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H: Find the sum $\sum_{n=1}^{\infty} \frac{3^n}{5^n-2^{2n}}$.
Can someone help me with this sum $\sum_{n=1}^{\infty} \frac{3^n}{5^n-2^{2n}}$. I can't find $S_n$
AI: $$\begin{eqnarray*}S&=&\sum_{n\geq 1}\frac{3^n}{5^n-4^n}=\sum_{n\geq 1}\left(\frac{3}{5}\right)^n\frac{1}{1-\left(\frac{4}{5}\right)^n}=\sum_{n\geq 1}\sum... |
H: Topology and algebraic topology have any application in biology?
If so,
In what area? evolution, genetecs ....
And how good do my computer skills need to be to work in these areas?
AI: Topology and knot theory are used to study DNA supercoiling and topoisomerases in molecular biology. There is a little elaboration ... |
H: Solution for $\beta$ in ridge regression
The RSS of the ridge regression in matrix form is:
$$RSS(\lambda) = (y−X\beta)^T(y−X\beta) +λ\beta^T\beta$$
the ridge regression solutions are easily seen to be
$$β_{ridge}= (X^TX+λI)^{−1}X^Ty$$
See page 64, https://web.stanford.edu/~hastie/Papers/ESLII.pdf
How is this deriv... |
H: Why the Spectral Theorem does not exist in a Euclidean Space when proving
When I read Algebra by Artin, I am confused why the proof can not be generalized to the Euclidean Space.(Since the fact that some orthogonal operator can not be diagbosed)
This is the proof:(In the picture)I can not find which point is broken... |
H: Prove that $\sin(nx) \cos((n+1)x)-\sin((n-1)x)\cos(nx) = \sin(x) \cos(2nx)$
Question:
Prove that $\sin(nx) \cos((n+1)x)-\sin((n-1)x)\cos(nx) = \sin(x) \cos(2nx)$ for $n \in \mathbb{R}$.
My attempts:
I initially began messing around with the product to sum identities, but I couldn't find any way to actually use the... |
H: $ (\prod_{i\in I}\kappa_i)^\mu = \prod_{i\in I}\kappa_i^\mu $, where $ \kappa_i ,\mu $ are infinite cardinals, $I$ an infinite set.
If $|B|=\mu$ and $|A_i|=\kappa_i \;\forall i\! \in\! I$, than $(\prod_{i\in I}\kappa_i)^\mu =|\text{Fun}(B,\prod_{i \in I}A_i)|$. Also, $ \prod_{i\in I}\kappa_i^\mu = |\prod_{i \in I}{... |
H: Counting Lattice Paths with Same Start/End Point
I want to find the number of paths of length $2n$ that start and end at $(0,0)$ in the diagram below (Just to be clear, each step is between connected nodes, so for example $(0,0)$ to $(0,2)$ is not allowed):
Clearly, any such path would have an even number of steps... |
H: An interesting limit
Let $x\in\mathbb{R}.$ For all $i,j\in\mathbb{N},$ define $a_{i0} = \frac{x}{2^i}, a_{ij} = a_{i,j-1}^2 + 2a_{i,j-1}.$ Find, with proof, $\lim\limits_{n\to\infty} a_{nn}.$
Below is my attempt.
Let for each $n, p_n(x) = a_{nn}$. Then observe that $a_{n+1,n} = p_n(\frac{x}2).$ As well, $p_{n+1}(... |
H: Why is $x^4+x^2+1$ over $_2$ a reducible polynomial? What do I misunderstand?
I don't quite understand when a polynomial is irreducible and when it's not.
Take $x^2 +1$ over $_3$.
As far as I know, I have to do the following:
0 1 2 using $x \in _3$
1 2 2 using $p(x)$
I calculated it like that:
$(0^2 + 1) \mod 3 = 1... |
H: Find an open set $U$ for which a function $f$ is one to one.
I was given a function $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ which is defined as follows:
$f(x,y) = (x+iy)^3$
We look at $\mathbb{C}\;$ as $\mathbb{R}^2$ with $(x,y) = x + iy$
I need to find an open set $U \subseteq \mathbb{C}\;$ for which $f$ is in... |
H: Point estimate for quadratic loss function
Suppose $X_1,\dots,X_n$ are IID from a distribution uniform on $\left(\theta-\frac{1}{2},\theta + \frac{1}{2}\right)$, and that the prior for $\theta$ is uniform on $(10,20)$. Calculate the posterior distribution for $\theta$ given $x = X_1,\dots,X_n$ and show that the poi... |
H: Prove that if $A \sim I_n$ and $A \sim I_m$ then $n=m$
Some definitions:
Definition of equinumerous sets
Two sets are equinumerous if there exists a bijection $f: A \rightarrow B$. We write $A \sim B $ if $A$ is equinumerous to $B$
Definition of finite set and cardinality
A set $A$ is said finite if $A \sim I_n$ w... |
H: Separability of group $C^*$-algebras
Let $A$ be the group $C^*$-algebra of free group $F_n$ of rank $n(n\geq 2)$. Is $A$ separable?
