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Why is cofinality required in the definition of a subnet? My professor mentioned that there are certain cardinality-related reasons for this requirement. Does this have anything to do with the confinality of cardinal numbers? |
In $\dfrac{365 \choose \#people}{365^{\#people}}$
this counts no repeats for the probability of none assuming Jan, Feb, March is the same thing as Feb, March Jan.
If
>We interpret the question as asking for the probability that **precisely two** people share a birthday.
>Imagine lining up our $n\ge ... |
Let $E=E(k), K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\int_0^1Kdk=G$ is the Catalan constant. We can also find, by switching integral trick that $$\int_0^1E(k)dk\\
=\int_0^{\frac\pi 2}\int_0^1\sqrt{1-k^2\sin^2\theta}d\theta\\
=\int_0^{\frac\pi 2}(\... |
Let $A_{ij}$ be a square, symmetric, and positive definite real matrix. I am interested in finding vectors $x_i$, with real components, that are solutions of the following equation:
$$b_{i}e^{ x_{i} } + \sum_{j}A_{ij}x_{j} = 0$$
where $b_i$ is another real vector of appropriate dimensions, with non-negative compo... |
Let $f(x)= sin ln ( 1+ \frac{x²}{2})$
Find the Mclaurin expansion of $f^{(6)}$ with order 3.
We know that :
$ f(x) = \Sigma^\infty _{k=0} \frac{f^{(6)} (0) }{k!} x^k $
In a neighborhood of 0, therefore
$f^{(6)}(x)=\frac{d^6}{dx} (\Sigma^\infty _{k=0} \frac{f^{(k)} (0) }{k!} x^k) = 6\Sigma^\infty _{k=0}... |
My problem concerns the big-O error after transforming a constraint optimization problem. I'm not really familiar in this field so sorry for any incorrect notation/logic in my reasoning. I will first explain the steps I make and ask the question in the end.
Let $n$ be a positive integer. Consider the following opt... |
According to everything you did up until the last sentence, the answer should be
$${4 \choose 2} \cdot 365 \cdot 364 \cdot 363$$
What makes you think otherwise?
If you used $364 \choose 2$ to represent the number of ways in which you can pick $2$ birthdays for the $2$ people with distinct birthdays (out of th... |
Let $f(x)= sin ln ( 1+ \frac{x²}{2})$
Find the Mclaurin expansion of $f^{(6)}$ with order 3.
We know that :
$ f(x) = \Sigma^\infty _{k=0} \frac{f^{(6)} (0) }{k!} x^k $
In a neighborhood of 0, therefore
$f^{(6)}(x)=\frac{d^6}{dx} (\Sigma^\infty _{k=0} \frac{f^{(k)} (0) }{k!} x^k) = 6\Sigma^\infty _{k=0}... |
Let $$f(x)= \sin \left(\ln \left(1+\frac{x^2}{2}\right)\right)$$
Find the Mclaurin expansion of $f^{(6)}$ with order 3.
We know that :
$$ f(x) = \sum^\infty_{k=0} \frac{f^{(6)} (0) }{k!} x^k $$
In a neighborhood of $0$, therefore
$$f^{(6)}(x)=\frac{d^6}{dx^6} \left(\sum^\infty _{k=0} \frac{f^{(k)} (0) ... |
Consider a strictly positive sequence $\{u_n\}$ of real numbers (i.e. $\forall n, u_n >0$). Suppose that $\displaystyle \lim_{n \to \infty} u_n = 0$ and that $\{u_n\}$ *is not decreasing*.
Consider the sequence $\{l_n\}$ defined by : $\forall n \in \mathbb N, l_n := \min\left\{i-n,\text{ such that } u_i\leq u_n,\for... |
Let $D = D_1 \cup D_2 \cup...\cup D_m$ be a subset of $\mathbb{R}^d$, where each $D_j$ is a (connected and) bounded domain with Lipschitz boundary, and $\min_{i,j} dist(D_i,D_j) \geq c > 0$ for some $c$. Let a function $f$ defined on $D$. Is there a Sobolev extension theorem for $f$ from $D$ to $\mathbb{R}^d$? |
Let $$f(x)= \sin \left(\ln \left(1+\frac{x^2}{2}\right)\right)$$
Find the Maclaurin expansion of $f^{(6)}$ with order 3.
