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H: Total Variation Distance between Two Matrices
I have two $n \times n$ matrices $P$ and $Q$. They are given as follows:
$$P = \begin{bmatrix}p_{11}&p_{12}&........&p_{1n}\\p_{21}&p_{22}&........&p_{2n}\\...&...&........&...\\...&...&........&...\\p_{n1}&p_{n2}&........&p_{nn}\end{bmatrix}$$
$$Q = \begin{bmatrix}q_{1... |
H: If $STU=Id_v$ find $T^{-1}$
Suppose $V$ is a vector space of finite dimension, $S,T,U:V\to V$ linear transformations. Suppose further that $STU=Id_v$. Show that $T$ is invertible and determine $T^{-1}$.
Show the statement is not necessarily true if the hypothesis that is finite is removed.
Well I know that S, T an... |
H: How to find the closed form of $\int _0^{\infty }\frac{\ln \left(x^n+1\right)}{x^n+1}\:\mathrm{d}x$
Is there a closed form for $$\int _0^{\infty }\frac{\ln \left(x^n+1\right)}{x^n+1}\:\mathrm{d}x$$
I tried multiple techniques such as the elementary ones but none really work out which leads me to think that it can m... |
H: Is this a lattice of $6$ elements?
I am reading "Set Theory & General Topology" by Fuichi Uchida.
In this book, the author wrote all lattices($15$ lattices) of $6$ elements.
I wonder the following is also a lattice of $6$ elements, but the following is not in the list written by the author.
Is the following a latt... |
H: Need help with Alternative Factoring method
I was working on some factoring, as I have always been terrible at it, when I found 3B1B's video on an easier method. There's a TL:DR at the bottom if you're familiar. The basics are as follows:
Imagine the graph of a quadratic. $x^2 - 1$ for example. It's got 2 roots $r$... |
H: simple method for expanding binomial with 3 or more terms?
I've seen
Binomial Theorem Question (Expansion of Three Terms)
Binomial Theorem with Three Terms
Expanding Equation with Binomial Theorem
but I'm not such a math expert, I need things explained in simple terms.
Basically I've heard that the solve
(x + y)^
... |
H: Can the interval defined in the fundamental theorem of calculus be $[-\infty,\infty]$?
With regards to the fundamental theorem of calculus, the statement defines a continuous function $f$ inside a closed interval $[a,b]$. Most examples I can find online uses finite numbers for $a$ and $b$. However, in problem 3 c) ... |
H: Prove that $n !$ is a divisor of $ \prod_{k=0}^{n-1}\left(2^{n}-2^{k}\right) $
Prove that for any natural number $n, n !$ is a divisor of
$
\prod_{k=0}^{n-1}\left(2^{n}-2^{k}\right)
$
i have already seen it here $\prod_{i=0}^{n-1}(2^n-2^i)$ can be divided by $n!$ but my doubt is different.
Solution -
for $p=2$ we... |
H: A property associated with restricted class of analytic functions
Question: Is my argument, described below, right?
Let $f(z)$ be an analytic function in a region containing the unit disc with $f(z)\neq 0$ in $|z|<1,$ and suppose for some fixed $M>0,$
$$
\Re \frac{zf'(z)}{Mf(z)}\leq \frac{1}{2}
$$ for all $z$ on $|... |
H: Integral of ydx + xdy
I know this is a very simple question but why is this wrong?$$\int(xdy+ydx)=\int xdy+\int ydx=x\int dy+y\int dx=2xy$$
I saw a similar question on Stack Exchange, but it was too complicated for me to understand. I am in 11th Grade and I have just done basic differentiation and integration for p... |
H: How to describe ceiling- and floor-like functions that round to a specific decimal place?
I am trying to describe floors and ceilings with non-integer factors.
Rather than rounding up or down to the nearest integer, I need to for example round to the nearest 0.1.
For example, in what I'm writing, $\lfloor3.21\rfloo... |
H: Proof verification of a number theory problem involving sequences.
$\textbf{Question:}$Does there exist an infinite sequence of integers
$a_1, a_2, . . . $ such that $gcd(a_m, a_n) = 1 $ if and only if $|m - n| = 1$?
$\textbf{My solution:}$Suppose we have a $n$ element sequence that satisfies the condition.say $a_1... |
H: Countability of the set
Let $f$ be differentiable function from $\mathbb{R}$ to $\mathbb{R}$. Consider the set
$$A_y=\{x \in \mathbb{R} : f(x)=y \}$$
I want to know whether $A_y$ is countable for each $y\in \mathbb{R}$. I can verify using simple function like polynomial , exponential function, sine, cosine function... |
H: Probability of a password not having 14 being the first two digits
We have a password that consists of $4$ digits, where the password is only numerical values from $0-9$. What is the probability that the first $2$ digits are not $14$?
