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H: Russian roulette statistics
I have the following probabilily problem.
"You're playing a game of Russian roulette, the revolver has 6 chambers in which you put 3 bullets completely random. What is the probability that all bullets are next to each other?"
The solution says 0.3
I have no clue how to start with this p... |
H: Solve: $\tan\frac x2 = x,\> x\in(0,\pi)$
I'm trying to calculate the shortest distance between two points on a sphere in terms of its radius and the angle they intend at the centre. I'm stuck at the equation:
$$\tan(x/2) = x,\ 0<x<\pi$$
I have no idea how to solve this.
AI: Given that there is no analytic solution... |
H: Finding the 2017th power of a nondiagonalizable matrix
Matrix image
The eigenvectors and eigenvalues of the matrix are as follows: https://matrixcalc.org/en/vectors.html#eigenvectors%28%7B%7B110,55,-164%7D,%7B42,21,-62%7D,%7B88,44,-131%7D%7D%29.
This matrix cannot be diagonalized since the eigenvectors are not lin... |
H: Volume of $CP^1$ wrt Fubini Study metric
I have shown that the FS form on an affine chart of $CP^1$ is $1/4$ times the usual area form (written in $\theta$,h coordinates).
So the area of $CP^1$ wrt FS form should be $1/4$ of usual area right?
But when I integrate the FS form that is $(dx\wedge dy)/(1+x^2+y^2)^2$ o... |
H: Show that when $n$ balls are thrown into $n^3$ cells, the chance that (at least) two balls will collide goes to zero as $n\to \infty$
Show that when $n$ balls are thrown into $n^3$ cells, the chance that (at least) two balls will collide goes to zero, as a function of $n$:
$$\lim_{n\to \infty }P(\exists\, x,y:x,y\... |
H: $M$ is projective iff $\operatorname{Ext}_R ^1 (M,P)= 0 $ for $P$ projective?
We know that an $R$-module $M$ is projective iff $\operatorname{Ext}_R ^1 (M,N)= 0 $ for every $R$- module $N$.
Is it true that:
$M$ is projective iff $\operatorname{Ext}_R ^1 (M,P)= 0 $ for every projective $R$- module $P$ ?
AI: No. Let... |
H: Proof: $x_n\to p \iff d(x_n,p)\to 0$
Let $(M,d)$ be a metric space, let ${x_n}\in M$ and $p\in M$
Prove: $x_n\to p \iff d(x_n,p)\to 0$
$\Leftarrow:$ be definition of a limit, for all $0 < \varepsilon$ there is $N\leq n$ such that $$|d(x_n,p) + 0|< \varepsilon \iff |d(x_n,p)|< \varepsilon \iff d(x_n,p)< \varepsil... |
H: $ X_n\stackrel{P}\rightarrow X\text{, }Y_n\stackrel{P}\rightarrow Y \Rightarrow X_nY_n\stackrel{P}\rightarrow XY $. How to show this?
Let $\{X_n\}$ and $\{Y_n\}$ be two sequences of random variables such that $X_n$ converges to $X$ in probability and $Y_n$ converges to $Y$ in probability. Show that $X_n Y_n$ conve... |
H: Let $f(x)=7-3x$, evaluate and simplify the following $\frac{f(1)}{x}$
Using this equation, how would I be able to simplify this? My initial thoughts are, this would be an inverse function and it would be as follows:
$f(x)=7-3x$
Then, multiply both sides by $-3x+7$, which would be
$-3fx^2+7fx=1$
Then, factoring it o... |
H: What does the solution space of differential equation mean?
I am trying to understand what I am actually getting when I solve a differential equation. For example, the last one I solved was
$$y^\prime + y x \sin(x) = 0$$
and I got the solution
$$y(x) = C e^{x \cos(x)- \sin(x)}$$
I checked its graph on GeoGebra bu... |
H: Relations of set operators: closure, interior, complement
For a proof related to one of the versions of Baire's category theorem I needed the following equality:
$$\operatorname{cl} \! \big( \!\operatorname{int} (A^c)\big) = \operatorname{int} \! \big( \! \operatorname{cl} (A)\big)^c \quad \text{ for a set } A.$$ I... |
H: Homework Problem, Implicit Function Theorem
Given the following equation:
$$
x^3y -y^3x -6 = 0
$$
determine using implicit differentiation $y'(2)$. (exact wording) Where $y' = \frac{\partial y}{\partial x}$
I called
$$
f(x,y) = x^3y -y^3x -6 = 0
$$
Here is what I have done so far:
$$
\begin{align*}
\frac{\partial... |
H: Understanding the disk model of projective space $\mathbb{P}^n(\mathbb{R})$
I'm working on Exercise 2.3.3 in Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Cavalieri and Miles:
In a previous definition, we realized $\mathbb{P}^n(\mathbb{R})$ as an identification/orbit space: let $\mathb... |
H: Counting pair in a poker hand - counting confusion
I want to be good enough to do math competitions, and one of the main topics in them is always combinatorics. So I'm going through a combinatorics book, "A Walk through Combinatorics."
