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H: In what type of series can we apply convergence tests?
Let $(a_n)_n \in \mathbb{R}$ and let $(z_n)_n \in \mathbb{C}$ be two sequences, and let $f_n(x) \in \mathbb{R}$ and $g_n(z) \in \mathbb{C}$ be two sequences of functions.
To check if $\sum a_n \in \mathbb{R}$ converges we can use tests such as the root test, t... |
H: Show that there exist $a_1,\ldots, a_{2n-1}$ such that $ a_{2n-1}J^{2n-1}+\cdots+a_1 J=I_n,$ where $J$ is a Jordan matrix
Let $J\in\mathbb{C}^{n\times n}$ be a Jordan normal form and assume that ${\rm tr~}J<2n$. Prove or disprove that there exist $a_1,\ldots, a_{2n-1}\in\mathbb{R}$ such that
\begin{equation}
a_{2n... |
H: Understanding some points in the proof that the Thomae function is continuous at irrationals (pg.74 in Petrovic book)
Here is the proof as it is given in the book:
My questions are:
1- Why the author assumed that $a \in (0,1),$ what about the case of $a=0$?
2- Why if $f(x) < \epsilon$ then $|f(x)| < \epsilon$ i... |
H: Addressing the probability of a category as a whole
I think this is rather an English language question, but I've asked this in ell.sx, and a person there insists that this is a concern of mathematics.
Let's say I have $k_g$ green, and $k_r$ red balls. I want to select one, randomly, but with a bias towards the re... |
H: Showing that the differential is an immersion
If $f: X \rightarrow Y$ is an immersion of smooth manifolds, then show that $df: TX \rightarrow TY$ is also an immersion.
The definition of immersion(when dim$X <$ dim$Y$) that I have is that for $f: X \rightarrow Y$, $f$ is an immersion if $df_{x}: TX \rightarrow TY... |
H: Efficient methods to calculate incomplete beta $B[a,b;x]$ for $b=0$
I am looking for an efficient numerical method (or a module) to calculate the incomplete $\beta-$function for $b=0$.
e.g. https://www.wolframalpha.com/input/?i=incomplete+beta%5B4%2F5%2C1.5%2C0.0%5D+
Most modules e.g. scipy.special.incbeta in Pyth... |
H: Orthogonality relation of eigenvectors for a self-adjoint operator
So everyone knows eigenvectors corresponding to different eigenvalues are orthogonal to each other, given that the operator is self-adjoint.
If we have a self-adjoint operator, say $L$, is it possible that $\exists u, v$ such that $Lu=\lambda u$, $L... |
H: Evaluating $\sum_{n=1}^\infty\frac{1}{4n(2n+1)}$
How to evaluate this sum, derived from "Lockdown math" by 3Blue1Brown?
$$\sum_{n=1}^\infty\frac{1}{4n(2n+1)}$$
AI: $$\frac{1}{4n(2n+1)}=\frac{1}{4n}-\frac{1/2}{2n+1}=\frac{1}{2}\left(\frac{1}{2n}-\frac{1}{2n+1}\right)$$
Hence
$$\sum_{n=1}^{\infty} \frac{1}{ 4n(2n+1)... |
H: Quantifier question $!\exists x ! \exists y \forall w(w^2>x-y)$
I have a question about the following quantified sentence if it is true or false.
$!\exists x ! \exists y \forall w(w^2>x-y)$ for the real numbers
I think this this is true because if take two negative numbers $x<y$ then it will work Like if you have ... |
H: The area of a triangle determined by two diagonals at a vertex of a regular heptagon
In a circle of diameter 7, a regular heptagon is drawn inside of it. Then, we shade a triangular region as shown:
What’s the exact value of the shaded region, without using trigonometric constants?
My attempt
I tried to solve it... |
H: Deep doubt on a double surface integral
I don't understand how to proceed with an exercise. I will write down what I have done so far.
The exercise is:
Evaluate the following integral $$\iint_{\Sigma}\dfrac{1}{x^2+y^2}\ \text{d}\sigma $$
Where $\Sigma = \{(x, y, z): x^2+ y^2 = z^2+1,\ 1\leq z \leq 2 \}$
My attemp... |
H: Prove that subset of complex plane is open or is closed (or both or neither )
I have subset
$$D = \{z: \text{Re}(z) > 2, \text{Im}(z) \leq1\}$$
So i think that set is not open because contain boundary points, but how prove this?
Let's say it's open. so by defenition we have: For all points of the set, they are cont... |
H: $P(X_1
We have two independent random variables $X_1$, $X_2$, with law $Exp(\rho_i)$ respectively.
