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H: Confusion about unjustified argument in solution to IMO 2018 algebra problem
The first algebra question in IMO 2018 is:
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions
$f\colon \mathbb{Q}_{>0} \to \mathbb{Q}_{>0}$ satisfying
$$f\left(x^2 f\left(y \right)^2 \right... |
H: Probability that exactly $k$ balls in each group are black and rest are white
Assume that we have a total of $n$ balls, $b$ of which are black and the remaining are white. I want to partition them into $g$ groups of size $r$ (i.e., $n=g \times r$) such that for $b/k$ of the groups exactly $k$ balls in each group ar... |
H: Finding $|\!\operatorname{Aut}(L(K_4))|$ using Orbit-Stabiliser Theorem
I know that you can find the size of an automorphism group of a simple graph $G$ by using the Orbit-Stabiliser theorem as follows: let $\DeclareMathOperator{Aut}{Aut}A = \Aut(G)$, and $v$ be a vertex of $G$, where $Av$ denotes the orbit of $v$,... |
H: Combinatorial identity: $\sum\limits_{k=0}^{i\land j}\binom ik(-1)^k\binom{i+j-k}i=1$
Let $i,j\in\mathbb Z_{\ge0}$ be nonnegative integers. How can we prove
$$\sum_{k=0}^{i\land j}\binom ik(-1)^k\binom{i+j-k}i=1?$$
(Here, $i\land j=\min(i,j)=\min\{i,j\}=\min(\{i,j\})$ is the minimum of $i$ and $j.$ This problem com... |
H: Interpretation of $\mathbb P(A|X=x)$ in two ways
Let $X:(\Omega,\mathscr A) \to (\mathbb R,\mathscr{B})$ be a random variable between two measurable spaces (the latter being the Borel measurable space over $\mathbb R$). Let $x\in \mathbb R$. Let $\mathbb P$ be a probability on $(\Omega,\mathscr A)$. Assuming $\math... |
H: If $f'(c)=0$ and $f''(c)\gt0$, then $f$ has a local minimum at $c$
Question: Let $f$ is differentiable on $I$. For $c\in I$, if $f'(c)=0$ and $\exists f''(c)\gt0$, then show that $f$ has a local minimum at $c$.
As you know, this is regarded as a fundamental theorem and is useful when graphing the function. But, I h... |
H: If $\lim (f(x) + 1/f(x)) = 2 $ prove that $\lim_{x \to 0} f(x) =1 $
Let $f:(-a,a) \setminus \{ 0 \} \to (0 , \infty) $ and assume $\lim_{x
\to 0} \left( f(x) + \dfrac{1}{f(x) } \right) = 2$. Prove using the
definition of limit that $\lim_{x \to 0} f(x) = 1$
Attempt:
Let $L = \lim_{x \to 0} f(x) $. Let $\epsilon ... |
H: Confusion on a lemma
So the lemma I had trouble understanding was this:
For all real numbers r, $−|r| ≤ r ≤ |r|$
The solution was something like this(used the definition of absolute values + division into cases)
Suppose r is any real number. We divide into cases according to
whether r ≥ 0 or r < 0.
Case 1 (r ≥ 0):... |
H: Poisson Distribution and Conditional Probability
I'm stuck with a problem for my Statistics class. The problem says that there is a hospital where patients arrive at a constant rate of 2 patients per hour, and there is a doctor that works 12 hours, from 6 am to 6 pm. If the doctor has treated 6 patients by 8 am, wh... |
H: Stirling's formula to bound infinite series
I'm trying to show the following bound, $$\sum_{n=k}^{\infty} \frac{e^{-n\alpha}{(n\alpha)}^{n-1}}{n!} \leq \frac{1}{\alpha}\sum_{n=k}^{\infty} \frac{e^{-kI}}{\sqrt{2\pi n^3}} e^{-{(n-k)I}}$$
where $I=\alpha-1-\log(\alpha)$. I understand that I'm supposed to use Stirling... |
H: Convert linear distance to steering angle
I need to calculate the angle of the front steering wheel using a collapsible piston(linear sensor). 'x' is used to represent the length in inches of the movable part of the sensor and is the independent variable.
θ represents the steering angle. The angle is 0 when the whe... |
H: Maximize $\sum\limits_{k =1}^n x_k (1 - x_k)^2$
Given problem for maximizing
\begin{align}
&\sum_{k =1}^n x_k (1 - x_k)^2\rightarrow \max\\
&\sum_{k =1}^n x_k = 1,\\
&x_k \ge 0, \; \forall k \in 1:n.