The confusion I have is with the following problem:
Find the prob that a poker h... |
H: Linear Functional is Discontinuous
I'm trying to show that the linear functional $f:(\ell^1, \|\cdot\|_\infty) \longrightarrow (\mathbb{R}, \|\cdot\|)$ given by $$f((x_n)_{n \in \mathbb{N}})=\sum\limits_{n=1}^\infty x_n$$ is discontinuous. Here, $\|(x_n)_{n \in \mathbb{N}}\|_\infty=\sup\limits_{n \in \mathbb{N}} |x... |
H: When a sigma algebra is closed under countable intersection?
In "Measure Theory and Probability Theory" by Krishna and Soumendra pag. 43 there is the following passage.
Let $(\Omega, \mathcal{F})$ be a measurable space and $\{f_n\}$ a
sequence of functions such that $f_n:\Omega \rightarrow \mathbb{R}$ is
a mea... |
H: Some intuition for $\sin\left(x+\frac\pi5\right)+\sin\left(x+\frac{3\pi}5\right)+\sin\left(x+\frac{7\pi}5\right)+\sin\left(x+\frac{9\pi}5\right)=1$
Can anyone give me some intuition how to start solving this equation please? Or maybe fastway to understand that this whole thing is equal to $\sin(x) = 1$.
$$\sin\le... |
H: Show that $\frac{(J-1)!}{2!(J-3)!}= \sum_{h=2}^{J-1}(h-1) $
Could you help me to show that
$$
\frac{(J-1)!}{2!(J-3)!}= \sum_{h=2}^{J-1}(h-1)
$$
where "$!$" denotes the factorial function?
AI: $$\sum_{h=2}^{J-1} (h-1) = \sum_{h=1}^{J-2} h = \frac{(J-2)(J-1)}{2}= \frac{(J-1)!}{2!(J-3)!}$$ |
H: Find a(n) with characteristic equation
Find a(n) with characteristic equation of recurrence: a(1)=3, a(2)=10, a(n)=6a(n-1)-3a(n-2) .
Started from equation t^2-6t+3=0 then find t1,t2 then take them in: a(n)=Ct1+ Dt2 then C and D have to find from a(1)=3 and a(2)=10 but I'm stuck here.
AI: $t^2-6t+3=0 \implies t=3 \p... |
H: Linear constrained form to free parameter form
I came across this fact that for a linear system of equation $Ax=b$ one can write a free parameter form as $x=Fz+x_0$, where the columns of $F$ span the null space of A, and $x_0$ is some specific solution.
I am wondering if this is always true? I am also wondering wh... |
H: Upper bound and limit of $\frac{x^5}{\sqrt{(1+\frac{x^2}{2})^5}}$
Let $Y=\sigma X$ be a scaled Student's t-distributed variabled with scale parameter $\sigma=1/\sqrt{2}$ and $4$ degrees of freedom.
I'm proving that for $k>0$
$$\frac{P(Y>x)}{kx^{-4}}\rightarrow k_0 \qquad\text{for}\ x\rightarrow\infty$$
where $k_0... |
H: The connectivity and diameter properties of a replacement product of graphs
I have difficulties with proving the properties of replacement product operation between two graphs. Actually, I'm not sure where to start on this proof.
Let G and H be connected graphs. Prove that replacement product of G and H be also co... |
H: Proof for any value of x greater than 1, the following statement is true: The limit of $\sum_{n=0}^{\infty} {\frac{1}{x ^ n}}$ is ${\frac{x}{x-1}}$
I've been looking into the following sum for values of x:
$$\sum_{n=0}^{\infty} {\frac{1}{x ^ n}}$$
And after plugging in different values of x, I became confident enou... |
H: Meaning of power matrix function
Suppose $t\in \mathbb{C}$, $\Lambda=\begin{bmatrix}
2 & 0 \\
0 & 1
\end{bmatrix}$, what does $t^\Lambda$ mean? Is it
$\begin{bmatrix}
t^2 & 0 \\
0 & t
\end{bmatrix}$?
AI: The easiest way to understand powers of matrices is through the exponential function $f(x)=e^x$. We know that f... |
H: Probability puzzle on Inference.
Couple of months ago, i was asked this weird probability puzzle in an interview.
Problem: There are two persons X and Y. Given a statement, X says it is true w.p $\frac{1}{3}$. and false w.p. $\frac{2}{3}$. Y always agrees with whatever X says.
