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H: Does $\{ (1, 2, 3, ..., n), (2, 3, ..., n, 1), ..., (n, 1, 2, ..., n-1) \}$ span $\mathbb{R}^n$?
$\{(1, 2), (2, 1)\}$ spans $\mathbb{R}^2$ and $\{(1,2,3), (2,3,1), (3,1,2) \}$ spans $\mathbb{R}^3$.
In general, does $\{ (1, 2, 3, ..., n), (2, 3, ..., n, 1), ..., (n, 1, 2, ..., n-1) \}$ span $\mathbb{R}^n$?
AI: I'll ... |
H: $\sup\{\omega,\omega_\omega,\omega_{\omega_\omega},...\}=\text{?}$
In Ordinal $\alpha$ such that $\alpha=\omega_\alpha$? the question is asked if fixed points exist for $\alpha=\omega_\alpha$.
I'm aware that $\varepsilon_0=\sup\{\omega, \omega^\omega, \omega^{\omega^\omega},\ldots\}$.
I'm not sure what the first ... |
H: Suppose that $x_{n}\to x\in X$ and $y_{n}\to y\in X$. Show that $\displaystyle\lim_{n\rightarrow\infty}d(x_{n},y_{n}) = d(x,y)$.
Let $x_{n}$ and $y_{n}$ be two sequences in a metric space $(X,d)$. Suppose that $x_{n}$ converges to a point $x\in X$ and $y_{n}$ converges to a point $y\in X$. Show that $\displaystyle\... |
H: Estimation of squared normal distribution
I am given a $w \sim N(0,I_n)$ and $w \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times d}$ such that $X_1,..., X_d \in \mathbb{R}^n $ of $X$ that satisfy $\|X_i\|^2 = n$ where $n$ is a scalar and real number.
From here I am supposed to derive that $z_i = (w^TX)_i \sim N(... |
H: Assigning tasks to workers - How many ways to do it?
Below is a problem I did. I would like somebody to check it for me.
Problem:
A manager needs four tasks of type A done. He also needs two tasks of type B done. He has 10 employees. The manager wants to assign one task each to six employees. This means that four e... |
H: $j$-invariants for elliptic curves over $\mathbb{F}_p$
I'm reading an article about elliptic curve volcanos. I know how to compute the $j$-invariant given a curve in Weierstrass form, but i don't have any idea on how to compute every possible $j$-invariant possible for curves defined over $\mathbb{F}_p$, other than... |
H: Prove that $u(x,y) = \frac{x}{x^2+y^2}$ is harmonic in $\mathbb{R}^2\setminus\{(0,0)\}$
I want to check if I did this right. I reached the conclusion that $u$ is not harmonic. We know that a function is harmonic if $$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$
For $u(x,y... |
H: Determine the convergence of the series $\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$
Does
$$\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$$
converge?
I tried the ratio test but I failed.
AI: Ratio test:
$$
\frac{\displaystyle\frac{5^{n+1}-2^{n+1}}{7^{n+1}-6^{n+1}}}{\displaystyle\frac{5^{n}-2^{n}}... |
H: Checking irreducibility of polynomials in two variables
There are a few exercises in Hartshorne about checking singularity of an affine curve. For example, $Y$ defined by $x^2 = x^4 + y^4$ over a field $k$ (with ${\mathrm{char}}k \neq 2$). This is easy.
However, I was wondering how easy it is to prove that it is a... |
H: $\operatorname{tr}(AB)$ in terms of $\operatorname{tr}(A)$
I have two symmetric and positive semi-definite matrices $A$ and $B$. I know $\operatorname{tr}(AB) \neq \operatorname{tr}(A)\cdot \operatorname{tr}(B)$.
Are there any ways to think of $\operatorname{tr}(AB)$ in terms of $\operatorname{tr}(A)$?
AI: No. Choo... |
H: How do I prove the floor identity $⌊x + n⌋ = ⌊x⌋ + n$ in a more precise way?
I am having trouble understanding the proof provided by the author for the property stated after "Goal:".
Except from the text here is a list of useful properties:
(PROPERTY 1a) $⌊x⌋ = n$ if and only if $n ≤ x < n + 1$
(1b) $⌈x⌉ = n$ if ... |
H: Use cases for interval notation
The notation $x \in [a,b]$ and $a \le x \le b$ are equivalent in how they describe the variable $x$. When writing mathematics, when does the use of one take precedence over the other? In the former case we explicitly point out that $x$ is the element of some set whose shorthand is gi... |
H: Find orthonormal $\{b_i\}$ such that $\mathcal{A} = \sum_i \sum_j \lambda_i \mu_j b_i b_j^T$
Suppose $\{b_i\}_{i = 1}^d \subset \mathbb{R}^n$ is an orthonormal set of vectors, $d \leq n$, and assume that a matrix $\mathcal{A} \in \mathbb{R}^{n \times n}$ has the following form:
$$\mathcal{A} = \sum_{i = 1}^d \sum_{... |
H: Given topological space X, t, is there an elegant way to express Int(X), the interior of X?