\end{align}
My attempt: first of all i tried AM-GM, or we can just say, that $x_k (1 - x_k)^2 \le x_kx_k^2 =x_k^3$, ... |
H: Calculating the number of poker hands that have exactly 4 different denominations
I'm been working through some examples of questions mentioned in the title and came across this one that I wasn't entirely if I was approaching this correctly.
My initial approach to solving this question was to calculate the probabil... |
H: A function that satisfies Cauchy-Riemann but is not holomorphic
I'm attempting Chapter 1, Exercise 12 in Stein & Shakarchi's Complex Analysis, which is as follows:
Consider the function defined by $$f(x+iy) = \sqrt{|x||y|}$$ whenever $x, y \in \mathbb{R}$. Show that $f$ satisfies the Cauchy-Riemann equations at th... |
H: Convergent Improper integral whose integrand tends to a non zero finite limit as x tends to infinity.
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $\int\limits_0^\infty f(x)dx$ exists. If $f(x)\ge 0 \,\forall x\, \in \mathbb{R}$, then prove or disprove that $\lim\limits_{x\to \infty}f(x)$ ex... |
H: Showing images and inverse images don't invert each other
While studying analysis I got the following:
If $f : Z → Z$ is the map $f(x) = x^2$, then $f^{−1}({0, 1, 4}) = \{−2,−1, 0, 1, 2\}$.
Note that $f$ does not have to be invertible in order for $f^{−1}(U)$
to make sense. Also note that images and inverse ima... |
H: Using implicit differentiation to find equation of tangent at arbitrary (a,b)
For the following equation $\sqrt x + \sqrt y = 2$
(1) Find equation of tangent at point (a, b) on curve
Using implicit differentiation: $$y' = - \frac{√y}{√x}$$
Equation at (a, b) is: $$y - b = - \frac{\sqrt b}{\sqrt a}(x - a)$$
$$y = -... |
H: Independent random variables - Finding $P(X = Y )$ and $P(X ≤ Y )$
Let $X$ and $Y$ be independent random variables in a probability space $(\Omega, P)$ with the following distributions:
Calculate:
(a) $P(X = Y ),$
(b) $ P(X ≤ Y ).$
Can you please check my solutions? I have the following:
(a) $P(X = Y )= 0.20 \... |
H: Hint as to how to formalize that the curvature of a curve inside the unit circle is bounded below by the curvature of the circle
I am trying to self teach differential geometry and to that effect I am trying to do the Homework in the MIT open course.
The specific question I am struggling with is:
Let $c$ be a regul... |
H: Why is the stereographic projection bijective?
I know this might be a very basic question, but I am just not able to wrap my head around it.
Why is the map
$$ S:\mathbb{S}^n-\{e_{n+1}\}\rightarrow \mathbb{R}^n \quad \textrm{such that } \bar{x}\mapsto (\frac{x_1}{1-x_{n+1}},...,\frac{x_n}{1-x_{n+1}})$$
a bijective ... |
H: Meaning of "extension field is a vector space"
For example it is clear that the case $\mathbb{Q}\big(\sqrt 2\big) = \{ a+b\sqrt2 \vert a,b \in \mathbb{Q}\}$
(Here the $a,b$ is a scalar)
Then if we more generalize this thought, we can get a well known fact that:
(*) Given the field extension L / K, the larger field ... |
H: Multinomial Coefficient of $x^{1397}$ in expansion of $(x^3+x^4+x^5+...)^6$
I have the following problem:
Find the Coefficient of $x^{1397}$ in expansion of $(x^3+x^4+x^5+...)^6$
I know how to solve these kind of questions using Multinomial Theorem but since the polynomial in this one is infinite I’m lost!
Thanks i... |
H: Logic and set theory books
I have studied real analysis, linear algebra, and how number sets are constructed from N, but now, i want to learn the foundations of math and some more advanced set theory (cardinals, ordinals), because my brain is full of questions such as "what is a property?, how does logic work in ma... |
H: If we have $f : x→ y$ with $S \subset x$ what can we say about $f^{-1}f(S)$ and $S$
Here is the full question:
Let $f : X → Y$ be a function from one set $X$ to another set $Y$, let $S$ be a subset of $X$, and let $U$ be a subset of $Y$. What, in general, can one say about $f^{−1}(f(S))$ and $S$? What about $f(f^{... |
H: Prove that $f(x) = \frac{x}{2} + c$ if $|f(x+y)-f(x-y)-y|\le y^2\ \forall \ x,y \in \mathbb{R}$
Problem Statement: Let $f:\mathbb{R} \to \mathbb{R}$ be a function such that $\forall \ x,y \in \mathbb{R}$,
$$|f(x+y)-f(x-y)-y|\le y^2$$
Prove that $f(x) = \frac{x}{2} + c$ for some $c\in \mathbb{R}$
My try:
$$|f(x+y)-f... |
H: Proof of limits of sequences tending to infinity
How can we prove this :
Prove that if $\quad\lim_{n\to+\infty} y_n = \lim_{n\to+\infty} z_n=+\infty$
then $\quad\lim_{n\to+\infty} v_n = \lim_{x\to+\infty} w_n=+\infty$ .