What is the probability that the state... |
H: Prove $\sum^{\infty}_{n=1} n^{13} q^{n}$ where $q \in \mathbb{R}$ converges if and only if $q \in (-1, 1)$
If the sum converges, we can argue that if $q \notin (-1, 1)$ then the necessary condition isn't met which is a contradiction.
For $q=0$ the sum is 0 and if $q \in (0, 1)$ then we can use comparison test. Ho... |
H: Simplify $(1-x)^3+2y^3-3(1-x)y^2=0$ to $-(x - 2 y - 1) (x + y - 1)^2 = 0$
How to simplify $(1-x)^3+2y^3-3(1-x)y^2=0$?
I tried it and the result I'm aiming for is $$-(x - 2 y - 1) (x + y - 1)^2 = 0.$$
AI: $$(1-x)^3+2y^3-3(1-x)y^2=0$$
$$\underbrace{(1-x)^3-(1-x)y^2}+\underbrace{2y^3-2(1-x)y^2}=0$$
$$(1-x)((1-x)^2-y^2... |
H: How do I show $\lim_{\epsilon\to0}\int_\epsilon^1x^n(\ln(x))^mdx=\frac{-m}{n+1}\lim_{\epsilon\to0}\int_\epsilon^1x^n(\ln(x))^{m-1}dx$
I have an assignemt where I'm to prove
$$\int_0^1x^{-x}dx=\sum_{k=1}^\infty k^{-k}$$
And one of the steps towards doing this is showing that
$$\lim_{\epsilon\to0}\int_\epsilon^1x^n(... |
H: Show that no entire function $f\big(\frac{1}{n}\big)= \frac{n}{2n-1}$ exists using the identity theorem.
Show that no entire function $f: \mathbb{C} \to \mathbb{C}$ exists with $$f\bigg(\frac{1}{n}\bigg)= \frac{n}{2n-1}$$ for $n \in \mathbb{N}$. What is the domain of a holomorphic function with such values?
I was t... |
H: Is the function $\phi:\Bbb C\to\Bbb C$ given by $\phi(z)=\bar{z}$ a ring isomorphism (where $\bar{z}$ is the complex conjugate of $z$).
Okay so I'm working on this problem:
The function $\phi:\mathbb{C}→\mathbb{C}$ given by $\phi(z)=\overline{z}$ is a ring isomorphism (where $\overline{z}$ is the complex conjugate ... |
H: Find the values $a$ and $b$ such that the function is differentiable at $x=0$
$\mathbf{Question:}$
Find the values $a$ and $b$ such that the function is differentiable at $x=0$
$$
f(x)=
\begin{cases}
x^{2}+1 &x≥0\\
a\sin x+b\cos x & x<0\\
\end{cases}
$$
$\mathbf{Solution:}$
$f(x)$ is differentiable at $x=0$ if $f'(... |
H: How to compute $\frac{\partial}{\partial X}tr(BXX^tA)$?
I know that $\frac{\partial}{\partial X}tr(BXX^t) = BX + B^tX$ according to the matrix cookbook equation 109.
However, I need to calculate $\frac{\partial}{\partial X}tr(BXX^tA)$. Is there a simple way to derive this formula from the previous one?
AI: The tra... |
H: Differential of Derivative
I have encountered with a problem, and I can't follow the intermediate steps. Consider the following differential equation:
$$ y^{\prime \prime} = -2y + f(y)-0.5y^3 $$
Here, the first derivative is expressed as the following integral:
$$ (y^{\prime})^2 = 2\int_0^{y^\prime} y^\prime dy^\pr... |
H: Let $n \geqslant 0$ be an integer. Show that $2^n | \lceil (3+\sqrt5)^n \rceil.$
Let $n \geqslant 0$ be an integer. Show that $2^n | \lceil (3+\sqrt5)^n \rceil.$
This turned out to be relatively hard. The idea that our lecturer gave was that one should work towards an linear recurrence relation here, but it does... |
H: How does this alternate form come about from Eulers identity?
I don't understand how this identity comes about, my tutor wrote it out and I checked it on wolfram so i know its not a mistake and now I'd like to understand how to derive it.
$sin(\theta)cos(\phi) -isin(\theta)sin(\phi) = sin(\theta)e^{-i\phi}$
I know... |
H: Are Quasi-Polynomial Time Functions Regularly Varying?
Is $$\lim_{x\to\infty} \frac{e^{(\ln ax)^k}}{e^{(\ln x)^k}}$$ finite for all positive real $a$ and $k$?
I have tried this on Desmos with $a$ and $k$ less than 2 where it seems to converge after around $10^{13}$, but with larger values the function is essentiall... |
H: Determine the edge-chromatic number of the next graph.