Given the topological space X, t, is there an elegant way to express Int(X), the interior of X - the entire space?
I know one way: Int(X) = $\bigcup Fi$ where $Fi$ is an open set in X.
But this comes by definition. Is there... |
H: Expected value of $X$ when an unbiased coin is tossed $5$ times
An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$
consecutive heads are obtained for $k = 3, 4, 5$ otherwise $X$ takes the value $–1$. Then the expected value of $X$, is :
For $k=0, X=-1, P(X)={5\... |
H: Rudin theorem 9.21 converse part proof
Theorem: Suppose f maps an open set $E \subset R^n$ into $R^m$. Then $f \in \mathcal{C'}(E)$ if and only if the partial derivatives $D_j f_i$ exist and are continuous on E for $1 \leq i \leq m, 1 \leq j \leq n$.
For the converse $\text{it suffices to consider the case m=1} \t... |
H: Distribution of the product of a standard normal and uniform distribution
Given that $X\sim N(0,1)$ and $Z\sim \operatorname{Unif}(\{\pm1\})$. Prove that $Y = XZ$ is a standard Gaussian distribution.
My approach:
I started trying to find the pdf of them. I found that:
$$f_Y(y)=\int_{-\infty}^\infty f_Z(z)f_X\left(\... |
H: What is the probability of 7 numbers picked from 1-30 ending with a different digit?
Between 1 and 30, each number with same last digit occurs 3 times.
i) 30C1, Last digit is 0..
ii) 27C1, Last digit is 1..
iii) 24C1, Last digit 2...
iv) 21C1,
v) 18C1,
vi) 15C1,
vii) 12C1,
Possible outcomes: 30C7
Favorable outcomes... |
H: Quadratics with Integer Intercepts and Extrema
I'm a math teacher and was creating an exam for my Algebra 1 students when I tried picking an equation that had integer Intercepts (both x- and y-), as well as extrema. I wanted to do so because I figured integer values would be most easily identifiable and comfortable... |
H: What is the interval of $U$ and $V$?
If the joint distribution of X and Y is given by
$$f_{x,y}(x,y)=2e^{-(x+y)} \hspace{.3cm} I_{(0,y)} (x) I_{(0,\infty)}(y) $$
find the joint distribution of $X$ and $X + Y$.Find the marginal distributions of
$X$ and $X + Y$.
Let $U=X$ and $V=X+Y$
doing the Jacobian
$|J|=1 $
so ... |
H: A linear algebra problem related to system of linear difference equations
In the teory of system of linear difference equations, I have the following fact:
If $A\alpha − B$ is singular for all $\alpha$, then its rows are linearly dependent, and
hence there exists a matrix polynomial $\psi(\alpha)$ such that:
$$\p... |
H: Confusion regarding cross product formula
I understand that the cross product formula is
$$\mathbf{A} \times \mathbf{B} =\left\|\mathbf{A} \right\|\left\|\mathbf{B} \right\|\sin(\theta)\ \mathbf{n},$$
where $\mathbf{n}$ is a unit vector perpendicular to the plane containing $\mathbf{A}$ and $\mathbf{B}$ in the dire... |
H: Spivak's Calculus Chapter 1, Question 19a
I found this post as solution to the question. Here's a quote for easy reference.
Supposing $y_1$ and $y_2$ are not both $0$, and that there is no number $\lambda$ such that $x_1=\lambda y_1$ and $x_2=\lambda y_2$, then $$\begin{array}{tcl}0 &<& (\lambda y_1-x_1)^2 + (\lam... |
H: Solution to the integral equation with constant parameter
I wanted to solve this integral equation, I'll be grateful for explanation with step by step solution.
$y(x)=1+α\int_{0}^xdp sin(x-p)y(p)$
AI: Hint for first steps.
\begin{align*}y(x)&= 1+\alpha \int_{0}^x \sin(x-p)y(p) dp \\
&= 1+\alpha \int_{0}^x (\sin x... |
H: How to find the probability of one sample variance is two times larger than another?