I want to find the probability of the following event $\{X_1<X_2\}$.
Is the following correct?
$P(X_1<X_2)=\int_0^\infty\int_0^{x_2=x_1}\rho_1\rho_2 e^{-(\rho_1x_1+\rho_2x_2)}dx_2dx_1=\int_0^\infty-\rho_1[e^{-(\rho_1x... |
H: If $\alpha$ is a cycle of length $k$. Then $o(\alpha)=k$
I found a proof of this theorem wich says:
$ Proof: $ If $ \alpha=(i_1,i_2,...,i_{k-1},i_{k}) $ . Then,
$\alpha^{k}(i_j)=\alpha^{j}\alpha^{k-j}(i_j)=\alpha^{j}(i_k)=i_j, \; \forall\; j $
Thus, $ \alpha^{k} =1$ and $ o(\alpha) \leq k $.
Now, if $ 1\leq s < k... |
H: Find the minimum value of $x+2y$ given $\frac{1}{x + 2} + \frac{1}{y + 2} = \frac{1}{3}.$
Let $x$ and $y$ be positive real numbers such that
$$\frac{1}{x + 2} + \frac{1}{y + 2} = \frac{1}{3}.$$Find the minimum value of $x + 2y.$
I think I will need to use the Cauchy-Schwarz Inequality here, but I don't know ho... |
H: Proof verification: the characteristic of an integral domain $D$ must be either 0 or prime.
Claim: the characteristic of an integral domain $D$ must be either 0 or prime.
Here is my attempt: Assume $D$ is an integral domain. Assume $k$ is the characteristic of $D$. Let $a \in D\setminus \{0\}$. Aiming for a contra... |
H: Distribution of $Y$ if $Y=X$ if $|X|\leq c$ and $Y=-X$ if $|X|>c$, $X\in N(0,1)$
Let $X\in N(0,1)$, and $c\geq 0$.
$Y$ is defined as
$Y=\begin{cases}X & \text{for}\quad |X|\leq c,\\
-X&\text{for}\quad |X|>c.\end{cases}$
What is the distribution of $Y$? I would guess it has a normal distribution given that
$X\ov... |
H: Same remainders of a sequence
Show that for the sequence $x_{1}=9$ and $x_{n+1} = 9^{x_{n}}$ the remainders for the third and fourth term are equal when divided by $100$. Determine this remainder.
So the second term seems to be $x_{2}=9^9$ and therefore the third and fourth $x_{3}=9^{9^9}, x_{4}=9^{9^{9^9}}$.
Is ... |
H: Regular closed subset of H-closed space
An H-closed space $X$ is a topological space which is closed in any Hausdorff space in which it is embedded. A well-known characterization is that $X$ is H-closed iff every open cover of $X$ has a finite proximate subcover, i.e. a finite subcollection whose union is dense.
... |
H: Convergence/dicergence
My series has a general term $\frac{(1+\frac{1}{n})^{n^2}}{e^n}$.
I found that the Root test is inconclusive here. Wolfram says to use "limit test". Is that the limit comparison test? Which series can I compare this one to? I know it should diverge.
AI: Let $u_n$ be the general term.
$$\ln(u_... |
H: Measure theory: motivation behind monotone convergence theorem
I am watching a very nice set of videos on measure theory, which are great. But I am not clear on what the motivation is behind the monotone convergence theorem--meaning why we need it?
The statement of the theorem is that given a set of functions $\\{... |
H: Clarification of finding this transition probability matrix
Let $X_n$ denote the two-state Markov chain with transition probability matrix
P=
$
\begin{bmatrix}
\alpha & 1-\alpha\\
1-\beta & \beta
\end{bmatrix}
$
given states 0 and 1. Let $Z_n=(X_{n-1},X_n)$ be a Markov chain over the four states (0,0), (0,1), (1,0)... |
H: Is $x^5-2x+4$ irreducible in $\mathbb{Q}[x]$?
I was asked this question on my exam but I didn't know how to solve it. Eisenstein criterion can't be applied, rational roots theorem shows that it doesn't have a linear factor because possible roots are $\lbrace 1,-1,2,-2,4,-4 \rbrace$ and it doesn't seem to be irreduc... |
H: Is X always in any open neighborhood system for the topological space X, t?
I have just been recently introduced to the concept of open neighborhood systems, but the fact that X is in any open neighborhood system $N_x$ in X, $t$ is not self-evident to me. Is this always true?
AI: Yes. By definition any topology on ... |
H: When the piecewise constant integral independs of the partition's choice?