We know that :
$$ f(x) = \sum^\infty_{k=0} \frac{f^{(6)} (0) }{k!} x^k $$
In a neighborhood of $0$, therefore
$$f^{(6)}(x)=\frac{d^6}{dx^6} \left(\sum^\infty _{k=0} \frac{f^{(k)} (0)... |
I have the following difference equation:
$x_{t+1}=f(x_t)=x_t+3.61*(1/x_t-1)$ where the domain is the closed interval E=[0.19,17.48]. I know the following facts for this map.
1. This map is smooth and unimodal (convex) with the modal point at x=1.9 (and $f(1.9)=0.19$).
2. $f|_{E}$ is a map to $E$.
3. $f|_{E}$ s... |
**Hint:**
Just notice that:
$$(-\dot{f} +af) = -e^{at}(e^{-at}f)'$$
Then, let us denote
$$g_1 = e^{-a_1 t}f_1$$
$$g_2 = e^{-a_2 t}f_1$$
The system of two equations is equivalent to
$$\begin{align}
&\iff \cases{a_1t+\ln(g_1)= A_{11}(a_1t + \ln(-\dot{g_1})) + A_{12}(a_2t + \ln(-\dot{g_2}))\\
a_2t+\ln(g_2)=A_{... |
If a multiple-choice test (where all questions are worth the same) has $n$ choices on each question, random guessing gets you a score of $\frac1n$ on average. But what if the number of choices increased as you went through the test? Let's say that the number of choices matches the question number, so question $\#3$ has... |
I'm writing a code in which
I have a point &
I want to identify if the points lies on front, back, top, bottom, left or right side of a cuboid.
Does anyone have an idea how could this be solved?
|
Calculate on which side of a cuboid is a given point located? |
Let $f$ be a continuous weakly-increasing function that maps $\mathbb{R}_+$ to $\mathbb{R}_+$. Define a function $g$ on $\mathbb{R}_+$ as follows: $g(x)$ is the largest $x'$ such that $f(x') = f(x)+1$.
For example, if $f(x) = x^2$, then $g(x) = \sqrt{x^2+1}$.
If $f$ is strictly-increasing, then it is possible to ... |
When I was reading complex analysis, I saw the definition for integration of a smooth curve, but late in a theorem, the proof is trying to integrate a triangle directly without proving that it can be smoothly parametrized by a interval. How can we determine this in general? Thanks! |
Are all symbol-wise optimal code Huffman code? |
Consider a maximin problem with $\lambda=[\lambda_1,\lambda_2,...,\lambda_n]$
$$\max_{\lambda}\min_{i\in\{1,2,3\}}{f_i(\lambda)}$$
subject to $\lambda$ is any vector locating in a standard probability simplex, that is $\sum_{i=1}^{n}\lambda_i=1$ and $\lambda_i\geq0, \forall i\in\{1,2,...,k\}$. Note that each function... |
how can I maximize a minimum function? |
I've encountered the following equations while studying complex exponentials:<br>
(1): $e^{j2\pi t} = \cos(2\pi t) + j \sin(2\pi t)$<br>
(2): $e^{j2\pi t} = (e^{j2\pi})^t = (\cos(2\pi) + j\sin(2\pi))^t = (1+0)^t = 1^t
$<br>
Upon comparing (1) and (2), it's evident that they represent different expressions. However,... |
For $ \overrightarrow{V} = V^i \overrightarrow{e_i} = V_i \overrightarrow{e^i} $,
$ \frac{\partial \overrightarrow{V}}{\partial x^j} = \frac{\partial V^i}{\partial x^j} \overrightarrow{e_i} + V^i \frac{\partial \overrightarrow{e_i}}{\partial x^j} = \frac{\partial V^i}{\partial x^j} \overrightarrow{e_i} + V^i (\Gamma... |
Why can we change indices when defining covariant derivative formula? |