So, I know that the sample space is $10 \cdot 10 \cdot 10 \cdot 10 = 10 ^4$
but I... |
H: Selection of four distinct non-consecutive natural number
Four distinct numbers are random. Four distinct numbers are randomly selected out of set of first 20 natural numbers. Find the Probability that no two of them are consecutive.
Let the sets A={1,2,3,4,...,19,20}
The number of ways of selecting 4 natural numbe... |
H: Will $\sqrt{h \sum_{i =0}^{N-1} （1 - u_i^2)^2} < C_1$ imply there exist $C_2$ satisfies $\sqrt{h \sum_{i=0}^{N-1} u_i^2} < C_2$
Assume the interval $[a, b]$ is divided by uniform grids $x_i = a + i * h, i = 0, 1, \cdots, N$, where $h = \frac{b - a}{N}$, $\mathbf u = [u_0, \cdots, u_{N-1}]^T$ is a grid funciton, wil... |
H: Find all the elements of order $63$ in $S_{50}$
Find all the elements of order 63 in the permutation group $S_{50}$.
I know that the number of elements of $S_{50}$ is $50!$, and $63=9\cdot7$, and that the number of cycles is $62!$, but I don`t quite know how to provide an answer.
I am new to this type of problems a... |
H: Find all of the x-intercepts using the Newton Method.
Use the Newton Method and find all of the x-intercepts of the function:
$$f(x)=x^3-4x^2+1$$
The Newton Method to finding the x-intercept is:
$$x_{i+1}=x_i - \frac{f(x_i)}{f'(x_i)}$$
Step $1$:
$$f(0)=(0)^3-4(0)^2+1=1 < 0$$
$$f(1)=(1)^3-4(1)^2+1=-2 > 0$$
Thus:
$... |
H: Is $U_{7^{n}}$ cyclic?
Let $(U,\cdot)$ be a group such that $U=\{z\in \mathbb{C}|\exists n \in \mathbb{N}, z^{7^{n}}=1\}$. Prove that any proper subgroup of $U$ is cyclic. Is $U$ cyclic?
I know that our group $U$ is $U_{7^{n}}$ and that any $z\in U$ can be written with deMoivre formula, but I am not sure how to h... |
H: Wrong arrangement leads to wrong answer
10 ambassadors are being arranged uniformly at random in a row. What is the
probability that: The French ambassador is next to the Russian ambassador?
The answer in the textbook is : "By viewing
the two ambassadors as one, we infer that there are exactly 2! · 9! possibilitie... |
H: Union of finitely generated submodule is a finitely generated submodule??
Let $A_i$ be a finitely generated submodule of $M$, for all $i\in\mathbb{N}$. Then $\bigcup_{i\in\mathbb{N}}A_i$
is a finitely generated submodule of $M$.
I know for normal submodule this is not true.
AI: $\Bbb Q^{(\Bbb N)}:=\{f:\Bbb N\to\Bbb... |
H: Prove that $\ker(T)$ of $T:V \rightarrow W$ is a subset of V
I understand how $\ker(T)$ would be a subspace of $V$ from the following post
Proof that a Kernel of a Linear Mapping is a Subspace
But how do we know that vectors in $\ker(T)$ would be in $V$ in the first place? Why is that a valid assumption?
AI: Based ... |
H: Finite simple group has order a multiple of 3?
Checking the list of finite simple groups, it seemed to me that all groups have order a multiple of $3$. This clear for alternating groups and checked case by case for sporadic groups. For groups of Lie type it looked like the orders are always multiples of $q(q^2 - 1)... |
H: Let $f$ be continuous. If $f(x) = 0 \implies f$ is strictly increasing at $x$, then $f$ has at most one root.
This is similar to this question I asked yesterday. I just need someone to check my proof (or offer an alternative proof) of the following statement
Let $f : \mathbb R \rightarrow \mathbb R: x \mapsto f(x)... |
H: Is there any closed curve whose area is proportional to its perimeter?
The question is in the title: Is there any closed curve whose area is proportional to its perimeter?
If not, why is it so? Can it be proved?
I tried all the simple shapes I know, but couldn't find a solution.
AI: You cannot get more proportional... |
H: Clarification regarding the outcome space of a stochastic process.
In his book Stochastic Differential Equations - An Introduction with Applications, Øksendal gives the following definition of a stochastic process:
A stochastic process is a parametrized colletcion of random variables
$$\{ X_t\}_{t\in T} $$
defined... |
H: Bourbaki's definition of function
I saw this definition and I got confused by it:
"Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E d a variable element y of F is called a functional relation in y if, for all x ∈ E, there exists a unique y ∈ F which is in the g... |
H: $Z$ has no accumulation point in $C$?