I have two normal distribution where $X\sim N(\mu_{x}, 40^{2})$ and $Y\sim N(\mu_{y}, 50^{2})$. 8 samples from X and 16 samples from Y is drawn. How to determine the probability that the variance of the first sample is more than tw... |
H: some confusion regarding empty set
let $X= \{ a, b, c\}$
Then choose the correct option
$1.$ $\varnothing \in X $
$2.$ $\{\varnothing\} \in X $
$3.$$\varnothing \subseteq X $
$4.$ $\{\varnothing\} \subseteq X $
My attempt : I think all option $1$, $2$, $3$,and $4 $are true because empty set contains no el... |
H: Prove that the following quadratic form is positive semi-definite. (Simplification)
The quadratic form that was given is below:
$$n\sum_{i = 1}^{n} x_i^2 - (\sum_{i = 1}^{n} x_i)^2 \quad (1)$$
So far, I've simplified it to
$$(n-1)x_1^2 - 2x_1x_2 - ... - 2x_1x_n \\
+ (n-1)x_2^2 - 2x_2x_3 - ... - 2x_2x_n\\
... |
H: How to find number of spanning tree?
Suppose $G$ is a $k$-regular graph with $n$ vertices and with eigenvalues
$k = λ_1 > λ_2 ≥ \cdots ≥ λ_n.$
Find the number of spanning trees in $G$.
AI: You can use Kirchoff's Theorem, which states that number of spanning trees,
$$ t(G) = \frac{1}{n} \lambda_1 ' \lambda_2 '... \l... |
H: Is there some relationship between a square constraint and $l_\infty$ constraint?
A square constraint is a constraint of the type $$\{x \in \mathbb{R}| x \in [a,b], a < b\}$$ (generalization to arbitrary dimension is using Cartesian products, with $<$ overloaded)
$$\{x \in \mathbb{R}^n| x \in [a,b]^n, a < b\}$$
W... |
H: Are null sets the only ones that are disjoint but not distinct?
While studying analysis (which includes a chapter about set theory) I read this:
Two sets $A,B$ are said to be disjoint if $A ∩ B = ∅$. Note that this is not the same concept as being distinct, $A \neq B$. For instance, the sets $\{1, 2, 3\}$ and $\{... |
H: Proof that two iid Gaussian random variables are conditionally independent of their mean given their sum
My question is a simplification of a statement in this book that i.i.d. Gaussian random variables $X_1, X_2, ..., X_n \sim \mathcal{N}(\Theta, 1)$ are conditionally independent of $\Theta$ given their sum $X_1 +... |
H: If $\lim_{x\to\infty} {\{x-f(x)\}}=2$, then find $\lim_{x\to\infty} \frac{\sqrt{x+1}-\sqrt{f(x)}}{\sqrt{x}-\sqrt{f(x)}}$
Question: If $\lim_{x\to\infty} {\{x-f(x)\}}=2$, then find $$\lim_{x\to\infty} \frac{\sqrt{x+1}-\sqrt{f(x)}}{\sqrt{x}-\sqrt{f(x)}}$$
I changed the form to use the condition:
$$\lim_{x\to\infty} \... |
H: In how many ways, Miyad can do the Trick?
The numbers 1 to 6 are written on six balls. Miyad took four balls from there such that if he multiplies the numbers of the balls and then say it to Mun, then Mun can’t say the number of the balls surely. In how many ways, Miyad can do this?
This is a question of Math Oly... |
H: Find $\lim \limits_{x \to 2} {sin(x)cos(x)e^{cos(x)}\over ln(x)}$
The limit can be found if I use a calculator but how do I find it without using one? I tried to use the Taylor' series and this is what I have so far but it doesn't look right:
$${(x-{x^3 \over 3} + o(x^5)) (1 - {x^2\over2} o(x^4))(1 + cos(x) + o(co... |
H: A problem in sequences and series
Three numbers in AP are removed from first $n$ consecutive natural numbers and average of remaining numbers is found to be $\frac{43}{4}$. Find $n$ as well as removed numbers if one of the removed number is a perfect square.
AI: If we let $k-d$, $k$, $k+d$ be the 3 terms in AP whic... |
H: Reciprocal of square root of a binomial to series.
by square root algorithm and long division (or by binomial theorem) it is simple matter to find $1/\sqrt{(1-x^2)} = (1+x^2/2 + 3x^4/8 + 5x^6/16 + ...)$ >
I am new to this kind of thing. Can someone explain what is the method (step by step) to convert this recipro... |
H: Approximate solution to a transcendental equation in the limit of a variable
I have the following transcendental equation:
$$2\cot{x}=\frac{kx}{hL}-\frac{hL}{kx}\tag1$$
I would like to inquire whether an approximate solution to $(1)$ can be developed in the limit $h\rightarrow0$.
Some typical values are: $k=16, L=0... |
H: What is fog(x),f(g(x)),f"(x),fof^(n-1)(x), fog^2012(x)?
I found a math assuming fog(x)=f(g(x)), f"(x)=fof^(n-1)(x), fog^2012(x)=0.