AI: Yes. So long as the group $\Gamma$ is countable, both $C^*(\Gamma)$ and $C_r^*(\Gamma)$ will be separable, as in both cases, those elements of $\mathbb C\Gamma$ with coefficients ... |
H: Proof to n-th order inhomogenous differential equation
Let be $ I \subset \mathbb{R} $ an intervall and $ s \in C^{ ( \infty ) }( I) $
How can I show that every solution $ y \in C^{ (n)} (I) $ of
$$ y^{ (n)} + \sum_{j=0}^{n-1} a_jy^{(j)} = s(x) $$
( $ a_0,...,a_{n-1} \in \mathbb{R} $ constant ) is in $ C^{( \infty... |
H: What is the derivative of the function $f(x)=ix?$ Is it $i$?
Why is this? How is $i$ the slope of the function? Where is it the slope?
I understand taking the derivative with the power rule in, for example, the parabola $x^2$ becoming $2x$ and seeing where that is the slope, but I don't understand how dividing two ... |
H: Testing convergence of a series using comparison test: $\sum_{k=0}^{\infty} \frac{\sqrt{k+1}}{2^k}$?
Can someone please explain to me why this series converged? In my textbook they compare it with geometric series that I don't understand. How am I supposed to come up with this? The series is:
$$\sum_{k=0}^{\infty} ... |
H: Theorem about SDR
Theorem Let $A_1,\ldots,A_n$ be subsets of a set $X$. Suppose that, for some positive integer $m$, we have $|A(J)|\ge|J|-m\mbox{ for all }J\subseteq\{1,\ldots,n\}$, where $A(J)=\bigcup\limits_{j\in J} A_j$. Then it is possible to find $n-m$ of the sets $A_1,\ldots, A_n$ which have a SDR. I know I ... |
H: For any $n \in \Bbb N$, for any representation $\phi:SL_2(\Bbb R) \to U(n)$ we must have $\phi \begin{pmatrix} 0 &1 \\ -1 &0 \end{pmatrix}= I_{n}$
Actually this is a continuation of this question, but am asking it separately as it deserves separate discussion as an independent problem. Thanks to comments by Exodd ... |
H: Evaluate $ \lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right) $
Evaluate
$$
\lim _{x \rightarrow 0}\left(x^{2}\left(1+2+3+\dots+\left[\frac{1}{|x|}\right]\right)\right)
$$
For any real number $a,|a|$ is the largest integer not greater than $a$.
I am getting no clue! from wh... |
H: Transformation of Uniform variable
I am looking for feedback on my solution to the following problem: A random variable X has a uniform distribution on $(c, 4c)$ with $c > 0$. $Y$ is given by $ \left\{
\begin{array}{ll}
x & x \in (2c,3c) \\
0 & \text{OW}
\end{array}
\right.$
Find the distribution of $Y... |
H: Is the following problem lacking more data?
I encountered the question below on a national-level high school test that took place today.
"Two ships, A and B, depart from the port at the same time. A sails at 8 km/h on a 120 degree course. B sails on a 195 degree course. After 90 min, the course from A to B is 255 ... |
H: Evaluation of $S_{k,j}=\sum_{n_1,\ldots,n_k=1}^\infty\frac{n_1\cdots n_j}{(n_1+\cdots+n_k)!}$ for $0\leqslant j\leqslant k>0$
For a positive integer $k$, and an integer $j$ with $0\leqslant j\leqslant k$, the problem of evaluating $$S_{k,j}=\sum_{n_1,\ldots,n_k=1}^\infty\frac{n_1\cdots n_j}{(n_1+\cdots+n_k)!}$$ app... |
H: True or False questions regarding $_9$ with the irreducible polynomial $x^2 +2x+2$
Let $_9$ be constructed with the irreducible polynomial $x^2 +2x+2$.
For $a,b \in _3$ we write $ax+b \in _9$ for $ab$.
In our exam we had to find out whether the following are true or false.
I know the answers, but I don't understand... |
H: Taking logarithm preserving asymptotic equivalence
Let for some $c \in (0,1)$, $d \in (0, \infty)$ and $j \in \mathbb{N}$: $$ f(n) \sim \frac{c^n \, d^{j-1}} { (c^n + d)^{j+1}} $$ as $n \to \infty$. "$\sim$" denotes asymptotic equivalence, i.e. the quotient of both sides converges to $1$ as $n \to \infty$. Now, I w... |
H: How many perfect square factors does $20^{20} $ have?
How many perfect square factors does $20^{20} $ have?
I found that $20^{20} = 5^{20}. 2^{40}$.
