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H: Set of numbers that are solutions for equations such as $ x = x + 1 $
Is there any number set that is used in that kind of equations without solution on the complex numbers?
AI: Extended real numbers. We add two numbers to the real numbers namely $-\infty,\infty$ with $c\infty=\infty$ if $c>0$ and $-\infty$ if $c<0... |
H: If $x$ is a root of $x^2 + x + 1$ over $GF(4)$, then so is $x + 1$
It's clear to me how $GF(4)$ can be gotten as the quotient ring of $GF(2)/X^2+X+1$. This leads to the four elements $0, 1, x, 1+x$. I'm struggling more as seen these ones as extending $GF(2)$ with the roots of the irreducible polynomial $x^2+x+1$ in... |
H: If $f:X\to X$, $f(f(x))=x$, is $f$ onto?
I have been trying following question and was unable to solve.
Let $f: X \to X$ such that $f(f(x)) = x$ for all $x\in X. \space$ Then:
Is $f$ 1-1?
Is $f$ onto?
Clearly $f$ is 1-1 . But I am unable to deduce why $f$ must be onto or not.
AI: If $f$ were not onto, since $f:... |
H: Double sum that grows at sublinear rate
Is there an example of a non-zero function $f: \mathbb{N} \to \mathbb{R}^+$ such that for any $n \in \mathbb{N}$, the following term is sublinear (or $o(n)$)?
$$\sum_{j=1}^n \sum_{i=1}^j f(i)$$
AI: Work the other way around: You want:
$\begin{align*}
\sum_{1 \le j \le n} s(... |
H: Proof of the Central Limit Theorem
I am reading the wikipedia article that proves the central limit theorem and had a question about one of the steps they take in the proof.(See the article here, the context is not really important to understand the question)
They prove that as $n$ approaches infinity, the characte... |
H: The orbit of a non zero vector $v$ in $\mathbb{R}^n$ is $\mathbb{R}^n \setminus \{0\}$
Prove that given any non zero vectors $v$ and $w$ in $\mathbb{R}^n$, there exists an invertible $n\times n$ matrix $A$ such that $Av=w$.
(I don't know where to start. Any hints will be appreciated)
AI: If $w$ is a scalar multiple... |
H: INMO $2020$ P1: Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$.
Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. T... |
H: Calculate $x$ from a given ratio calculation with a known answer
I have the below ratio calculation, and I need to find the value of $x$:
$ \dfrac{181.5 + 16.5x}{181.5 + 11x} = 1.251 $
Can I use this to find the value of $x$?
Let me know if I need to provide more info.
Cheers
AI: Assuming you meant $(181.5+16.5X)/(... |
H: What is the probability function of the sum of $N$ categorical distribution experiments
Let's say I have a categorical distribution defined by
$$P(X)=0.5\delta(X+1)+0.25\delta(X)+0.25\delta(X-1).$$
Suppose I want to repeat the experiment $N$ times and sum all the $X$ values
The resulting value would fit another cat... |
H: If $S$ is the unit square with opposite corners $(0, 0)$ and $(1, 1)$, show $\iint_S \frac{y^2}{\sqrt{(x^2+y^2)^3}}=\log(1+\sqrt{2})$
Let $S$ be the unit square with opposite corners $(0, 0)$ and $(1, 1)$. Show $$\iint_S \frac{y^2}{\sqrt{(x^2+y^2)^3}}=\log(1+\sqrt{2}).$$
What I tried:
Since $\iint_S \frac{y^2}{\sq... |
H: Need help in understanding parametric equation of a circle in 3D
$ = \cos _1 + \sin W_1$
Where V1 is a unit vector on the circle and W1 is a unit vector perpendicular to V1
I am currently working on finding an equation for a unit circle in 3D. I have come across the above equation many times but I have no idea wh... |
H: True or false: infinite sequence in a compact topological group is dense.
This is from an exercise in Bredon's Topology and Geometry:
Let $G$ be a compact topological group (assumed to be Hausdorff). Let $g\in G$ and define $A=\{g^n:n=0,1,2...\}$. Then show that the closure $\bar{A}$ is a topological subgroup.
No... |
H: c.l.u.b. set and type of $T$
In Shelah's book Proper and Improper forcing , what is the type (i.e. what is the underlying set of $T$ and $T_\alpha$, respectively) and relationship of $T$ and $T_\alpha$
for $\alpha\in \omega_1$ in the following snippet ? I'd know though, what is $\Gamma$, if I knew
what is $T_\alpha... |
H: If $\overline B\subseteq\overset{Β°}{A}$ then is possible that $\partial (A\setminus B)=\partial A\cup\partial B$?