What do they mean actually? Please describe in a way that I understand.
AI: $(f$ o $g)(x)$ is the concatenation of two functions. It is described by the first equation
$$(f \text{ o } g... |
H: Let $z, w \in \mathbb{C}$. Prove that if $zw$ and $z + w \in \mathbb{R}$, then $z = \bar{w}$
Let $z, w \in \mathbb{C}$. Prove that if $zw$ and $z + w \in \mathbb{R}$, then $z = \bar{w}$.So I have started to see complex numbers in class and by reading I find this problem. So I know that $$Re(z)= \frac{z + \bar{z}}{2... |
H: Proving doubly stochastic matrix limit
Let n > 0 and Xn be an irreducible aperiodic Markov chain having a doubly stochastic transition matrix.
By definition, $\sum_{y∈S} P(x,y) = 1$ and $\sum_{x∈S} P(x,y) = 1$ for all x y ∈ S.
I want to show that if |S| < $\infty$ then $P^{n}(x,y)\rightarrow 1/|S|$ as $n\righta... |
H: Where is the loophole in this homotopy argument?
Suppose $f,g:[0,1]\to [0,1]$ are two maps then we have that $f$ is homotopic to $g$ since $[0,1]$ is a convex space and if $h:[0,1] \to X$ is a path in an arbitrary space $X$, it implies $h \circ f$ homotopic to $h \circ g$ since homotopy is invariant under compositi... |
H: Circular Bayes Theorem?
I came across a seemingly simple Bayes Theorem question, but I am going in circles currently to try to figure it out.
I am given only that $$P(A) = .78$$ $$P(B|A) = .66$$ $$ P(B'|A') = .45$$
I am trying to find P(A'|B'). Obviously the first step is to find P(B).
So far I have shown that $$P(... |
H: Are computer integers a finite group (under addition with overflow)?
The integers and the integers modulo a prime are both groups under addition.
What about the computer representation of integers (e.g. int64)?
It's closed under addition, since a sum that is too large wraps around to the negatives. It also inherit... |
H: If A is a non-proper subset of B, can we deduct A=B?
I have the following definition:
Let $A,B$ be sets. We say that $A$ is a subset of $B$, denoted $A ⊆ B$, iff every element of A is also an element of B, i.e.
For any object $x$, x$ ∈ A ⇒ x ∈ B$.
We say that $A$ is a proper subset of $B$, denoted $A \subsetne... |
H: Can the following conditional value be expressed using the min and max functions?
For example, consider $y$ such that
$$
y=\cases{c, & if $c\le0$,\\0, & if $c>0$.}
$$
Then, it can be easily expressed as $$y=\min\{c, 0\}.$$
Using the above trick, can I express the following $y$ using $\max$ or $\min$ ?
$$
y = \case... |
H: measurable finite-valued function
I am reading the measurable function in the book of stein's real analysis. It is stated that
The finite-valued function $f$ is measurable if and only if $f^{-1}(O)$ is measurable for every open set $O$.
I am not sure whether the 'finite-valued' condition is necessary. Is there ... |
H: Is this version of the dominated convergence theorem true?
Let $I$ be a real interval, $t_0 \in I,$ $E$ a measurable set in $\mathbb{R}$ and $f \colon I \times E \to \mathbb{R}$ a function such that:
$f(t, \cdot) \in L^1(E)$ for every $t \in I;$
There exists $\lim_{t \to t_0} f(x, t) =: F(x) \in \mathbb{R}$ for al... |
H: $x^2+3x+3$ is irreducible in $\mathbb{F}_{25}[x]$
Give an example of an irreducible non-linear polynomial in $\mathbb{F}_{25}[x]$.
I know that $x^2+3x+3$ is irreducible in $\mathbb{F}_{25}[x]$ but I know no shorter proof then the exhaustive search (since $\mathbb{F}_{25}[x]$ is not a factorial ring we cannot use ... |
H: finding $h$ that make absolute instability for Euler method
I have an Euler method that has this form:
$$\hat{I}(t_{n+1}) = \hat{I}(t_{n})+h\beta \hat{I}(t_{n})[1-\frac{\hat {I}(t_{n})}{N}]$$
which can also be written like
$$\hat{I}(t_{n+1})=\phi (\hat{I}(t_{n}))$$
where $\phi (x)$ is the iteration function down b... |
H: Lebesgue integral, Is the solution right?
I'am trying to understand Lebesgue integration
Compute
$\int_{0}^{\pi}$ f(x)dx
Where
$f(x) = \begin{cases}
sin x & \text{ if } x \in \mathbb{I} \\
cosx & \text{ if } x \in \mathbb{Q}
\end{cases}$
I tried this
(L)$\int_{[0,\pi]} f(x) dx = (L)\int_{[0,\pi] \cap \... |
H: RVs that are uncorrelated but not independent
Given that $X\sim N(0,1)$, $Z\sim Unif(\{\pm 1\})$, and $Y=XZ$.