We say that $z_0$ is an accumulation point in a domain $D$ if there exists a sequence $(z_n)_{n\in N}\subset D$ s.t $z_n$ converges to $z_0$. I would like to know, using this definition, why $\Bbb Z$ has no accumulation point in $\Bbb C$ ?
AI: You are using the wrong definition... |
H: Sum $ \sum_{k=0}^\infty \frac{k^2}{4^k}$
I am an Economics undergraduate who was reading through a textbook on statistical theory.
On one of the questions, I had to find the Variance of $X$ the joint probability distribution,
$f(x,y)=\frac{1}{4^{x+y}}$, where $x$ and $y$ were discrete random variables $x=0,1,2,...$... |
H: Finding the maximum volume of a tetrahedron with 3 concurrent edges
Can someone help me out? I am not good at math. Thank you.
Find the maximum volume of the tetrahedron that have three concurrent edges and satisfy the following condition:
The sum of three edges is constant.
One edge is double the length of anothe... |
H: Can I solve this equation, that always gives me square root?
I have this equation:
Fig.1
I need to solve it for b, so I can square it:
Fig.2
and use:
Fig.3
But problem is, that I still have a square root there and I can't do anything more than just square it over and over again.
Does anyone have an idea how to solv... |
H: How is the sum of two Lebesgue integrable functions?
I'm practicing for the final exam in real-analysis and I am at the chapter Measure Theory and Integration. I found this exercise, but I don't know how to solve it...Could you please help me?
Let $(X, \cal{A}, \mu)$ be a measure space and $f, g : X → \mathbb{R}$. ... |
H: Applications of dominated convergence theorem for Lebesgue integrals
I have been working through measure theory, specifically the dominated convergence of Lebesgue integrals and its applications such as differentiating under the integral sign.
There I came across the following example
For $t>0$ it holds $\int_{-\i... |
H: Proving that $f(x)=rx+x_0 $ is open in $(\mathbb{R}^n,\varepsilon_n)$
I have the following example in my lectures notes:
In $(\mathbb{R}^n,\varepsilon_n)$ let $ B=B(O,1)$ be the open ball with center the origin $O$ and radious $1$.
$B$ is homeomorphic to any open ball $ B(x_0,r)$ with rispect to the euclidean metri... |
H: Prove that there a non-zero continuous function $f$ on $[-1,2]$ for which $\int_{-1}^2 x^{2n} f(x) \; dx = 0$ for all $n \geqslant 0$.
I've been trying to find an example of a function $f \in \mathcal{C}[-1,2]$ with $f \neq 0$ such that $\int_{-1}^2 x^{2n} f(x) \; dx = 0$ for all $n \geqslant 0$, but I'm finding it... |
H: How is the product of two Lebesgue integrable functions?
Let $(X, \cal{A}, \mu)$ be a measure space and $f, g : X → \mathbb{R}$. Determine if the following implications hold in general:
(i) both functions $f$ and $g$ are integrable $⇒ f \cdot g$ is integrable;
(ii) $f \cdot g$ is integrable $⇒$ at least one of the ... |
H: PDF goes unbounded. Is probability of event infinite?
This is follow up from here: Curve above $x$ axis but area is negative?
I have a PDF which has unit area but it goes unbounded to infinity at $x=b$ (please refer to attached link). Does it mean the probability of events near $b$ is infinite?
AI: No. Assume $$f_X... |
H: How do I find the integer solutions that satisfy $xyz = 288$ and $xy + xz + yz = 144$?
Find all integers $x$, $y$, and $z$ such that $$xyz = 288$$ and $$xy + xz + yz = 144\,.$$
I did this using brute force, where $$288 = 12 \times 24 = 12 \times 6 \times 4$$ and found that these set of integers satisfy the equati... |
H: Show that a partial derivative exists in $(0,0)$
$f: \mathbb{R}^2 \rightarrow \mathbb{R}$
$f(x,y)=\begin{cases}
xy \frac{x^2-y^2}{x^2+y^2} & ;(x,y)\neq(0,0) \\
0 & {; (x,y)=(0,0)}
\end{cases}$
Show that all partial derivatives of $f$ exist everywhere and calculate these. Distinguish between $(x,y)=(0,0)$ and $(x,... |
H: How to solve $\omega^4-[(\frac{eB}{m})^2+2\omega_0^2]\omega^2+\omega_0^4=0$ in the simplest way
I was solving a normal-mode problem and got a different result for the quadratic equation. The book provides a simpler solution than mine so I suspect I am the one who's wrong. Let's check it out.
