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H: Inverse of an $n\times n$ matrix
I'm curious to know the matrix form of the inverse of an $n\times n$ matrix. Also, how many operations will be needed to compute it?
AI: Like Git Gud said, a closed form would be $$A_{ij}=\frac{(\textrm{adj } A)_{ij}}{\det A}$$
Using the notation $A_{ij}$ for the element at the i-th... |
H: Inverse Laplace Transform of Discontinuous Function
I'm currently studying transform of discontinuous and periodic functions (Differential Equations.) I was presented with the following question.
$$\dfrac{se^{-3s}}{s^2+4s+5}$$
(Sorry, I couldn't get the formatting to work properly. Feel free to fix it. )
I've iden... |
H: Finding all $\mathbb{C}$-algebra homomorphisms $\mathbb{C}[x^2, y^2]\to\mathbb{C}$
I am interested in calculating all $\mathbb{C}$-algebra homomorphisms from $\mathbb{C}[x^2, y^2]\to\mathbb{C}$. I understand that every $\mathbb{C}$-algebra homomorphism will fix $\mathbb{C}$ and send $x^2$ and $y^2$ somewhere in $\m... |
H: Lagrange multiplier problem of looking for the point on $\frac1x + \frac1y + \frac1z =1$ closest to the origin
Use Lagrange multipliers to find the point on the surface
$$\frac1x + \frac1y + \frac1z =1$$ which is closest to the origin.
I was wondering if I would start off by using the distance formula, $$d=\sqr... |
H: Do countable $\cup$ and countable $\cap$ always exist?
Let $\Omega$ be a set and $(A_n)_{n\geq 1}$ a sequence of subsets of it. Does the countable union $\cup_{n=1}^{+\infty} A_n$ always exist? I understand that letting $B_n = A_1 \cup A_2 \cup \dots A_n$, $(B_n)_{n\geq 1}$ is an increasing sequence under the rel... |
H: How many times is the print statement executed?
Hello,
I've gotten far on this exercise, with the following insight:
Here is a matrix of examples (vertical-axis is n=1,2,3,4,5,6,7,8; horizontal-axis is k=1,2,3,4)
1: 1 1 1 1
2: 2 3 4 5
3: 3 6 10 15
4: 4 10 20 35
5: 5 15 35 70 ... |
H: how to choose positive symmetric matrix?
What are the ways to find a positive symmetric matrix $P$ such that
$
A^{T}P+PA=-Q$
where $Q$ is also positive symmetric matrix, $A=\left[\begin{array}{cc}
0 & I_{n}\\
-K_{v} & -K_{p}
\end{array}\right]_{2n\times2n}$, $K_{v}=k_{v}I_{n}$, and $K_{p}=k_{p}I_{n}$? Here $I_n$ st... |
H: accurancy of Tertiary bisection method
The question is:
Assume we change the bisection method into "tertiary bisection" which divides the interval into 3 parts and chooses the one from the left which is minimal and changes sign. e.g in the following sketch
if the first interval is $[a,b]$ and we assign
$h1=a-\fra... |
H: Software for visualizing partial derivatives?
I'm whipping up a set of notes, and I want to include a diagram or two showing some partial derivatives. Specifically, a diagram would include: a 3D surface of the form z=f(x,y), a plane of the form y=c, the intersection of the curve and plane shown as a highlighted cu... |
H: Why orientation on $S^1$ is the one that counter-clockwise-pointing vectors are positive.
I am totally disoriented by orientation..
So in the text, it says
The closed unit vall $B^2$ in $\mathbb{R}^2$ inherits the standard orientation from $\mathbb{R}^2$. The induced orientation on $S^1$ is the one for which the ... |
H: How to find number of squares in a chess board
Problem :
An $ n\times n$ chess board is a square of side $n$ units which has been sub-divided into $n^2$ unit squares by equally spaced straight lines parallel to the sides. Find the total number of squares of all sizes on an $n \times n$ chess board.
I have been t... |
H: Are composite fields unique?
Suppose for $i=1,2$ that $\Omega_i$ is a field containing fields $K_i$ and $L_i$, with $K_1 \cong K_2$ and $L_1 \cong L_2$.
Is it then true that there is an isomorphism $K_1L_1 \cong K_2L_2$ of composite fields?
AI: Consider e.g. non-Galois situations, like
$\Omega_1={\bf C} \quad K_1=... |
H: Continuous open maps on compact sets are surjections.