$Z$ is independent of $X$. After calculation we found that $Y\sim N(0,1)$.
Show that X and Y are uncorrelated but not independent.
I can only assume that since $Y=XZ$ then, $Y$ must be dependent of $X$.
AI... |
H: Vector triple product: BAC-CAB rule
I am currently studying Introduction to Electrodynamics, fourth edition, by David J. Griffiths. Chapter 1.1.3 Triple Products introduces the vector triple product as follows:
(ii) Vector triple product: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$. The vector triple produc... |
H: Show that the set in the ring of polynomials in $\mathbb Q[x]$ is an ideal of $\mathbb Q[x]$.
In the ring of polynomials with rational coefficients $\mathbb Q[x]$, let
$$I = \{h(x) \in \mathbb Q[x] : (x^2 - 2) \ \text{ divides }\ \ h(x)\}$$
Show $I$ is an ideal of $\mathbb Q[x]$.
I know I must show that the se... |
H: Each face of a rectangle Q ha measure zero in Jordan measure
Definition
Let $A$ a subset of $\Bbb{R}^n$. We say $A$ has measure zero in $\Bbb{R}^n$ if for every $\epsilon>0$, there is a covering $Q_1,Q_2,...$ of $A$ by countably many rectangles such that
$$
\sum_{i=1}^\infty v(Q_i)<\epsilon
$$
Definition
If $Q=[... |
H: Convex conjugate of a function
I am working on the convex conjugate, defined as $\forall y \in \mathbb{R}^n, f^*(y)=\sup_{x \in dom(f)}\langle x,y\rangle-f(x)$, where $f:\mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\} $.
I have to show that for $f(x)=\frac{1}{2}\langle Ax, x \rangle-\langle b,x \rangle, $ wh... |
H: Where is $f(x) = |4x-4-x^2|$ differentiable and where is it $C^1$ and $C^2$.
I'm a university student taking a real analysis paper. I'm currently working down my problem sheet and have arrived at a series of questions reguarding smoothness and the like. I'm still getting the hang of these types of question so if an... |
H: Summation of binomial-like terms
Simplification of the two expressions
$$S_1=n\sum_{k=0}^{n-1} {n-1 \choose k} \frac{(-1)^k}{k+1}$$
and
$$S_2=n\sum_{k=0}^{n-1} {n-1 \choose k} \frac{(-1)^k}{k+3}$$
It seems that $S_1=1$, for example; are these familiar forms,
available in a list of summations?
AI: Hint:
$$\binom{n-1... |
H: Nine cards $1$ to $9$, drawn twice simultaneously , whats the expected value of the score?
Nine cards $1$ to $9$, drawn twice simultaneously, $S$ denote sum of $2$ numbers drawn. $S \leq 5$ the score is $10-S$, and when the $S>5$ the score is $2$. whats the expected value of the score?
I don't understand the questi... |
H: Help on calculating this integral and changing variables
Calculate the integral $I=\iint_D e^\frac{x^3+y^3}{xy}dA$ where D is the surface bounded by $x=y^2$,$x^2=y$ and $x^2=2y$.
Attempt: Let $u=\frac{x^2}{y}$ and $v=\frac{y^2}{x}$. Then we have $1≤u≤2$ and $0≤v≤1$.
$$\frac{∂(u,v)}{∂(x,y)}=A = \begin{vmatrix} \frac... |
H: Solving $C^{12}_{x} + C^{12}_{x+1} = C^{13}_{2x}$.
$C^{12}_{x} + C^{12}_{x+1} = C^{13}_{2x}$
I did find by brute force the solutions $n=1$ and $n=4$, through the inequalities $2x \le 13, x \ge 0 \implies x \in \{0,1,2,3,4,5,6\}$
But is there a more analytical way to solve this?
Here is my attempt:
$C^{12}_{x} + C^... |
H: Convergence of random variables series
Let $\xi_1, ... \xi_n, ...$ - are independent random variables such that $\mathbb{P}(\xi_n=\pm 1/n)=1/2 $
I want to prove that if $\phi : \mathbb{N} \rightarrow \mathbb{N}$ is any bijection of natural numbers, then the series
$\sum_{n=1}^{\infty} \xi_{\phi(n)}$ converges a.s.... |
H: exponentiation of cardinal number
How to prove that $a^{x}\leq b^{y}$ for any cardinal number a,b, $x,y$ with a$\leq$b , $x\leq y$?