Let us start from the f... |
H: Problem with summation by method of difference
Question: What would be the result of: $$\sum_{k=1}^{n}\frac{1}{n(n+2)}$$
My Approach:
Let $T_n$ denote the $n^{th}$ term of the given series. Then we have
$$T_1=\frac12 \left(\frac11-\frac13\right)$$
$$T_2=\frac12 \left(\frac12-\frac14\right)$$
$$T_3=\frac12 \left(\fr... |
H: How to Calculate center of mass for 20 dimensions particles
I have a problem with calculation the center of mass of 20-dimension particles,
some thing like this:
A = [1 6 8 54 6 8 5 4 8 9 6 4 7 9 6 6 3 8 43 9] , Mass = 0.25
B = [2 6 3 4 6 8 4 4 8 5 6 4 2 2 6 6 3 8 1 1] , Mass = 0.6
C = [4 3 4 53 6 2 5 21 8 1 6 2 37... |
H: If a finite sum is a unit, then it has a term that is a unit.
Source:
Theorem 19.1 (A First Course in Noncommutative Rings by T.Y. Lam)
Local Ring on Wikipedia
Theorem 19.1
For any nonzero ring R, the following statements are equivalent:
(1) $R$ has a unique maximal left ideal.
(2) $R$ has a unique maximal right ... |
H: Check if a sequence converges.
Suppose that a sequence $(x_n )$ in $R$ satisfies
$x_{n+1} = 1 −\sqrt{1 − x_n}$
for all n ∈ N. Show that $(x_n )$ converges. To what does it converge? Does
$(x_{
n+1}/
x_{n})
$
converge?
I have solved the first and found that except for $x_1=1$, all other initial values makes the sequ... |
H: Implicit function theorem for $f(x,y,z)=z^2x+e^z+y$
$f: \mathbb{R}^3 \rightarrow \mathbb{R}$, $f(x,y,z)=z^2x+e^z+y$
Show that an neighboorhood $V$ of $(1,-1)$ in $\mathbb{R}^2$ and a continuous differentiable function $g:V \rightarrow \mathbb{R}$ with $g(1,-1)=0$ and $f(x,y,g(x,y))=0$ for $(x,y) \in V$ exists.
Ca... |
H: Finding the total number of possible matches
Consider six players $P_1, P_2, P_3, P_4, P_5$ and $P_6$. A team consists of two players.
(Thus, there are $15$ distinct teams.) Two teams play a match exactly once if there is no common player. For example, team $\{P_1, P_2\}$ can not play with $\{P_2, P_3\}$ but will p... |
H: Density of $\sqrt{Z}=\sqrt{X+Y}$
Let $(X,Y)$ be a random variable with density $f_{XY}(x,y)=\frac{1}{2}(x+y)e^{-(x+y)},x>0,y>0$.
Verify that it is indeed a density i.e :
$\rightarrow \frac{1}{2}\int_{0}^{+\infty}[\int_{0}^{+\infty}(xe^{-x}e^{-y}+ye^{-x}e^{-y})dy]dx=1$
Find the marginal densities and the expected... |
H: A quadratic form problem.
Given a symmetric $n\times n$ real matrix $A$,if we have for all $x\in \mathbb{R}^n$ ,$\|x\|_2=1$,and $x^tAx = c$ for some constant $c$.
Prove that $A = \lambda I$ for some $\lambda$.
My solution is since $A$is symmetric,we can take some orthogonal transformation $Q$ that makes $Q^tAQ$ as ... |
H: Evaluating series using operator
Consider,
$$ S= \sum_{k=0}^{k=\infty} \frac{ k!}{x^{k}} (-1)^k$$
now this is, $$ S = ( 1 +D+D^2 +D^3...) ( \frac{1}{x})$$
using geometeric series
$$ 1+D+D^2.. = \frac{1}{1-D}$$
So, $$ S= \frac{1}{1-D} \frac{1}{x}$$
$$ S = \frac{1}{x-1}$$
Therefore , for x<1
$$ S= \sum_{k=0}^{k=\inft... |
H: if I have $n$-bit binary number $x$, if add 1 at its $m$-bit ($m>n$), how would the counterpart base-10 number change?
If $x$ (base-10) is an $n$-bits number in binary, such as $(x)_{10}=\underbrace{11\cdots 1}_{n\text{}}$, if I add $1$ in $m$-bit position, it becomes $(y)_{10}=\underbrace{10\cdots0011\cdots 1}_{m\... |
H: Smallest residue over $\Bbb Z[\omega]$
I'm asked to prove that $\Bbb Z[\omega]$, where $\omega^2+\omega+1=0$, is a Euclidean domain. The norm is $N(a+b\omega)=(a+b\omega)(a+b\omega^2)$.