With $$w_n=\frac{n}{\sum_{i=1}^n\frac{1}{z_i}},\quad v_n = \frac{y_1+y_2+....y_n}{n} $$.Can we... |
H: Find $\sup\left\{ \frac{m}{|m| + n }\right\} $ where $m \in \Bbb{Z}$ and $n \in \Bbb{N}$
Im trying to do $\sup A $ and $\inf A$ if $A= \left\{ \frac{m}{|m| + n } : m \in \Bbb{Z} , n \in \Bbb{N}\right\} $ rigoriously.
my try:
Let $f(m,n) = \dfrac{m}{|m|+n} $. Notice that $f(1,1) = \dfrac{1}{2}$ and $f(-1,1) = \df... |
H: Convergence weak * (star) in a metric space $X$
Let $X$ be a normed space and $\{x^*_n\} \subseteq X^*$,$\{y_n\}\subseteq X$
a) if $\{x^*_n\} \rightharpoonup x ^*$ then $\{x^*_n\}$ is strongly bounded and $\|x^*\| \le \liminf ||x^*_n||$?
b) if $\{x^*_n\} \rightharpoonup x ^*$ and $\|y_n-y\|_X \to 0$ then $(y_n,x^*_... |
H: are there rules that cant be rigorously explained?
this is sort of a shower thought question that came to mind, and I would prefer if it wasn't taken violently seriously, but are their rules that cant be rigorously explained. like for example the limit of a constant times a function is equal to the constant times t... |
H: Express a Probability through $\Phi(t)$
Using Poisson i.i.d. random variables and having $Z$ as standard Gaussian r.v. $~N(0,1)$,
I am struggling to express $P(|Z|≤t)$ in terms of $Φ(r)=P(Z≤r)$ for $t>0$.
I understand that the absolute value of Z means non-negativity and therefore the range is bound by 0 and t, but... |
H: Every irreducible is product of irreducibles
I'm studying some commutative algebra and learning some of the bases. While taking a look at the proof of Theorem 1, proposition 2, i've stumbled upon the following: "Suppose $d$ is not a product of irreducible elements. Hence, $d$ isn't irreducible". I'm trying to under... |
H: How to show that a (metric) space having a countable dense subset is a topological property?
I have to show that a (metric) space having a countable dense subset is a topological property.
Given that A property P of a space is said to be a topological property if home-omorphic spaces share the same properties.
I t... |
H: Find the solutions to the $w''-z^2w=3z^2-z^4$ as Taylor series where $w(0)=0$ and $w'(0)=1$
We need to find the solutions of the
$w''-z^2w=3z^2-z^4$
where
$w(0)=0;w'(0)=1$
I wrote down the series that we can use to find the answer ($w$ as Taylor series):
$w=\sum_{n=0}^\infty C_nz^n$
$w'=\sum_{n=0}^\infty nC_n... |
H: Square root with rational exponent
It might seem very stupid question.
If $x^2=9$ then to solve for $x$ we take both principal $n$-th root of $9$, i.e. $3$ and the negative $n$-th root of $9$, i.e. $-3$. This is right until I found about rational exponents.
If I try to solve the same equation using rational exp... |
H: MLE for an undirected network degree distribution
I have an empirical undirected network. I assume, that a degree distribution is $ F(k) = 1 - e^{1 - \frac{k}{m}} $. and would like to estimate $m$.
The only method I'm aware of for such task is MLE.
If I write a likelihood function $ L = \prod_{i=1}^{N}{p(x_i)} $ (... |
H: Connectedness of a union of convex sets
Let $p_1,...,p_k$ be points in $\mathbb{R}^n$, let $S_1,...,S_m$ be subsets of $\{p_1,...,p_k\}$, and let $V_i$ be the convex hull of $S_i$. Is it true that \begin{equation}
V:=\bigcup_{i=1}^m V_i
\end{equation} is path-connected if and only if any two $p_i,p_j\in V$ can be j... |
H: Radius of convergence for n^n
Given is a power series. We need to find the radius of convergence for this series. The series given is:
$$ \sum_{n=0}^{\infty}n^n(x-1)^n$$
To find the radius of convergence, I have first tried to substitute $y = x - 1$, since this was explained the the course notes. Then I took the li... |
H: Probability - Bayes' Theorem
99% of the restaurants practice good hygiene. Each time you eat in a clean restaurant, there is a $1\%$ chance that you will get sick, independent of your previous visits. Each time you eat in a restaurant that does not practice good hygiene, on the other hand, there is a $50\%$ chance ... |
H: What is the integral of the fractional part of a variable -- related to integration by parts?