Proposition
Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ be a function. Suppose that $\textbf{P}$ and $\textbf{P}'$ are partitions of $I$ such that $f$ is piecewise constant both with respect to $\textbf{P}$ and with respect to $... |
H: Right Triangle within another right triangle inside a square
As you can see in the below picture, We have a right triangle inside a big square, and within the triangle there is another small right triangle.
The question as follows: Find the length of AB "With" using BE and FGin your solution.
Well I came up with th... |
H: Differentiation of a function with 2 variables
I have a function:
$$
f: R^2 \to R
$$
Which satisfies:
$$
x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = 0, \forall (x,y) \in \mathbb{R}^2
$$
Now the answers get in the end to:
How did they conclude that the right side sums to zero? I assume that... |
H: Prove that if a sequence converges (in a metric space), then every subsequence does always converge as well.
Let $(x_{n})_{n=m}^{\infty}$ be a sequence in $(X,d)$ which converges to some limit $x_{0}$. Then every subsequence $(x_{f(n)})_{n=m}^{\infty}$ of that sequence also converges to $x_{0}$.
My solution
Let $\... |
H: When does a substructure of an algebraic structure exist? (from Fraleigh)
I read in Fraleigh, A first course in Abstract Algebra, that
If we have a set, together with a certain type of algebraic structure (groups, rings, integral domains, etc.), then any subset of this set, together with a natural induced algebrai... |
H: Does this strategy of characterizing poles always work?
I stumbled upon a fast way to characterize poles of order $m$ of a meromorphic function $f$ (on some open set $\Omega$) in this answer here. My question is, does this general strategy always work?
Here's why I think it doesn't always work. Consider
$$f(z) = \... |
H: Understanding Fraleigh's proof of: Every finite integral domain is a field
Here's how Fraleigh proves: Every finite integral domain is a field in his book:
Let
\begin{equation*}
0, 1, a_1, \dots, a_n
\end{equation*}
be all the elements of the finite domain $D$. Now, consider
\be... |
H: t statistic formula
I just have a quick question about t statistic. Of these two, which formula is correct?
$$t(x) = \frac{\bar{X} - \mu }{\sqrt{\frac{1}{n}\sum \left ( X_{i} - \bar{X}\right )^{2}}}\sqrt{n-1}$$
$$t(x) = \frac{\bar{X} - \mu }{\sqrt{\frac{1}{n-1}\sum \left ( X_{i} - \bar{X}\right )^{2}}}\sqrt{n}$$
I... |
H: On a bus, 1/10 of the total number of passengers get off at one spot, then 1/3 of the rest...
On a bus, 1/10 of the total number of passengers get off at one spot, then 1/3 of the rest (the passengers left on the bus) get off at another point. There are 42 passengers left on the bus. What's the total number of pass... |
H: Show that $\mathbb{I}_A$ is continuous at $x_0$ if and only if $x_0$ is not a boundary point of $A$
Let $A \subset\mathbb{R}$ and $\mathbb{I}_A:\mathbb{R}\longrightarrow \mathbb{R}$ the indicator function of A. Show that $\mathbb{I}_A$ is continuous at $x_0$ if and only if $x_0$ is not a boundary point of $A$. If $... |
H: How do we prove that compact spaces in metric spaces are bounded?
Let $(X,d)$ be a compact metric space. Then $(X,d)$ is both complete and bounded.
My solution
The space $(X,d)$ is indeed complete. This is because every Cauchy sequence which admits a convergent subsequence is also convergent.
Now it remains to prov... |
H: Maximization inequality for Frobenius norm after adding orthogonal matrix
Let $A$ be a matrix and $Q$ be an orthogonal matrix such that $AQ^T$ is symmetric, positive semidefinite. Show that $$||A+Q||_F\geq||A+P||_F$$ for any orthogonal matrix $P$. Here, $||\cdot||_F$ is the Frobenius norm.
AI: Let $AQ^T=VDV^T$ be a... |
H: Find a number $n \neq 2017$ such that $\phi(n) = \phi(2017)$
Find a number $n \neq 2017$ such that $\phi(n) = \phi(2017)$, as above. I know the formula for a general $\phi$ function, but I cannot see how this is helpful here. Any help would be appreciated!
AI: Hint: $2017$ is a prime number.
Solution: Fortunately, ... |
H: What does it mean to take the “ gradient with respect to the position $r_ i$”?
Let’s say we have a number of particles (charged, massive or anything that can create potential energy). The total potential energy of any particle can be given by $$U_i (\vec r_1, \vec r_2, ... \vec r_N) = \sum_{j\gt i}^{N} U_{ij}(|r_i... |
H: Two players put fill $1$ and $0$ in a $3\times 3$ matrix and compute its determinant when it is full. Can Player $0$ win if $1$ starts at the center?