Lemma
If $X$ is a topological space then
$$
\partial(A\cap B)\subseteq[\overline{A}\cap\partial B]\cup[\partial A\cap\overline{B}]
$$
for any $A,B\subseteq X$.
Corollary
If $X$ is a topological space t... |
H: Theorem on GCD of polynomials
Let
$F$ be a field and suppose that
$d(x)$ is a greatest common divisor of two polynomials $p(x)$ and $q(x)$ in $F[x]$. Then there exist polynomials $r(x)$ and $s(x)$ such that $d(x)=r(x)p(x)+s(x)q(x).$
Furthermore, the greatest common divisor of two polynomials is unique.
Let
$d(x)$... |
H: closed subspaces in $\ell^p$
Given the operator: $T: \ell^1 \rightarrow \ell^2$ with $Tx = x$ which of the following subsets of $\ell^1 \times \ell^2$ are closed?
$U = \ell^1 \times \{0\}$
$V = \Gamma_T$ the Graph of T
$U + V$
Defining the operator $A: \ell^1 \times \ell^2$ and $Ax = 0$ we have $U = \Gamma_A$.
$A$ ... |
H: Arc length of function - confused about particular u-substitution in integral
I am trying to teach myself how to compute the arc length of a function through a textbook, and I am stuck on how they did a particular integral substitution in a worked example.
I'll provide the specific line of working I am stuck to pro... |
H: Relation between $f(x)$ and $f(\sqrt{x})$
This might be silly, but if $f(\sqrt{x})=\frac{0.1}{a}x$, is $f(x)=\frac{0.1}{a}x^2$?
AI: The argument of a function is a dummy variable, meaning that it is a placeholder for a number. Consider $g(x)=x^2$; we could just as equally have written $g(y)=y^2$. The function is ab... |
H: Let $A$ be a square matrix and $A=BC$ be its rank factorization. Show that rank $(A)$=rank $(A^2)$ if and only if $CB$ is non-singular.
Let $A$ be a square matrix and $A=BC$ be its rank factorization. Show that rank $(A)$=rank $(A^2)$ if and only if $CB$ is non-singular.
please give a hint.
My approach : Suppose th... |
H: Confused by an example of generalization of a point
Suppose $A=\operatorname{Spec}\mathbb{C}[x,y]$. We say that a point $P\in\operatorname{Spec}A$ is a generalization of a point $Q\in\operatorname{Spec}A$ if $Q\in\overline{\{P\}}$. So let $P=[(y-x^2)]$ and $Q=[(x-2,y-4)]$. The claim is that $P$ is a generalization ... |
H: Orthocenter, Circumcenter, and Circumradius
In triangle $ABC,$ let $a = BC,$ $b = AC,$ and $c = AB$ be the sides of the triangle. Let $H$ be the orthocenter, and let $O$ and $R$ denote the circumcenter and circumradius, respectively. Express $HO^2$ in terms of $a,$ $b,$ $c,$ and $R.$
I know what orthocenter, circum... |
H: Largest Component Interval Proof
I want to check whether the proof for the following is correct:
Every point in an open set $S\subset \mathbb{R}$ belongs to one and only one component interval of $S$, where an open interval $I$ is a component interval of $S$ if and only if $I\subset S$ and there exists no open int... |
H: Question about the proof of convergence in probability implies convergence in distribution
I am following the notes for my master's program, trying to prove conv.in probability implies conv. in distribution. The proof goes:
Let $F_n$ be the distribution function of $X$. Fix any $x$ s.t. $F$ is continuous at $x$, an... |
H: Question regarding algebraic proof for Pascal's identity.
I was looking at the proof for
$\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$, where $n$ and $k$ are each $\ge 1$.
According to the proof, expressing the right-hand side in terms of factorials, we get
$$\frac{(n-1)!}{k!(n-k-1)!}+\frac{(n-1)!}{(k-1)!(n-k)!}.$... |
H: Multinomial coefficient of a sequence in specific form
I came across a question which asked me to find the coefficient of $x^{2n}$ in the following polynomial:
$$(\sum\limits_{i=0}^{n-1} x^i )^{2n+1}$$
My approach was to isolate every term, i.e. if we choose $x^2$ , n times and 1 n times again, we get a part of the... |
H: Prove that $b^2-4ac$ can not be a perfect square
Given $a$,$b$,$c$ are odd integers Prove that $b^2-4ac$ can not be a perfect square.
My try:Let $a=2k_1+1,b=2n+1,c=2k_2+1;n,k_1,k_2 \in I$
$b^2-4ac=(2n+1)^2-4(2k_1+1)(2k_2+1)$
$\implies b^2-4ac=4n^2+4n+1-16k_1k_2-8k_2-8k_1-4 $
AI: Assume $b^2-4ac=d^2$.
