singularity, then calculate the limit; if it is a pole, then give the order of the pole and the main part.
$$
f(z)=\frac{1}{1-e^{z}}, z_0=0~~~~~~~~~~~~~~~~~~~~g(z)=\frac{... |
H: A proof in circles
The question is this:
Show that the equation of a straight line meeting the circle $ x^2 + y^2 = a^2 $ in two points at equal distances $d$ from a point $(m, n)$ on its circumference is
$ mx + ny - a^2 + \dfrac {d^2} {2} $
I am confused. Hints please!
EDIT:
I tried to relate the perpendicular di... |
H: Convergence of dual product of weakly and strongly convergent sequences
Say $V$ is a reflexive> Banach space, so that we can identify the dual of the dual $V'$ with $V$, i.e. $(V')'=V$.
Consider two sequences $\{v_k\} \subset V$ and $\{f_k\} \subset V'$ that converge to $v\in V$ and $f\in V'$ in the strong and in ... |
H: A group of order $8$ has a subgroup of order $4$
Let $G$ be a group of order $8$. Prove that there is a subgroup of order $4$.
I know that if $G$ is cyclic then there is such a subgroup (if $G=\langle a\rangle$ then the order of $\langle a^2\rangle$ is $4$). But how do I prove this when $G$ is not cyclic? Also, I k... |
H: Convergence in probability of the sum of scheme of series
Could you please help with this one. It looks like smth simple but I can't figure it out.
Let $\{x_{in}\}, \ i=1,\dots, n, \ n=1,\dots,\infty$ be the scheme of series of random variables. For each $n$ random variables $\{x_{in}\}, \ i=1,\dots,n,$ are i.i.d. ... |
H: Potential function & constant
$\mathbf{Question:}$
Give a piar functions $Φ:\Bbb R^2 \to \Bbb R $ and $Ψ: \Bbb R^2 \to \Bbb R$, it is often useful to known that there exists some contiunously differentiable function $f:\Bbb R^2 \to \Bbb R$ such that $\partial f/\partial x(x,y)=Φ(x,y)$ and $\partial f/\partial y(x,y... |
H: Process of finding a solution for a problem
I have a question that doesn't need an answer, just an explanation.
Prove that if $A$ is nilpotent, then $I+A$ is invertible.
The solution is:
Given that $A^k=0$ for some integer $k$, you can say that
$I - A^k = I$
you can factor this as
$(I + A)(I - A + A^2 - A^3 +\cdots... |
H: e-notation scientific notation
Hey all I asked this over at StackOverflow =) and I got a good answer but I still have no idea whats going on. I want to know how the expression got to the answer and maybe the math behind how it was reasoned.. the steps if you would be so kind. The likelihood is I don't understand th... |
H: When can a non-autonomous system NOT be re-written as an autonomous system?
Consider Duffing's equation
$\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$
where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real parameters, $t$ represents time and $\dot x := dx/dt$.
Since there is an expl... |
H: Dirichlet integral
I want to prove
$$\displaystyle\int_0^{\infty} \frac{\sin x}x \,\mathrm{d}x = \frac \pi 2$$
and
$$\displaystyle\int_0^{\infty} \frac{|\sin x|}x \,\mathrm{d}x \to \infty$$
And I found in wikipedia, but I don't know, can't understand. I didn't learn differential equations, Laplace transforms, an... |
H: Probability of guessing two out of five answers in multi-choice test
I am doing some exercises for a probability class I am taking, but the answer of one confuses me quite a bit. The problem is to calculate the probability of getting two of fewer correct answers on a multi-choice test with three alternatives per qu... |
H: Can $\sum_{x \in [0,1]} e^x$ be represented as an integral?
In $$\sum_{x \in [0,1]} e^x,$$ $e^x$ is summed over all values in the interval $[0,1]$.
Am I right to say that $$\sum_{x \in [0,1]} e^x = \int^{x=1}_{x=0} e^x \, \mathrm dx?$$
AI: No, they're not equal; the sum of uncountably many positive numbers is nece... |
H: Does $f(n)\sim g(n)$ imply $\lim_{k\to\infty} \frac{1}{k} \sum_n f(n)/g(n) = 1$?
Is it true that
$$\lim_{k\to\infty}\frac{1}{k}\sum_{n=1}^k \frac{f(n)}{g(n)} = 1 \leftrightarrow f(n)\sim g(n).$$
My thought: $f(n)\sim g(n) \to \frac{1}{k}\sum \frac{f(n)}{g(n)} = 1$ since we can find an arbitrarily small $\epsilon$ w... |
H: How to simplify a sum with factorial?
I have the following sum:
$$\sum_{n=1}^k \frac{k!}{n!(k-n)!}, \quad k=9$$
wolfram alpha
It got simplified to $2^k-1$. How can I do it with math formulas?
Thank you!
AI: Use the binomial theorem, which states:
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$$
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