Consider a maximin problem with $\lambda=[\lambda_1,\lambda_2,...,\lambda_n]$
$$\max_{\lambda}\min_{i\in\{1,2,3,...,M\}}{f_i(\lambda)}$$
subject to $\lambda$ is any vector locating in a standard probability simplex, that is $\sum_{i=1}^{n}\lambda_i=1$ and $\lambda_i\geq0, \forall i\in\{1,2,...,k\}$. Note that each fu... |
The other answers give the theoretical reasons why this doesn't contradict the axioms of probability. In practice, what goes wrong is as follows. Let's assume you have assigned random reals to each player, that is, each player has an infinite string of digits, and they're all different. You also have chosen a well-orde... |
I've encountered the following equations while studying complex exponentials:
$$\begin{align}
e^{j2\pi t} &= \cos(2\pi t) + j \sin(2\pi t) \tag1\\[4pt]
e^{j2\pi t} &= (e^{j2\pi})^t = (\cos(2\pi) + j\sin(2\pi))^t = (1+0)^t = 1^t \tag2
\end{align}$$
Upon comparing (1) and (2), it's evident that they represent differ... |
According to everything you did up until the last sentence, the answer should be
$${4 \choose 2} \cdot 365 \cdot 364 \cdot 363$$
What makes you think otherwise?
If you used $364 \choose 2$ to represent the number of ways in which you can pick $2$ birthdays for the $2$ people with distinct birthdays (out of th... |
Given an ordinal $\alpha$, define $\alpha !$ as it follows:
$$
\alpha! := \begin{cases}
0! = 1 \\
(\alpha + 1)! = \alpha ! \cdot (\alpha + 1) \\
\lambda! = \left(\sup_{\gamma < \lambda} \gamma !\right) \cdot \lambda &\text{$\lambda$ limit}
\end{cases}
$$
I was trying to evaluate $\varepsilon_0 !$, where $\var... |
Does this simple model have a name? |
Let us define the complete elliptic integral of the first kind as follows:
$$
K(k)=\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2t^2} }
\text{d}t,
$$
where we restrict the modulus $k$ to $|k|<1$ with its complementary $K^\prime(k)=K\left ( \sqrt{1-k^2} \right )$. They are shortened as $K,K^\prime$ respectively.
... |
Given an ordinal $\alpha$, define $\alpha !$ as it follows:
$$
\alpha! := \begin{cases}
0! = 1 \\
(\alpha + 1)! = \alpha ! \cdot (\alpha + 1) \\
\lambda! = \left(\sup_{\gamma < \lambda} \gamma !\right) \cdot \lambda &\text{$\lambda$ limit}
\end{cases}
$$
I was trying to evaluate $\varepsilon_0 !$, where $\var... |
In a 2015 paper titled *Lecture on the abc conjecture and some of its consequences* by Michel Waldschmidt it is stated that:
It can be easily seen that the $abc$-conjecture is equivalent to the following statement:
For each $\epsilon > 0$, there exists $\kappa(\epsilon)$ such that, for any abc triple $(a, b, c)... |
I have angle between two vectors:
pseudocode for func angleBetween:
dotProduct = v1.x * v2.x + v1.y * v2.y;
determinant = v1.x * v2.y - v1.y * v2.x;
angle = atan2(determinant, dotProduct);
and compare angleBetween(v1,v2)<angleBetween(v3,v4)
How to compare using only determinant,dotProduct (o... |
How compare angles without arcus tangens? |
Given tall matrix $B \in \mathbf{R}^{n \times r}$, where $n \gg r$, let
$$ A := B B^T $$ and the columns of $B$ are in general not orthogonal to each other.