Let $\{x\}$ denote the fractional part of a variable, i.e. $\{x\}=x-\lfloor x\rfloor$. Would the integral of $\{x\}-\frac{1}{2}$ from $0$ to $1$ evaluate to $1$? That is
$$
\int_{0}^1 \{x\} -\frac{1}{2} = 1
$$
Am asking th... |
H: If $\vert x \vert > 1$ then $x > 1$ or $x < –1$
Prove that if $\vert x \vert > 1$ then $x > 1$ or $x < –1$ for all $x\in\mathbb{R}$.
I can't wrap my head around as to how it is provable, I could just put some values but that wouldn't be a concrete generalized proof
AI: Since$$|x|=\begin{cases}x, &\text{ if } x\ge... |
H: I need to find the spectrum of an operator.
I need to prove that the spectrum of operator A in $L_{2}[0,1]$ is [-1;1] where $Ax(t) = \sin(\frac{1}{t})x(t)$ if $ t > 0$ and$ Ax(0) = 0$
Elementary - norm of $A$ is less or equal to $1$. Hence, all $\lambda$ from spectrum is such that $|\lambda| \leq$ 1.
$g(t)$ := $\si... |
H: Distribution of $\Big(Y_1+Y_2\Big)^2$ and $\Big(Y_1-Y_2\Big)^2$ where $Y_i \sim N(0,1)$
Does anyone know what is the distribution of $(Y_1+Y_2)^2$ and $(Y_1-Y_2)^2$ where $Y_i \sim N(0,1)$ are independent variables? I have tried to go through the joint pdf, but when trying to change variables I have multiples cases... |
H: The set of real numbers whose product is rational is Borel in $\mathbb{R}^2$
Let $A=\{(x,y)|xy\in \mathbb{Q}\}$ be a set in $\mathbb{R}^2$. Show that A is a Borel set, and find its Lebesgue measure $m_2(A)$.
This is an exercise of the chapter of Fubini and Tonelli's theorem, so I wonder if we consider $f(x,y)=xy$... |
H: Dice rolled 6 times. What is the probability to roll 2 twice, 4 twice, and 6 twice? Also, what is the probability to roll 2 thrice, and roll 4 thrice?
Here's what I currently have, however im not sure its correct.
a) $\mathbb{P}(2 \text{ twice}, 4 \text{ twice}, 6 \text{ twice}) = \{a, a, b, b, c ,c\}$ where $a$, $... |
H: Proving $\sqrt 1+\sqrt 2+..+\sqrt n \approx\frac{2}{3} n^{\frac{3}{2}}$ asymptotically
Let $n$ be a positive intiger, prove this asymptotic formula for large $n$
$$\sqrt1+\sqrt2+\cdots +\sqrt n=\frac{2}{3}n^{\frac{3}{2}}+\text{lower order}$$
using a Riemann sum.
AI: We can obtain your approximation as follows:
$$\s... |
H: Textbooks in which determinant is defined as an alternating multilinear map
I'm interested in this abstract definition of determinant, i.e. determinant is defined as an alternating multilinear map. Could you please suggest me some Linear Algebra textbooks that define determinant in such way.
Thank you so much for y... |
H: What number's factorial is $i$?
I am trying to find the solution to the equation- $$\Gamma(z)=i$$
I have tried doing it the following way- LHS is-
$$\displaystyle \int_{0}^{\infty}t^ze^{-t}\ dt$$
Taking $z=a+ib$, we get-
$$\displaystyle \int_{0}^{\infty}t^{a+ib}e^{-t}\ dt$$
or
$$\displaystyle \int_{0}^{\infty}t^{a}... |
H: Radius of convergence and uniformly convergence
I think the radius of convergence for $\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)x^n$, $x\in \mathbb R$ is:
$r^{-1}$=$\lim_{n\to \infty}$$|\frac{a_{n+1}}{a_n}$|=1 so we get that $r$=1.
But how can I show it formally?
Afte... |
H: Why is $\Big\{\frac{1}{n^2}\Big\}_{n \in \mathbb{N}} \bigcup \hspace{0.2cm} \{-1,0 \}$ compact?
We have a set on Real numbers:
$$\Big\{\frac{1}{n^2}\Big\}_{n \in \mathbb{N}} \bigcup \hspace{0.2cm}\{-1,0 \}$$
I can't seem to understand why this is a compact set as if we look at the interval I think that there are ti... |
H: $A$ and $B$ are two subnormal $p$-subgroups of $G$, how to show that $\langle A,B\rangle$ is a $p$-subgroup of $G$?
