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H: Determine the matrix of a quadratic function
I'm given a quadratic form $\Phi:\mathbb{R}^3\longrightarrow\mathbb{R}$, for which we know that:
$(0,1,0)$ and $(0,1,-1)$ are conjugated by the function
$(1,0,-1)$ belongs to the kernel
$\Phi(0,0,1)=1$
The trace is $0$
From here, I know the matrix must be symmetric, so... |
H: Finding covariance of two negative binomial distributions
Take a set of random variables $X_n$ which indicate number of tosses of a coin ($P(H)=p$) to achieve $n$ heads. What is $\operatorname{cov}(X_1,X_5)$?
I reasoned that the number of tosses till the first heads is not going to affect the number of tosses for t... |
H: Does $f(\epsilon)=o(\epsilon\ln(\epsilon))$ imply $\frac{f(\epsilon)}{\epsilon}=o(1)$?
I have the following homework question:
Does $f(\epsilon)=o(\epsilon\ln(\epsilon))$ imply
$\frac{f(\epsilon)}{\epsilon}=o(1)$ ?
It doesn't seem correct to me, using the definition I could only get
$$\frac{f(\epsilon)}{\epsilo... |
H: Is a linear map of norm $1$ always an isometry?
Let $(E,\|.\|_{E})$ and $(F,\|.\|_{F})$ be two normed spaces and let $f:E\longrightarrow F$ a linear map such that $\|f\|=1$. I don't know if this means that $f$ is an isometry $(\|f(x)\|_{F}=\|x\|_{E}, \,\, \forall x\in E)$. Thanks
AI: It does not. Consider the funct... |
H: How to solve $e^{ix} = i$
I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary parts. What are the steps to solving such equations? I am familiar with Taylor series and the E... |
H: linear map of linear in/dependent vectors
L a linear map of $\mathbb{R}^m$ in $\mathbb{R}^n$.
1)Does a linear map of two linear dependent vectors $\underline{x}$,$\underline{y} \in \mathbb{R}^m$ to two linear independent vectors $\underline{u}$,$\underline{v} \in \mathbb{R}^n$ exist/is possible?
2)a)Does a linea... |
H: What does 'i-th' mean?
I have seen a problem set for the tower of hanoi algorithm that states:
Each integer in the second line is in the range 1 to K where the i-th
integer denotes the peg to which disc of radius i is present in the
initial configuration.
What does i-th actually mean? If i have a line conform... |
H: Subtracting roots of unity. Specifically $\omega^3 - \omega^2$
This is question that came up in one of the past papers I have been doing for my exams. Its says that if $\omega=\cos(\pi/5)+i\sin(\pi/5)$. What is $\omega^3-\omega^2$. I can find $\omega^3$ and $\omega^2$ by De Moivre's Theorem. But I cant make much he... |
H: How do I prove Binet's Formula?
My initial prompt is as follows:
For $F_{0}=1$, $F_{1}=1$, and for $n\geq 1$, $F_{n+1}=F_{n}+F_{n-1}$.
Prove for all $n\in \mathbb{N}$:
$$F_{n-1}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n\right)$$
Which, to my understanding, is Bi... |
H: Quotient group of normal subgroups is cyclic if quotient group of intersection is cyclic
Let $M$, $N$ be normal subgroups of a group $G$.
Prove that if $G/M\cap N$ is cyclic, then $G/M$ and $G/N$ are cyclic. Give a counter example to show that the converse is not always true.
I proved until now that if $G/M\cap N$ ... |
H: Help me solve this equation: $x^\frac23 - 9x^\frac13+8=0$
Not sure what to call this type of equation so please let me know. I'm having trouble solving it though.
Solve the equation:
$$x^\frac23 - 9x^\frac13+8=0$$
AI: Hint: try the substitution $y = x^{\frac{1}{3}}$. This should give you a quadratic. |
H: What should be proved in the binomial theorem?
I'm following Cambrige mathematics syllabus, from the list of contents of what should be learned:
Induction as a method of proof, including a proof of the binomial theorem with non-negative integral coefficients.
I know what it is, but I'm not sure of what should be ... |
H: What is the names of $A\vec{x}=\vec{b}$ linear equation system components?
Having $A\vec{x}=\vec{b}$ .
What is the names of $A\vec{x}=\vec{b}$ linear equation system components?