Then $d$ is o... |
H: Prove that $T$ is normal if and only if $\|Tv\| = \|T^*v\|$ for all $v$
I'm going through the proof of the above theorem in Linear Algebra Done Right (3rd ed). It goes as follows:
\begin{align}
T \text{ is normal } &\iff T^*T - TT^* = 0\\
&\iff \langle (T^*T - TT^*)v, v \rangle = 0, \text{ for ... |
H: Show that $\sum_x \Big\lfloor\sqrt[m]{\frac{n}{x}}\Big\rfloor=\lfloor n\rfloor$
This seems to be a trivial fact however, I can't find a satisfactory proof for the following statement:
For any integer $n>0$, and $m>0$,
$$
\sum_x\left\lfloor\sqrt[m]{\frac{n}{x}}\right\rfloor=\lfloor n\rfloor
$$
where the sum is over... |
H: Let $R$ be the Ring of all functions in $\mathscr{C}^{0}[0, 1]$
$\bullet~$ Problem: Let $R$ be the Ring of all functions in $\mathscr{C}^{0}[0, 1]$. Prove that the map $\varphi : R \to \mathbb{R}$ defined by
$$ \varphi(f) = \int_0^1 f(t) dt \quad \text{for any } f \in \mathscr{C}^{0}([0, 1])$$
is a Homomorphism ... |
H: Defining a topology on $\mathbf{Q}$
We may define a topology on the set $\mathbf{Q}$ of rational numbers
by taking for opens sets all unions of bounded open intervals.
I am not sure I understand this. Do they mean that we take
$$\mathfrak{D}:=\left\{A\ \middle|\ (\exists x)(\exists y)\left(x\in\mathbf{Q}^{\mathbf... |
H: Question regarding Triangle Inequality.
Working on the book: Lang, Serge & Murrow, Gene. "Geometry - Second Edition" (p. 23)
If two sides of a triangle are 12 cm and 20 cm, the third side must be larger than __ cm, and smaller than __ cm.
The answer given by the author is:
$20 - 12 < x < 12 + 20$, where x is the ... |
H: Expected number of coin flip
I need to calculate the expected number of coin flips needed to get two consecutive heads.
Below is my approach -
Expected number 1. Probability is 0 (because I need atleast 2 tosses)
Expected number 2. It is HH. Probability is .5^2
Expected number 3. It is THH. Probability is .5^3
So... |
H: Probability in Poisson distribution.
Q: The location of trees in a park follows a Poisson distribution with rate $1$ tree per $300$ square meters. Suppose that a parachutist lands in the park at a random location. What is the probability that the nearest tree is more than $4$ meters away from where he lands?
Is thi... |
H: How should I notate this function?
My apologies for the vague, non-descript title, I couldn't come up with a concise way to describe what I mean.
Basically, I have a sequence $A$ such that $\forall \;x \in A: x \in \{0, 1, 2, ..., n\}$, where $n \in \mathbb{N}$. Let's take $n = 6$ as an example. $A$ could look like... |
H: Uniform convergence of sequence on interval $[-b,0]$
I am aiming to prove that there exists a continous function $\exp:\mathbb{R}\rightarrow\mathbb{R}$ using the sequence $$e_n^x=\sum_{k=0}^{n}\frac{x^k}{k!}$$
So far: defining the sequence $e_n^x=\sum_{k=0}^{n}\frac{x^k}{k!}$ given that this is a polynomial, I hav... |
H: Notation for a coordinate grid
Consider the coordinate grid of points between $(-5, -5)$ and $(5, 5)$, inclusive, spaced 0.3 units apart in both directions. Is there a more compact way of notating this without writing something long, such as $\{(-5, -5), (-5, -4.7), \dots, (-5, 5), (-4.7, -5), \dots, (5, 5)\}$?
AI:... |
H: To what degree does a centered random walk $S_n$ behave asymptotically similar to $-S_n$?
Let $S_n$ be a centered random walk, i.e. increment mean $EX = 0$. For convenience let us say that $EX^2 < \infty$ and it only takes two values $\{a, -b\}$, $a,b > 0$. Considering objects such as $$ E ( e^{S_n}) \quad \text{ ... |
H: Fourier transform of $e^{-at}u(t-b)$
I have this signal $ s(t) = e^{- \frac{t}{RC} } [ u(t) - u(t - T_c) ] $ and I have to calculate the Fourier transform. I obtained $$ S(f) = \frac{ RC( 1 - e^{-i2 \pi f T_c } )}{1 + i 2 \pi f RC } $$ but I should obtain $$ S(f) = \frac{ RC( 1 - e^{-i2 \pi f T_c } e^{\frac{-T_c}{... |
H: Prove $x^n-p$ is irreducible over $Z[i]$ where $p$ is an odd prime.