Determine the edge-chromatic number of the next graph.
For Vizing`s Theorem $\chi'(G)=4$ or $\chi'(G)=5$. I have try show that $\chi'(G)=4$ but I cannot do it. I think that $\chi'(G)=5$ and I suppose that G has a proper edge coloring with 4 colors, but I dont... |
H: Sum $\sum_{n = 1}^{\infty}\left[\frac1n\sin nx + \frac{1}{n^2}\cos nx\right]$
I want to find the following sum by using the complex methods for series ($z = \cos nx + i \sin nx$).
$$
\sum_{n = 1}^{\infty}\left[\frac1n\sin nx + \frac{1}{n^2}\cos nx\right]
$$
Here is my attempt:
$$
S_N = \sum_{n = 1}^{N}\frac1n\sin ... |
H: Combinatorial argument for the number of pairs whose sum is less than or equal to k
Example: How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$.
My solution:
If we fix x then we can choose y in $(k - x + 1)$ ways (i.e. $y = 0,1,2,3 .... (k-x)$).
Therefore, the final answer... |
H: Show that $\|A\|_{\infty} \leq \sqrt n \|A\|_2$
Question:
Let $A \in \Bbb R^{m\times n}$. Show that $$\|A\|_{\infty} \leq \sqrt n \|A\|_2$$
Attempt:
First, I tried invoking the SVD (Singular Value Decomposition) of $A$:
$$\|A\|_\infty = \|UDV\|_\infty \leq \|U\|_\infty\|D\|_\infty\|V\|_\infty \leq \sqrt m \|A\|_2\... |
H: Prove that the number of subsets of [n] equals the number of n-digit binary numbers.
So, I am going through David Mazur’s Combinatorics: A Guided Tour and I came across this exercise question:
Give a bijective proof: The number of subsets of [n] equals the number of n-digit binary numbers.
So, I know how to go ab... |
H: How to calculate expected code length?
How to calculate expected code length?
AI: Should be:
$$L = \sum p_i l_i = (0.25 + 0.25 + 0.2) \times 2 + (0.15 + 0.15) \times 3 = 2.3 $$ |
H: Finding a general formula for the function $f$
I need some help! During a moment of research, I wanted to find a general formula with a function $f$.
$$f(x)=\begin{cases}
0&& \text{if $x\in\mathbb R^*$}\\
1&&\text{if $x=0$}
\end{cases}$$
The domain of $f(x)$ is $\mathbb R$.
Try to determine an explicit formula for ... |
H: Locally constant sheaf on irreducible space is constant
Let $X$ be an irreducible space, and let $\{U_{i}\}_{i\in I}$ be an open covering of $X$. Let $\mathcal{F}$ be a sheaf on $X$ such that the restriction of $\mathcal{F}$ to each open $U_{i}$ is constant. I want to show that $\mathcal{F}$ has to be constant.
No... |
H: $f$ entire and $f(n) = n$ with $\lim_{|z| \to \infty} |f(z)| = \infty$, show $f(z) = z$
The question is: $f$ entire and $f(n) = n$ with $\lim_{|z| \to \infty} |f(z)| = \infty$, show $f(z) = z$.
I know that $f$ must be a polynomial which can be deduced from just the entire and $\lim_{|z| \to \infty} |f(z)| = \infty$... |
H: Spheric equation to Cartesian equation
I have this equation $$\rho^2=2 \sin(2\theta)\cos(\phi)$$ and I want to convert it into a Cartesian equation. I’ve already applied an trigonometric identity and I have this$$\rho^2=4\sin(\theta)\cos(\theta)\cos(\phi)$$ and I don’t know what else to do
AI: I suppose, that you h... |
H: Using the remainder theorem to prove a quadratic is a factor of a polynomial
For example, if I have $P(x) = 3x^4 + 5x^3 -17x^2 -13x + 6$ then to show that $x^2 + x - 6$ is a factor I individually show that $x+3$ and $x-2$ are factors using the factor theorem (i.e. $P(-3) = 0$ and $P(2) = 0$).
However, if $x+3$ and ... |
H: Conditions under which the torsion subgroup of an abelian group is a direct summand.
Hi: Let $T$ be the torsion-subgroup of an abelian group $G$. If $T$ is the direct sum of a divisible group and a bounded group, then T is a direct summand of $G$.
The author says "This follows easily from 4.1.3 and 4.3.8".