$5^{2}, 5^{4}, 5^{6}, ... , 5^{20}$ (10 perfect square factors)
$2^{2}, 2^{4}, 2^{6}, ... , 2^{40}$ (20 perfect square factors)
$5^{2}.2^{2}, 5^{2}.2^{4}, ..., 5^{2}... |
H: Will $ -a e^{2x} + b e^{x} - cx + d$ always have a root for positive $a,c$?
Consider the following exponential polynomial
$$p(x) = -a e^{2x} + b e^{x} - cx + d,$$
with $a>0,c>0$ and $b,d$ arbtirary. My question is, how could one check whether this always has a root regardless of the particular choice of $b,d$?
I di... |
H: Vector subspaces F and W such that F + W = F
If $F$ and $W$ are vector subspaces of the vector space $E$, can $W$ be any other vector subspace besides $\{ 0_{E} \}$ such that $F + W = F$?
At first I thought that if $W \subset F$, then $F + W = \{ u + v: u \in F \wedge v \in F \}$, so $\forall (u+v) \subset F: u,v \... |
H: Is partial trace additive in sense of direct sum?
I have an intuition, but not sure exactly, whether the partial trace is additive in the sense of direct sum. My intuition is that partial trace is additive and direct sum acts as a sum but with orthogonality (I hope I understand that right).
I mean: $Tr_{E_1} V_1 \o... |
H: For every natural number $n$, $f(n) =$ the smallest prime factor of $n.$ For example, $f(12) = 2, f(105) = 3$
QUESTION: Let $f$ be a continuous function from $\Bbb{R}$ to $\Bbb{R}$ (where $\Bbb{R}$ is the set of all real numbers) that satisfies the following property:
For every natural number $n$, $f(n) =$ the s... |
H: prove that for any prime $p≥3$ the following divisibility holds $p|11…122…233…3…99…9-123456789$
prove that for any prime $p$ the following divisibility holds:
https://photos.app.goo.gl/1P1RSiUobgAJbbGr9
$p|11…122…233…34…445…556…667…778…899…9-123456789 $
for each different digit of the minuend being used p times.
I ... |
H: Why doesn't adjoining $\sqrt{3}$ to $\mathbb{F}_{11}$ return $\mathbb{F}_{11}$?
I am confused about a particular instance where adjoining an element of a field to itself makes it not equal to itself and I am asking for clarification. I can see the result is true, but I can not see why. We are not introducing any ne... |
H: Why is $a$ the derivative of $f(x)=ax$?
I thought there was some kind of process to calculate a derivative. Can this be graphed? I know about the power rule, the chain rule, etc. but I don't know what is happening here.
AI: You could look at this a few ways...
For instance, consider what the derivative "means." $f'... |
H: Counting the number of solutions of $x^2\equiv 1 \text{ (mod n)}$ for even $n\geq 4$?
I am trying to solve the following problem:
Given the context in the book, I have noticed the following: Suppose $n=30$ then we write the following system of equations:
$$x^2\equiv 1 \text{ (mod 2)}\\x^2\equiv 1 \text{ (mod 3)}... |
H: Is there an entire function with domains for which $f(A)=B$ and $f(B)=A$?
Let $f$ be an entire function. Suppose that there exist two nonempty disjoint, open, connected non-empty sets $A,B$ in the plane such that $f(A)=B$ and $f(B)=A$.
Does it follow that $f$ is linear?
Equivalently, if a meromorphic function sat... |
H: How to prove a tighter bound $|\lambda_3-1| \leq \epsilon^2$ for an eigenvalue of $A$ with Gerschgorin's theorem and similar matrices?
Given the following matrix
$$A = \begin{bmatrix}
8 & 1 & 0\\
1 & 4 & \epsilon\\
0 & \epsilon & 1\\
\end{bmatrix}, |\epsilon| < 1.$$
Gerschgorin's theorem states that each of the $\l... |
H: Game theory:- value of a game?
I haven't found any suitable explanation or even definition for this concept. What is the value of game in game theory? Can anybody explain it to me with an example.
AI: The value of a game is the expected value to a given player. For example, a game where you flip a coin and win $2$... |
H: Closed graph theorem between Banach spaces: sufficiency of null sequence
For for a linear operator between Banach spaces $T:E \rightarrow F$, why does the seemingly weaker implication $$x_n \rightarrow 0,\quad Tx_n \rightarrow y \implies y = 0$$
yield
$$x_n \rightarrow x,\quad Tx_n \rightarrow y \implies y = Tx.$... |
H: If $S$ is simple a module over a ring $R$ which is noetherian, hereditary and every simple module is injective, then $S$ is finitely presentated.