Prove $x^n-p$ is irreducible over $Z[i]$ where $p$ is an odd prime.
By gausses lemma this is equivalent to irreducability over $\mathbb{Q}(i)$. Using field extensions this is easy. $[\mathbb{Q}(i,\sqrt[n]{p}):\mathbb{Q}(i)][\mathbb{Q}(i):\mathbb{Q}... |
H: Show that $y'(t)=y^{2/3}(t) \text{ with } y(0)=0$ has infinitely many solutions
Show that the problem
$$y'(t)=y^{2/3}(t) \text{ with } y(0)=0$$
has infinitely many solutions.
Explain why the existence and uniqueness theorem does not apply here
My attempt
By solving the differential equation by the variable separati... |
H: Convergence/Divergence of Complex Series $\sum\limits_{n=1}^{\infty} \frac{n(2+i)^n}{2^n}$
$$\sum\limits_{n=1}^{\infty} \frac{n(2+i)^n}{2^n}$$
My Attempt: I am new to analyzing complex series, so please forgive me in advance. I apply the ratio test:
$$\lim_{n \to \infty}\frac{|a_{n+1}|}{|a_n|} = \lim_{n \to \infty}... |
H: Derivative Greater Than 0 Implies One-To-One Function In Neighborhood
Let $f: \textrm{dom}(f) \rightarrow \mathbb{R}.$
Let $x_0 \in \mathbb{R}.$
Assume $f'(x_0) > 0$.
i.e. $~ \displaystyle\lim_{h \rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h} > 0$
i.e. $~ \exists l > 0 \textrm{ s.t. } \forall \varepsilon_1 > 0, \exis... |
H: Classifying the cyclic extensions of a fixed degree
This is the following result from Milne's Fields and Galois Theory (page 72):
Milne only showed the "only if" part but I admit that I cannot give the reason why the "if" part is true.
My attempt: In case we have $a= b^r c^n$, it is $a^{1/n}/c$ a root of $X^n - b^... |
H: Rudin's Functional Analysis Theorem 7.23
At the end of (b) of Theorem 7.23 in Rudin's Functional Analysis, it says
"Since (1) holds for $z \in \mathbb{R}^n$ (by the choice of $u$), Lemma 7.21 completes the proof of (b)."
Here, (1) is $\mathbf{f(z) = u(e_{-z})}$, where $u \in \mathscr{D}'(\mathbb{R}^n)$. The latte... |
H: Properties of distribution function
Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X$ a random variable and $F(x) = P(X^{-1}(]-\infty, x])$. The statement I am trying to prove is
The distribution function $F$ of a random variable $X$ is right continuous, non-decreasing and satisfies $\lim_{x \to \infty}F(... |
H: Combinatorics of a Tournament
$8$ people participate in a tournament, so that each person plays with all the other people once. If a person wins against another, then the winner gets $2$ points, while the losing team will get none. If they tie, they each get $1$ point, respectively. When finished, the people are r... |
H: Are there any elementary functions $\beta(x)$ that follows this integral $\int_{y-1}^{y} \beta(x) dx =\cos(y)$
Are there any simple functions $\beta(x)$ that follows this integral $$\int_{y-1}^{y} \beta(x) dx =\cos(y)$$
I think there is an infinite amount of solutions that are continuous everywhere but how can I fi... |
H: Integral of modulus is the modulus of the integral iff arguments are constant.
I'm sure this has been asked before but I can't find it and it's bugging me. I'm reading Voison's Hodge theory book and I ran into this elementary inequality that I realize I don't understand as well as I should: if $f:\mathbb C\to \math... |
H: Dual image map restricts to open sets?
A book I'm reading on category theory says that if $A$ and $B$ are topological spaces and $f:A\to B$ is continuous, then the "dual image" map
$$f_*(U)=\{\,b\in B\mid f^{-1}(b)\subseteq U\,\}$$
restricts to open sets; that is, $f_*:\mathcal{O}(A)\to\mathcal{O}(B)$. (So then it'... |
H: How to find the intersection of the graphs of $y= x^2$ and $y = 6 - |x|$?
I was trying to solve this question in preparation for the Math subject GRE exam:
The region bounded by the graphs of $y=x^{2}$ and $y=6-|x|$ is revolved around the $y$ -axis. What is the volume of the generated solid?
(A) $\frac{32}{3} \pi$... |
H: Why don't these two different methods of counting give the same result?
We have 4 bananas, 5 apples, 6 oranges. How many ways can we choose 7 fruits with at least 4 oranges?