Could someone help me to show that if $X\subset \mathbb{R}^m$ is compact, then every continuous open map $f:X\to S^n$ is surjective?
This question was taken of an Analysis book (the subject of section is connectedness)
Thanks.
AI: As $f$ is open and $X$ is open ... |
H: Shifting Eigenvalues
Let $\mathbf{A}$ be a hermitian positive semi-definite matrix and $\mathbf{B}$ be a hermitian positive definite matrix. Then I am interested in the eigenvalues of matrix $\mathbf{C}(t)=\mathbf{A}-t\mathbf{B}$ where $t$ is a real parameter. Note that when $t=0$, $\mathbf{C}$ is a positive semi-d... |
H: Existence of minimal $\sigma$-algebra and transfinite induction
It is well-known that:
Given a set $X$ and a collection $\cal S$ of subsets of $X$, there exists a $\sigma$-algebra $\cal B$ containing $\cal S$, such that $\cal B$ is the smallest $\sigma$-algebra satisfying this condition.
Certain texts, Lieb and L... |
H: Orthogonal matrices form a compact set
Prove that the set of all $n \times n$ orthogonal matrices is a compact subset of $\mathbb{R}^{n^2}$.
I don't know how it can be done. Thanks.
AI: Let $M:=\mathbb{R}^{n\times n}$ be the set of all matrices and $\mathcal{O}$ the subset of orthogonal matrices. Define $f\colon ... |
H: the positive semidefiniteness of a Hermitian Toeplitz matrix
For a square matrix $\mathbf{A}$ whose $(m,n)^{\text{th}}$ element is defined as $\mathbf{A}_{mn}=\int_{-\pi}^{\pi}e^{\imath(m-n)\pi\cos\phi}d\phi$ where $\imath=\sqrt{-1}$. It is obvious that $\mathbf{A}$ is Toeplitz and Hermitian. But whether $\mathbf{A... |
H: Does the infinite product $\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$ diverge to zero or some other finite value.
Does the infinite product diverge to zero or some other value?
$$\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$$
AI: Consider $$P_m=\prod_{n=1}^m\frac{2^n}{3^n}=\left(\frac{2}{3}\right)^{m(m+1)/2... |
H: What is basis of $\mathbb{R}$
I think it is just 1; but I am also under the impression that it is just any open interval on $\mathbb{R}$.
Furthermore, I am trying to figure out how a compact interval $X = [0,1]$ inherite standard orientation from $\mathbb{R}$, from Guillemin and Pollack's Differential Topology.
To ... |
H: Find $ \bigcup_{i = 1}^\infty A_{i} $ and $ \bigcap_{i = 1}^\infty A_{i} $ if for every positive integer $i$, $A_{i} = \left\{ {-i, i}\right\}$.
This is a question from Rosen's Discrete Mathematics and Its Applications. It asks:
Find $ \bigcup_{i = 1}^\infty A_{i} $ and $ \bigcap_{i = 1}^\infty A_{i} $ if for ever... |
H: A function question; inverses
If $f(g(x)) = x$ for all $x$ and $f$ and $g$ are continuous. Does it necessary follow that $g = f^{-1}$? Or do we need $g(f(x)) = x$ as well?
AI: If "all $x$" means all real $x$, and "continuous" means "continuous on $\mathbb R$", then yes. It's clear that $g$ is one-to-one.
$g({\math... |
H: Geometric Distribution and roulette wheel question
Suppose that your bank roll is \$75.00 and your goal is to win \$5. Your strategy is to spin the roulette wheel and wager \$5 on black (18 red numbers, 18 black numbers, and 2 green house numbers). If you lose, you will double your wager until you eventually win... |
H: Does $S^1$ has boundary?
According to Guillemin and Pollack's Differential Topology:
The sum of the orientation numbers at the boundary points of any compact oriented one-manifold $X$ with boundary is zero.
By The Classification of One-manifold, every compact, connected, one-dimensional manifold with boundary is ... |
H: decoupling and integrating differential equations
I am having trouble with the process of decoupling. If I have
$$\frac{dx}{dt}=-x+y$$
$$\frac{dy}{dt}=-x-y$$
I am trying to figure out how to solve for $x(t)$ and $y(t)$ by decoupling the system so that I only have one variable but I can't seem to get anywhere
AI: Co... |
H: How a compact interval $X = [0,1]$ inherite standard orientation from $\mathbb{R}$.