In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty 3 × 3 matrix. Player 0 counters with a 0
in a vacant position, and play continues in turn until the 3 × 3 ... |
H: Meaning of "$\exp[ \cdot ]$" in mathematical equations
I am reading book "Fuzzy Logic With Engineering Applications, Wiley" written by Timothy J. Ross. I am reading chapter 7 and in this chapter, "Batch Least Squares Algoritm" has been defined. It illustrates the development of a nonlinear fuzzy model for the data ... |
H: Closed ball is weakly closed
The problem is in a Banach space, if $||x_n||\leq 1$ and $x_n\to x$ weakly, then $||x||\leq 1$
This question has an answer here: math.stackexchange.com/questions/714049/closed-unit-ball-in-a-banach-space-is-closed-in-the-weak-topology
But for convenience, I will repost the answer.
"If $... |
H: When the increasing of sequences $a_n$ leads to increasing of $\frac{1}{a_n}$?
I was asked the following question:
When the increasing of sequences $a_n$ leads to increasing of $\frac{1}{a_n}$?
I could not think of an example. It looks like if $a_n$ is increasing then $\frac{1}{a_n}$ is decreasing. Is it true? If... |
H: Opposite determinant in Autonne-Takagi factorization
Let us consider a complex symmetric matrix in $M_2(\mathbb C)$
\begin{equation}
A = \begin{pmatrix}
x_1+ix_2 & x_3 \\
x_3 & -x_1+ix_2
\end{pmatrix}
\end{equation}
where the $x_i\in \mathbb R,\;\;i=1,2,3$. The Autonne-Takagi factorization theorem tells that a unit... |
H: Do I have the chain rule right?
I was revising chain rule and I made up a problem to write down in my notes that uses it at least two times. Here it is, if a function $\zeta(x) = (z(x))^2$ where $z(x) = x + f(x), f(x) = \ln(g(x))$ and $g(x) = \frac{1}{2}x^2$ then $\zeta'$ or $\frac{d\zeta}{dx}$ is defined as,
\begi... |
H: How to write numbers in the language of first-order set theory.
I saw this Numberphile video (link at bottom), and at around 10:10 they talk about writing numbers in the language of first-order set theory. For example, to write $0$, it showed the empty set:
$$\exists x_1\neg\exists x_2(x_2\in x_1)$$
And to write $1... |
H: An application of Uniform Boundedness Theorem
I did this problem before
"Let $X$ be a compact Hausdorff space and assume that $C(X)$ is equipped with a norm $||.||$ with which this is a Banach space. For each $x\in X$, define $\lambda_x:C(X)\to \mathbb{R}, f \mapsto f(x)$. Prove that $\sup_x ||\lambda_x||$ is bound... |
H: if $B$ is countable, then the following are equivalent
Suppose $B \neq \varnothing$. Prove the following are equivalent
${\bf A.}$ B countable
${\bf B.}$ there is a surjection $f: \mathbb{Z}_+ \to B$
${\bf C.}$ there is an injection $g: B \to \mathbb{Z}_+ $
Attempt:
(I already proved $A \implies B$) First we prove ... |
H: Find all positive integers $n$ such that $\varphi(n)$ divides $n^2 + 3$
This was the Question:- Find all positive integers $n$ such that $\varphi(n)$ divides $n^2 + 3$
What I tried:-
I knew the solution and explanation of all positive integers $n$ such that $\varphi(n)\mid n$ .
That answer was when $n = 1$, or $n$ ... |
H: Calculating the mean and standard deviation of a Gaussian mixture model of two curves
An ELO rating is a Gaussian curve with a mean and a standard deviation. Assuming there are two such ratings that belong to the same player (he's using two separate online identities so he has two separate ratings) - How would I be... |
H: Find the $n$th term of sequence in the form of $a_{n+2}=ba_{n+1}+ca_n+d$
$a_1=1$, $a_2=3$, $a_{n+2}=a_{n+1}-2a_n-1$
How do you solve this? I only solve the sequence in the form $a_{n+2}=ba_{n+1}+ca_n$ before by writing it in $x^2-bx-c=0$ but for this I don't know how to. Please help
AI: The method is fairly simil... |
H: For convex functions $f(x)>1,g(x)>1, x\in \mathbb{R}$, is the product $(f\cdot g)(x)$ necessarily convex?