My strategy is to write $\alpha=\beta\gamma+\rho$, then look at $$\frac{\alpha}{\beta}=\gamma+\frac{\rho}{\beta}$$
for $\alpha,\be... |
H: The correlation between the Residuals and the prediction $Cov(e,\hat{Y}) =0 $
assume a linear regression model: $y_i$ = $\beta_0$ + $\beta_1x_{i1}$..... + $\beta_px_{ip}$+ $\epsilon_i$
I'm asked to prove that:
$Cov(e,\hat{Y})$ = $0$
where: $e$ = the residuals vector
$\hat{Y}$ = the predicted vector of Y
Hint: use ... |
H: Show that the set $\{(x,-2x)\mid x \in \mathbb Z\}$ is denumerable.
Show that $A = \{(x,-2x)\mid x \in \mathbb Z\}$ is denumerable.
I know that I have to show that a bijection between $A$ and the set of natural numbers (or the set of integers since both are known to be denumerable) exists but I'm not sure what th... |
H: What are the odds of drawing the same card $3$ times in a row in a $4$ card deck ( $3$ of the same card and $1$ joker )
I made the question simple but there are $2$ things that i'd like to know:
In a deck of $4$ randomly shuffled cards with $3$ aces and $1$ joker, what are the odds of drawing $3$ aces in a row, and... |
H: Implicit differentiation of $x^2+y^2-1$
$f(x,y) = x^2+y^2-1$
$0=x^2+y^2-1 \Rightarrow y=\sqrt{1-x^2}$
I differentiaded $g(x)=\sqrt{1-x^2}$ with the chain rule and got $g'(x)=-\frac{x}{\sqrt{1-x^2}}$.
Can someone tell me how to do it with implicit differentiation?
I tried this formula $y'(x)=-\frac{f_x}{f_y}=-\fra... |
H: Reference Request: $H^1(\mathfrak g, V)=0$ for semisimple Lie algebra $\mathfrak g$ and $\mathfrak g$-module $V$
I read the following theorem in the lecture note of Victor Kac. Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero.
Theorem(Vanish... |
H: Commutativity of multiplication for natural numbers (Terence Tao's Analysis I exercise 3.6.5)
Exercise 3.6.5: Let $A$ and $B$ be sets. Show that $A\times B$ and $B\times A$ have equal cardinality by construction an explicit bijection between the two sets. Then use Proposition 3.6.14 (the one about the cardinal arit... |
H: Contexts in Natural Deduction
This is my first post. I have a basic question about the use of context in natural deduction. If $A$ is true in an empty context, written as
$\vdash A$
then, by monotonicity, in any context $\Gamma$, $A$ is true as well, written as
$\Gamma\vdash A.$
However, the interpretation of $\Gam... |
H: Probabilities of bivariates
$$ ( ≥\frac12| ≥\frac12)
$$
¿is it correct using the intervals 1/2 to infinity?
I don´t get it
AI: $$P[X>\frac{1}{2}|Y>\frac{1}{2}]=\frac{\int_{\frac{1}{2}}^{2}dx \int_{\frac{1}{2}}^{1}\frac{3}{2}y^2 dy}{\int_{\frac{1}{2}}^{1}3y^2dy}=\int_{\frac{1}{2}}^2\frac{1}{2} dx=P[X>\frac{1}{2}]=... |
H: Show that $x^4 + 8x - 12$ is irreducible in $\mathbb{Q}[x]$.
Is there a nice way to show that $x^4 + 8x - 12$ is irreducible in $\mathbb{Q}[x]$?
Right now I'm going with the rational root theorem to show there are no linear factors and this result, involving the cubic resolvent, to show there are no irreducible q... |
H: How to show that the following set is connected?
Let $X$ be a (metric) space. Let $S$ and $L_i$ ($i\in I$) be connected subsets of $X$. Assume that $S\cap L_i \neq \phi$. Show that $S\cup (\cup_i L_i)$ is a connected subset of $X$.
My work: I know that union of two connected set is connected if the intersection ... |
H: A quiz question in real analysis
I am trying to solve this quiz questions of senior batch in Real Analysis:
For disproving (A) (B) option $f(x) =x^{6}$ was sufficient.
But I am unable to think how to prove/ disprove (C) , (D) the problem arising due to function being given bounded in (C) and infinitely differentia... |
H: There are only two six-digit integers $N$, each greater than $100,000$. for which $N^2$ has $N$ as its final six digits
There are only two six-digit integers $N$, each greater than 100,000 for which $N^2$ has $N$ as its final six digits (or $N^2-N$ is divisible by $10^6$). What are these two numbers?
Is the problem... |
H: For a $2 \times 2$ matrix having eigenvalues 1,1 will the matrix satisfy a two degree monic polynomial other than characteristic polynomial?