AI: According to Elementary Linear Algebra by Venit and Bishop, we have:
$$A\vec{x}=\vec{b}$$
$A$ is the coefficient matrix of the system... |
H: Simple trigonometry question
I am just wondering how can you get from
$\cos(\pi t)=1/2 $ or $\cos(\pi t)=-1$
for $0<t<4$
to
t = 1/3, 1
t = 5/3, 7/3, 11/3 ,3
I got
$(\pi t) = \pi/3 +2k\pi $ and
$(\pi t) = 5\pi/3 + 2k\pi $
$t = 1/3 + 2k $ and $t = 5/3 +2k$
but i couldn't quite get t = 3, and t =1.... i'm n... |
H: Finding the value of $c$ that makes this system of equations have a solution
Find the value of $c$ that makes the system of equations below have a solution.
$$\begin{align*}
u + v + 2w &= 2 \\
2u + 3v - w &= 5 \\
3u + 4v + w &= c
\end{align*}$$
I have taken a suggestion from a similar question asked 2 months ... |
H: Showing existence of an element with order $p$
If a group $G$ has order $p^n$, where $p$ is prime and $n \geq 1$, does there exist some element $a\in G$ s.t. the order of $a$ is $p$?
I happen to know that this is true by Cauchy's theorem, but that theorem has not been presented yet in the book. I only have this s... |
H: Properties of Entropy
When someone writes $H(X_1, X_2, X_3) = H(X_1) + H(X_2\mid X_1) + H(X_3\mid X_2, X_1)$, how should that last term be interpreted/read? As the joint entropy between 2 variables where variable 1 is $X_3\mid X_2$ and variable 2 is $X_1$? Or As the entropy of $X_3$ conditioned on both $X_2$ and $X... |
H: How can I prove $2\sup(S) = \sup (2S)$?
Let $S$ be a nonempty bounded subset of $\mathbb{R}$ and $T = \{2s : s \in S \}$. Show $\sup T = 2\sup S$
Proof
Consider $2s = s + s \leq \sup S + \sup S = 2\sup S $. $T \subset S$ where T is also bounded, so applying the lub property, we must have $\sup T \leq 2 \sup S$.
... |
H: Algebraic problem for satisfying a given equation
I'm trying to solve the following exercise:
My backward equation looks like:
$P_{i,j}'(t) = i\lambda P_{i+1,j}(t) - i\lambda P_{i,j}(t) $
So i started with differentiating $P_{i,j}(t)$:
${j-1 \choose i-1}(e^{-i\lambda t}(1-e^{-\lambda t})^{j-i})'$
By using $(uv)' =... |
H: $f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
The function $f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
Can anyone give me any tips how to work on this question?
AI: Hint When is it true that $(ax)^2=a^2x^2$? That is $$axax=aaxx$$
Further hint ... |
H: Nilpotent operator / Orthogonal projection
If you have a nilpotent operator $A \colon V \to V$, $V$ being $A$-cyclic (meaning that $V$ is generated by a single vector $v$ in $V$), is it true that the minimal polynomial of the Gram operator $P = A^{T} A$ is $p(t) = t^2 - t$, i.e., $P$ is the orthogonal projection?
A... |
H: Estimate the scale of the power series with Poisson pdf-like terms
Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue.
I would like to have an estimate for the series
$$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^... |
H: Spin group without Clifford algebras
I have to build the spin group $Spin(n)$ without use Clifford algebras. Can I find a complete description of spin group with a topological method? How can I build $Spin(n)$ as the double covering of $SO(n)$?
AI: Define $Spin(n)$ as the universal cover of $SO(n)$. The universal c... |
H: What are the cluster points of this filter?
Let X be a topological space, $A\subset X$ and $\mathcal{F}=\{F\subset X|A\subset F\}$. Then $\mathcal{F}$ is a filter on X. I would like to know what are the cluster points of this filter.
I am preparing for an exam and I need help. Thank you in advance.
AI: HINT: The c... |
H: Simplifying a radical equation?
I'm lost on how to simplify this.
Simplify:
$$\frac{\sqrt[\large 4]{144x^9y^8}}{\sqrt[\large 4]{9x^5y^{-3}}}$$
OK So I think I got it.
$$\frac{\sqrt[\large 4]{144x^9y^8}}{\sqrt[\large 4]{9x^5y^{-3}}} = \left(\frac{{144x^9y^8}}{{9x^5y^{-3}}}\right )^{1/4}$$
You can break that down... |
H: Simple Graph Transformation Question $\rightarrow$ $1/f(x)$
for the graph:
such that the function is : $ y = \frac{a+x}{b+cx} $ where a = -2, b = 1 and c = 1/2
how do you sketch the graph of $ y = |\frac{b+cx}{a+x}| $ ??
i got that the VA of the new graph is "$+2$" and the root of the new graph is "$-2$", but th... |
H: Eigenvalue of f and df
Given 1 is not an eigenvalue of $df$ at $x_0$, take a chart $(U,\phi)$ around $x_0.$ Then in this coordinate neighborhood, think of $f$ as a map from open ball in $\mathbb{R}^n$ (say $B$), to itself with $f(0)=0.$ Now consider we have a function $f:B\rightarrow B$ such that $f(0)=0.$ Then $d... |
H: Strange AP Calculus BC question help?