AI: We can assume that $b > 0$. The case of $a = b = 0$ holds trivially.
Let $A, B, X, Y$ be sets with cardinalities $a, b, x, y$. We may identify $A$ and $X$ with subsets of $B$ and $... |
H: How would you simplify the following boolean expression $(!A B)+(B !C)+(BC)+(A !B !C)$?
How would you simplify the following boolean expression $(!A B)+(B !C)+(BC)+(A !B !C)$?
I factorised B and managed to get $B(!A+!C+C)+(A !B !C) = B+(A !B !C)$, but I do not know how to continue.
Using a K-map, I managed to get t... |
H: pigeonhole principle cube problem
i got this problem that we need to prove that we cannot pick 28 points that are 1.75cm(atleast) from eachother in a cube where each edge is 3cm long .
i tried $dividing$ each square in the cube to $4$ , that way we will get that each edge is $1.5cm$ now and the hypotenuse will be ... |
H: Given an affine space, will an orthogonal projection have 0 inner product?
I know that if we have a vector space $V$ and orthogonally project $x$ onto it to get $\hat x$, then we have $\langle v, x-\hat x \rangle$ for any $v \in V$. However, I do not know whether the same holds for an affine space $S$. It seems tha... |
H: Wave Equation: what do Characteristic Curves mean?
This is my question:
Compute the characteristic curves of the following wave equation
$$
\frac{\partial^{2} u}{\partial t^{2}}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0
$$
and draw them on an $x-t$ coordinate system.
I couldn't figure out what "characht... |
H: Positive continuous supermartingale is a proper martingale
Let $M$ be continuous positive supermartingale with $\mathbb{E}[M_0]< \infty$. By the supermartingale convergence theorem $M_\infty = \lim M_t$ exists almost surely. How do I show that, if $\mathbb{E}[M_0]=\mathbb{E}[M_\infty]$, $M$ is a proper martingale?
... |
H: Is the set $I$ a maximal ideal?
If $I = \{f(x) ∈ Q[x] \ | \ \text{f(x) is divisible by} \ x^2 - 1 \}$
is an ideal.
Is $I$ a maximal ideal?
I know the definition of a maximal ideal, but I am struggling to understand how to prove it.
An ideal of a ring R is maximal if and only if
i) I ⊄ R; and
ii) there does no... |
H: Is there a general solution to $ \int{f \cdot g } $ if I know that $ \dfrac{d^n}{dx^n}f = $ c$ \cdot f $ $, $ for some $n \in \mathbb{N} $
These type of integrals seem to pop up a lot in physics and it got me wondering whether a closed form expression exists, even if it has some limitations (only for even n-s for e... |
H: Sum $\sum \frac{1}{(4k-3)(4k-2)(4k-1)(4k)}$
I am stuck on this problem for quite a while now, and I don't seem any closer to the solution. So, here it is:
$S = 1/4! + 4!/8! + 8!/12! + 12!/16! + ......$
I crossed out the factorials first, and it could be easily represented by the general term,
$T = \frac{1}{(4n-3)... |
H: Any reference(a book) that defines the $n$-dimensional rotation matrix?
I want to refer to a mathematics book that explains the n-dimensional rotation matrix or rotation transformation.
Wikipedia concentrates most on 2D or 3D.
There are things that one can say definition here and there, but I think it is not a good... |
H: How can i prove Delta function property with derivative?
How can i proof below delta function property?
$$\int_{-\infty}^{\infty}f(t)\delta^{(n)}(t-a)dt=(-1)^{n}f^{(n)}(a)$$, where (n) denotes nth derivate.
AI: If you don't know the formal definition of distributions and derivatives of such, just do integration by... |
H: Matrix multiplication commutativity
We know that if $A$ is $2\times 2$ square matrix\begin{bmatrix}a&b\\c&d\end{bmatrix}, such that $A$ is commutative over multiplication with any $2\times 2$ matrix, then $A$ is a scalar matrix.
To prove that I tried to rely on the matrix multiplication and then I got $4$ equations... |
H: Union of arbitrary path connected sets is necessarily a path connected?
Assume $X$ is a topological space that does not have a countable dense set. Let $\{ A_\alpha \}$ be an arbitrary set of path connected sets such that if $\alpha$, $\beta$ are two different indices then $A_\alpha$ is different from $A_\beta$ and... |
H: If R is a local ring, then its maximal left-ideal is a left and right ideal
I have been trying to prove the statement in the title, however I seem to get stuck at a certain point. Let $M$ be the maximal left-ideal of $R$. Then consider $M.r$ for $r \in R$. If $M.r \neq R$, we have that $M.r \subseteq M$. Now if thi... |
H: Probability solution doubts
I have a doubt on an exercise and this is the request:
A family would have 5 children, what is the probability to have all the children female (so 5 female children) OR all the children male (so 5 male children)?