$A$ and $B$ are two subnormal $p$-subgroups of $G$, how to show that $\langle A,B\rangle$ is a $p$-subgroup of $G$?
It is not true in general if $A$ and $B$ are not subnormal. For example, $A:=\langle... |
H: Does the convergence in probability imply the following limit is $1?$
Let $X_n \in \mathbb{R}$ be a sequence of non-constant random variable with continuous PDF converging in probability to $c,$ but not necessarily convergence almost surely, i.e.
$$\lim\limits_{n \to \infty}P[|X_n - c| \le \epsilon]=1 \forall \epsi... |
H: Flat extension of local rings with a specified extension of residue field
Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$.
Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: R\to S$ such that $f(\mathfrak m_R)S=\math... |
H: $f(x) = ax^3 + bx^2 + cx + d,$ with $a > 0. $ If $f$ is strictly increasing, then the function $g(x) = f′ (x) −f′′(x) + f′′′(x)$ is
QUESTION: Consider the function $f(x) = ax^3 + bx^2 + cx + d,$ where $a, b, c$ and $d $ are real numbers with $a > 0. $ If $f$ is strictly increasing, then the function $g(x) = f′ (x)... |
H: Why is set $ \bigcup\limits_{n = 1}^{\infty} \{(\frac{k}{n},\frac{1}{n}):k=0,1,\dots,n\} $ closed.
We have a set:
$$ \bigcup_{n = 1}^{\infty} \left\{\left(\frac{k}{n},\frac{1}{n}\right):k=0,1,\dots,n\right\} $$
I don't understand the notation, does it mean that when $n=1$ then $k=0$ and both raise equivalently or c... |
H: The "o" notation and limit
I just want some explanation about this notation ,it's very new to me ,and I've been also been given this exercise :
"Is this true or false : $x=o(\sqrt x)$ , $x\to 0$"
And it is true in the hint of the exercice ,and here is the hint : $\lim_{x\to 0} $=$\frac{x}{\sqrt x}$=$0$
How do we ... |
H: Circular track problem (LCM)
A question in my text book:-
A circular field has a circumference of $360 \;km$. Three cyclists start together and can cycle $48$, $60$ and $72$ km a day, round the field. When will they meet again?
Solution: We first find out the time taken by each cyclist in covering the distance.... |
H: $1^{-1}+2^{-1}+\dots+\Big(\frac{p-1}{2}\Big)^{-1} \equiv -\frac{2^p - 2}{p} \mod p$ for an odd prime $p.$
I've reduced a problem down to proving this identity. Unfortunately, I don't know where to even start. There has to be some way of expanding the RHS or combining terms on the LHS, but I don't see it. Any hints?... |
H: Showing the collection of subsets is the neighborhood system of a given topology
Suppose that we have a topological space $(X,\tau)$, a subset $A\subseteq X$ is a neighborhood of $x \in X$ if there exists an open subset $U_x$, containing $x$ such that $U_x \subseteq A$. Also, suppose that the collection of all neig... |
H: How does a transitive extension differ from a transitive closure?
Quoting an example from C.L Liu's Discrete Mathematics: Let R be a binary relation on A. The transitive extension of R (let's denote it as $R_1$) is a binary relation on A such that $R_1$ contains R. Doesn't that make $R_1$ the transitive closure to ... |
H: Is this weighted average smaller than the corresponding arithmetic average?
Is it true that the following weighted-average is smaller than the respective arithmetic average
$$
\sum_{n=1}^{N}\left(\frac{b_{n}}{\sum_{m=1}^{N}b_{m}}\right)\frac{a_{n}}{b_{n}}\overset{?}{\le}\frac{1}{N}\sum_{n=1}^{N}\frac{a_{n}}{b_{n}}
... |
H: Decide whether this is correct, using the method of resolution. If not, provide a counter example.
I am new to logic and wanted to decide whether the following is correct using the method of resolution: |= p → ¬ (p → (p ∧ (p ∨ q)))
My attempt to this I answered that the conclusion is incorrect, though the premises ... |
H: Holomorphic function, existence of a sequence
Suppose $f$ is holomorphic on $D$. Prove that there exists a sequence $\lbrace z_n\rbrace$ such that $|z_n|\rightarrow 1$ and $\lbrace f(z_n)\rbrace$ is bounded.
Now, what I did is
$f$ is analytic on $D$ and $|f(z)|<1$. Let $z_0,z_1,z_2,...$ be zeros on $D$ then each $... |
H: Closely minimize error bound $\frac{4 \pi \exp(\cosh(a))}{\exp(a N) -1}$
I'm trying to minimize an error bound $$\frac{4 \pi \exp(\cosh(a))}{\exp(a N) -1},$$ where $N$ is the step size for the trapezoidal rule and $-a < Im < a, a > 0$ is a strip bound which may be adjusted for different $N$. Now the paper with thi... |
H: Which matrices can or cannot be obtained by matrix multiplication?