From what I understand, this is how it is:
assume $f\cdot g$ is concave.
then,
$$(f\cdot g)(0)>1$$
by this and the assumption, the product must be less than $1$ for some real $x$. thus, at least one of the func... |
H: Counter-example of Jacobi identity for antisymmetric bilinear operation
For a bilinear, antisymmetric, alternating operator to be a Lie bracket, it must satisfy the Jacobi identity. I assume this is because a bilinear, antisymmetric, alternating operator does not always satisfy the Jacobi identity.
If I consider th... |
H: Show that $KL$ is parallel to $BB_1$.
The incircle of a triangle $ABC$ has center $I$ and touches $AB , BC , CA$ at $C_1 , A_1 , B_1$ respectively. Let $BI$ intersects $AC$ at $L$ and let $B_1I$ intersects $A_1C_1$ at K. Show that $KL$ is parallel to $BB_1$.
I’ve draw some lines and try to find angles but it didn’t... |
H: Two questions of the proof of $\mathbf{z}^{\oplus n}$ is a free abelian group on $A={1,...,n}$
I am reading algebra chapter 0 and have two questions about the proof:
$\mathbf{z}^{\oplus n}$ is a free abelian group on A={1,...,n} .
Proof: we frist define a function $j:A\rightarrow $$\mathbf{z}^{\oplus n}$ by $j(i)... |
H: limit of $2^n\arcsin\frac{\sqrt{y_0^2-x_0^2}}{2^ny_n}$
I want to know why $\lim_{n\rightarrow \infty}2^n\arcsin\frac{\sqrt{y_0^2-x_0^2}}{2^ny_n}=\frac{\sqrt{y_0^2-x_0^2}}{y}$ when $y_n\rightarrow y$ as $n\rightarrow \infty$. Here $y_0$ and $x_0$ are constants. I thought about using a theorem $\lim ab=\lim a\lim b$.... |
H: How many degrees of freedom are there when generating a joint probability function for $n$ different binary random variables?
To generate a proper probability function one should assign a probability to all $2^n$ possible events.
One constraint is that the sum of the probabilities of all events must be equal to one... |
H: Why does $\sum_{i=1}^n a = (n+1)a$
I'm currently working my way through Eccles's "An Introduction to Mathematical Reasoning" and an alternative proof of $\sum_{i=0}^n (a+ib) = \frac 12 (n+1)(2a+bn)$ states that $$\sum_{i=0}^n (a+ib) = \sum_{i=1}^n a + b\sum_{i=1}^n i = (n+1)a + \frac 12n(n+1)b$$ After some careful ... |
H: Why am I getting the wrong answer when I factor an $i$ out of the integrand?
Consider the following definite integral:
$$I=\int^{0}_{-1}x\sqrt{-x}dx \tag{1}$$
With the substitution $x=-u$, I got $I=-\frac{2}{5}$ (which seems correct).
But I then tried a different method by first taking out $\sqrt{-1}=i$ from th... |
H: Solve equation: $\log_2 \left(1+ \frac{1}{a}\right) + \log_2 \left(1 +\frac{1}{b}\right)+ \log_2 \left(1 + \frac{1}{c}\right) = 2$
$$
\log_2 \left(1 + \frac{1}{a}\right) + \log_2 \left(1 + \frac{1}{b}\right)+ \log_2 \left(1 + \frac{1}{c}\right) = 2 \quad \text{where $a$, $b$, $c \in N$.}
$$
Apparently, the answer... |
H: dice question : what is best?
got into a discussion what is best if you have :
- 5 dices
- 3 throws
You aim for getting 12345 or 23456 - doesnt matter which one of the 2 combinations.
First throw : 12356.
So we are missing a 4 to get either 12345 or 23456.
What is best for 2nd throw ?
To keep 235 - and throw 2 d... |
H: Determine convergence annulus of Laurent series without computing the Laurent series
Take the holomorphic function: $$\mathbb{C} \setminus \{2k \pi i \ \mid \ k \in \mathbb{Z} \} \ni z \mapsto \frac{1}{e^z - 1} \in \mathbb{C}. $$ How can we determine the annulus of convergence of the Laurent series around the poin... |
H: Doubt on conditional expected value
Let $S$ and $T$ two random indipendent variables with exponential distribution, and let $\mathbb{E}(S)=\alpha,\mathbb{E}(T)=\beta$ .
1) Find the distribution of $Y=\min(S,T)$.
$\rightarrow Y\sim Exp(\frac{1}{\alpha}+\frac{1}{\beta})$
2) Find the probability of event $\mathbb{... |
H: Prove that $AB*CD+AD*BC\ge2*A(ABCD)$
Let $ABCD$ be a quadrilateral. Then prove $(AB\cdot CD+AD\cdot BC)\geq 2A(ABCD)$, where
$A(ABCD)$ means area of $ABCD$.