Let $R$ be a left noetherian and left hereditary ring , also suppose every simple left module $M$ over $R$ is injective. Prove that a simple left module $S$ over $R$ is f... |
H: Compactness of $GL_n\left (\cal{K}\right)$ where $\cal{K}$ is cantor set
Consider the following functions
$f:GL_n(\mathbb{C})\to \mathbb{C}\backslash \{0\},f(A)=det(A),\forall A\in GL_n(\mathbb{C})$;
and $g:\mathbb{R}\to M_2(\mathbb{R}),
g(x)=\begin{pmatrix} \cos x & -\sin x\\ \sin x & \cos x \end{pmatrix}, \foral... |
H: If $M$ is a compact Riemannian manifold and $g$ and $\tilde{g}$ are metrics on $M$, then $\frac{1}{C} g \leq \tilde{g} \leq C g$ for $C > 1$
I am reading Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities by Emmanuel Hebey and he stated on page $22$:
Let $M$ be a compact manifold endowed with two Rie... |
H: When will an ellipse ‘fall’ into a parabola?
Consider the parabola $y=x^2$ and an ellipse which ‘rests’ on it, given by the equation $$\frac{x^2}{a^2} +\frac{(y-h)^2}{b^2}=1$$The goal is to find all ordered pairs $(a,b)$ for which the ellipse doesn’t fall to the origin, namely it touches the parabola at two distin... |
H: Derivation of pressure using partition function
expression for the pressure due to a molecule in state number $i$
$$P_{i}=-\frac{d \varepsilon_{i}}{d V}$$
To find pressure
$$\begin{aligned}
P=\frac{N}{z} \sum_{i} P_{i} e^{-\varepsilon_{i} / k_{\mathrm{B}} T} &=-\frac{N}{z} \sum_{i}\left(\frac{d \varepsilon_{i}}{d V... |
H: Two definitions of Strong Markov property for Brownian motion
Strong Markov property for Brownian motion:
(Def 1)For every almost surely finite stopping time $T$, the process
$$\{B(T+t)-B(T): t\geq 0\}$$
is a standard Brownian motion independent of $\mathcal{F}(T)$.
(Def 2) $$\mathbb{E}_x[f(B(t))|\mathcal{F}(T)]... |
H: Relation for Bessel functions
I have function $$Q_{n}(z)=\frac{J_{n+1}(z)}{zJ_{n}(z)},$$where $J_{n+1}(z)$ and $J_{n}(z)$ are Bessel functions of the first kind. I need to prove
$$\frac{dQ_{n}}{dz}=\frac{1}{z}-\frac{2(n+1)}{z}Q_{n}(z)+zQ_{n}^{2}(z)$$
but I don't know where to start.
AI: The shown identity is more o... |
H: Given $ax^2+bx+c=0$ with two real roots, $x_1>x_2$, find a quadratic equation whose roots are $x_1+1$ and $x_2-1$ without solving the first equation
Roots of the equation $ (1): ax^2+bx+c=0$ are $x_{1}$ and $x_{2}$. They are both real.
Without solving first equation, make up new quadratic equation such that one of ... |
H: Residue of $\frac{1}{\cosh(z)}$.
When looking at $f(z)= \frac{1}{\cosh(z)}$, I found a singularity at $i \frac{\pi}{2}+i \pi k$ with $k \in \mathbb{Z}$ which has to be a pole of order 1. Now, when looking for the residue at that pole, is it enough to look at $\lim_{z \to i \frac{\pi}{2}} \frac{z-i \frac{\pi}{2}}{\c... |
H: Solving the system $x\sqrt{y} + y\sqrt{x} = 30$, $x\sqrt{x} + y\sqrt{y} = 35$
I'm stuck on this problem:
$$ \begin{cases} x\sqrt{y} + y\sqrt{x} = 30 \\ x\sqrt{x} + y\sqrt{y} = 35\end{cases} $$
I've not solved this kind of problem before. I tried formula for square of sum and also sum of cubes to possibly isolate an... |
H: How do I define inverse isomorphisms between Hom-sets?
Let $S_q(X;R)$ denote the free $R$-module with basis the singular $q$-simplices $\{\sigma:\Delta^q\to X\}$. I am trying to prove that $S^q(X;R)\cong Hom_\mathbb{Z}(S_q(X;\mathbb{Z}),R)$. We have that $S^q(X;R)=Hom_R(S_q(X; R),R)$. We claim that $Hom_\mathbb{Z}... |
H: Calculate total no . of case per category, given case rate per $100,000$ and total no. of cases
I have the information on case rate per category
Eg - $$A \to 97 \text{ per } 100,000$$
$$B \to 169 \text{ per } 100,000$$
$$C \to 189 \text{ per } 100,000$$
$$D \to 234 \text{ per } 100,000$$
$$E \to 241 \text{ per } 10... |
H: What is the limiting probability distribution of a prime random walk
This random walk has an infinite amount of possibility. these are the moves ranked most common to least $(\times 0+1,+1,\times2,\times3,\times5,\times7,\times11,\times13,\times17,\times19,\times23,\times29,...,\times P(n),...)$
the most common ope... |
H: Distribution of $X_{N(t)+1}$ in poisson process
Assume $\{N(t)\}_{t\geq 0}$ is a poisson process with parameter $\lambda$, $X_n$ is the $n^{th}$ interarrival time, $n \in \{1, 2, 3, ...\}$, which means $X_n$ is exponential distribution with parameter $\lambda$.