Probability female = 0.55
Probability male= 0.45
Any help is apprecited.
T... |
H: Russian Roulette Probability Puzzle understanding
"We are to play a version of Russian Roulette, the revolver is a standard six shooter but I will put one bullet in the gun. I spin the chambers, put the gun to my head, pull the trigger and survive. I hand you the gun and give you a choice...
You may put the gun str... |
H: Prove ${\bf u}.{\bf Tv} = ({\bf u} \otimes {\bf v}) : {\bf T}$
Example 8 (on Page 6) of these lecture notes appear to suggest that for any vectors $\bf u$ and $\bf v$ and tensor $\bf T$
$
{\bf u}.{\bf Tv} = ({\bf u} \otimes {\bf v}) : {\bf T}
$
I'd appreciate help proving this.
AI: To reframe this in terms of matri... |
H: An interesting property of a particular set of triples - multiplying two and adding the other always gives 1
Find all triples of real numbers such that multiplying any two in a
triple and adding the third always gives $1$.
When will this be the case? How can we find all such triples?
So far, I've let the number... |
H: In the ring $\mathbb Z_n$,we have $kx=k.x$.
Let $\mathbb Z_n$ be the ring of integers modulo $n$.Now,suppose for $x\in \mathbb Z_n$,define $kx=x+x+...+x(k$ times$)$.And $k.x$ denote the product of $k\in \mathbb Z_n$ and $x\in \mathbb Z_n$.Is it true that $kx=k.x$?It is although very trivial still relevant thing to ... |
H: $\operatorname{tg}2x=-1$ find $x$
$\operatorname{tg}2x=-1$
$x∈[\pi/2,\pi]$
I tried expressing $\operatorname{tg}2x=\sin2x/\cos2x$ but is there any elegant other method?
AI: Compute the reference angle (first quadrant) by ignoring the sign of the tangent first. Let $y=2x$.
$\tan y = 1 \implies y =\frac{\pi} 4$
Since... |
H: A trigonometric definite integral with a parameter $\int_0^\pi\frac{dx}{1+\alpha^2\sin^2(x)}$
Give the expression of the integral
$$\int_0^\pi\frac{dx}{1+\alpha^2\sin^2(x)}$$
where $ \alpha \in (0,+\infty)$.
I tried the substitution $ t=\tan(x)$ but both bounds become zero.
I used $t=\tan(\frac x2) $ but it became
... |
H: Integer side length right triangles with area = perimeter
How many right triangles with integer side lengths (up to congruence) are there with the property that the area of the triangle is the same as the perimeter?
I found that $5, 12, 13$ has both an area and perimeter of 30 but I'm not sure if there are any more... |
H: What is the total number of positive integers <300 whose digit sum is a multiple of 5?
As stated in the title
For one digit number there is $1$
For two digit numbers there are $27(=4+(4+5)+(4+5+5)) (sum
=5n, n=1,2,3)$ $since (5n<18)$
But I'm struggling to find the ones that are of three digits
$(sum)=5n, n=1,...,4... |
H: Prove that $f_n \to f$ in the metric of $C(R)$, iff $f_n \to f$ uniformly on every compact subset of $\mathbb{R}$
Let $f_n$ be a sequence in $C(R)$. Then $f_n$ converges to $f$ in the metric of $C(R)$, iff $f_n$ converges uniformly to $f$ on every compact subset of $\mathbb{R}$.
Here, the metric on $C(R)$ is $$d(... |
H: Express in terms of Euler integrals
Express in terms of Euler integrals:
$$
\int_{0}^{+\infty}\frac{x^{m-1}}{(1+x)^n} dx
$$
AI: Let $ t=\frac{1}{1+x} $
\begin{align}
I&=\int_0^1 t^{n-m-1}(1-t)^{m-1} \,dt \\
&=B(n-m,m) \\
&=\frac{\Gamma(n-m) \Gamma(m)}{\Gamma(n)}
\end{align} |
H: Show inequality $a\sqrt{b - 1} + b\sqrt{a - 1} \le ab$
Given numbers $a$ and $b$; $a, b \ge 1.$
I'm trying to prove
$$a\sqrt{b-1} +b\sqrt{a - 1} \le ab.$$
Also conditions for turning it to equality.
I tried to use AM-GM to the $(a - 1)(b - 1)$, which less than ab, but got nothing.
Applying AM-GM to $a\sqrt{b-1}$ a... |
H: Matrices how to prove $A^{-1} = A$
Apologies mix up from earlier the wrong values where placed in $x_2$ and $x_3$.