I would like to get the eigenvectors and values of $A$ efficiently. I think the number of eigenvalues $\lambda_n > 0$ and the rank of $A$ is at a maximum $n... |
Suppose you have the coordinates of three points $A,B,C$. You want to construct a plane $\pi$ such that
$ d(\pi, A) = r_1 $
$ d(\pi, B) = r_2 $
$ d(\pi , C ) = r_3 $
What are the conditions on $r_1, r_2, r_3$ so that a solution exists? And what is the equation of this plane in terms of the coordinates of... |
Let's consider a smooth function $f:\mathbb{R}\to \mathbb{R}$ and the quantity
$$
q(f):= \frac{|f(0)|}{\lVert f\rVert_{L^1(-1,1)}}.
$$
If $q(f)$ is rather large it means that $f$ is relatively small at most points in the interval $(-1,1)$ but not at the midpoint $0$.
In this case $f$ must be somewhat spiky at the... |
Question: I'm currently delving into the realm of Markov models to understand how they can be applied to determine the cost of car insurance. Thus far, I've come across the bonus-malus system and hidden Markov chains. However, I'm curious to learn about other Markov models that could be relevant in this context. Could ... |
As a developer building a chart application, I'm using linear interpolation to map time values to pixel positions on the screen. My current approach utilizes the following function:
fun interpolatePixelByTimestamps(
startPixel: Double,
endPixel: Double,
startTimestamp: Long,
... |
There is no right answer for this question. And the answer I am about to give is solely based on my personal experience as a Mathematics student.
Self studying Mathematics is not an easy task. I was lucky enough to have great professors at university. Most of them had a vast teaching experience. After teaching somet... |
What is wrong with the following reasoning? <br/><br/>
The statement is true for $n=2$:$$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2= p^2 + q^2$$
So you can introduce a new pair $(r^2+s^2)$ and write:$$(p^2+q^2)(r^2+s^2)=(a^2+b^2)(c^2+d^2)(r^2+s^2)$$ etc.
|
As a developer building a chart application, I'm using linear interpolation to map time values to pixel positions on the screen. My current approach utilizes the following function:
fun interpolatePixelByTimestamps(
startPixel: Double,
endPixel: Double,
startTimestamp: Long,
... |
As far as I remember, The proof of Pelerman of the Geometerization Conjecture took a year to check by the experts. The claimed proof of the ABC conjecture Mochizuki took few years to check, and Peter Scholze another great mathematician rejected the proof while Mochizuki who is also another respected mathematician still... |
Are there attempts to use machine learning to settle disputes over long mathematical proofs? |
When self studying a text where it is not expected to be able to solve all (or most) of the problems, what are the appropriate criteria to use for advancement?
> A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all.... Many are introduced n... |
Let us consider an essentially bounded function $f:[0,1]\to \mathbb{R}^d$, that is $f \in L^{\infty}([0,1],\mathbb{R}^d)$. Let us assume that $f(x) \in A$ for almost every $x\in [0,1]$, where the set $A\subset \mathbb{R}^d$ is compact.
I try to prove that the set
$$
\{x\in [0,1] \mid x \text{ is a Lebesgue point ... |
In writing a proof
My example would be
$(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$
The goal is $(0\le\sqrt{a}\lt\sqrt{b})$
But here the $0\le\sqrt{a}$ is like an assumption or a definition
So the proof of $0\le\sqrt{a}$ is ignored
Could we add these extra assumptions or definitions in the proof ... |
In writing a proof how are we certain whether some part of a goal in the proof can be ignored or not? |
Consider a strictly positive sequence $\{u_n\}$ of real numbers (i.e. $\forall n, u_n >0$). Suppose that $\displaystyle \lim_{n \to \infty} u_n = 0$ and that $\{u_n\}$ *is not decreasing*.