The question is $\int_0^3\frac{1}{(1-x)^2}$. I got an answer (from u-substitution) however the solution manual says that the integral does not converge. Someone told me that the integral is undefined at $x = 1$, but if we look at $\frac {1}{x}$ that function is undefined at $x=... |
H: Find the critical points and say whether they are maxima, minima or saddle points
I have this problem:
Find the critical points and say whether they are maxima, minima or saddle points
$$f(x,y)=x^2y(2−x−y)$$
My answer:
$f_x = xy (4-3 x-2 y) $
$f_y = -x^2 (-2+x+2 y)$ then
$xy (4-3 x-2 y)=0 , -x^2 (-2+x+2 y)=0$ ... |
H: Infinite linear ordered set which doesn't include infinte well-ordered subsets
Need an example of such set.
Thank you for your time
AI: What about the negative integers, usual order? Verification that this has no infinite well-ordered subset should not be difficult. |
H: Cardinality and Measurability
We can show that $\mathbb{R}$ and $\mathbb{R}^2$ or ($\mathbb{R}^n$) have same cardinality using the following one-to-one and onto mapping:
Say x = (0.123456789....)
Then f(x) = {(0.13579...),(0.2468..)}
My question is can we claim that the Borel sigma algebra in $\mathbb{R}$ has a cor... |
H: AP Calculus multiple choice question
The position of a particle along a line is given by $z(t) = 2t^3 -24t^2 + 90t +7$ for $ t \geq 0$. For what values of $t$ is the speed of the particle increasing?
a) $3 < t < 4$ only
b) $t > 4$ only
c) $t > 5$ only
d) $0 < t < 3$ and $t > 5$
e) $3 < t < 4$ and $t > 5$
For this p... |
H: $R_n = 3(2^n)-4(5^n)$, $n \geq0$, prove $R_n$ satisfies $R_n = 7R_{n-1}-10R_{n-2}$
So the question is:
$R_n=3(2^n)-4(5^n)$ for $n\ge 0$; prove that $R_n$ satisfies $R_n=7R_{n-1}-10R_{n-2}$.
I don't really know what to do from here. If I substitute
$$R_n = 3(2^n)-4(5^n)$$
into
$$Rn = 7R_{n-1}-10R_{n-2}$$
I end u... |
H: Solving For Variables In Simultaneous Equations
I'm doing some work in linear algebra and these came up and I realized I don't know how to solve them as they have quadratics in them. I'm sure I've done this before but if someone could give me a crash course on how to find the values of the variables, it would be mo... |
H: Flux across surface
Find A uniform fluid that flows vertically downward is described by the vector field F (x, y, z) = (0, 0, −1).
Find the flux through the cone z =
$z= \sqrt{x^2 + y^2}$,
$x^2 + y^2 \le 1$.
I attempted this question with spherical coordinates and i don't know why it didn't work out. I used that $... |
H: Group acting on a set
Let $A$ be a set, and let $G$ be any subgroup of $S_A$. $G$ is a group of permutations of $A$; we say it is a group acting on the set $A$. Assume here that $G$ is a finite group. If $u \in A$, the orbit of $u$ (with respect to $G$) is the set
$$O(u)=\{g(u): g \in G\}.$$
Prove that $G_u =... |
H: Can you recommend some books on elliptic function?
I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve？Many thanks in advance!
AI: McKean and Moll have written the nice book Elliptic Curves: Function Theory, Geometry, Arithmetic t... |
H: How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?
I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$,
where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$.
Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ where $s_k$ satisfies the recurrence relation
\begin{align}
&... |
H: Inner product polynomials
Let $V$ be the vector space of real polynomial $\mathbb{R}[x]$ endowed
with the inner product
$\langle f,g \rangle = \displaystyle\int_{-\infty}^{\infty}
e^{-|x|}f(x)g(x) \ dx$
By considering the sequence of subspaces $\{V_n\}$ where
$V_n = \{f(x) \in \mathbb{R}[x] : \deg f \leq n \}$... |
H: Problem related with solving ODE
I was solving old exam papers and am stuck on the following problem:
Consider the system of ODE
$\frac {d}{dx}Y=AY,Y(0)=\begin{pmatrix}
2\\
-1
\end{pmatrix}$ where $A=\begin{pmatrix}
1 &2 \\
0&-1
\end{pmatrix},Y=\begin{pmatrix}
y_1(x)\\
y_2(x)
\end{pmatrix}$. Then I have... |
H: intuition for the closed form of the fibonacci sequence
I'm trying to picture this closed form from Wikipedia visually:
The idea is, if you take $\phi^n / \sqrt{5}$ and round it to the nearest integer, you'll get the $n$th Fibonacci number. I see how it works out on paper, but is there an intuitive way to unders... |
H: Find all polynomials $P(x)$ satisfying this functional equation
Find all polynomials $P(x)$ which have the property
$$P[F(x)]=F[P(x)], \quad P(0) = 0$$
where $F(x)$ is a given function with the property $F(x)>x$ for all $x\geq 0$.
This is an exercise from my homework. I would appreciate any kind explanations... |
H: Help with Combination
Guys need help to solve this one..
How will we arrange Red balls in '$N$' places , so that if you choose any '$M$' consecutive places, there should be at least '$K$' Red balls among this '$M$' chosen places.And we should use minimum number of Red balls.
Now, If $N = 6$ and $M = 3$ and $K = 2$... |
H: the partial derivatives computing
Given that:
$u_{xx}^{''}=u_{yy}^{''} \;, \;
u(x,2x)=x \;, \;
u_{x}^{'}(x,2x)=x^{2}\;, \;
$
How to find the following values?
$u_{xx}^{''}(x,2x)=?\;
u_{xy}^{''}(x,2x)=? \;
u_{yy}^{''}(x,2x)=?\; $
Thanks a lot.
AI: As we have $u(x, 2x) = x$, the chain rule gives
$$ u_x(x,2x) ... |
H: Spherical coordinate system
I can easily write $z$ axis value is $r\cos\theta$ but what will be for $x$ and $y$ axis, explain a bit please.
From the above how can I write the area element as $d\vec{a} = r^2\sin\theta d\theta d\phi\hat{r}$?
AI: The piece of the sphere with radius between $r$ and $\mathrm{d}r$ etc. ... |
H: A few questions on nonstandard analysis
I know that nonstandard analysis is analysis plus the existence of infinitesimal numbers. Does it mean that nonstandard analysis is the same theory as $ZF+\exists$infinitesimal numbers?
From what I read about it on Wikipedia there seem to be a few different approaches to it.... |
H: I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.
My understanding from the definition in my book (Rudin) is this.
A seq. $\{p_n\}$ in a metric space $X$ (I only really know $\mathbb R^k$) is said to be a Cauchy sequence if for any given $\epsilon > 0$, $\exists N\in \... |
H: Fourier series for $f(x)=(\pi -x)/2$
I need to find the Fourier series for $$f(x)=\frac{\pi -x}{2}, 0<x<2\pi$$
Since the interval isn't symmetric over $0$, I guess I need to consider $f$'s periodic extension to $\mathbb R$. let's call it $g$. Then $$g(x)=\begin{cases}\frac{\pi -x}{2}, \text{ if } 0<x<\pi\\ \frac{-... |
H: homomorphisms of infinite groups
Prove that each of the following is a homomorphism, and describe its kernel:
the function $f: \mathbb{R}^*\to\mathbb{R}_{>0}$ defined by $f(x)=|x|$
My proof step: The kernel of $f$ is the set $$k=\{x\in\mathbb{R}^*:f(x)=e\}.$$
Let $a,b \in k$, $f(a)f(b)=f(ab)=ee=e$. I think I'm ... |
H: Why is this polynomial a function of $X^3$?
In studying that recent question, I noticed that curious (or perhaps
not so curious) property : if $x,y$ are rational numbers and $a$ is the real
part of a cubic root of $x+iy$, then $Q(a^3)=0$ where $Q$ is a polynomial
of degree three with rational coefficients.
This is ... |
H: $1+1+1+1+1+1+1+1+1+1+1+1 \cdot 0+1 = 12$ or $1$?
Does the expression $1+1+1+1+1+1+1+1+1+1+1+1 \cdot 0+1$ equal $1$ or $12$ ?
I thought it would be 12 this as per pemdas rule:
$$(1+1+1+1+1+1+1+1+1+1+1+1)\cdot (0+1) = 12 \cdot 1 = 12$$
Wanted to confirm the right answer from you guys. Thanks for your help.
AI: Your a... |
H: Finding a palindromic number which is the difference of two palindromic numbers
Let $X$ and $Y$ be two $4$-digit palindromes and $Z$ be a $3$-digit palindrome. They are related in the way $X-Y=Z$. How can we figure out $Z$?