AI: Define $D'$ s.t. $|A-D|=|C-D'|,\ |C-D|=|A-D'|$ so that ${\rm area}\
ABCD = {\rm area}\ ABCD'$
Further \begin{align*} 2{\rm area}\ ABCD' &=|B-C||C-D'|\sin\ ... |
H: $A$ is a matrix of 3 rows and 2 columns while $B$ is a matrix of 2 rows and three columns. If $AB=C$ and $BA=D$, then pick the correct option
Options
A) determinant of C and D are always equal
B) determinant of C is zero
C) determinant of D is zero
The reasoning should be really simple.
$$|C|=|A||B|$$
$$|D|=|B||A... |
H: What is the history of the different logarithm notations?
There are two different notation of writing logarithms. In my country (Indonesia), the notation of writing logarithms is
$$
^a\log b
$$
but the most commonly used notation is
$$
\log_{a}b
$$
I'm used to the widely used notation. I've searched the web for why... |
H: If densities converge then the corresponding RV converge in distribution
I tried to prove the following
Theorem: Given $(X_n)_{n\in\mathbb{N}}$ iid. random variables with $\mathbb{E}[X_i^2]<\infty$. If the rv's have respective densities $(f_n)_{n\in\mathbb{N}}$ and $f_n\rightarrow f$ pointwise, it follows that $X_... |
H: Prove big O for a recursive function
Let
$t(n):=\begin{cases} \frac{2+\text{log}n}{1+\text{log}n}t(\lfloor\frac{n}{2}\rfloor) + log ((n!)^{\text{log} n}) \hspace{1cm} \text{if}\hspace{0.5cm} n>1 \\
1 \hspace{0.5cm} \text{if}\hspace{0.2cm} n=1
\end{cases}$
We need to prove that $t(n) \in O(n²)$, thus $t(n... |
H: Confusion regarding $\models \forall x A \equiv \forall y A[y/x]$
I have some confusion regarding the following statement given in Jean H. Gallier's book "Logic for Computer Science". It says,
For every formula $A$ $$\models \forall x A \equiv \forall y A[y/x]$$
Now if I take $A = \forall y \Phi(x,y)$, then $A[y/... |
H: Probability Question using series
There are $n$ dice with $f$ faces marked from $1$ to $f$, if these are thrown at random,what is the chance that the sum of the numbers exhibited shall be equal to $p$?
Please help me understand the first part of the solution - the number of ways in which the numbers thrown will hav... |
H: If $A,B$ and $AB$ are symmetric matrices then is $A^{-1}B^{-1}$ also symmetric.
if $ A,B$ and$ AB$ are symmetric matrices then is $A^{-1}B^{-1}$ also symmetric ?
my approach:
since $$AB=BA$$,
pre multiply by $A^{-1}$ and then post multiply by $A^{-1}$ to get
$$BA^{-1}=A^{-1}B$$
hence I was able to prove that $BA^{-... |
H: Conditionalish/joint probability?
What is the overall probability of some event happening if it has 15% chance of happening, but can only happen if another event with 40% of happening takes place ?
AI: 0.40*0.15=0.06. So, for the event 2nd to occur you need the 1st to occur. So 60 percent of time neither occurs.... |
H: Hodge star operator and exterior calculation
I am learning complex geometry by D. Huybrechts. Here is a formula that I can't understand $$\omega \wedge \beta\wedge \star \alpha=\beta\wedge(\omega \wedge \star \alpha)\tag 1$$
Here $\omega$ is the fundamental form which is a $2$-form actually. And I try to expand bot... |
H: 'Completeness' of ordered topological space
This is a follow-up question to an answer given by Henno Brandsma in this thread How to prove ordered square is compact.
In the answer it is shown that:
A non-empty LOTS (linearly ordered topological space) $(X,<)$ is compact iff every subset $A\subseteq X$ has a supremu... |
H: Finding coefficients on a complex Taylor series
I don't see a result that my book say it's straightforward. Here's my try:
Prove that the coefficients of the Taylor series of the function $$f(z)=\frac{1}{1-z-z^2}$$around $z=0$ verify $$c_0=1,\\ c_1=1, \\ c_{n+2}=c_{n+1}+c_n, n\geq 0.$$
From here, what I've done ... |
H: If $B\subset B(H)$ is a C*-subalgebra and $T\colon B''\to B''$ is linear, bounded and weakly continuous, then $\|T\|=\|T|_{B}\|$
Let $H$ be a Hilbert space and let $B\subset B(H)$ be a C*-subalgebra. Suppose that $T\colon M\to M$ is linear, bounded and operator-weakly continuous, then I want to prove that $\|T\|=\|... |
H: Why can we distribute the complex modulus? (looking for intuition/proof)
if you have,
$$ z= z_1 \cdot z_2 ... z_n$$
then,
$$ |z| = |z_1 \cdot z_2 ... z_n| = |z_1| |z_2| |z_3| ... |z_n|$$
Now, why is this equality true?