According to Guillemin and Pollack's Differential Topology:
A compact interval $X = [0,1]$ inherite standard orientation from $\mathbb{R}$.
To my understanding, "inherit from $\mathbb{R}$" means a linear transformation between $X$... |
H: Why is empty set an open set?
I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition.
The definition of an open set: a set S in n-dimensional space is called open if all its points are interior... |
H: Solving a simple exponential equation
How can I solve this logarithm:
$$ e^{2x} - 3 e^x + 2 = 0. $$
I think it should be re-written as a quadratic equation in $e^x$.
AI: $$e^{2x}-3e^x+2=0$$
let $e^x=y$
$$y^{2}-3y+2=0$$
$$y^2-2y-y+2=0$$
$$y(y-2)-1(y-2)=0$$
$$(y-2)(y-1)=0$$
$$(y-2)=0\;,(y-1)=0$$
$$y=2,1$$
$$e^x=2... |
H: solving an equation about the number of divisors
Find all natural numbers $n$ such that $n=d(n)+1$
($d(n)$ is the number of positive divisors of $n$)
Any ideas on how to solve this ???
AI: If $n=p_1^{a_1}\cdots p_m^{a_m}$ with $a_i\ge1$, then $d(n)=(a_1+1)\cdots (a_m+1)$.
We have $p^a\ge a+1$ for all $p\ge 2, a\... |
H: Continuous function with linear directional derivatives=>Total differentiability?
As in the title:
If $f\colon\mathbb{R}^n\to\mathbb{R}$ is continuous in $x$ and has directional derivatives $\partial_vf(x)=L(v)\,\forall v\in\mathbb{R}^n$, where $L$ is linear, does this imply that $f$ is totally differentiable?
AI:... |
H: Equilateral and equiangular polygon
Can we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-gon possible with ruler and compass?
AI: For the first part, note that a regular pe... |
H: Parametric Equations (Basic) - Cartesian equation of curves
$x = 2 \cos t$, $y = 2 \sin t$, $0 \le t \le 2\pi$
Find the Cartesian equation of the curves.
Please help i know it's basic but my problem is that $2 \cos t$ doesn't equal $1 - \sin^2 t$ and if it does how? (maybe i just need a re-freshener).
Help would be... |
H: Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .
Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ?
How to find if it is convergent or not?
Thanks!
AI: Hints:
$$z=\sqrt{i+\sqrt{i+\ldots}}=\sqrt{i+z}\implies z^2-z-i=0$$
and you're left with a simple complex quadratic to solve
Note: Formally , you first sho... |
H: Hartshorne's weird definition of right derived functors and prop. III 2.6
There is something very weird with the way Hartshorne defines right derived functors.
Hartshorne p 204 Let $\mathfrak A$ be an abelian category with enough injectives, and let $F \colon \mathfrak A \to \mathfrak B$ be a covariant left exact ... |
H: A technical question about the Lebesgue measure
Let $U$ be an open set in $\mathbb{R}^2$. How to prove that the boundary of the CLOSURE of $U$ has Lebesgue measure 0 ? Thanks.
AI: It appears that the statement in your question is not always true. See https://mathoverflow.net/questions/25993/sets-with-positive-lebes... |
H: $f^{-1} $ is continuously differentiable.
Let $f(x,y)=(x^3+y^2, xy+y^4)$.
I am trying to show that $f^{-1}$ is continuously differentiable at $(1,-1)$.
Solution:
$$Df(x,y)=\begin{pmatrix}3x^2 & 2y \\ y & x+4y^3\end{pmatrix} $$
$$Df(-1,1)=\begin{pmatrix}3& -2\\ -1 &-3 \end{pmatrix}$$
$$\lvert Df(1,-1)\rvert=-9-(2)=... |
H: Find the vector by the following criteria
I want to find the vector that meets the following:
$$X\parallel (2,1,-1)$$
$$X*(2,1,-1)=3$$
what I did so far is :
$$2x+y+z=3$$
I know that parallel vectors the angle is $180$ or $0$.
how to continue from here?
Note: X is vector.
Thanks!