4.1.3 A... |
H: Local characterization for convex $C^1$ functions
Let $f:\mathbb R^n\to \mathbb R$ be $C^1$ and satisfy following condition: For every $x\in\mathbb R^n$ there exists some $\varepsilon_x > 0$ such that for every $y$ with $\|y - x\| < \varepsilon_x$ it follows
$$ f(y) \ge f(x) + \nabla f(x)^T (y - x). $$
Is $f$ conve... |
H: How to sketch the level curves of the following function
Consider the function f defined by $f(x,y)=ln(x-y)$
How do I sketch the level curves for this function for the values of $k=-2,0,2$?
AI: Observe that the level curves of $f(x, y)$ are given by $f(x, y) = k$ for some constant $k.$ Considering that $f(x, y) = \... |
H: Constant functionals on a set generated by one elements are linear?
In functional analysis we say that if $X$ is a normed space and $f\in X'$(dual of $X$).(Say real normed space.)
f is $f:X\to \mathbb R$ to satisfy $$f(x+y)=f(x)+f(y),\\f(\alpha x)=\alpha x \\ \forall x,y \in X, \forall \alpha \in \mathbb R$$
But If... |
H: If $f: X \to Y$ and continuous in $D \in \mathcal{B}_{X}$, then $V$ open implies $f^{-1}(V) = U \cap D$ with $U$ open
I am reading through Yeh's Real Analysis 3rd edition, pp. 21-22. Yeh proves the following theorem:
Given two measurable spaces $(X, \mathcal{B}_X)$ and $(Y, \mathcal{B}_Y)$ where $X$ and $Y$ are to... |
H: Determine whether a function on the even integers is injective and surjective
Let $E$ be the set of even integers. Define $f:E \rightarrow \{0, 1, 2, 3, 4\}$ by $f(n) = n \mod 5$.
a) Determine whether $f$ is one-to-one (injective).
b) Determine whether $f$ is onto (surjective).
AI: Let $m=4 \in \mathbb{N}$ then $f(... |
H: Equinumerosity: Proof Validation
Synopsis
In my studying of Set Theory, I've had the privilege of being able to compare much of my work to a solutions manual. This solutions manual, however, ended after the chapter on the various number systems. As such, I'm going to start posting a lot of my own personal solutions... |
H: Find solution set of $\dfrac{8^x+27^x}{12^x+18^x}=\dfrac{14}{12}$
What I've done is factoring it.
$$\dfrac{2^{3x}+3^{3x}}{2^{2x}\cdot 3^{x}+3^{2x}\cdot{2^{x}}}=\dfrac{7}{2\cdot 3}$$
This looks like it can be factored more but it doesn't work from my attempts.
AI: Well you've reached a good point:
$$\dfrac{2^{3x}+3^... |
H: How to prove that the limit is equal to the function?
I have the following statement to prove:
Prove that if $g$ is derivable, therefore $\lim_{h\to 0}\frac{1}{g(x+h)}=\frac{1}{g(x)}$
My attempt was:
If $g$ is derivable, then is continuous on its domain and the limit of continuous function in a point of its domain ... |
H: Why did Spivak choose this definitions? ("Calculus on Manifolds", the definition of the norm, the definition of open sets)
I am reading "Calculus on Manifolds" by Michael Spivak.
In this book, the norm of $x \in \mathbb{R}^n$ is defined by $|x| := \sqrt{(x^1)^2 + \cdots + (x^n)^2}$.
In this book, a set $U \subset... |
H: Composition of Irreducible Representation and Non-surjective homomorphism: counterexample
I came across this problem where it proved that if $\phi: G\rightarrow H$ be a epimorphism and $\psi: H \rightarrow GL (V)$ is an irreducible representation, then $\psi \circ \phi$ is an irreducible representation of . I learn... |
H: To solve polynomial equation of a special form.
It is known that there is no general method to solve polynomial with order greater than 5. However, some equations of the special forms have been knowns to solve.
While I was studying today, I faced the following problems.
$Ax^{n}+Bx^{m}=C$
where $n > m >0$, $A$, $B$ ... |
H: Prove or disaprove that proposition
Let $l_1=(Cos\alpha_1,Cos\beta_1,Cos\gamma_1)$,$l_2=(Cos\alpha_2,Cos\beta_2,Cos\gamma_2)$,$l_3=(Cos\alpha_3,Cos\beta_3,Cos\gamma_3)$ \, three points that are perpendicular in the space ($l_1\perp l_2 \perp l_3$), and $u=f(x,y,z)$.
Prove or disaprove that $\\
$
$(\frac{\partial u}... |
H: Associative Law - Number Sequence
I'm stuck in a problem, need Help.
Associative property States that
(a + b ) + c = a + (b + c)
Which is true
but what if i change the position of the numbers
(a + c) + b = a + (c + b)
Does it hold true for the associative Law because the answer will still be same..
Stuck in thi... |
H: Function from $\mathbb{Z^+}$ to $\mathbb{Z^+}$ that is neither one-to-one nor onto?