I was reading this post, Can you transpose a matrix using matrix multiplication?, and I thought it was interesting that we can't get a "transpose matrix" $B$ such that $BA = A^T$, at least not for all $A$. My questions are:
Given a matrix $A$, what... |
H: Winning strategy for player $0$ in the game $G_3(\mathfrak{A_1},\mathfrak{A_2})$ $\mathfrak{A_i}=(\mathbb{N},\{(n,n+i),n\in \mathbb{N}\})$
This is an Ehrenfeucht–Fraïssé game
Is it possible that player $0$ or the attacker/competitor can win against the defender/player $1$/duplicator in $3$ rounds ?
I have thought a... |
H: Test if a point on a hexagonal lattice falls on a specified superlattice?
Based on previous answers (1, 2, 3) integers $i, j$ produce a hexagonal lattice using
$$x = i + j/2$$
$$y = j \sqrt{3} / 2.$$
From a point $k, l$ I can make a superlattice from integers $I, J$ using
$$i_{sup} = I k + J (-l)$$
$$j_{sup} = I ... |
H: How to write a matrix from $SU(2)$ in terms of one angle and one complex number $z$ , where $z$ is from sphere $S^{2}$
For given a matrix from $SU(2)$ , how can represent it in terms of two parameters: one angle and one complex number $z$ from the sphere $S^{2}$ ?
Does this have any links with : $\mathrm{SU}(2)$ ax... |
H: Prove $\left|\frac{a_1 + ... + a_n}{b_1 + ... + b_n} - c \right| \le \max\limits_{k \in 1:n}\left|\frac{a_k}{b_k} - c\right|$
Given two sets of numbers - ${a_1, ..., a_n}$ and ${b_1, ..., b_n},b_i \ge 0 \; \forall i \in 1:n$ and some constant $c$.
I'm trying to prove that
$$\left|\frac{a_1 + ... + a_n}{b_1 + ... + ... |
H: Finding $a$ such that $a^2x^2+3x-5\frac{1}{a}=0$ has exactly one solution
For what value of $a$ would the following function have exactly one solution?
$$a^2x^2+3x-5\frac{1}{a}=0$$
I know that it needs to become
$$\frac{3}{2}x^2+3x+\frac{3}{2}=0$$
but how can one find value of parameter $a$ for this to happe... |
H: Show that $f(A ∩ B) ⊆ f(A) ∩ f(B)$, can the relation be improved to equality?
I have the following question:
Let $A,B$ be two subsets of a set $X$, and let $f : X → Y$ be a function. Show that $f(A ∩ B) ⊆ f(A) ∩ f(B)$. Is it true that the $⊆$ relation can be improved to $=$?
While I know that in these type of que... |
H: Marginal distributions for a standard bivariate Cauchy distribution
Consider the following:
Let $X$, $Y$ be two jointly continuous random variables with joint PDF
$$f\left(x,y\right)=\frac{c}{2\pi}\frac{1}{\left(c^{2}+x^{2}+y^{2}\right)^{\frac{3}{2}}}\quad\text{(Standard Bivariate Cauchy Distribution)}.$$
Find the ... |
H: Is $\sum\limits_{n=0}^{\infty} \frac{n}{n+1} (-1)^{n}$ convergent?
I will like to know whether $\sum\limits_{n=0}^{\infty} \frac{n}{n+1} x^{n}, \: x \in \mathbb{R}$ is convergent for $x=\pm1$ or not.
I found the radius of convergence to be $r=1$. So I have to check each ($x=1$ and $x=-1$) case separately. $x=1$ is ... |
H: Is there a real valued positive function such that it and its square integrate to $1$
Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f > 0$ and
$$
\int_{-\infty}^\infty f(x) dx = \int_{-\infty}^\infty f(x)^2 dx = 1.
$$
I suspect the answer is yes. I have looked at taking $f$ to be th... |
H: Variation of a functional
I have to find the variation of the following functional:
There are two conditions a > 0 and b > 0.
The question is "find the differential equation with respect to x(t), so that the functional is minimized".
AI: Set $y(\tau) = x(\tau) + \varepsilon \eta(\tau)$ where $\eta(\tau)$ is an arb... |
H: Do there exist sudoku's with clues that only contain $1$'s and $2$'s
I read this article saying it was mathematically proven that if a sudoku has a unique solution, then it has at least $17$ initial hints.