Consider the sequence $\{l_n\}$ defined by : $\forall n \in \mathbb N, l_n := \min\left\{i-n,\text{ such that } u_i < u_n,\foral... |
Suppose you have the coordinates of three points $A,B,C$. You want to construct a plane $\pi$ such that
$ d(\pi, A) = r_1 $
$ d(\pi, B) = r_2 $
$ d(\pi , C ) = r_3 $
What are the conditions on $r_1, r_2, r_3$ so that a solution exists? And what is the equation of this plane in terms of the coordinates of... |
# Question
I'm exploring the interplay between probability distributions and entropy, focusing on finding a distribution that maximizes entropy under a specific constraint. The core of my question involves the sigmoid function, defined as $\sigma(x) = \frac{1}{1 + e^{-x}}$, which is widely used in various fields suc... |
In writing a proof
My example would be
$(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$
The goal is $(0\le\sqrt{a}\lt\sqrt{b})$
But here the $0\le\sqrt{a}$ is like an assumption or a definition
So the proof of $0\le\sqrt{a}$ is ignored
Could we add these extra assumptions or definitions in the proof ... |
In writing a proof
My example would be
$(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$
The goal is $(0\le\sqrt{a}\lt\sqrt{b})$
But here the $0\le\sqrt{a}$ is like an assumption or a definition
So the proof of $0\le\sqrt{a}$ is ignored
Could we add these extra assumptions or definitions in the proof ... |
*Correct me if I'm using incorrect terms, I'm not well-versed with geometrical terminology*
I'm writing a code in which
I have a point &
I want to identify if the points lies on front, back, top, bottom, left or right side of a cuboid.
For Example,
in the 1st image, the point lies on 'top' side of the cu... |
In case you’d want to know: I’m a 6th grade student and I am self-learning probability (that’s one of the things).
I know Bayes’ theorem:
$$ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} $$
Here’s an example:
$A$: The chance you get sick, which could be $P(A) = 0.000219$.
$B$: The chance you test negative,... |
How to sum the series $$\frac{C_0}{1.3}+\frac{C_1}{2.4}+...+\frac{C_n}{(n+1).(n+3)}$$ where $c_i$ are the usual binomial coefficients in expansion of $(1+x)^n$.
I know how to sum such series when the denominator has consecutive multiplicands (by repeatedly integrating the expansion of $(1+x)^n$).
Does there even ... |
Could anyone tell me what this transformation looks like?
$T: [0,1)\to [0,1), T(x)= 10x\quad (\text{mod} \quad 1)$
If $x=0.31$ then $T(x)=3.1$, now modulo $1$ means what? Thank you. |
Hint:
$$\dfrac{\binom nr}{(r+1)(r+2)}=\dfrac{n!}{(n-r)! r!(r+1)(r+2)}=\dfrac1{(n+1)(n+2)}\cdot\dfrac{(n+2)!}{(r+2)!(n+2-(r+2))!}=?$$ |
How to find $P(A|B)$ when we only know $P(A)$ and $P(B)$? |
(1) "Given the success of ChatGpt in dealing with human natural language" ??
Maybe Chatgpt can itself answer this question , no humans needed here !!
(2) Currently , Chatgpt is a little toy with no understanding of logic & mathematics.
What output it gives is not mathematically valid beyond what inputs it ... |
Problem. There's $n$ coins, $n-1$ are fair and one always shows heads, but all coins look the same. You are allowed to flip $m$ coins, where $m < n$ and the choice of the $i$th flip can depend on previous outcomes. After $m$ flips you give a bet on which coin you think is special. What strategy should be used to maximi... |
Hint:
$$\dfrac{\binom nr}{(r+1)(r+3)}=\cdots=\dfrac1{(n+1)(n+2)(n+3)}\cdot\dfrac{(n+3)!(r+2)}{(r+3)!(n+3-(r+3))!}=?$$
Again,$\displaystyle\dfrac{(n+3)!(r+2)}{(r+3)!(n+3-(r+3))!}=\dfrac{(n+3)!(r+3-1)}{(r+3)!(n+3-(r+3))!}=\cdots=(n+3)\binom{n+2}{r+2}-\binom{n+3}{r+3}$ |
How to sum the sequence $$\frac{C_0}{1.3}+\frac{C_1}{2.4}+...+\frac{C_n}{(n+1).(n+3)}$$ where $c_i$ are the usual binomial coefficients in expansion of $(1+x)^n$.