I am thinking of something like $f(x) = 8$
Does this make sense? It seems a bit simple to me so I'm not sure.
My reasoning is that this function is not one-to-one because f takes same value for all domain.
It's also not onto becaus... |
H: $\sum_{n=1}^{\infty} \frac{d_{n}(x,y)}{2^{n}}$ is a metric
Could you help me with the following please, I have a question with the following:
If $d_1,d_2,...$ is a succession of metrics in a space $X$ such that $d_{n}(x, y) \leq 1$ for each $x, y \in X$ and each $n$. Prove that the function $d: X\times X \rightarro... |
H: Why these different forms of Triangle Removal Lemma are equivelent?
Triangle removal lemma can be stated as follows.
$\forall \epsilon>0,\exists \delta>0$ such that any graph on $n$ vertices with $\leq \delta n^3$ triangles can be made triangle-free by removing at most $\epsilon n^2$ edges.
I can also find anothe... |
H: Application of finite intersection property
If $\mathscr{F}$ be a family of compact sets with finite intersection property in a metric space (X, d). Then show that $\cap \mathscr{F} \neq \phi$.
My thinking: I want to prove by contradiction.
Let, $\cap \mathscr{F} = \phi$ Then {$X-F: F\in \mathscr{F}$} is an op... |
H: Design a rank one matrix $\boldsymbol{C} = \boldsymbol{a} \boldsymbol{b}^T$ to make there exists no same value in the matrix $\boldsymbol{C}$.
I wanto to design a rank one matrix $\boldsymbol{C} = \boldsymbol{a} \boldsymbol{b}^T \in \mathbb R^{m \times n}$. The matrix $C$ should not have same values, the vector $ \... |
H: Is terminal object the kernel of identity morphism?
Let's say that there is a category $\mathbf{C}$ with $A$ being an object of that category and a zero object exists in that category. If we have an identity morphism ${id}_A: A\to A$, is the kernel of this morphism a terminal object of the category? My reasoning fo... |
H: Prove that $\lim_{n \to \infty} \sqrt{a_n} = \sqrt{L}$
$1.$ Let $\{a_n\}$ be a sequence of positive terms such that $\lim_{n \to \infty} a_n = L$ where $L > 0$. Prove that $\lim_{n \to \infty} \sqrt{a_n} = \sqrt{L}$.
Proof of $1:$
\begin{align*}
\forall \epsilon > 0\: \exists N >0\:\: s.t\:\:n>N
&\implies |a... |
H: Simply way to count points in finite Projective plane
What is an easy way to count points and lines in a finite Projective Plane over finite field $\mathbb F_q$, accessible for student with minor knowledge of linear algebra (spaces sub-spaces, e.c.t).
Points in Projective Plane over finite field $\mathbb F_q$ are... |
H: Investigate the solution (no solution, a unique solution, or infinitely many solutions) according to the constant $a$ and $b.$
Investigate the solution (no solution, a unique solution, or infinitely many solutions) of the following system of differential equations according to the constants $a$ and $b$.
$$\begin{a... |
H: There are $20$ females and $15$ males in a party. In how many ways can $15$ couples be created?
There are $20$ females and $15$ males in a party. In how many ways can $15$ couples be created?
I think it is $P^{20}_{15}$ but I feel I'm wrong.
AI: Similar to @David G. Stork, I'm assuming you define "couple" as one ... |
H: Geometric interpretation to a system of equations
For part (a) these are clearly two parallel lines so no points of intersection.
For part (b) this has one point of intersection because these two lines cross at exactly one point.
For parts (c) and (e) we have $z=0$ and $x=2y+1$ but what does this mean geometricall... |
H: Questions About Combinatorial Games
I've been reading about combinatorial games, on an article by Brilliant: https://brilliant.org/wiki/combinatorial-games-winning-positions/#chomp-and-strategy-stealing.
The article makes 4 statements about combinatorial games, which I am confused about:
Due to the deterministic... |
H: Evaluating $\lim_{n\to\infty} \frac{S_1(n)S_5(n)-S_3(n)²}{S_7(n)}$
If $S_k(n)=\sum_{r=1}^n r^k$, then compute
$\lim_{n\to\infty} \frac{S_1(n)S_5(n)-S_3(n)²}{S_7(n)}$
($n$ defined on all natural)
What I did -
$S_1(n)$ is a term of degree 2. $S_5(n)$'s last would be $n^5$ so in the product of $S_5(n)$ with $S_1(n)$... |
H: Property of set of positive Lebesgue measure
Let $E\subseteq \mathbb{R}$ be a compact set and positive Lebesgue measure. Does there exits $a>0$ such that $$\cap_{0\leq x\leq a}E+x$$ is a set of positive Lebesgue measure?