AI: Hint: Write $X = 1001a + 110b$, $Y = 1001c + 110d$ with $a,b,c,d \in \{0,\ldots, 9\}$, w... |
H: Proof of the Riesz Representation Theorem
Theorem: Let $F$ be a continuous linear functional on the Hilbert space $H$, then $\exists !$ (exists one and only one) $y \in H$ such that $F(x) = (x,y)$ for $x\in H$.
Proof:
Uniqueness: $$F(x)=(x,y)=(x,y') \Rightarrow (x,y-y')=0 \space \forall x \in H \Rightarrow ||y-y'... |
H: On Absolutely Continuous Functions
I would like to know if we can extend the concept of absolute continuity to functions $f:[a,b]\to X$, where $X$ is a topological vector space. I browsed some books on Topological Vector Spaces but can't find the definition of absolutely continuous functions defined on $[a,b]$ and ... |
H: Can $\frac{n!}{(n-r)!r!}$ be simplified?
I'm trying to calculate in a program the number of possible unique subsets of a set of unique numbers, given the subset size, using the following formula:
$\dfrac{n!}{(n-r)!r!}$
The trouble is, on the face of it, you may need an enormous structure to hold the dividend (at le... |
H: Second Derivative of a matrix
Pardon me for not knowing LateX representation,
I have following function, where $\mu$ and $\Sigma$ are both Matrices.
$$ h = \mu^T \Sigma \mu $$
which is a function of $\alpha$, such that its derivative can be written as
$$ \def\p#1#2{\frac{\partial #1}{\partial #2}}
\p h\alpha = \le... |
H: Is it possible to find function that contains every given point?
Let say we have a arbitrary number of given points and there is at least one function, for which every point lies on its graph. Is it possible to find that function using only X and Y coordinates of every given point?
Example:
We are given points $A(0... |
H: Tangential Space of a differentiable manifold is always $\mathbb R^n$?
Let $\mathcal M$ be a differential manifold with a point $p$. Let U be an open set, $p\in U$, on $\mathcal M$ and let $\phi,\psi:U\to \mathbb R^n$ be a charts on $\mathcal M$. I'm having diffculties arranging all the concepts of a differential m... |
H: Principal value of $\int_0^\infty \frac{x^{-p}}{x-1}dx$ for $|p|<1$
If $|p|<1$, how to find the Cauchy Principal Value of $$\int_0^\infty \frac{x^{-p}}{x-1}dx$$
I tried spliting the integration from $0\to 1$ and $1 \to \infty$ and switching $x = 1/u$, but no luck getting desired result which the book says is $ \pi ... |
H: generalized derivative of Wiener process
Defined a standard Wiener process $W = (W_t , \mathcal F_t)_{t≥0}$ and a deterministic, continuously differentiable function $f : [0, ∞) → \mathbb R$. Prove that
$$f(t)W_t=\int_0^tW_sf'(s)ds+\int_0^tf(s)dW_s$$
AI: If you know Ito Lemma, then you just consider $F(t,x) = f_t\c... |
H: Pigeon Hole Principle; 3 know each other, or 3 don't know each other
I found another question in my text book, it seems simple, but the hardest part is to prove it. Here the question
There are six persons in a party. Prove that either 3 of them recognize each other or 3 of them don't recognize each other.
I heard... |
H: Projective general linear groups of order 2 = projective special linear group of order 2
If $q\geq 5$ is a prime and we consider the group PGL(2,q) the projective general linear group of order 2 and the group PSL(2,q) the projective special linear group of order 2.
Can I then conclude PSL(2,q) = PGL(2,q)?
AI: The c... |
H: Find derivatives of functions with respect to $ x$
Can someone help me with these.
Find the derivatives of the following functions with respect to $x$: here $a$ is an arbitrary
(fixed) real number.
$(a)$ $\displaystyle\int_{a}^{x^4} t^3\ \mathrm dt$
$(b)$ $\displaystyle \int_{-x^3}^{a^3} \dfrac{\mathrm dt}{1+t^2}$
... |
H: Matrix operations - equivalent operation for a given operation
This is the given problem, I need to write a code for this:
$(M*Q) \circ (N*Q) $
where $M,Q,N$ are known matrices, "$\ast$" denotes matrix multiplication and "$\circ$" denotes elementwise division.