For a 2 X 2 matrix (except the identity matrix) having eigenvalues 1,1 is it necessary for the matrix to satisfy a two degree monic polynomial (X-1)(X-K) for some real K (K is... |
H: Find upper bound for $P(X>Y+15)$
Let $E(X)=E(Y)=75$ and $Var(X)=10$ and $Var(Y)=12$ and $Cov(X,Y)=-3$. Then find upper bound for this values
a) $P(X>Y+15)$
b) $P(Y>X+15)$
I tried to solve this question by calculate $E(X^2) , E(Y^2) , E(XY)$ but i havn't find the upper bound with this datas.
AI: You can find a nont... |
H: Given a probability density, how can I sample from the induced distribution?
Let $f$ be an integratable function such that $\int f(x) dx=1$. If we want to take random samples from this, using whatever programming language one pleases, we should compute $F(t)=\int_{-\infty}^t f(x) dx$, invert this function and feed ... |
H: If every $x \in \mathbb{R}^{n}$ has a neighbourhood whose intersection with the set $A$ is closed, then $A$ is closed.
How would I go about proving this statement?
Denoting the each neighbourhood by $V_{x}$, I have tried to use the following facts:
$$
\bigcup V_{x} = \mathbb{R}^{n}
$$
and thus
$$
\bigcup \left(A \c... |
H: Is the radius of convergence related to the ratio limit or half of the interval of convergence?
I have a series $S$ with general terms $a_n=\frac{(-1)^n(x-1)^n}{(2n-1)2^n}$, $n\ge 1$:
$$S = \sum_{n=1}^\infty \frac{(-1)^n(x-1)^n}{(2n-1)2^n}$$
Finding the ratio $\left|\frac{a_{n+1}}{a_n}\right|$ and then finding the ... |
H: A $m$-dimensional differentiable manifold that has a non zero continuous $m$-form is orientable.
If $M$ is a $m$-dimensional differentiable manifold and $\omega$ is a continuous $m$-form on $M$ such that $\omega(x) \neq 0$ for every $x \in M$, then $M$ is orientable.
The author takes $A$ as the set of parameteriz... |
H: Cardinality of the set of all the subsets of $X$ which have cardinality less than $|X|$
Let $X$ be an infinite set of cardinality $|X|=\kappa$, and let $\mathcal{P}_{< \kappa}(X)$ be the set of all subsets $S$ of $X$ such that $|S| < \kappa$.
Is it true that $|\mathcal{P}_{< \kappa}(X)| < 2^{\kappa}$?
I do not know... |
H: Rank of matrix over $GF(2)$ whose rows have exactly $k$ elements $1$
Consider the $\binom{n}{k}\times n$ matrix $A$ whose rows have $k$ $1$'s and $n-k$ $0$'s. There are no repeated rows. What is the rank of $A$ over $GF(2)$?
AI: You’re asking for the dimension over $\mathbb{F}_2$ of the span of the functions $\{1,\... |
H: Convergence to 0 of certain integral by DCT
I need to prove the following property:
Let $f:\mathbb{R}^N\to \mathbb{R}$ a integrable function in $B(0,1)$. Then it is satisfied that
$$\lim_{\varepsilon\to 0}\int_{|x|<\varepsilon}f(x)dx=0.$$
My attempt consists in trying to use the Dominated Convergence Theorem. I... |
H: $\int_0^\infty \frac{1}{(x^p+2020)^q} \,dx $
Let $p,q>0$, when $$\int_0^\infty \frac{1}{(x^p+2020)^q} \,dx $$
converges?
I know when $q=1$, $p\ge2$ is the condition, and if $p\ge2$, $q\ge1$ is the condition, but in other case, I have no idea. Thank you for your help in solving this.
AI: Hint
$$ \int_0^\infty \fra... |
H: Finding the determinant of a matrix by using the adjoint
Problem:
Find the inverse of the following matrix by finding its adjoint:
$$
\begin{bmatrix}
-1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{bmatrix} $$
Answer:
The first step is to find the determinant of the matrix.
\begin{align*}
\begin{vmatrix}
-1 & 2 & 3 \\... |
H: $\inf{d(x,y);x∈A, y∈B}>0$
Let $A$ be a subset of sequence of points which converges to point a∈$R^n$. For a closed subset $B$ of $\mathbb R^n$ satisfying closure of $A$ and $B$ has no intersection,can we say $\inf{d(x,y);x∈A, y∈B}>0$?
I guess true, but I cannot proof this..
Any help would be appreciated, thank you... |
H: Are projective modules over $\mathbb{Z}[x_1,...,x_m]$ free?