I proved that above is true if $E$ is an interval.
Edit: I changed the question and add the co... |
H: Find all functions $f : \mathbb R \to \mathbb R$ such that: $f\left(x^3\right)+f\left(y^3\right) = (x + y)f\left(x^2\right)+f\left(y^2\right)-f(xy)$
Here is the question:
Find all functions $f : \mathbb R \to \mathbb R$ such that:
$$f\left(x^3\right)+f\left(y^3\right) = (x + y)f\left(x^2\right)+f\left(y^2\right)-f... |
H: Is $\binom{n}{n/2}\sim 2^n?$
We say $a_n\sim b_n$ if $\lim_n(a_n/b_n)=1$.
Is $$\binom{n}{n/2}\sim 2^n?$$
Actually, we know $n!\sim n^{n+1/2}e^{-n}\sqrt{2\pi}$. So
$$\binom{n}{n/2}\sim \frac{n^{n+1/2}e^{-n}\sqrt{2\pi}}{(n/2)^{n+1}e^{-n}2\pi}=c2^{n+1}n^{-1/2}?$$
How about the next step?
AI: Since you've already got... |
H: How To Represent Draws In Combinatorial Games
I've been reading about combinatorial games, specifically about positions in such games can be classified as either winning or losing positions. However, what I'm not sure about now is how I can represent draws using this: situations where neither player wins or loses. ... |
H: Point of Inflection Inequality in a Quartic
The following is a multiple choice question from a workbooklet (year 12):
The family of graphs of the form $y=x^4+ax^3+bx^2+cx$ has two points of inflection if and only if:
A. $b<\frac{3a^2}{8}$
B. $b^2>3c$
C. $a<b$
D. $b^2<4c$
E. none of these
Answer: A
My question is ho... |
H: If $L(y)=\min_{x\in \mathbb{R}}h(x,y)$ is well defined in $(a,b]$ then is it also well defined on $a$?
Let $h:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that $\lim_{x\to \pm\infty}h(x,y)=0$ for all $y$. Then we know that if $L(y)=\min_{x\in \mathbb{R}}h(x,y)$ exists, it must be $L(y)\l... |
H: How do I calculate the price of 50g of a product, if the total grams of the product cost a certain amount
I am looking for a formula that can calculate cost per 50g. e.g. How do I calculate the price of 50g of Pepper, if 300g costs 132? or How do I calculate the price of 50g of Salt, if 1000g costs 100?
What standa... |
H: Limit $\lim_{n\to \infty} \frac{1}{n^{n+1}}\sum_{k=1}^n k^p$
The evaluation of $$\lim_{n\to \infty} \frac{1}{n^{n+1}}\sum_{k=1}^n k^p$$
as an integral ( evaluating this as a right Riemann sum ) requires the form $f(a + k\cdot \Delta x_i)$, where $a$ is our lower bound of integration, and $\Delta x_i$ is the length ... |
H: If $a$, $b$, $c \geq 1$; $y \geq x \geq 1$; $p$, $q$, $r > 0$ Prove the insane inequality
$$\left(\frac{1+y\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}{1+x\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}\right)^{\frac{p+q+r}{\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}}\left(\frac{1+xa}{1+ya}\right)^{\frac{p}{a}}\left(\frac{1+x... |
H: Diagonalizable projection operator
How to say that a projection mapping is digonalizable.further it can be represented in the diagonal matrix having a block of identity matrix of order r inside it.where r is the rank of the operator.
AI: In my book a linear projection mapping is by definition one that is diagonalis... |
H: Does the analytic function have a square root
Let $f:U\to \mathbb{C}\setminus(-\infty,0]$ be a holomorphic function. Does it always have a sqare root function $g^2=f$ which is also analytic in the same region $U$?Note that $U$ represents the open unit disk in $\mathbb{C}$.
I think yes, but am not even able to provi... |
H: How to determine the degree of these polynomials
I have two functions, $p_1,p_2\in\mathcal{H}(\mathbb{C})$ verifying $p_1(p_2(z))=z^2$. I have to prove that $p_1$ and $p_2$ are polynomials and, in addition, show that one of them has degree equal to 1 and the other equal to 2.
After reading this question I know how ... |
H: GCD of Gaussian Integers $\text{gcd}(4, 36+18i)$
I have to compute $\text{gcd}(4, 36+18i)$. I computed the norms: $16$ and $1620$.
I am sure $2$ is the gcd. Is there any method to prove $2$ is the gcd, other than using the Euclidean Algorithm (which I don't know how to use)?
Or if it couldn't be proved directly, ca... |
H: Is this proof of a harmonic progression question correct?
This question is from RMO 2017, the second round of selection for the IMO in India.