Then how to compute the distribution of $X_{N(t)+1}$ ... |
H: Using Fermat's Little Theorem for remainders
Using Fermat's little theorem, we know that
$$k^{p-2} \cdot k \equiv 1 \pmod p.$$
To find the multiplicative inverse of $6$ modulo $17$, we need to calculate $6^{15} \pmod {17}$. It's supposed to be all congruences hold modulo $17$.
$$6^{15} \equiv 6^8 \cdot 6^4 \cdot 6^... |
H: How to calculate $ \int_{0}^{2K(k)} dn^2(u,k)\;du$?
How to calculate $$ \int_{0}^{2K(k)} dn(u,k)^2\;du?$$ Where $dn$ is the Jacobi Elliptical function dnoidal and $k \in (0,1)$ is the modulus. I know from the Fórmula $(110.07)$ of [1] (see page 10) that
$$ \int_{0}^{K(k)} dn(u,k)^2\;du=E(k),$$
where $E$ is the norm... |
H: Fermat's little theorem $a^y \pmod{p}$ when $y
I have a problem where I need to guess $425^{17} \pmod{541}$
$p=541$ is prime so, applying Fermat's little Theorem $a^{p-1} \equiv 1 \pmod{p}$ we got
$425^{540} \equiv 1 \pmod{541}$
But how should I continue?
I am trying $\frac{17}{540}= 0*540 + 17$ but nothing to do w... |
H: Equilibrium Points Help
Hi I'm doing work over the summer for my Differential Equations Module. Finding the equilibrium points here is important for all follow on questions and wanted to check to see if I'm wayyy out? Please could someone let me know if this is okay or whether I need to go back to the drawing board... |
H: $\lim_{\lambda \to \infty}\dfrac{1}{\lambda}\int_0^{\lambda}yf(y)dy = 0$?
Assume that $f : [0,\infty) \to \mathbb{R}$ is a Borel--measurable function. Assume also that is integrable with respect to the Borel measure on $[0,\infty)$.
Is it true that:
$$\lim_{\lambda\to\infty}\dfrac{1}{\lambda}\int_0^{\lambda}yf(y)dy... |
H: Show that $\binom{n}{1}-3\binom{n}{3}+3^2\binom{n}{5}\cdots=0$
Show that if $n\equiv 0\pmod 6$ (although the statement holds true for $n\equiv 0\pmod 3$)
$\binom{n}{1}-3\binom{n}{3}+3^2\binom{n}{5}\cdots=0$
I am having trouble finding the appropriate polynomial to resolve this sum. Any hints? I prefer hints to comp... |
H: $T:X\to Z$, $S:Y\to Z$ be given linear maps and $X,Y,Z$ be given Banach spaces, if $\forall x\in X$, $Tx=Sy$ has unique solution y.
$T:X\to Z$, $S:Y\to Z$ be given linear maps and $X,Y,Z$ be given Banach spaces, if $\forall x\in X$, $Tx=Sy$ has unique solution y. Then $M:X\to Y$, $Mx=y$ is continuous.
The intuitio... |
H: $M_x$ is free $\Rightarrow \widetilde{M}$ is locally free at $x$
Let $X=\text{Spec}(A)$ where $A$ is noetherian. Suppose $M$ is a finitelly generated $A$-module and that $M_x$ is a free $A_x$-module with finite rank for some $x\in X$. Show that there exists an open neighbourhood $U\subset X$ of $x$ such that $\wid... |
H: Prove $x^4 + x^2 +1$ is always greater than $x^3 + x$
Let's say P is equal to $x^4 + x^2 +1$ and $Q$ is equal to $x^3 + x$.
For $x <0$, $P$ is positive and $Q$ is negative. Hence, in this region, $P>Q$.
For $x=0$, $P>Q$.
Also, for $x = 1$, $P>Q$.
For $x > 1$, I factored out $P$ as $x^2(x^2+1) + 1$ and $Q$ as $x(x^2... |
H: Quotient of continuous local martingale with quadratic variation
Consider a local martingale $(M_t)_{t\ge 0}$ with continuous paths and $\lim_{t\rightarrow\infty}[M]_t=\infty$ a.s.