AI: The easiest way is to notice ... |
H: I need clarification on $\delta$ - derivative
Please can someone tell me more about $\delta$ -derivative ($\delta=x\dfrac{d}{dx}$) as it appears in the Hadamard definition of frational derivative or elsewhere. Why, when or where we use it. Does anything the usual derivative ($\dfrac{d}{dx}$) does that the $\delta$ ... |
H: Complex exponents and matrices
If the matrix $A$ is defined as:
$$A_{m,n} =
\begin{pmatrix}
a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m,1} & a_{m,2} & \cdots & a_{m,n}
\end{pmatrix}$$
is it possible to define the following ma... |
H: Iterated conditioning
I understand how this holds:
$\mathbb{E}(\mathbb{E}(X|\mathit{G})|\mathit{H}) = \mathbb{E}(X|\mathit{H})$
where $\mathit{H} \subset \mathit{G} $.
Then, how does this hold:
$\mathbb{E}(\mathbb{E}(X|\mathit{G})) = \mathbb{E}(X)$
as here, $\mathit{H} = \Omega$ and now $\mathit{G} \subset \Omega$... |
H: Confused with natural logarithms
How can we solve the following natural logarithms? I'm confused with this stuff:
$\ln(x+1) - \ln x = \ln 3$
$\ln(x+1) + \ln x = \ln 2$
AI: Here is a worked example slightly different to your question; solve the following for $q$:
$$\ln(q+2)-\ln(q)=\ln(5)\tag{1}$$
Using the laws for... |
H: Calculate $\bigtriangleup$ ABC where $A(-2,-3,0)$,$B(-1,0,5)$,$C(4,2,2)$
I want to calculate $\bigtriangleup$ ABC where $A(-2,-3,0)$,$B(-1,0,5)$,$C(4,2,2)$
What I did was to mark the triangle vertices randomly
1) calculate the middle of AB ( I call it G ) to find the vertical vector CG then what I do is to calcula... |
H: How to proof homeomorphism between open ball and normic space
How can I prove that an open ball $B$ in a normed vector space $X$ is homeomorphic to $X$?
AI: Without loss of generality take the ball to be the unit ball $B = \{x ∈ X;\; \lVert x \rVert < 1\}$ around Null. (Translations and homotheties are homeomorphis... |
H: Calculating half-life?
After taking a tablet, a patient has 10 units/ml in a sample of blood taken soon after, and this decreased to 6 units/ml 9 hours later.
What is the half-life of the tablet?
How long will it take decrease to 20% of its original value?
AI: Assuming exponential decay, you model the amount of d... |
H: Evaluating: $\lim\limits_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$
I am trying to evaluate the following but without result.
$$\lim_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$$
Can you please give me some hints? I have tried to put log to both sides but it hasn't lead me somewhere... Thanks a lot
... |
H: integral computed with respect to a sub-$\sigma$-algebra
Let $\mathcal M_0$ be a $\sigma$-algebra that is contained in a $\sigma$-algebra $\mathcal M$ of subsets of a set $X$, $\mu$ a measure on $\mathcal M$ and $\mu_0$ the restriction of $\mu$ to $\mathcal M_0$. Let $f$ be a nonnegative real-valued function that i... |
H: Type of solutions for the linear equation AX=B
I have a problem and a proposed solution. Please tell me if I'm correct.
Problem: For A,B real matrices, prove that if there is a solution in
the complex numbers then there is also a real solution.
Solution: A and B are real matrices. Therefore, they are not define... |
H: Topology exercise - open, closed and bounded sets in $\mathbb{R}^n$
Would someone check my solution to this exercise:
Exercise. Determine whether the following subsets of $\mathbb{R^2}$ are open, closed, and/or bounded.
$A=\{\|x\|\le1\}$
$B=\{\|x\|=1\}$
$C=\{\|x\|\lt1\}$
$D=\{\text{the x-axis}\}$
$E=\mathbb{R^2}-\... |
H: boolean algebra: DeMorgan's law confusion
the following function should be put into table values:
$$y = \overline{(a*b*d+c)}$$
So the first thing i am doing is using DeMorgan to get rid of the "whole-term-negation":
$$y = (\tilde a + \tilde b +\tilde d * \tilde c)$$
from here all i do is making a 4 variables truth ... |
H: How to calculate $2^{mn-1}/(2^n-1) \bmod{(10^9+7)}$
I was trying to solve Magical Five problem on codeforces.
I have correctly formed an equation which I need to solve via program such that resulting number don't overflow. Answer can be Python or C++ specific.