$$|z_1 \cdot z_2 ... z_n| = |z_1| |z_2| |z_3| ... |z_n|$$
AI: Think about the polar form of a com... |
H: $\lim_{x \to 0+}\log_{10}{\tan^2x}$
$\lim_{x \to 0+}\log_{10}{\tan^2x}$
What is the value of this one?
Could you please give me some hints on this problem?
AI: Let $f(x) = \log_{10}(\tan^2(x))$. Since $f(x) = f(x+\pi)$ and $f$ isn't constant, $f$ is a periodic function, so $\displaystyle \lim_{x \to \infty} f(x)$ d... |
H: When can we say that a linear transformation is equivalent to a change of basis?
I'm aware that every change of basis is a linear transformation, but the converse isn't true. What conditions must a linear transformation $T$ satisfy for us to call it a change of basis? One condition that I can think of is that $T$ s... |
H: Computing $\int_{0}^{\infty} \frac{x}{x^{4}+1} dx$ using complex analysis.
I want to compute $$\int_{0}^{\infty} \frac{x}{x^{4}+1} dx$$ using complex analysis. Now the first thing that strikes me is that $f(x)$ is not an even function. So this troubles me a bit since I would normally use
$$\int_{0}^{\infty} f(x)dx... |
H: Justification of restricting to functions with finite support in the definition of ordinal exponentiation
For ordinals $\alpha$ and $\beta$, the ordinal $\beta^\alpha$ is defined to be the epsilon-image of the well-ordered structure $\langle F,<\rangle$, where $F$ consists of the functions $f:\alpha\rightarrow\beta... |
H: What does this theorem statement mean?
This is from Royden's Real Analysis book (4th edition). On page 17, Proposition 9 says:
Every nonempty open set is the disjoint union of a countable collection of open intervals.
At first, I thought it meant that:
if $X$ is a non-empty open set then there is a set of interv... |
H: Find the number of symmetric closure
There is a set A with n elements. R is relation of set A.
R has 3 elements. When n ≥ 4, the symmetric closure of the R was obtained. Find a minimal and maximum value of number of R elements.
I want to figure out this question.
I think I can draw a matrix to prove this, but I don... |
H: When is a function not a local homeomorphism?
my background is engineering and I am very new to the topology.
I think I got the concept of a local homeomorphism, but I cannot come up with a concrete example that is a continuous surjection but not a local homeomorphism.
For example, if I have a map $f:\mathbb{R}^n\t... |
H: How to figure the next number in this sequence?
This question has been asked as a puzzle in a forum of an online game, but I coudn't solve it neither any of the members of the forum, so I asked for help here.
giving this sequence of numbers
1 > 170
2 > 344
3 > 520
4 > 698
5 > 875
6 > 1052
8 > x
16 > 2811
20 > y
Wh... |
H: Ways for the differential equation $y' + {{y\ln(y)}\over x}= xy$
I've been stuck on one of my homework numbers.
The number precise that the following equation is a non-linear equation of order 1 with x>0.
$$y' + {{y\ln(y)}\over x}= xy$$
So far, I tried 2 different methods to solve them. As suggested by internet (l... |
H: Floating Numbers in Combinations
What could be the answer to
${\displaystyle {\binom {2.5}{2}}}$
is it defined or considered as $0$ or $1$?
AI: The answer could use the gamma function:
$$\binom {2.5} 2=\dfrac{2.5!}{0.5!\times2!}=\dfrac{\Gamma(3.5)}{\Gamma(1.5)2}=\dfrac{2.5\times1.5}2=1.875.$$ |
H: Degree of pole of $\frac{1}{\cosh(z)}$
I'm having difficulties with calculating the singularity of $\frac{1}{\cosh(z)}$. So far, I have the complex zero at $z = i \frac{\pi}{2}+i \pi k$ with $k \in \mathbb{Z}$ from which it would follow that that is the only singularity. My problem is, how would I show that it is a... |
H: For (Lebesgue) measurable functions $f$ and $g$, if $f=g$ a.e., then $ \int_{E} f=\int_{E} g. $
Problem
Let $f$ and $g$ be bounded (Lebesgue) measurable functions defined on a set $E$ of finite measure. If $f=g$ a.e., then
$$
\int_{E} f=\int_{E} g.