I know how to sum such sequence when the denominator has consecutive multiplicands (by repeatedly integrating the expansion of $(1+x)^n$).
Does there e... |
As far as I remember, The proof of Perelman of the Geometerization Conjecture took a year to check by the experts.
The claimed proof of the ABC conjecture Mochizuki took few years to check, and Peter Scholze another great mathematician rejected the proof while Mochizuki who is also another respected mathematician st... |
How to sum the sequence $$\sum_{k=0}^n \frac{\binom n k}{(k+1)(k+3)} $$
where $n \choose k$ are the usual binomial coefficients in expansion of $(1+x)^n$.
I know how to sum such sequence when the denominator has consecutive multiplicands (by repeatedly integrating the expansion of $(1+x)^n$).
Does there even exi... |
Hint:
Let $\displaystyle\dfrac{\binom nr}{(r+1)(r+3)}=a\binom{n+3}{r+3}+b\binom{n+2}{r+2}+c\binom{n+1}{r+1}$ where $a,b,c$ are arbitrary constants
On simplification,
$$\dfrac1{(r+1)(r+3)}=\dfrac{a(n+3)(n+2)(n+1)}{(r+3)(r+2)(r+1)}+\dfrac{b(n+2)(n+1)}{(r+2)(r+1)}+\dfrac{c(n+1)}{r+1}$$
$$\iff r+2=a(n+3)(n+2... |
Let $A(0,0,0),$ $B(0,b,0),$ $C(a,b,0),$ $D(a,0,0)$ and $V(x,y,z)$. Then, the perimeter condition $VA+VD=VB+VC$ and the area condition $||VA\times VD||=||VB\times VC||$ give
$$\sqrt{x^2+y^2+z^2}+\sqrt{(x-a)^2+y^2+z^2}=\sqrt{x^2+(y-b)^2+z^2}+\sqrt{(x-a)^2+(y-b)^2+z^2}$$
and
$$y=\frac b2 ?.$$ |
**Let f and g be functions such that for all real numbers x and y,
(( + )) = () + ( + )() . The value of (0) + (1)+. . . (2024) is?**
My attempt:
When x=0 and y=0 then, g(f(0))=f(0)..........(i).
When x=0 and y=1 then, g(f(1))=f(0)+g(1)
or, g(f(1)) =g(f(0))+g(1) [because f(0)=g(f(0)) fr... |
# Question
I'm exploring the interplay between probability distributions and entropy, focusing on finding a distribution that maximizes entropy under a specific constraint. The core of my question involves the sigmoid function, defined as $\sigma(x) = \frac{1}{1 + e^{-x}}$, which is widely used in various fields suc... |
I have learned the component formula for the scalar product for quite a while, but really, it doesn't seem intuitive at all. I have 2 questions that confuse me while imagining what the scalar product really is, which are:
1- Why is the scalar product defined as $\vec{u} . \vec{v} = \sum a_{i}b_{i}$ (It doesn't make ... |
Background
---
So I was watching this [youtube video][1]. It poses a problem solve:
$$ 2^x = x^4$$
My Method
---
$$2^x = x^2 x^2$$
Or:
$$ 2^{x/2} 2^{x/2} = x^2 x^2$$
By prime factorization theorem and symmetry:
$$x/2 = 2$$
Extending this method to other problems
---
We only restrict ourselves t... |
Applications of this factorization method? |
Meaning of $\,r\bmod 1?\ $ [circle group $\Bbb R/\Bbb Z = $ reals $\!\bmod 1$] |
Background
---
So I was watching this [youtube video][1]. It poses a problem solve:
$$ 2^x = x^4$$
My Method
---
$$2^x = x^2 x^2$$
Or:
$$ 2^{x/2} 2^{x/2} = x^2 x^2$$
By prime factorization theorem:
$$x/2 = 2$$
Extending this method to other problems
---
We only restrict ourselves to natural num... |
Claim: If for any $\epsilon >0$, there exists a $\delta >0$, such that for any $|x-x_0| < \delta$, it holds
$$
(L-\epsilon)^k \leq f(x) \leq (L+\epsilon)^k,
$$
where $k$ is a positive constant. Then we have
$$
\lim_{x \rightarrow x_0} f(x) =L^k.