I want to show, that
$$\lim_{t\rightarrow\infty}\frac{M_t}{[M]_t}=0\quad\text{a.s.}$$
I tried using fatou's lemma giving
\begin{align}
... |
H: Giving an proof on a combinatorial statement
Prove with a combinatorial argument that $\displaystyle\binom{a+b}{2}-\binom{a}{2}-\binom{b}{2}=ab.$
I'm assuming we can give a committee forming argument, but I'm not sure how to start.
AI: Rewrite as $$\binom{a+b}{2}=\binom{a}{2}\binom{b}{0}+\binom{a}{1}\binom{b}{1}+... |
H: Find the PDF, and the conditional PDFs of $Y$ when $Y = X + Z$, where $X$ and $Z$ are exponential functions.
$Y = X + Z$
$X$ and $Z$ are independant, and are exponentially distributed with parameters: $\lambda_X=5$ and $\lambda_Z=1$
a) Find the PDF of $Y$.
b) Find the conditional pdf of $Y$ when $X = 2$, and also t... |
H: Basis for polynomials of degree k or lower
Is there a simple way to show that $\{(x-i)^n(x+i)^{k-n}\}_{n=0,...,k}$ is a basis of $\mathbb{C}_k[x]$ (space of polynomials of degree $\le k$) for $k\ge 2$ even? And likewise that $\{(x+w)^n(x+w^2)^{k-n}\}_{n=0,...,k}$ is a basis, where $w$ is the third root of unity.
AI... |
H: What is the codomain of the function which inputs a set and outputs a vector whose entries are elements of that set?
Consider the mapping $m$ whose domain is a totally ordered set $S=\mathcal{P}(\{1,\pi,e\})\backslash \varnothing$ (where $\mathcal{P}$ represents the power set) and whose output is a vector where eac... |
H: Why the variance of uniformly distribution is like that?
According to several reference,
The variance of uniformly distribution is like below:
$$\frac{1}{12} (b-a)^2$$
However, after calculating the variance from scratch:
$$\sum _{x=a}^b \frac{\left(x-\frac{a+b}{2}\right)^2}{b-a}$$
The result is this:
$$\frac{1}{12... |
H: Verifying the definition of convergence, or showing it does not converge.
Can someone please help me prove whether this sequence converges or not? I am having trouble figuring it out. Should I find some $\epsilon$ such that its greater than our sequence? Thank you for your time and help!
For the following sequence ... |
H: Prove that 9 divides $7\cdot5^{2n}+2^{4n+1}$
We have to prove that the following statement is true for all non zero natural numbers:
$$9|7\cdot5^{2n}+2^{4n+1}$$
AI: with congruence:
$7\cdot5^{2n}+2^{4n+1}\equiv$
$(-2)\cdot(-4)^{2n} + 2^{4n}\cdot 2\equiv $
$-2\cdot 16^n + 16^n \cdot 2 \equiv 0\pmod 9$.
Oh... I didn'... |
H: Inclusion–exclusion principle for probability
Inclusion–exclusion principle for probability is as follows: https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
(1)How to use this principle to show that:
$$\sum_{i=1}^nP(\{x_i\}\subset X)+\sum_{1\leq i_1<i_2\leq n}P(\{x_{i_1}, x_{i_2}\}\subset X)+\dots... |
H: What is this decomposition?
What is the name of the decomposition shown below?
AI: It is technically a QR factorisation (https://en.wikipedia.org/wiki/QR_decomposition#Rectangular_matrix).
It is quite trivial, but you can indeed observe that on the Right Hand Side, the matrix on the left is orthogonal, and the matr... |
H: How is the rank of a matrix affected by centering the columns of a matrix?
For some $n$ by $p$ matrix $X$, I'm trying to figure out how the rank of $X$ is affected if each column in $X$ is centered by the mean of that column (call the centered design matrix $Z$).
If $p < n$ and $X$ is full column rank, $Z$ is full ... |
H: True or false: continuous image of convex is convex
Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function.
Is it true that $f(A)$ is convex given that $A$ is convex?
This claim seems to be quite intuitively true, but maybe I am using my intuition too much for $\mathbb{R} \to \mathbb{R}$ type functions. I ... |
H: Open submanifolds; why does $\mathcal{A}_U$ cover $U$?
I am currently reading Lee's Introduction to Smooth Manifolds, and have come across open submanifolds. Suppose that $M$ is a smooth manifold, and that $U \subseteq M$ is an open set, and define $\mathcal{A}_U := \{\text{smooth charts } (V, \varphi) \text{ of } ... |
H: Why are there $p+1$ solutions to a projective line over a finite field of order $p$
Let $\mathbb{F}_p$ be a finite field with $p$ elements, and let
$$x+y+z=0$$
be a projective line with $x,y,z \in \mathbb{F}_p$. In a book I am currently reading about elliptic curves, it uses the fact that this projective line obvio... |
H: Dimension of an open subset of a submanifold?