Dimensions of the matrices:
\begin{align*}
M: &a\times ... |
H: Proof of the Lebesgue-Radon-Nikodym Theorem
Theorem: Let $\lambda, \mu$ be $\sigma$-finite measures defined on the $\sigma$-algebra $\mathcal{A}$ of the space $X$. Then,
a) Lebesgue decomposition: $\lambda=\lambda_a+\lambda_s$ where $\lambda_a \ll \mu$ and $\lambda_s \perp \mu$
b) $\exists h \in L^1(\mu)$ such tha... |
H: Extrapolating an abstract algebra proof, arriving upon an incorrect conclusion.
Could you kindly point out what is wrong with my reasoning?
EDIT: What I have unintendedly proven through my reasoning is that every field can only have one automorphism- the identity mapping. Hope this helps in navigating the mess belo... |
H: Spliting Field over $\mathbb{F}_3$
How to find the splitting field of $f(x)=x^3-x+1$ and $g(x)=x^3-x-1$ over $\mathbb{F}_3$ and how to construct a isomorphism between them?
AI: I'll try to help you with one, you do the other one and try to build an isomorphism between the corresponding fields (knowing it exists mus... |
H: 1D Green's function: from interval to infinite line
Let's consider two problems for diffusion equation.
The first one:
$$
u_t = a^2u_{xx},\qquad 0<x<l,\quad 0<t\leq T
$$
$$
u(x,0) = \phi(x), \qquad 0 \leq x \leq l
$$
\begin{equation}
u(0,t)=0,\quad u(l,t)=0, \quad 0 \leq t \leq T
\end{equation}
and the second one:... |
H: derivative of a numerical expression
I have a function $$g(y) = \int{f(y, t)dt}$$ and I am currently able to evaluate the integral numerically (doesn't seem like it is possible to find an analytical solution...), which I have also plotted for various values of $g(y)$. All good so far. However, I am also interested ... |
H: About the connection of $L^2$-convergence and convergence in distribution.
Let $(T_{1,n})_{n\in\mathbb{N}}$ and $(T_{2,n})_{n\in\mathbb{N}}$ two sequences of real valued random variables in $L^2(\Omega,\mathbb{A},P)$. Suppose that
$\|T_{1,n}\|_{L^2(\Omega)}\xrightarrow{n\rightarrow\infty} 1$,
$\|T_{2,n}\|_{L^2(\Om... |
H: Roots of a polynomial with integer coefficients
Let $f(x)$ be a polynomial in the ring $\mathbb{Z}[x]$. Suppose that $f(i)=0$, where $i^2=-1$.
Can I conclude that $x^2+1$ is a factor of $f(x)$? If so, how can be proven?
AI: Hint: Apply the remainder factor theorem.
Hint: Apply the conjugate root theorem. |
H: Half-line - open or closed set
Half-line is given by
$$A=\{x\in R^n \mid x=x_0 + t(a-x_0), \forall t\geq 0\}$$
Is it open or closed set?
AI: Clearly the complement of $A$ is open (since the point $x_0$ is not in the complement of $A$). This $A$ is closed.
On the other hand, the set $A$ is not open, so it must be ... |
H: Deferred Annuity not working
A simple financial math problem:
Mack obtains $500\ 000$ repayable over $20$ years. If interest is compounded monthly at $9.25\%$ per annum, determine the monthly repayments if the repayment begins in $6$ months time.
I used the formula:
$$P_v = x[(1-(1+i)^{-n})/i]$$
but I'm not gett... |
H: Wirtinger derivative of composition of functions
So I have a very basic question : let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^1$ function, and let $g : \mathbb{C} \rightarrow \mathbb{R}$ be defined by $g(z)=h(z \overline{z})$. I want to compute $\frac{\partial g}{\partial \overline{z}}$ (which is defined b... |
H: Ordinal arithmetic $3+\omega^2 = \omega^2$
Which one of the following equalities is false:
a) $2\cdot \omega = 3\cdot\omega$
b) $3+\omega+\omega^2 = \omega+3+\omega^2$
c) $\omega^2 + 3 = 3+\omega^2$
d) $12\cdot(5+\omega)=60 \cdot\omega$
I think the wrong one is c):
In a) and d) both sides are $\omega$. And for b) ... |
H: Cardinality of a power set (cartesian product)
$A = \{0,1,2\}$ and $C = \{1,2\}$
$|P(A \times C)| = ?$
The answer states $|P(A \times C)| = 2^{3×2} = 2^6 = 64$
What formula/logic is used to obtain this answer please?