Now, the equation is:
$$\frac{2^{mn-1}}{2^n-1} \bmod{p}... |
H: Evaluating $\cos(\alpha+\beta+\gamma)$
I am trying to evaluate $\cos(\alpha+\beta+\gamma)$
This is what I have done so far:
I know $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$
and $\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$
Treating $\cos(\alpha+\beta+\gamma)$ as $\cos[(\alp... |
H: Can we derive that $A$ commutes with $B$ from this?
Based on some Physics backgrounds, I want to confirm the following thing.
Let $[A,B]:=AB-BA$, where $A,B$ are matrices. Now the question is as follows:
If for any real number $\lambda$, $[A,e^{\lambda B}]=0$, then is $[A,B]=0$ true? Where $A,B$ are matrices.
If t... |
H: Addition formulas for Jacobi amplitude function
Are there any known summation formulas for the Jacobi amplitude function? I need a formula like $\mathrm{am}(t+x)=\mathrm{am}(t) + f(x)$. I have plotted some graphs and it seems that $f(x)$ is periodic but I wasn't able to figure out what this function is.
Maybe someo... |
H: Number of optimal paths through a grid with an ordered path constraint
I found, but the awesome explanation of Arturo Magidin: Counting number of moves on a grid
the number of paths for an MxN matrix. If I am thinking about this correctly (please say something if I am wrong), but the number of optimal/shortest pat... |
H: Short way? Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$
Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$
Does there exist any short way? I have to calculate all partial dervatives. Is it?
AI: The short way is to use the well known one-dimensional result: The 4th dergree taylor polynomial of... |
H: Find the greatest value of $x$ satisfying $21\equiv 385\pmod x$ and $587\equiv 167\pmod x $
Problem: Find the greatest value of $x$ satisfying $21\equiv 385\pmod x$ and $587\equiv 167\pmod x $.
Solution: I think $21\equiv 385\pmod x$ is not possible.
There is some printing mistake in the question
Am I right?
AI: ... |
H: Definition of $0$?
In real number axioms, it is defined that there is $0$ such that $x+0=x$ for all $x.$ I was wondering an example is there any other algebraic structure than real numbers which satisfy the real number axioms other than $x+0=x$ but where $x\cdot 0=0\cdot x =0$ for all elements $x$.
This question po... |
H: Holomorphicity of $\zeta(s)$
Let $\zeta(s)=\sum_{n=1}^{\infty}1/{n^s}$ be the Riemann zeta function, $s\in \mathbb{C}$. I can show that it converges absolutely and uniformly on the right half-plane Re$(s)>1$.
How can I show that it is holomorphic there?
For example, is there a theorem which states that a uniform ... |
H: If $i\colon A\to X$ is a cofibration then $1\times i\colon B\times A\to B\times X$ is a cofibration for any space $B$. Is that true?
In Algebraic Topology (Hatcher, pg 14)
I find: A pair $\left(X,A\right)$
has the homotopy extension property if and only if $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$
is a ret... |
H: Relationship between the rows and columns of a matrix
I am having trouble understanding the relatioship between rows and columns of a matrix.
Say, the following homogeneous system has a nontrivial solution.
$$
3x_1 + 5x_2 − 4x_3 = 0 \\
−3x_1 − 2x_2 + 4x_3 = 0 \\
6x_1 + x_2 − 8x_3 = 0\\
$$
Let A be the coefficient... |
H: Proving Uniform Convergence
ST the sequence $f_n$ where $f_n(x)=e^{-nx}$ is point wise but not uniformely convergent in $[0,\infty[$. Also show that the convergence is uniform in $[k,\infty[$, $k$ being a positive number.
I am able to show that point wise limit is $f(x)=0 \ \ \forall \ x$
Let $\epsilon>0$ be given ... |
H: Fast legendre symbol calculation
Let's say that I would like to calculate all legendre symbols from $1$ to $p-1$ $\pmod{p}$, is there a way to calculate them in an incremental way?. For example, an incremental table of legendre symbols could help to calculate them in a memoized algorithm, but let´s assume we can't ... |
H: Why is $\int_{\partial D}x\,dy$ invalid for calculating area of $D$?