The straight forward method is to divide this into cases with 4, 5, or 6 oranges and then picking from bananas and apples, so that we have cho... |
H: Constructing A Truth Table
How do I construct a truth table with a formula that has 3 logical operators that lack a parentheses?
$$P \lor Q \land \neg(R \lor \neg S)$$
AI: Generally
$$P \lor Q \land \neg(R \lor \neg S)$$ is same as $$P \lor (Q \land (\neg(R \lor (\neg S))))$$
Can you proceed now? |
H: List all the pairs $(x,y)$ s.t. $x^2 - y^2 = 2020$
List all of the pairs $(x,y) \in \mathbb{Z}^2$ s.t. $^2 - ^2 = 2020$.
The prime factorization of $2020$ is $2^2 \cdot 5 \cdot 101$. I used the fact that there exists a solution $(x,y) \in \mathbb{Z}^2$ to the Diophantine equation $x^2 - y^2 = n$ if and only if $n$ ... |
H: Konig's lemma proof
Konig's lemma
In this proof of Konig's lemma taken from Kaye's book "The mathematics of logic: A guide to completeness theorems and their applications", I am having a lot of trouble understanding the "We are going to find a sequence of $s(n)$ of elements of $T$ such that..." portion. We are tryi... |
H: ISL 2006 G3:Prove that the line $AP$ bisects the side $CD$.
Let $ABCDE$ be a convex pentagon such that
$$ \angle BAC = \angle CAD = \angle DAE \qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.$$ The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$.
My Proof: Note that ... |
H: Finding a closed formula for a sum
I wrote this sum (out of the blue) and wondered if it has a closed form:
$$\sum_{k=1}^{\infty} L^{\frac{1}{k}} \cdot(-1)^{k+1}$$ where $L \in \mathbb{N}$
I thought of a sum that would use "$\text{k-root}$" but with alternating sign ($+$ to $-$ etc..)
I couldn't find a way to do so... |
H: Understand a derivation
I am learning formal logic through the book An Exposition of Symbolic Logic; in chapter 1, section 10, I am asked to derivate the following argument:
(PβQ) β S
S β T
~T β Q
β΄ T
I couldn't work out the solution, so I saw the answer the book tells:
Show T
~T ass id
Q 2 pr3 mp
Show PβQ
... |
H: Expressing a vector space over a finite field as a finite union of proper subspaces.
Am trying to solve the following exercise which appeared in an abstract algebra textbook :
Assume that $V$ is an $n$-dimensional vector space over a finite field ${\bf F}_{q}$ which consists of $q$ elements. If $V$ is a finite set ... |
H: Is $f(x)=\frac{x^{2}-1}{x-1}$ continuous at $x=1$?
Given $f(x)=\frac{x^{2}-1}{x-1}$. The function is said to be discontinuous at $x=1$ but since we can simplify it and rewrite $f(x)=x+1$, this removes the discontinuity. So is the function continuous or discontinuous at $x=1$
How do the two forms of $f(x)$ differ as... |
H: Moore-Penrose Pseudoinverse: Because $(AA^+)^T = AA^+$, we have that $A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$
I am studying this answer by user "Etienne dM". They claim that, because $(AA^+)^T = AA^+$, we have that $A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$. However, I do not understand how $A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb... |
H: Solving Laplace Equation using Separation of Variables inside an annulus
Question: find the bounded function $u(x,y)$ that satisfies the following conditions
$$\nabla^2u(x,y)=0, \qquad 4<x^2+y^2<16$$
$$u(x,y)=x, \qquad x^2+y^2=4$$
$$u(x,y)=y, \qquad x^2+y^2=16$$
First of all I transform this problem to polar coor... |
H: Projection matrix.
Suppose $P=X(X'X)^{-1}X'$ and $X$ can be descomposed as $X= [X_1 X_2]$, where $X$ is a matrix. Then is true that $PX_1=X_1$.
My proof:
$X'X=[X_1^2+X_2^2]$ Then $[X_1^2+X_2^2]^{-1}=1/(X_1^2+X_2^2)$. Thus $(X_1^2+X_2^2)X_1/(X_1^2+X_2^2)=X_1$
Is this proof correct?
AI: ok, So $X$ is tall skinny ma... |
H: Prove $\mathcal{R}:=\left\{\left(a,b\right)\mid a\le b\right\}$ is not symmetric
Given a homogeneous binary relation $\mathcal{R}$ over a set $A$,and is defied as:
$$\mathcal{R}:=\left\{\left(a,b\right)\mid a\le b\right\}$$
Prove $\mathcal R$ is not symmetric.