In which case, is it possible to have a sudoku with $17$ initial hints that consist of nine $1$'s and eight $2... |
H: Discrete measure and Lebesgue measurability
According to wikipedia
https://en.wikipedia.org/wiki/Discrete_measure
a driscrite measure is defined in the following way:
Let's consider a real line $\mathbb{R}$. For some (possibly finite) sequences $s_{1}, s_{2}, \dots$ and $a_{1}, a_{2}, \dots$, s.t. $a_{i}>0$ and $\s... |
H: Prove that $U(E_{\lambda})=E_{\lambda}$ and $U(K_{\lambda})=K_{\lambda}$.
Let T be a linear map on a finite-dimensional vector space V , and let $\lambda$ be an eigenvalue of T with corresponding eigenspace and generalized eigenspace $E_{\lambda}$ and $K_{\lambda}$. Let U be an invertible operator on V that commun... |
H: complex non algebraic manifold local ring of holomorphic functions is noetherian?
Consider $X$ a complex manifold. Denote $x\in X$ a point and $O_x$ as the local holomorphic function ring at $x$.
Assume $X$ is not algebraic.
$\textbf{Q1:}$ Is $O_x$ Noetherian? If it is Noetherian, then my guess is that it can be l... |
H: Confusion with mathematical objects (sets and multisets in particular)
The two sets $\{1,2,3\}$ and $\{1,1,2,3\}$ are equal, but the two multisets $<1,2,3>$ and $<1,1,2,3>$ are not equal.
Multisets are generalizations of sets that allow repeated elements. A natural mapping exists from sets to multisets: we can wri... |
H: What does the notation of $2^{\mathbb{N}}$ mean?
I've learned over the course of the last years that some mapping $\lambda$ denoted :
$$ \lambda : \mathbb{N} \rightarrow 2^{\mathbb{N}}.$$
essentially means that for every natural number, you assign (or map) some set of natural numbers to it. At multiple times, I've... |
H: How many ways to pick pairs from a group of six?
How many ways can we group six people into pairs?
I expect the answer to be $6!/(2!)^3$ but the textbook gives the solution as 6!/(2!)33!
I'm confused, as the way I got my answer was to do
(6 pick 2) * (4 pick 2) * (2 pick 2)
(6!)/(2!4!) * (4!/(2!2!))
AI: There ar... |
H: For angles $A$ and $B$ in a triangle, is $\cos\frac B2-\cos \frac A2=\cos B-\cos A$ enough to conclude that $A=B$?
Brief enquiry:
$$\cos\frac B2-\cos \frac A2=\cos B-\cos A$$
Optionally
$$\sqrt\frac{1+\cos B}{2}-\cos B=\sqrt\frac{1+\cos A}{2}-\cos A$$
Is above equality sufficient to prove that it implies $A=B$?
... |
H: Verifying a Topological Property
Let (X,T) be any Topological Space. Verify that Intersection of any finite number of members of T is a member of T.
I tried to prove using that intersection of any two sets of T belongs to T. So the result of the intersection can Intersect with any other subset of T, the result of w... |
H: Negative-base logarithm, where's the issue here
Here we go again, basics of the basics. Faced with the following question.
Definitions
Logarithm base a of x is by definition a number such as:
$$a^{\log_a x} = x$$
i.e. that answers the question "what power do I have to raise a in order to get x". I've read this answ... |
H: Cauchy sequence is bounded? (Do we need any element in a sequence to be finite?)
This question is related to the question Is every cauchy sequence bounded?
The sequence $\{a_n\}$ used in that question
$$a_n=\frac{1}{n-1}$$
has the first element $a_1\rightarrow\infty$. As shown in the answer of that qu... |
H: Represent $\sum_{k=1}^{n} k^{2}$ in terms of binomial coefficient.
Came across a probability problem that is sort of challenging for a beginner in a sense that I may have not seen or came across a lot of binomial identities.
What I am looking for is to see if there is any way to represent $\sum_{k=1}^{n} k^{2}$ in... |
H: Evaluate: $\int_0^1 \frac{e^x(1+x) \sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx $
Evaluate: $\int_0^1 \frac{e^x(1+x)\sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx $
My assumption put $t= x e^x $ then $\int_0^e \frac{\sin^2(t)}{\sin^2(t)+ \sin^2(e-t)} \,dt$ How can I proceed from here? Thanks in advanc... |
H: How to prove if $(A \cup B) - (A \cap B) = A$ is true, then $B = \emptyset$?
I have figured out that this is true, but I'm not quite able to prove it.
I have tried direct proof, contrapositive and contradiction.
For contradiction, I assumed that B is not the empty set.
Then, I wrote that the difference between A an... |
H: How do I finish deriving this property of normal subgroups?
Socratica has a fantastic video on normal subgroups and quotient groups, but there’s one part of which I can’t convince myself.