$$
Attempt
Let $f,g:E\to \mathbb{R}$ be a (Lebesgue) measurabl... |
H: If $\text{adj}A=\left[\begin{smallmatrix}1&-1&0\\2&3&1\\2&1&-1\end{smallmatrix}\right]$ and $\text{adj}(2A)$ is $2^k\text{adj}(A)$, find $k$
I have thought of a solution for this, but I know it’s wrong. I don’t know what’s wrong with the procedure, I just solved it instinctively.
$$\text{adj} A =A^{-1}|A|$$
So for... |
H: Minimizing costs of a specific geometry shape
I have geometry mathematical problem. I have a shape that is made of a cylinder and two half spheres as to top and bottom. How can I minimize the cost of this shape when the volume is known and the cylinder part costs $k_1$ dollar per square meters and the sphere part c... |
H: Improving the Cauchy's bound on the absolute values of the roots of the monic polynomial $x^n=m \times \sum_{k=0}^{n-1} x^k$
Given a polynomial $x^n=m \cdot \sum_{k=0}^{n-1} x^k$ (for all $m,n \in \mathbb{N}, m \geq 2,n\geq 2$), the numerical computation of roots for different $n$ and $m$ shows, that the absolute v... |
H: Common eigenvectors implies simultaneously block diagonalizations?
Suppose $A,B$ are symmetric matrices of the same size. If $A,B$ have a common eigenvectors, can we block diagonalize them at the same time?
More specifically, suppose $A$ and $B$ have one common eigenvector $v$ , and $Av = \lambda v$ and $Bv =\mu ... |
H: How to solve $x^{17}\equiv 37$ in $\mathbb{Z}/101\mathbb{Z}$?
I need to solve the equation $x^{17}\equiv 37$ in $\mathbb{Z}/101\mathbb{Z}$.
I've looked into these topics (the calculation of the primitive root is missing,
n is not prime) but couldn't derive a solution.
So summarize what I know:
101 is prim... |
H: It is based on set theory. Here I wan to know that how can I derive the modulus of P(A) is equal to 2ⁿ.
How to proof /P(A)/=2ⁿ
/ / For modulus sign
A: Given set
P(A):Power of set A
2ⁿ: The number of possible elements.
Hints: In the question it says that to prove it by induction method . /P(A)/=2ⁿ is possible when ... |
H: The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$
Problem:
Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that
$$\|S-I \|_F \leq \|S-Q\|_F$$
for all orthogonal matrices $Q$, where $I$ is the identity matrix.
Attempt:
So from what I know, th... |
H: Prove that a function is uniformly continuous in $[a,\infty)$
Let $f:[a,\infty)\to\mathbb{R}$ be a continuous function.
For every $\varepsilon>0$ there exist $0<\delta_{\varepsilon}$ and $a<c_{\varepsilon}\in\mathbb{R}$ so that for every $x_{1},x_{2}>c_{\varepsilon}$ so that $|x_{1}-x_{2}|<\delta_{\varepsilon}$ it... |
H: How to calculate percent dynamically?
$65$ is $100 \%$ and $55$ is $0 \%$. How can I calculate the percentage of all other numbers, , starting from 55 to 65?
AI: If you consider the points $(65,100)$ and $(55,0)$,then the line joining them is $$ \frac{y-0}{x-55}=\frac{100-0}{65-55}.$$ This gives us $$ y=10(x-55).$$... |
H: Integrate $\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$
How to integrate
$\int_0^{2\pi} e^{\cos\theta} \sin(\sin\theta-n\theta)d\theta$
$\int_0^{2\pi} e^{-\cos\theta} \cos(\sin\theta+n\theta)d\theta$
AI: The first integral is$$\Im\int_0^{2\pi}e^{\cos\theta+i(\sin\theta-n\theta)}d\theta=\Im\int_0^{... |
H: convergence of $\sum_{n=1}^\infty \frac {\sin (nx)}{n^{\alpha}}$
I'm trying to understand why the series $$\sum_{n=1}^\infty \frac {\sin (nx)}{n^{\alpha}}$$ converges for $\alpha > 0$.
At the end of the prof for $0 < \alpha \le 1 $ it is not clear to me two passages.
My book says that this result is obvious for $\... |
H: Is this a property of isomorphisms?
In a homework problem I was doing, I was trying to show that $U(8)$ is not isomorphic to $U(10)$. They used that, supposing $f: U(10) \rightarrow U(8)$ was an isomorphism, $|f(3)| = |3| = 4$ and there are no elements of order $4$ in $U(8)$, thus disproving that any such isomorphi... |
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