I don't understand why $\mathcal R$ is not symmetric,i... |
H: Show that $X \in L^p$
Let $X$ and $Y$ be independent variables and $p \geq 1$. Show that $X + Y \in L^p$ $\implies$ $X, Y \in L^p$. I tried using the inequalities
$$|X + Y|^p \leq 2^{p}(|X|^p + |Y|^p),$$
and
$$|X|^p \leq 2^{p}(|X + Y|^p + |Y|^p),$$
but I didn't get anything useful.
AI: This is proved using Fubini's... |
H: Don't understand the difference between $P(a \le X \le b)$ and $P(X\in B)$
By def, $P(a \le X \le b)=\int_a^b f_X(x) dx$ and $P(X\in B)=\int_B f_X(x) dx$.
I think I understand $P(a \le X \le b)$ well enough: it tells you that if $X$ is the random variable that maps the height of someone to $x$, then $P(a \le X \le ... |
H: Compute $\int xy dx +(x+y)dy$ over the curve $Ξ$, $Ξ$ is the arc $AB$ in the 1st quadrant of the unit circle $x^2+y^2=1$ from $A(1,0)$ to $B(0,1)$.
Compute $\int xydx+(x+y)dy$ over the curve $Ξ$, where $Ξ$ is the arc $AB$ in the first quadrant of the unit circle $x^2+y^2=1$ from $A(1,0)$ to $B(0,1)$.
I solved this ... |
H: Show that $(A\wedge Bβ) \vee B β‘ A \vee B$.
How do I prove this using laws of logical equivalency?
$$
(A\wedge Bβ) \vee B β‘ A \vee B
$$
What I tried. $(A \vee B') \wedge (A \vee A)$ I started by applying the distributive laws. Is this the right path?
AI: $%
\require{begingroup}
\begingroup
\newcommand{\calc}{\begin... |
H: Result of $\int_0^1 Tr(e^{-a(1-t)T}A^{β}e^{-atT}B)) dt , a>0$
$T$ is a positive selfadjoint densely defined on $H$ (Hilbert), only has point spectrum $\sigma (T)=\{\lambda_k \}_{k\in\mathbb{N}}.\lambda_k < \lambda_{k+1}\forall k$ and $\{u_k\}_{k\in \mathbb{N}}$ is the respective orthonormal base of eigenvectors of ... |
H: Probability of being sick after three positive tests
Question: The probability of getting sick is 5%. The probability of correct detection after getting sick is 80%. The probability of detection error in healthy people is 1%. Three consecutive tests are all getting sick. What is the actual probability of getting si... |
H: MLE of $(\theta_1,\theta_2)$ in a piecewise PDF
I am trying to find the MLE of $\theta=(\theta_1,\theta_2)$ in a random sample $\{X\}_{i=1}^n$ with the following pdf
$$f(x\mid\theta)= \begin{cases}
(\theta_1+\theta_2)^{-1}\exp\left(\frac{-x}{\theta_1}\right) &, x>0\\
(\theta_1+\theta_2)^{-1}\exp\left(\frac{x}{\th... |
H: Name and proof of the general form of ${a_1}{b_1} + {a_2}{b_2} = \left( {{a_1} - {a_2}} \right){b_1} + {a_2}\left( {{b_1} + {b_2}} \right)$?
I was running into a strange identity that is
Given ${x_1},{x_1},...,{x_n}$ and ${y_1},{y_1},...,{y_n}$ are all real number.
Denote ${c_k} = {y_1} + {y_2} + {y_3} + ... + {y_k... |
H: Better methods to approximate $2^{2\over 3}$
Recently while solving a problem on thermodynamics I ended up with $2^{2\over 3}$ .
Now the problem was on a test where no calculators were allowed and answer was required upto $2$ decimal digits.
I then resorted to binomial theorem for help (for $x\lt 1$) $$\left. \begi... |
H: Concrete Mathematics: Chapter 1: Generalised Josephus Recurrence: Understanding Radix 3 to 10 digit-by-digit replacement
Summary
When converting $(201)_3$, specifically, converting the $2$, I am not entirely sure how they select $5$ (from $f(2) = 5$) and not $8$ (from $f(3n+2) =
10f(n)+8$) when doing the radix repl... |
H: Why the improper integral $\int_{e}^{\infty} \frac{d x}{x(\log x)^{n}}$ converges iff $ - n + 1 < 0$?
I was trying to answer this problem in preparation for GRE exam:
And the answer was given below:
But I do not understand why the improper integral converges iff $ - n + 1 < 0,$ could anyone explain this for m... |
H: Is there a basis $\beta$ for $V$ such that $\langle \mathbf{v}, \mathbf{w}\rangle=\langle [\mathbf{v}]_{\beta}, [\mathbf{w}]_{\beta}\rangle$.