AI: There are two theorems at play here:
Theorem 1 If $|A|=n$ and $|B|=m$ then $|A \times B|= n\c... |
H: Minimize $(x+y)(x+z)$ with constraint without calculus
Let $x,y,z \in \mathbb R^+$ such that $xyz\cdot(x+y+z) = 1$
Find $\min\{(x+y)(x+z)\}.$
Using calculus, and Lagrange multipliers, I get:
$(x+y)(x+z) \ge2$ (with the equality occurring if and only if $y=z=1,\ x=\sqrt{2} - 1$).
But I want to solve it in an easy ... |
H: Confirmation of correctness in proof regarding norm preserving operator
I just want to know if my solutions are correct for the following problem (euclidean norm assumed):
A linear transformation $T:\Bbb R^n \to \Bbb R^n$ is norm preserving if $|T(x)| = |x|$ for all $x \in \Bbb R^n$, and inner product preserving ... |
H: In a integral domain every prime element is irreducible
I'm trying to understand a proof of Hungerford's book which says that in a integral domain every prime element is irreducible:
I didn't understand why this implication $p=ab\implies p|a$ or $p|b$, is not the contrary $p=ab\implies a|p$ and $b|p$ ?
I'm a littl... |
H: Prime ideals in quotients of polynomial ring over finite field
Reading a book, i found this argument ($\mathbb{F}_2$ is the field with 2 elements):
consider the quotient ring:
$$\mathbb{F}_2[x]/(x+1)^2$$
Then it has only one prime ideal, namely the following:
$$(x+1)\mathbb{F}_2[x]/(x+1)^2$$
I've some questions:
w... |
H: Is there an image to the point at infinity through this map?
I encountered a conformal mapping on the complex plane:$$z\rightarrow e^{i\pi z}$$
and I am not sure about where it does send the point at infinity. If I could say something along the lines: $$\text{Im}(\infty) = \infty$$
Then it would map it to the origi... |
H: $\mathbb N$ a Banach space?
Is $\mathbb N$ a Banach space with the norm $|x-y|$ from $\mathbb R$? I think is Banach space because there is no convergent sequence that is not constant after some $N$. Then all limit points are in the space. But I am not sure.
AI: It surely is a complete metric space, your proof is co... |
H: $p$ is irreducible if and only if the only divisors of $p$ are the associates of $p$ and the unit elements of $R$
Let $R$ be an integral domain and $p ∈ R$ be such that $p$ is nonzero and a nonunit. Then $p$ is irreducible if and only if the only divisors of $p$ are the associates of $p$ and the unit elements of $R... |
H: A branch cut problem
In Ahlfors' Complex Analysis text, chapter 3, section 4 the transformation $z=\zeta+\frac{1}{\zeta}$ is discussed. The author notes that for every $z$, there exists 2 solutions for $\zeta$ and they are inverses of each other. In order to get a unique $\zeta$ he suggests the restriction $|\zeta|... |
H: How prove this $\sum_{k=1}^{m-1}\dfrac{(-1)^{k+1}}{C_{m}^{k}}=\dfrac{(-1)^{m}+1^m}{m+2}$
prove that
$$\sum_{k=1}^{m-1}\dfrac{(-1)^{k+1}}{C_{m}^{k}}=\dfrac{(-1)^{m}+1^m}{m+2}$$
where $C_{m}^{k}=\dfrac{m!}{(m-k)!k!}$
This problem is my frend ask me,I think we can prove by case$:m=2n,m=2n+1$
Thank you everyone can nic... |
H: expanding convoluted integrand
I have a function on the form $$g(y) = \int_{-\infty}^{\infty}{e^{-v^2}f(y-v)dv}$$ I know that $g(y)$ is linear around $0$, $g(y\approx 0)\approx yG$, and I am interested in finding this gradient $G$.
For this reason I thought that the integrand could be expanded around $y=0$. In gen... |
H: How to prove there are no more positive integers that are products of 2 and 3 consecutive numbers?
$6$ and $210$ share the property that both are the products of both two and three consecutive numbers. $6$ is $2\times3$ and $1\times2\times3$ and $210$ is $14\times15$ and $5\times6\times7$. It was easy enough to wri... |
H: A definite integration problem on law of large number
The problem is given as follows:
If $g(x),h(x)$ are continuous function on $[0,1]$, satisfying $0\le g(x) <M h(x)$, where $M$ is a nonzero constant. Prove that
$$\lim_{n\to \infty} \int_{0}^1 \int_0^1\cdots\int_0^1\frac{g(x_1)+g(x_2)+\cdots +g(x_n)}{h(x_1)+h... |
H: Show that $n^2\log\left(1+\frac{1}{n}\right)$ does not converge to $1$
How to show that $n^2\log\left(1+\dfrac{1}{n}\right)\to 1$ is false?