Suppose that $S$ is an embedded/regular submanifold of $M$ with $\mathrm{dim}\ S = s < \mathrm{dim}\ M$. If $U$ is an open subset of M, then $S' = U \cap S$ is an open subset of $S$ in the subspace topology.
Question: If $S' \neq \varnothing$, is the dimension of $S'$ k... |
H: In which direction is the directional derivative of $ f(x, y) = (x^2 − y^2 )/(x^2+ y^2 )$ at $(1, 1)$ equal to zero?
I tried to use the definition of directional derivative and I think I need to solve vector v which gives the address but I don't know if it's okay,
AI: The directional derivative in direction $v \in ... |
H: Sufficient conditions for De Morgan's law in intuitionistic logic
What are the sufficient conditions for De Morgan's law $\lnot(P\wedge Q)\Rightarrow \lnot P \vee \lnot Q$ in intuitionistic logic?
If $P\vee \lnot P$ and $Q\vee \lnot Q$ are true, is it true?
AI: Yes. Under that assumption, you can examine the four c... |
H: Showing $\frac{d\theta }{ d \tan \theta}=\frac{ 1}{ 1+ \tan^2 \theta}$
I suppose that
$$
\frac{d\theta }{ d \tan \theta}=\frac{d \arctan x }{ d x}= \frac{1}{1+x^2}=\frac{ 1}{ 1+ \tan^2 \theta}
$$
So is
$$
\frac{d\theta }{ d \tan \theta}=\frac{ 1}{ 1+ \tan^2 \theta}
$$
correct? And
$$
\frac{d (\theta) }{ d \tan... |
H: Prove or disprove when for any r,x $\in$ R and rx $\in$I where I is an ideal, then x $\in$ I?
Recall the definition of ideal I in a ring R;
I is a subgroup of R under addition
For any x $\in$ I and any r $\in$ R, rx $\in$ I and xr $\in$ I
My question is : Change the order
If for any r,x $\in$ R and rx $\in$I w... |
H: Find all the values of $y$ so that $\min\limits_{[1, 2]}\left | x^{3}- 3x+ y \right |= 6$ .
Find all the values of $y$ so that $\min\limits_{[1, 2]}\left | x^{3}- 3x+ y \right |= 6$ .
By Desmos https://www.desmos.com/calculator/i3cesnjguw , I see that the blue line $x= 2$ meets $\left | x^{3}- 3x+ y \right |\leq ... |
H: Evaluating $\int\frac{1}{x\sqrt{x^2+1}}dx$
I am very confused by this. I am integrating the function;
$$\int\frac{1}{x\sqrt{x^2+1}}dx$$
And Wolfram alpha is telling me, the result is;
$$\log{\left(\frac{x}{\sqrt{x^2+1}+1}
\right)}$$
However, Wolfram Mathematica is telling me that the answer is;
$$\int\frac{1}{x\sqr... |
H: definite integration of a function in terms of a composite function over a log-transformed domain
Let $f(x) = g(w)$, where $w=\log(x)$. Can the definite integral $F(b) - F(a) = \int_a^b f(x) \,dx$ be expressed as an integral involving $g(w)$ over the corresponding log-transformed interval (that is, from $\log(a)$ ... |
H: Pair of linear equation in two variables
This is from a text book:-
"The general form of a linear equation in two variables is $ax + by + c = 0$ or, $ax + by = c$ where $a, b, c$ are real numbers such that $a ≠ 0$, $b≠0$ and $x, y$ are variables.
(we often denote the condition $a$ and $b$ are not both zero by $a^2... |
H: Largest number of different values in $f(0),f(1),..,f(999)$ given $f(x)=f(398-x)=f(2158-x)=f(3214-x)$
I am having trouble trying to understand the Solution (question is also linked here). The solution states that $GCD(1056, 1760) = 352$ implies that $f(x)=f(352+x)$. However we also know that $GCD(398, 2158)=2$. Wou... |
H: Why does the DFT matrix in numpy differ from the math definition?
I want to understand the DFT matrix better (starting with the real part first).
I'm first computing a DFT matrix by calculating the FFT of an identity matrix ( I can do sp.linalg.dft and I get the same result anyway while the former is faster )
dft... |
H: Vector space and "linear structure"
This question really only concerns terminology. In the linear algebra lectures that I am watching, the professor refers to the "linear structure" of a vector space. I know the definition of linearity in the context of a linear transformation, but that's a map between vector space... |
H: Let $A$ & $B$ be sets. Prove that $\{A,B\}$ is a set.
Here are the axioms that I'm allowed to use.
Axiom of Existence:
There exists a set.
Axiom of Belonging:
If $x$ is an object and $A$ is a set, then $x \in A$ is a proposition.
Axiom of Extension:
Two sets are equal iff they have the same members.