Question 1
Proof that the following is true for matrix $A$, $A^{-1}$ = $A^{T}$ = $A$
$A$=
$$
1/7 \begin{pmatrix}
2 & 3 & 6 \\
3 & -6 & 2 \\
6 & 2 & -3 \\
\end{pmatrix}
$$
$A^T$=
$$... |
H: How to compute channel capacity?
I am working on the following exercise:
Let $\mathcal{C} = (\mathcal{X}, P, \mathcal{Y})$ be the following channel:
$$\mathcal{X} = \{0,1,2,3\}$$
$$\mathcal{Y} = \{0,1,2,3,4,5,6,7\}$$
$$P = \begin{bmatrix}
0 &1/8 &1/8 &0 &1/4 &0 &1/2 &0 \\
0 &1/8 &0 &1/2 &1/4 &1/8 &0 &0 \\
0 &1/8... |
H: Ring of locally finite matrices
Let $I$ be a countable set. A matrix $A=(a_{ij})_{i,j \in I}$ is locally finite if for every $i \in I$, the families $(a_{ij})_{j\in I}$ and $(a_{ji})_{j\in I}$ have finite support . Denote by $M_{I}(\mathbb{Z})$ the ring of locally finite matrices with integral entries indexed by $I... |
H: Continuity of functions containing greatest integer function
I need help with this problem:
For all $x \ge \frac{1}{2}$ let $f(x)=[x]+(x-[x])^{[x]}$.
Prove that $f(x)$ is continuous and increasing on $[1,+\infty[$.
I've been given this hint but still I don't really understand it:
We got :
\begin{equation}
... |
H: $n^2$ coins on a board flipping game
This is something I needed a little help with, I've been stuck on this problem for a while now.
Suppose you have $n^2$ coins on an $n \times n$ grid, $\left( n \gt 0\right)$, each with their heads side up. In each move, you can pick one of the $n$ rows or columns and flip ove... |
H: Difference between Fourier series on $(0,l)$ and Fourier series on $[0,l]$
I'm currently using Strauss's partial differential equations book and there is something that confuses me. The Fourier series of $f(x)=x$ on $(0,l)$ is not the same on $[0,l]$.
For $(0,l)$ the Fourier series is
$$x=\frac{2l}{\pi} \sum_{n=1}^... |
H: How to find d(A,B)?
Let A is the rectangular hyperbola xy=1 and B is the union of axes xy=0.find d(A,B).
Now if A and B be two non-empty subset of a metric space X then d(A,B)=inf{d(a,b): a $\in$ A and b $\in$ B}.
Honestly I didn't understand this question and have no idea how to proceed.
AI: For $x>0$, $(x,\frac{1... |
H: Order of elements in an external direct product
Can anyone explain why the largest order of an element in $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_4 $ is 12?
Also, why does $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_3$ have no elements ... |
H: Polynomial Division Under Certain Remainders
Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-17$, the remainder is $14$, and when $P(x)$ is divided by $x-13$, the remainder is $6$. What is the remainder when $P(x)$ is divided by $(x-13)(x-17)$?
Here was my process, that I'm not sure if it's righ... |
H: Find a first integral ODE
I am having trouble finding a first integral for the following system:
$$x'=x\left (1-x^2-\frac{y^2}{2}\right)$$
$$y'=y\left(1-\frac{x^2}{2}-y^2\right)$$
Could someone please help me?
AI: This is a planar gradient system. Let $f(x,y)=x(1-x^2-y^2/2)$ and $g(x,y)=y(1-x^2/2 -y^2)$. Notice tha... |
H: An example of a local ring of $V$ at $p$ not ufd
Let be $O_{V,p}$ the local ring of an irreducible variety $V$ at point $p$, I would like to know an example where $O_{V,p}$ is not UFD.
AI: Take the zero locus in the plane of $y^2 = x^3$. At the origin there are two factorizations of $y^2$. |
H: Odd prime $p$ implies positive divisors of $2p$ are $1,2,p,$ and $2p$
$1,2,p,$ and $2p$ are indeed divisors of $2p$. I want to show these are the only positive divisors. Is there a more elegant or concise way to prove this besides the proof I have below?
Suppose that positive $a \in \left([3,2p-1] \cap \mathbb{N}\... |
H: $f(x)=x+2x^2\sin(1/x)$ increasing?
The function is as follows.
$$f(x)= \begin{cases} x+2x^2 \sin(\frac{1}{x}) &\text{, if $x\neq 0$}\\ 0 &\text{, otherwise} \end{cases}$$
i solved the problem as follows,
$f$ is not increasing for any neighborhood of $0$
By using the fact,
For $x_n={1\over2n\pi}$,
$f'(x_n)=-1$ then... |
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