I am just learning about differential forms, and I had a question about employing Green's theorem to calculate area. Generalized Stokes' theorem says that $\int_{\partial D}\omega=\int_D d\omega$. Let's say $D$ is a region in $\mathbb{R}^2$. The f... |
H: Simplifying the expression $(\sqrt{5}+\sqrt{7})/(\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21})$
Alrite guys, this question might sound stupid, but I can't find a way to simplify this complicated expression:
$$\frac{\sqrt{5}+\sqrt{7}}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}$$
I can't take the conjugate, nor I can factor... |
H: Boas 17.28 - simplify a complex expression
This is a question from "Mathematical Methods in the Physical Sciences" (Boas, 3rd Ed), Question 17.28:
Express the following expression in terms of a hyperbolic function:
$$ \left|\frac{(a+bi)^2e^b-(a-bi)^2e^{-b}}{4abie^{-ia}}\right|^2$$
Here is my attempt: by using the ... |
H: Use implicit function theorem and show $u(x,y)^2+v(x,y)^2=\frac{16}{x+y}$
I am attempting exercise 11.6.4 on p. 368 of Wade's Introduction to Analysis. It asks:
"Find conditions on a point $(x_0, y_0, u_0, v_0)$ such that there exist functions $u(x,y)$ and v(x,y) that are C1 at $(x_0,y_0)$ and satisfy the simultane... |
H: Simplifying the expression $(1+\sqrt[4]3)/(1-\sqrt[4]3)+1/(1+\sqrt[4]3)+2/(1+\sqrt{3})$
Can anyone give provide me some help to simplify this expression?
The three denominators are pretty much different, and I can't find a common denominator.
$$\frac{1+\sqrt[4]3}{1-\sqrt[4]3}+\frac1{1+\sqrt[4]3}+\frac2{1+\sqrt{3}}$... |
H: How many different events can occur if a die is rolled for 10 times?
By saying "events" I mean we don't count the permutations of same numbers. For example, if a die is rolled for 2 times only, 21 different events can occur.
We consider $\{4_{(\text{die #1})}, 4_{(\text{die #2})}\}$ and $\{4_{(\text{die #2})}, 4_{(... |
H: Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$
Solving for $x,y\in\mathbb{N}$:
$$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$
This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very complicated fraction on the left hand side, and I don't even ... |
H: The union of a sequence of infinite, countable sets is countable.
While reading Walter Rudin's Principles of Mathematical Analysis, I ran into the following theorem and proof:
Theorem 2.12. Let $\left\{E_n\right\}$, $n=1,2,\dots$, be a sequence of countable sets, and put
$$
S=\bigcup_{n=1}^\infty E_n.
$$
Then $S$ ... |
H: Uniqueness of meromorphic continuation
Let $\Omega$ be a non-empty region of $\mathbb{C}$ and suppose $f$ is a holomorphic function on $\Omega$.
How can one show that a meromorphic continuation of $f$ to all of $\mathbb{C}$ is unique, if it exists?
By a meromorphic function $f$ on $\mathbb{C}$ I mean a function ... |
H: Why does zero derivative imply a function is locally constant?
I've been trying to prove to myself that if $\Omega$ is an open connected set in $\mathbb{R}^n$, then if $f\colon\Omega\to\mathbb{R}^m$ is a differentiable function such that $f'(x)=0$ for all $x\in\Omega$, then $f$ is constant.
I've reduced the proble... |
H: what is the maximum number of roots of quadratic function with 3 variables?
Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$
I would like to ask what is the maximum number of roots of the provided equation.
Thanks in... |
H: A method for calculating this integral hermite polynomials
I need proof this,
$\int_{-\infty}^{\infty}e^{-x^2}H_n^{2}(x)x^2dx=2^nn!\sqrt{\pi}(n+\frac{1}{2})$
This is the idea: Multiply $(1-t^2)^{-1/2}e^{2x^2t/(1+t)}=\underset{n=0}{\overset{\infty}\sum}\frac{H_n^{2}(x)}{2^nn!}t^n,~~~|t|<1,$ by $x^2$, integrate from ... |
H: A question about bilinear maps.
This wikipedia article says:
Let $B:V\times W\to X$, and $(v,w)\in V\times W$. If $v$ in $(v,w)$ remains fixed, then $B$ becomes a linear mapping. I quote "In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linea... |
H: The complement of the poles of a meromorphic function
Why does the complement of the set of poles of a meromorphic function form a connected subset of $\mathbb{C}$?
Thanks for helping.