Axiom Schema of... |
H: Find all nonconstant polynomials P such that P({X})={P(X)}
Find all nonconstant polynomials $P$ which satisfy $P(\{X\})=\{P(X)\}$, where $\{x\}$ is the fractional part of $x$.
I've tried to prove that the polynomial in question is linear, but I can't think of how to prove it, especially since we don't know anything... |
H: Show that $S$ is subset of f$^{-1}(f(S))$
Here is the full question:
Let $f : X → Y$ be a function from one set $X$ to another set $Y$ , let $S$ be a subset of $X$, and let $U$ be a subset of $Y$. Show that $S \subset f^{-1}(f(S))$
My main problem is that I am not able to translate my reasoning to a formal Mathem... |
H: maximize $v_0 x+ v_1 y$ s.t. $ (x/a)^2+(y/b)^2 =1$
How to maximize the dot product of two vectors, one is fixed, the other is constrained on an ellipse?
i.e., how to maximize
$$
v_0 x+ v_1 y
$$
s.t.
$$
\left(\frac{x}{a} \right)^2 +\left(\frac{y}{b} \right)^2=1
$$
intuitively, let $x=a \sin t, y= b \cos t$, then the... |
H: What are the differences between $\mathbb{R}^{k+m}$ and $\mathbb{R}^{k}×\mathbb{R}^{m}$
For $k,m \in \mathbb{N}$ are the two sets exactly the same? Or they are same only for $k = m = 1$?
AI: Formally, the two sets are different: the left one consists of elements of the form $\;(x_1,...,x_k,x_{k+1},...x_{k+m})\;$, w... |
H: For a topological manifold $X$ is it true that $X$ is a covering of $X\lor X$?
Here is my question:
Let $X$ be a topological manifold. Is it true that $X$ is a covering
of $X\lor X$ and $X\lor X\lor X$ and, so on.
I have a intuition, $\pi_1(X\lor X)=\pi_1(X)*\pi_1(X)$.
AI: It is not true. Consider $X=\mathbb R$,... |
H: How to evaluate : $\lim_{n \to \infty}\sum_{k=0}^{n} \frac{{n\choose k}}{n^k(k+3)}$
Usually I would write the given sum in the form
$$lim_{n \to \infty}\frac{1}{n}\sum_{r=o}^{n}{f(\frac{r}{n})}$$
and then approximate it with the integral
$$\int_{0}^{1}f(x)dx$$
but it doesn't seem so easy to do with this question.
T... |
H: Norm of functional on $L^4[0, 1]$
I am trying to calculate the norm of the operator
$$
\begin{align}
f: L^4[0, 1] &\rightarrow \mathbb{R} \\
x &\mapsto \int_0^1 t^3x(t) dt
\end{align}
$$
I started off by estimating
$$
||fx||
= \left| \int_0^1 t^3x(t) dt \right|
\le \int_0^1 |t^3x(t)| dt
\stackrel{Hölder}{\le}... |
H: Is the or statement always inclusive in Mathematics?
My question is about when the statement has a potential of inclusivity, for example a statement like "It's either day time or night time" will obviously be exclusive as it's a logical contradiction if we are in day time and night time simultaneously, however I ha... |
H: Statistics - Bootstrap Method
After scouring the internet and reference books for a couple of days I couldn't really find an answer to the current problem I am trying to solve. Lets say that I want to construct a confidence interval of a mean for a sample using the bootstrap method. The mean will represent the expe... |
H: Every Cauchy sequence in $A$ converges in $X$, where $A$ is dense. Show that $X$ is complete.
My question is: Let $( X, d)$ be a metric space and $A$ a dense subset of $X$ such that every Cauchy sequence in $A$ converges in $X$. Prove that $( X, d)$ is complete.
Solution:
Case 1: If $X = A$ then it's trivial.
Case ... |
H: Proving $r \binom{n}{r}=n\binom{n-1}{r-1}$ combinatorially. (Advice on combinatorial proofs in general?)
How do you combinatorially prove the following?
$$r \binom {n}{r} = n \binom {n-1}{r-1}$$
I find it easy to prove such equalities algebraically, but have a hard time finding the right combinatorial intuition.... |
H: why $ f(x,y) =-g(x,y)?$
I have some confusion in Apostol calculas book Page no $: 369$
Books Pdf link : https://www.academia.edu/4744309/Apostol_-_
My confusion is marked in red circle ,given below
why $ f(x,y) =-g(x,y)?$
why negative sign come ?
AI: Let $(x,y) \in S$. Let $E$ be the solid in question. Then:... |
H: Determining moment generating function $\sum_{i=1}^n iX_i$
Let $X_i \sim Ber(0.5)$ and $X_i$'s independent. Let $Y$ be a random
variable with the same distribution as $\sum_{i=1}^n iX_i$. Determine
the moment generating function of $Y$.
I figured the moment generating function of $iX_i$ would be
$$M_{iX_i}(t) = 0... |
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