Let $G$ be a commutative group under juxtaposition, let $N$ be a normal subgroup of $G$, and let the quotient group $G/N$ be equ... |
H: A variation in the construction of the tensor product of modules
Let $A$ be a ring, $E$ a right $A$-module and $F$ a left $A$-module. Consider the free $\mathbf{Z}$-module $\mathbf{Z}^{(E\times F)}$ which comes with the injective canonical mapping $\phi:E\times F\rightarrow\mathbf{Z}^{(E\times F)},\,(x,y)\mapsto e_... |
H: To what extent does Goedel's 2nd incompleteness theorem extend?
In chapter 8 of Shoenfield's matheamtical logic[1967], He proves that The formula of P which states that P is consistent is not a theorem of P, where P stands for Peano Arithmetic. And then He says, without a proof, that this result can be extended to ... |
H: Confusion about A1, IMO 2002
The following is question A1 from the 2002 IMO:
$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k<n$. Each element of $S$ is colored red or blue such that if $(h,k)$ is red, and $h'\leq h,k'\leq k$, then $(h',k')$ is also colored red. A Type 1 subset of $S$ ... |
H: How long to catch up to a stream started 1 hour ago at 1.5x speed?
I opened a stream that started an hour ago. Not wanting to miss anything, I started from the beginning and set it to 1.5x speed. How long will it take for me to catch up?
I know that it will take 40 minutes to watch the hour that I missed ($ \frac ... |
H: Show $f:\mathbb{Z}\to\mathbb{Z}[i]/(3+2i)$ via $c\mapsto c+(3+2i)$ is surjective
Write $x=a+bi+(3+2i)\in\mathbb{Z}[i]/(3+2i)$. I want to find a $c\in\mathbb{Z}$ such that $f(c)=x$. I think we need to use the fact that there exist $m,n\in\mathbb{Z}$ so that $2m+3n=1$, but I am not sure how. I'd like to do something ... |
H: Is this switch of order of summation legal?
$$\sum_{k=1}^{\infty} \frac{1}{k^3} \sum_{m=0}^{\infty} \frac{(-1)^m}{k^{3m}} \to \sum_{m=0}^{\infty} (-1)^m \sum_{k=1}^{\infty} \frac{1}{k^3k^{3m}} \to \sum_{m=0}^{\infty} (-1)^m \sum_{k=1}^{\infty} \frac{1}{k^{3(m+1)}} \to \sum_{m=0}^{\infty} (-1)^m \sum_{k=2}^{\infty} ... |
H: Finding residues of $\frac{z}{e^z+e^{-z}}$.
I am having trouble finding the residues of
$$\frac{z}{e^z+e^{-z}}.$$
I have found the poles (at $z= ±i(\frac{\pi}{2} +n\pi)$) but I cannot find a way to expand the function into a Laurent series to find the residues.
AI: You don't need such a Laurent expansion, since\be... |
H: Is $f(a_1 + p\mathbb{Z},a_2 + p^2 \mathbb{Z}, \ldots) = (0,pa_1 + p^2 \mathbb{Z},\ldots)$ surjective?
Let $p$ be prime and the $\mathbb{Z}$-module $M=\prod_{n=1}^\infty\mathbb{Z}_{p^n}$ where $\mathbb{Z}_{p^n}$ is $\mathbb{Z}$ modulo $p^n$. Define a map $f : M\to M$ by $f(a_1 + p\mathbb{Z},a_2 + p^2 \mathbb{Z},a_3... |
H: Polynomials that form $1+xy+x^2 y^2$
Show that there is no polynomials $a(x), b(x) \in R[x]$ and $c(y), d(y) \in R[y]$ such that $1+xy +x^2 y^2 = a(x) c(y) + b(x) d(y) $
AI: Expanding on @MikeDaas's comment we have$$\begin{align}1&=a(x)c(0)+b(x)d(0),\\1+x+x^2&=a(x)c(1)+b(x)d(1),\\1-x+x^2&=a(x)c(-1)+b(x)d(-1)\\\impl... |
H: Finding the asymptotic of an integral
I came across the following exercise on asymptotic behavior of integrals:
$$I(a) = \int_0^\infty\frac{\cos x}{x^a} \, dx, \text{ where } a\to0^+.$$
I have tried integrating by parts or replacing $\cos x$ with the first summands of its Taylor series, but I end up with something ... |
H: Find 5 zero divisors of $\mathbb{Z}_6 \times \mathbb{Z}$
I want to find 5 zero divisors of $\mathbb{Z}_6 \times \mathbb{Z}$.
So far, I have found the zero divisors of $\mathbb{Z}_6$ as follows:
First, note that the non-zero elements of $\mathbb{Z}_6$ are $\{1, 2, 3, 4, 5\}$. Now, check for the various possibilitie... |
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