AI: Angelic answer: because the complement of a closed discrete subset of $\mathbb C$ is connected.
Diabolical answer: because th... |
H: Number of radians in one steradian cross-section?
What is the conversion between degrees and steradians? That is, if you rotate a two-dimensional angle around one side, what is the equivalent in solid angle?
I specifically need the number of degrees in radians that, when rotated about the axis, form a cone with a s... |
H: Induction for statements with more than one variable.
I'm going through the first chapters of Tao's Analysis text and I'm not entirely sure about one thing, namely why we're allowed to 'fix' variables when inductively proving statements pertaining to more than one variable. This is not explained in the text. For in... |
H: Proving convexity of $X=\{(x,y) \in \mathbb R^2 ; ax + by \le c\}$
Given $a,b,c \in \mathbb R$, how can I prove that $X=\{(x,y) \in \mathbb R^2 ; ax + by \le c\}$ is convex in $\mathbb R^2$? I know that $X$ is convex when $u,v \in X \rightarrow [u,v] \subset X$, where $[u,v]$ is the line segment whose extremes are ... |
H: Solving equation with logarithms
I happen to use this heavy math for the first time for a long time (if ever) and don't know how to bite it.
Given:
$$\begin{align}
A &= 1.45\\
B &= 4.1\\
C &= 14\\
\frac1A + \frac1B + \frac1C&=100\%\\\
\end{align}
$$
I want to find $X$, $Y$ and $Z$ such that
$$
... |
H: What does log likelihood in dfittool in Matlab means?
I have a data set (400 real valued numbers between zero and one) which I need to fit an appropriate distribution to this data set. I used "dfittool" of Matlab and tried different distribution. For each distribution "dfittool" gives me a "log likelihood" of that ... |
H: Whats the diffrence between Products and Coproducts
So I just started in on Category theory (reading the quintessential text, "Categories for the Working Mathematician"), and I am trying to get my head around the difference between Products and Coproducts. Can someone explain or discuss the conceptual and practical... |
H: Is this notation for the set of limit points a standard notation?
Well, this doubt is probably silly. We have a standard notation for closure of a set $E$, we denote it $\bar{E}$ or $\operatorname{cl}{E}$ and we have a notation for the interior of a set $E$ we denote it $E^\circ$ or $\operatorname{int}{E}$. Now, wh... |
H: How many rounds of blackjack do you have to simulate to get accurate outcome probabilities?
Say you play blackjack a lot and record how often each outcome occurs (win, push, lose, win $2x$, lose $2x$, $+3x$, $-3x$, $+4x$, $-4x$, $+1.5x$) (keep in mind you can double and split).
How many rounds would you have to pla... |
H: scalar curvature
I am studying scalar curvature. It is the trace of the Ricci operator. I read that its geometric meaning follows from this formula
$\frac{Vol_M(B(p,t))}{Vol_{\mathbb{R}^n}(B(o,t))}=1-\frac{1}{6(n+2)}k(p)t^2+o(t^2)$
where $k$ is the scalar curvature. From this formula one knows that if the scalar cu... |
H: How to solve $ \begin{cases} \cos (z_1 +iz_2) = i\\ |z_1|=|z_2| \end{cases} $?
How to solve
$ \begin{cases}
\cos (z_1 +iz_2) = i\\
|z_1|=|z_2|
\end{cases}
$?
where $z_1, z_2$ are complex variables
Rectangular form is convenient for the first equation, and polar form is suitable for the second one! What to do?
AI: ... |
H: Calculating the norm of an infinite vector
I'm reading "Introduction to Hilbert Spaces" by N. Young. Right in the first chapter, after introducing inner products and norms in general linear spaces, it asks to show that the norm of the vector:
$$x = \left(\frac{1}{n}\right)_{n=1}^\infty$$
is $\frac{\pi}{\sqrt 6}$, s... |
H: How many ways to arrange a group of 10 R's and 10 D's?
Given 10 letter R's and 10 letter D's, how many distinct sequences of length 20 can be formed? A justification of your answer would be much appreciated.
AI: If you have decided on which places you want to place the D's, you know the entire sequence, right? This... |
H: How to show that the distance between these sets is positive?
Let $T_i=\{(1-t)x_i+ty_i;\;0\leq t\leq1\}$, where $x_i,y_i\in\mathbb{R}^n$; $i=1,2$.