The Quillen-Suslin theorem states that any finitely generated projective module over $\mathbb{k}[x_1,...,x_m]$ is free, for $\mathbb{k}$ a field.
Is it known whether this statement is true in the case that $\mathbb{k}=\mathbb{Z}$, rather than a field?
Alte... |
H: Prove that $a_{n}\to L$ as $n\to\infty$ iff $\limsup_{n\to\infty}(a_{n}) = \liminf_{n\to\infty}(a_{n}) = L$
Suppose that $(a_{n})_{n=0}^{\infty}$ is a bounded sequence and that $L\in\textbf{R}$. Then $a_{n}\to L$ as $n\to\infty$ if and only if
\begin{align*}
\limsup_{n\to\infty}(a_{n}) = \liminf_{n\to\infty}(a_{n})... |
H: Clarifying why compactness in a topology, implies compactness in a coarser topology
If $(X,\tau) $is compact and $\tau'\subseteq \tau$, then $(X,\tau')$ is compact.
I have already read several posts on the subject, but it is still unclear to me. The usual argument is:
"In a coarser space, more sets are compact, ess... |
H: Prove that the following maps are group homomorphisms. Show that this map is (not) injective and/or (not) surjective.
$ \mathbb{R}^+ \to \mathbb{C}^{\ast}$ with $x \mapsto e^{2\pi i x}$.
To prove that this is a group homomorphism, we prove $f(a+b) = f(a) \cdot f(b)$.
$$f(a+b) = e^{2\pi i (a+b)} = e^{2\pi i a + 2 \p... |
H: $f:\aleph_{\omega_1}\to\aleph_{\omega_1}$ strictly increasing and continuous with $\aleph_1$ fixed points. Can $f$ exist?
Continuous function: $\forall \lambda$ limit ordinal $f(\lambda)=\underset{\gamma<\lambda}\bigcup{f(\gamma)}$
If I prove that $Fix(f):=\{\alpha\in\aleph_{\omega_1}|f(\alpha)=\alpha\}$ is unlimi... |
H: Given $n$ slots and $k$ objects to fill the slots, what is the probability of a given slot to be filled.
Problem
Given:
$n$ slots, numbered from $1\ldots n$
$k$ objects
a slot can be filled by one object
what is the probability that a slot at some position $i$ to be filled?
Some Notation
For a visual representati... |
H: Do we need gammas to determine $\nabla$?
I know that something must be wrong with the following calculation - otherwise, the covariant derivative could be defined intrinsically on a differentiable manifold - but I don't seem to be able to find the mistake.
Let $(M,\mathcal{O},\mathcal{A},\nabla)$ be a differentiabl... |
H: Calculate partial derivatives of $f(x,y) =\begin{cases}0, & xy\neq0 \\ 1, & xy=0\end{cases}$
Calculate the partial derivatives of:
$$ f(x,y) = \begin{cases}0, & xy\neq0 \\ 1 , & xy=0\end{cases} $$
I'm not sure how to evaluate it. Can anyone give me a direction?
AI: So $xy = 0$ is only true if $x = 0$ or $y = 0$ o... |
H: Finding $a$ such that these two vectors are orthogonal
Suppose we have an inner product space V, with inner product $<x,y>$. In this space, we have two nonzero vectors u and v.
I am trying to find an arbitrary, real $a$ for which the following two vectors are orthogonal: $av-u$ and $v$.
I know two vectors are ortho... |
H: Proof Verification: $\text{Hom}(\mathbb Z[x],S)=S$ (as rings)
How to prove $\text{Hom}(\mathbb Z[x],S)=S$ (as rings), where S is any ring?
My attempt: took an element $b$ in $S$, defined a map , $b: \mathbb Z[x]\to S$ which maps $f(x)$ to $f(b)$. Clearly $b$ is a ring homomorphism, hence we proved one side inclusio... |
H: Gauss curvature derived from unit normal vector
I want to know more about the differential geometry of surfaces, especially Gaussian curvature. Obviously, we can get the mean curvature of a surface from the divergence of the unit normal vector of the surface. However, can the Gaussian curvature be derived from the ... |
H: Subgradient of argmax and chain rule
Let $\mathcal{X} \subset \mathbb{R}^n$ and $c \in \mathbb{R}^n$. Moreover, define
$$
f(c) := \max_{x\in\mathcal{X}} \ x^\top c \quad \text{and} \quad \bar{x}_c := \text{arg}\max_{x\in\mathcal{X}} \ x^\top c.
$$
In this paper, Proposition 3.1, it is argued that $\bar{x}_\hat{c}$ ... |
H: What does positive dimensional variety mean?
I recently heard the term 'positive dimensional variety'. Does this simply mean that the variety is nonempty and not a point? Or am I misunderstanding this?