Given an inner product product $\langle \cdot, \cdot\rangle$ on a finite dimensional vector space $V$ over $F$, $F=\mathbb{R}$ or $F=\mathbb{C}$.
My Questions
Is there a bas... |
H: Proving the equation of an ellipse
If $(c, 0)$ and $(-c, 0)$ are the foci of an ellipse, and the sum of the distance of any point on the ellipse with the foci is $2a$ I am asked to prove thath the equation of the ellipse is:
$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
$$
where $b^2=a^2-c^2$.
I tried to first write the def... |
H: When is it necessary to use proof by induction?
Suppose $T \in \mathcal{L} (V)$ and $U$ is a subspace of $V$ invariant
under $T$. Prove that $U$ is invariant under $p(T)$ for every
polynomial $p \in \mathcal{P} (\mathbb{F})$.
$\underline{\textrm{My attempt at the solution:}}$
Suppose $p (z) \in \mathcal{P}(\mathb... |
H: How to find quantity of subsets generated from subsets of a set?
What is the maximum number of subsets that can be formed from $n$ subsets of a fixed set using intersection, join, and complement operations? Answer is $2^{2^n}$, but can you explain why?
AI: Suppose the fixed set is $A$ with subsets $B_1,B_2,\ldots,B... |
H: Combinations with restrictions in a set
Q1. Alex and Erin are two of the eight students trying out for a certain five-person chess team. If Alex and Erin must make it to the chess team, in how many different ways can the chess team be populated?
To solve for the above we can assume that two players Erin and Alex ar... |
H: Tangent to an ellipse
I want to find the tangent on point $(x_0, y_0)$ to an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
We assume $y_0$ is positive. We can derive $y=\frac b a \sqrt{a^2-x^2}$.
In order to find the slope I use the limit
$$
\lim_{x \to x_0}\frac b a \frac{\sqrt{a^2-x^2}-\sqrt{a^2-x_0^... |
H: Relation between diagonal entries of $A^{-1}$ and inverse values of $a_{ii}$ for positive definite $A$.
I'd like to expand upon this question. Namely, it says that if $A$, $A=A^T$, is a positive definite matrix, then it holds that \begin{equation}\tag{*}(A^{-1})_{ii}\ge \frac1{A_{ii}}.\end{equation}
Can we prove th... |
H: There is only one disc with center i0 for which exists a holomorphic bijection from a given domain
Let D be a domain which is not the entire complex plane.
and let f a holomorphic bijection to a disc with center at 0, that satisfies $f(z_0)=0;f'(z_0)=1$ for some $z_0$ on a disc.
Prove that there is only one such di... |
H: Coset Enumeration: Defining Cosets
I have a problem with understanding the initial step in the Todd-Coxeter coset enumeration algorithm. One needs to define a few cosets when you start off, but I'm not sure how to define them.
As an example, I found the following example: For the presentation $\left\langle {x,y\;\... |
H: Expectation $E[e^{\lambda B_{T}}]$ where $T$ is a stopping time w.r.t. Brownian Motion
Consider a $1$-dimensional Brownian motion started from $0$. Compute $E[e^{\lambda B_{T}}]$, where $\lambda>0$ and $T$ was the first time $t$ for which $B_t=1$.
If this were $E[e^{\lambda {T}}]$, then I would know how to comput... |
H: Proving that $\int_{-1}^{1} \frac{\{x^3\}(x^4+1)}{(x^6+1)} dx=\frac{\pi}{3}$, where $\{.\}$ is positive fractional part
Here, $\{-3.4\}=0.6$.
The said integral can be solved using $\{z\}+\{-z\}=1$, if $z$ is a non-zero real number;
after using the property that $$\int_{-a}^{a} f(x) dx= \int_{0}^{a} [ f(x)+f(-x)] d... |
H: Measurability of composition of random vector w.r.t. $\sigma$(X)
Here is a problem from Resnick - Probability Path (3.3) :
Let $f:\mathbb{R}^k \rightarrow \mathbb{R}$, and $f \in \mathscr{B}(\mathbb{R}^k) / \mathscr{B}(\mathbb{R}) $.
Let also $X_1,...,X_k$ be random variables on $(\Omega,\mathscr{B}) $ .
Show that ... |
H: Values of $p$ for an improper integral to converge
I am trying to find values of $p \in \mathbb{R}$ such that $\displaystyle\int_0^{+\infty} x^p\sin(e^x)$ converges. All I have managed doing is using reduction formulas but I couldn't reach a result. Any ideas?
AI: Near $+\infty$ one has by partial integration
\beg... |
H: Let $M$ be a non-empty set whose elements are sets. What are $F=\{AΓ\{A\} : AβM, Aβ β
\}$ and $βF$?