Question: Prove that there cannot exist a strictly increasing infinite harmonic progression consisting of terms which are positive integers.
My proof:
Let the first term of a... |
H: For $A, B \subset \mathbb{R}^+$, $\sup(A \cdot B) = \sup A \sup B$.
I am trying to prove that:
For nonempty subsets of the positive reals $A,B$, both of which are bounded above, define
$$A \cdot B = \{ab \mid a \in A, \; b \in B\}.$$
Prove that $\sup(A \cdot B) = \sup A \cdot \sup B$.
Here is what I have so ... |
H: $y \prime=10^{x+y}$ how should I proceed from here?
$$y \prime=10^{x+y}$$ $$\int10^{-y}dy=\int10^xdx$$ $$-\frac{10^{-y}}{\ln(10)}=\frac{10^{x}}{\ln(10)} + C$$ $$10^{-y}=-10^x -C\cdot \ln(10)$$
How I should get $y$ from here? And also, can I call $C\cdot \ln(10)$ just another constant $D$? Is that fine to do such a ... |
H: Divergence of $\int_{0}^{1}\frac{\sin^{2}x}{x^{2}}dx$
Does $\int_{0}^{1}\frac{\sin^{2}x}{x^{2}}dx$ diverge or converge?
Symbolab says it diverges, and I get why, but I don't get the logic behind this because you can clearly see that the graph is bounded and continuous (also $\underset{x\to0^{+}}{\lim}\frac{\sin^{2}... |
H: Consider a matrix $A$ with $n=2m$ and $a_{ii}=a_{n+1-i,n+1-i} = a_{i,n+1-i}=a_{n+1-i,i} = i$ for $i=1,\dots, m$. Find the eigenvalues of $A$.
Consider a matrix $A$ with $n=2m$ and $a_{ii}=a_{n+1-i,n+1-i} =
a_{i,n+1-i}=a_{n+1-i,i} = i$ for $i=1,\dots, m$. Find $||A||_2$.
My attempt:
An example of such a matrix wou... |
H: Does expectation inequality imply conditional expectation inequality?
Given a probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$ and two random variables defined on it, does it hold true that
$$
\mathbb{E}\left(X\right)<\mathbb{E}\left(Y\right)\hspace{0.5cm}\Rightarrow\hspace{0.5cm}\math... |
H: Let $N$ be a normal subgroup of $G$ and $H$ be a subgroup of $G$ such that $ H N =G$. Prove that $H$ is a system of representatives of $N$ in $G$.
Let $G$ be a group and let $N \subseteq G$ be a normal subgroup.
Suppose there is a subgroup $H \subseteq G $ such that :
$HN=\{ hn : h \in H , n \in N\} = G$
$ H \cap N... |
H: Finding the ratio between the area of a circle inscribed by a kite and a circle inscribing the kite
In the following problem, $\angle DAB = 2\alpha$, and $ABCD$ is a kite ($AD=AB, DC=CB$).
I need to prove the ratio between the circle inscribed by the kite to the area of the circle inscribing the kite is $\frac{\sin... |
H: Sum of elements in two different subsets of a set
Let $a$ be any positive integer. Consider set $$S=\{ a^i:i\in \mathbb N \cup \{0\} \}.$$ Let $S_1$ and $S_2$ be any two different finite subsets of $S$. Then show that sum of elements in $S_1$ and $S_2$ are different.
Attempt
The case $a=1$, which is very trivial be... |
H: Finding a recurrence relation for the number of ternary strings of length $n$ containing $11$
Find the recurrence relation for
$$a_n = \text{number of ternary strings of length $n$, containing $11$}$$
Some of these strings are: $11$, $112$, $1102$,... .
I'm thinking that is same if the question was for $00$ ... |
H: $\sigma$-algebra over a set of measures
In Theorem 6.6 of the book Probability Theory by Varadhan, he proved the existence of a probability measure over a set $M_e$ of ergodic measures. The context is as follows.
Let $(\Omega,\mathcal{F})$ be a complete separable metric space with its Borel sets. And let $T:\Omega... |
H: Can an object be a proper direct summand of itself?
Let $\mathcal{A}$ be an additive category, $A,C$ two objects in it. If $A\oplus C\cong A$, is it true that $C=0$?
It seems rather clear, but I am not finding anything on the web and can't really understand how to prove/disprove it.
AI: Consider category of Abelian... |
H: Are these conditional probabilities related?
Knowing $P(A|B)$ & $P(A|C)$, would it be possible to infer $P(A|B,C)$?
I looked at this other question: Any mathematical relation between these conditional probabilities but it doesn't solve my problem as he only uses mutually exclusive probabilities.
AI: Just look at si... |
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