AI: Yes, a variety is zero dimensional if it's a finite set of points (over an algebraically closed field), and po... |
H: Interpretation of zero angle between two elements in a inner product space
Take $f,g \in V$, where $V$ is an inner product space. Let $\langle \cdot, \cdot \rangle : V \times V \to [0,\infty)$ denote the inner product operator in $V$. Let the "angle" $\theta$ between $f$ and $g$ be defined through the rule
$$
\cos(... |
H: Is it true for all values in probability? - Intersection of $2$ sets
I know that B can have values only from $0$ to $1$ .
If I have this probability: $\text{Pr}(A=a, B \leq 1)$
Is it true to say that: $\text{Pr}(A=a, B \leq 1) = \text{Pr}(A=a)$
It seems trivial but I am not sure.. Thank you!
AI: In general if $\mat... |
H: Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin^3{(2x)}}{\ln{\left(\csc{x}\right)}} \mathop{dx}$
Challenge problem by friend is $$\int_0^{\frac{\pi}{2}} \frac{\sin^3{(2x)}}{\ln{\left(\csc{x}\right)}} \mathop{dx}$$
I know you can write $\ln{\left(\csc{x}\right)}=-\ln{\sin{x}}$ and $\sin{(2x)}=2\sin{(x)}\cos{(x)}$. I tr... |
H: Can a finite set have a topology with an infinite number of open sets?
Can a finite set have a topology with an infinite number of open sets? ..(1)
The question originated when my professor gave us as an example that if $X$ is finite or $\tau$ is finite, $(X, \tau)$ is compact
And that that was so, even if, in the... |
H: Evaluating: $\lim_{t\to\infty}\frac1t\int_0^t\sin(\alpha x)\cos(\beta x)dx$
I tried evaluating the integral but maybe there's an easier way. Please help.
Here is what I did:
$\begin{aligned}\lim_{t\to\infty}\frac1t\int_0^t \sin(\alpha x)\cos(\beta x)dx&=\lim_{t\to\infty}\frac1t\int_0^t\frac12(\sin(\alpha x+\beta x)... |
H: Not getting the right answer in this limit with absolute value
$\lim_{x \to a} \dfrac{\sqrt{ax}-|a|}{ax-a^2}$ , a<0
im getting:
$\lim_{x \to a} \dfrac{1}{\sqrt{ax}+a}$
So my final answer is:
$\dfrac{1}{|a|+a}$
But the right answer is:
$\dfrac{1}{2|a|}$
Im not sure why, can you help me please?
AI: It should be
$$
\l... |
H: If matrix $A-I$ is positive semidefinite, does $\lambda_{\inf} \geq 1$ hold?
If matrix $A-I$ is positive semidefinite, does the following hold?
$$\lambda_{\inf} \geq 1$$
where $\lambda_{\inf}$ is the infimum of the set of all eigenvalues of $A$. If so, why?
Thanks in advance.
AI: HINT: Suppose you had an eigenvecto... |
H: Proof of $n-$dimensional Brownian motion identities for components of $B_t$
The person in this post (Proving Kolmogorov's continuity condition holds for Brownian motion?) used the following two identities in their proof for an $n$ dimensional Brownian motion:
$$ \mathbb{E}((B_{t,i}-B_{s,i})^4) = 3(t-s)^2$$
and
$$ \... |
H: Meromorphic function with a removable singularity and a few poles
Here is the question:
Let $f$ be a meromorhic function on $\mathbb{C}$, having poles at the following three points: $z=5$, $z=1+3i$ and $z=3-4i$. Also, let $f$ have one removable singularity at $z=3$. For the following, find the value or explain wh... |
H: Calculating lim, why not infimum?
why the lim of the following is 1 and not infinity when n goes to infinity?
Why I believe so?
The greatest power in numerator is -2.5 and in denominator it's 1/17 -5 since the greatest power is in the numerator then the limit is infinity
AI: When $n \to \infty$, we need to find the... |
H: Copula with a certain correlation
What does it mean for values to be "drawn from a normal copula with correlation $\rho\in [0, 1]$"? Is that a normal distribution with a covariance matrix whose entries are uniform random in $[0, 1]$?
Would sampling from a multivariate normal distribution using a probability package... |
H: Convergent sum of an Harmonic-like series
Let $b$ be an integer greater than $1$ and let $d$ be a digit $0\leq d<b$. Let $A$ denote the set of all $k\in \mathbb{N}$ such that its $b$-adic expansion of $k$ fails to contain the digit $d$. If $a_k=1/k$ for $k\in A$ and $a_k=0$ otherwise, prove $\sum_{k=1}^\infty a_k<\... |
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