I have to show that $\left(1+\dfrac{1}{n}\right)^{n^2}$ doesn't tend to $e.$
AI: $$\log\left(1+\frac1n\right)=-\log\left(1-\frac1{n+1}\right)\geqslant\frac1{n+1}$$ |
H: Matrix equation involving a Pauli matrix
I should solve the following problem:
find the matrix $A$ that satisfies the following equation:
$$\sin(\pi A)+\cos(\pi A)^2= \left( \begin{array}{ccc}
0 & 1 \\
1 & 0 \end{array} \right)$$
How can I solve the problem?
Thanks in advance.
AI: Note: diagonalization is essential... |
H: Calculate interior, closure and boundary
As part of an exercise, I want to calculate the interior, closure and boundary of the following sets in $\mathbb{R}^2$ (with the standard topology).
1. $\mathbb{Z} \times \mathbb{Z}$
2. $\mathbb{Q} \times (\mathbb{Q}\cap]0,+\infty[)$
I found the following solution and wou... |
H: Choosing a bound when it can be plus or minus? I.e. $\sqrt{4}$
My textbook glossed over how to choose integral bounds when using substitution and the value is sign-agnostic. Or I missed it!
Consider the definite integral: $$ \int_1^4\! \frac{6^{-\sqrt{x}}}{\sqrt x} dx $$
Let $ u = -\sqrt{x} $ such that $$ du = - \... |
H: Modulo question about equality
True or false?
$$24 \equiv 77 \mod 16 $$
$1.$ $77/16 = 4.8125 $
$2.$ $4.8125 - 4 = .8125$
$3.$ $0.8125 \times 16 = 13$
$4.$ $24 != 13$
So the answer is false? Am I right?
AI: A simpler approach: $a\equiv b\pmod{n}$ if and only if $n$ divides $(b-a)$.
In this problem, $77-24=53$, which... |
H: Isomorphism between two finite fields
We have $k_1:= \mathbb F_7(\alpha)$ and $k_2 := \mathbb F_7(\beta)$ where $\alpha^2 = 3$ and $\beta^2 = -1$ in $\mathbb F_7$. I have to show that these two are isomorphic.
Let $\phi:k_1 \rightarrow k_2$ be a homomorphism which preserves $1 \in k_1$. Then
$$\phi(\alpha^2)= \ph... |
H: Is it possible to isolate p in $"a = 2bpq + 2apq + a(p^2) + 2aqq + bqq"?$
Is it possible to simplify this equation so that $p$ is isolated on one side instead of $a$? I tried factoring out $p$ on the right side, but I get stuck with the $ap^2$.
$$a = 2bpq + 2apq + ap^2 + 2aq^2 + bq^2$$
AI: By solving p in this equa... |
H: addition with a variable (mod)
Given $2+x \equiv 7 \pmod 3$.
$2 + 0 = 2$
$2 + 1 = 3$
$2 + 2 = 4$
.
.
.
$2 + 5 = 7$
so, the answer will be $x = 5, 8, 11, 14, 17,\dots$
Is this correct? Because somebody told me the answer should be $x = 2, 5, 8, 11, 14, 17,\dots$
AI: Hint: $\ x\equiv 7-2 \equiv \color{#c00}5 \pmod ... |
H: What is the Idea of a Relative Open Set?
My real-analysis text gave the following defintion:
Let U be a subset of E. U is open relative to E if
for $\forall t \in U$, $\exists \epsilon$ such that $N_\epsilon(t) \cap E \subset U$.
Although the idea that U is open in $\mathbb R$ follows the definition,
I normall... |
H: How does one derive $O(n \log{n}) =O(n^2)$?
I was studying time complexity where I found that time complexity for sorting is $O(n\log n)=O(n^2)$. Now, I am confused how they found out the right-hand value. According to this $\log n=n$. So, can anyone tell me how they got that value?
Here is the link where I found o... |
H: If I have three points, is there an easy way to tell if they are collinear?
Points $(a,b)$, $(m,n)$, and $(x,y)$ are selected at random. What is the quickest/easiest way to tell if they are collinear? At first I thought it was a matter of comparing slopes but that doesn't appear to be enough.
AI: "At first I though... |
H: Why $v_1 \in L$ and $v_2 \in L^\perp$?
Let $L$ be a one dimensional subspace of $R^2$ (we may view $L$ as a line in the plane through the origin).
Suppose $\alpha$ is the angle from the positive x axis to $L$.
Let $v_1=(\cos \alpha, \sin \alpha)$ and $v_2=(-\sin \alpha, \cos \alpha)$.
Then $v_1 \in L$ and $v_2 \in ... |
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