I think it's not so difficult but I struggling a little to figure it out, I want to make sure I'm correct, is $F$ a set of the form:
$F=\{ \{(a, A), (b, A), β¦\}, \{(Ξ±, B), (Ξ², B), β¦\}, ...\}$ for all $a,b,...βA$ and $Ξ±... |
H: Taylor/Maclaurin series of $\arctan(e^x - 1)$
Right, so this keeps bugging me, and I'm probably stuck in some tunnel trying the same thing over and over again.
Give the Maclaurin series of the function $\arctan(e^x - 1)$. up to terms of degree three. Since I try to be as lazy as possible and differentiating this th... |
H: Simplifying expectation of square of sum
I am working on a problem (not for homework) where one step involves simplifying an expectation. The solution looks like this:
Call $p_\mu(X)$ the PDF of the variable distributed as $N(\mu, \sigma^2I)$. Let $E_0$ denote the expectation under $N(0, \sigma^2I)$.
Then $E_0\left... |
H: Why Can a Plane Not Be Defined Solely With a Vector
Why can a vector not be used to define a plane, why does it have to be a vector and a point. Couldn't you just take a vector and draw a plane at the tip which is perpendicular to the "stem" of the vector in all directions?
AI: A plane through the origin can always... |
H: Commutative von Neumann regular ring with more than two prime ideals.
Is there an example of commutative von Neumann regular ring which has more than 2 prime ideals?
Matrix rings are not commutative so I couldn't find any, please help.
AI: It is well-known that the quotient of a commutative ring by is nilradical... |
H: Is it possible for a sequence of rational numbers to have a non rational limit?
I'm solving an exercise, and at some point I have to prove that any inteval $(a,b)$, with $a,b \in \mathbb R$, is the union of intervals $[\alpha,\beta)$, with $\alpha \in \mathbb R$ and $\beta \in \mathbb Q$. The values of $\alpha$ ar... |
H: Integral of $\int\limits_0^{2\pi } {\operatorname{erfc}\left( {\cos \left( {a + \theta } \right)} \right)d\theta } $?
I am sorry if it does not fit here. I found some of the integral for the complementary error function e.g.
So far I did not find any integral regarding,
$\int\limits_0^{2\pi } {\operatorname{erfc}... |
H: Triangle inside an open disk
Given an open disk $D=\{x \in \mathbb{R}^2 \mid |x-x_0|<r\}$ and fixed three points $x_1,x_2,x_3 \in D$, how can I show that the triangle (of vertices $x_1,x_2,x_3$) $\Delta=\{t_1x_1+t_2x_2+t_3x_3 \in \mathbb{R}^2 \mid t_1,t_2,t_3 \ge 0 \quad \land \quad t_1+t_2+t_3=1\}$ is all inside $... |
H: Maximum angle of separation between $n$ vectors in $m$ dimensions
My question is exactly as above. If there are $n$-vectors having the same origin in an $m$-dimensional Euclidean space then what is the maximum angle of separation that you can achieve.
Example:
In $2D$ Given $2$ vectors the max angle of separation ... |
H: Combinations of towers
You have 25 red blocks and 25 blue blocks.
You can stack the blocks in any order into 5 towers with 5 blocks maximum in each tower.
You do not have to use all the blocks and blocks cannot float in mid-air.
How many combinations of towers can you make?
The base being 0 0 0 0 0 as 1 of the comb... |
H: Given $n \in \mathbb{N}$, find the number of odd numbers among ${n}\choose{0}$,${n}\choose{1}$,${n}\choose{2}$, $...,$ ${n}\choose{n}$ .
So here is the Question :-
Given $n \in \mathbb{N}$, find the number of odd numbers among ${n}\choose{0}$,${n}\choose{1}$,${n}\choose{2}$, $...,$${n}\choose{n}$ .
What I Tried ... |
H: Suppose that $f(x) \to\ell$ as $x\to a$ and $g(y) \to k$ as $y \to\ell$. Does it follow that $g(f(x)) \to k$ as $x \to a$?
If $f$ and $g$ are continuous the limit of the composition is the composition of the limit, so the implication follows.
But what if $f$ and $g$ are not continuous? There exist counter examples ... |
H: How to prove that $\frac{\cos(x)-\cos(2x)}{\sin(x)+\sin(2x)} = \frac{1-\cos(x)}{\sin(x)}$ in a simpler way.
EDIT: Preferably a LHS = RHS proof, where you work on one side only then yield the other side.
My way is as follows:
Prove: $\frac{\cos(x)-\cos(2x)}{\sin(x)+\sin(2x)} = \frac{1-\cos(x)}{\sin(x)}$
I use the ... |
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