Could someone help me to prove that if $x_2= (1+\varepsilon)x_1$ and $y_2= (1+\varepsilon)y_1$ for some $\varepsilon>0$ then $d\left (T_1,T_2\right)>0$?
Thanks.
AI: Ho... |
H: Rounding to the nearest term in a geometric progression
Consider the following progression:
R(i) = 5*10^(i/30)
where i is ith number within the progression.
I would like to devise an equation that will round input value to the nearest number from this progression.
For example 5th, 6th, and 7th numbers in this pro... |
H: Double Integrals over Rectangles
Estimate the volume of the solid that lies below the surface $z=xy$ and above the rectangle
$$R=\{(x,y)\mid 0\le x \le 6, 0 \le y \le 4\}$$
Use a Riemann sum with $m=3, n=2$, and take the sample point to be the upper right corner of each square.
I'm sure it's really simple, but I'm... |
H: Positive rationals satisfying: $a^2+a^2=c^2$?
If there are none why not?
Thanks in advance.
AI: Note that this can be written as $$2=\left(\frac{c}a\right)^2$$
Is $\sqrt 2$ rational? |
H: Let F be a function from $N ^{n} \longrightarrow N$. Show that if F is computable/ recursive then its graph is computable
Let F be a function from $N ^{n} \longrightarrow N$. Show that if F is computable then its graph is computable.
According to the definition of computable/recursive I am looking at, a relation i... |
H: Prove that $(n+1)(n+2)(n+3)$ is $O(n^3)$
Problem
Prove that $(n+1)(n+2)(n+3)$ is $O(n^3)$
Attempt at Solution
$f(n) = (n+1)(n+2)(n+3)$
$g(n) = n^3$
Show that there exists an $n_0$ and $C > 0$ such that $f(n) \le Cg(n)$ whenever $n > n_0$
$f(n) = n^3+6n^2+11n+6 = n^3(1 + 6/n + 11/n^2 + 6/n^3)$
$f(n) \le C*g(n)$ ... |
H: History Question on Continued Fractions
I worked out the periodicity of some infinite continued fractions last night by hand. (Don't ask me why)For example, $\sqrt{13}= [3,1,1,1,1,6,1,1,1,1,6,\ldots]$. Last night I worked out the first period of this continued fraction and the algebra was a little meh.
I was wond... |
H: Are there any algorithms to check if a big number is a prime number?
I want to check if a given number is a prime number. Are there other ways than brute-force? It should be fast and work with bigger numbers (>1.000.000).
AI: If your number is big enough (absurdly big), and you need a deterministic algorithm (unlik... |
H: LRT and Power for a Normal Distribution
For one observation $Y$ from a normal distribution with variance $1$ and mean $0$ or $2$, consider $H_{0}:\mu=0$ and $H_{1}:\mu=2$. Suppose first that we observe only $Y$. Construct a size $\alpha$ likelihood ratio test. Give explicitly the rejection region in terms of $Y$, a... |
H: A elementary question about cardinal arithmetic
My question is: Let $X$ be a finite set and let f be a function, $f: X \rightarrow Y$ show that $ \# f[X] \le \#X$.
Here is my attempt:
As X is a finite set, therefore there exist a bijective map $ g: X \rightarrow n$ where $ n \in \omega$. And for each $y \in f[X]$... |
H: Die Roll Probability
If Zachary rolls a fair die five times, what is the probability that the sum of his five rolls is 20?
1st I did:
Patterns of 5 that can give us 20
66611, 66521, 65531, 65522, 64442, 64433, 64415,
Where do we go from here?
AI: Assuming that you know generating functions.
We are interested in th... |
H: Applying the Compositional Formula in Enumerative Combinatorics
Following up on this question:
I understand the theorem itself, but I'm not sure how to apply the theorem in order to complete the proof.
Can someone provide me some assistance, or hints, to prove that $\forall k,n\in\mathbb{N}_{>0}$, the following ide... |
H: Differentiability and Convergent Sequence
Let $f$ be differentiable on $\mathbb{R}$ with $a= \sup\{|f'(x)| : x \in \mathbb{R} \} < 1$. Let $s_0 \in \mathbb{R}$ and let $s_n = f(s_{n-1})$. Show that $(s_n)$ is convergent.
Hint from my book: Show $(s_n)$ is Cauchy by showing that $|s_{n+1} - s_n| \leq a|s_n-s_{n-1}|$... |
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