$$
-------------------------------------
The following i... |
Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be a $2 \times 2$ matrix with integer entries and $\det(A) = \pm 1$. Consider the toral automorphism defined on 2-torus $f_A : T^2 \to T^2$,
$$f_A (x, y) = (a x + b y \;\text{mod}\;1, c x + d y \;\text{mod}\; 1).$$
I don't understand what does it mean $a x + b y ... |
I ask this question because I probably meet the same obstacle when writing other proofs.
In writing a proof
My example would be
$(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$
The goal is $(0\le\sqrt{a}\lt\sqrt{b})$
But here the $0\le\sqrt{a}$ is like an assumption or a definition
So the proof of $0... |
Background
---
So I was watching this [youtube video][1]. It poses a problem solve:
$$ 2^x = x^4$$
My Method
---
$$2^x = x^2 x^2$$
Or:
$$ 2^{x/2} 2^{x/2} = x^2 x^2$$
By prime factorization theorem and symmetry:
$$x/2 = 2$$
Extending this method to other problems
---
We only restrict ourselves t... |
Let $f$ and $g$ be real nonzero polynomials with *nonnegative* coefficients such that
- the degree of $g$ equals the degree of $f$ plus one, that is $\deg(f+g) = \deg(f) + 1$,
- $(f+g)(0) \neq 0$.
Let
$$
h = \frac{f}{f+g}.
$$
In particular, it follows that $h$ is well defined on the interval $[0,+\infty)$, and... |
Solve for $n$ in $4\log_2(n)=n,\,$ i.e. $\,n^4 = 2^n$? |
Here is Exercise I.2 of R. Friedman's Algebraic Surfaces and Holomorphic Vector Bundles.
>Let $X$ be a smooth surface in $\mathbb{P}^3$ of degree $d$. Show using standard facts about the cohomology of $\mathbb{P}^3$ that $q(X)=0$. ... Using the fact that $K_X=\mathcal{O}_X(d-4)$ by adjunction and the fact that $q(X)... |
Is there a closed-form solution for the recurrence relation $x_{t+1} = x_{t}^\alpha + 1$? If it were $x_{t+1} = x_{t}^\alpha$, then taking the Log of both sides would make it linear and easy to solve. However, the same thing does not work in the example.
|
Let $S_6$ be equipped with the natural action on $\{1,2,\dots,6\}$. For $1\leq i,j\leq 6$, define subsets:
$$X_{j\to i}:=\{\sigma\in G\,\mid \sigma(j)=i\}.$$
Let $X:=\{X_{j\to i}\mid 1\leq i,j\leq 6\}$. Suppose that $\psi\in \mathrm{Aut}(S_6)$. Consider the set:
$$\psi X_{j\to i}:=\{\psi(\sigma)\mid\sigma\in X_{j\to... |
Not decreasing sequence that converges to 0 : can the sequence of the indexes for which the sequence decreases be bounded? |
Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Suppose we use the notation $a_{i}$ for the $i$-th row (or column, which would be the same given its symmetry), with $i\in\{1, ... , n\}$.
Suppose, in addition, that $A$ has the property that $\Vert a_i \Vert_2^2 < 1$, for every $i$.
I have the following que... |
Question:
>Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any non-constant sequence?
**Motivation.**
Suppose $X$ is a topological vector space over $\mathbb{R}$ or $\mathbb{C}$. If we define a *sequentially separated* set $S$ of... |
Let $(X, \mathcal{A},\mu)$ be a measure space and $f : X → [0,∞]$ be measurable. Show that there exist measurable sets $A_k, k = 1,2,...$ such that
$$f =
\sum_{k = 1}^{\infty} \frac{1}{k} \chi_{A_k}$$
Where $\chi_{A_k}(x) = 1$ if $x \in A_k$ and $0$ otherwise.
How do I even start to proceed with this problem? ... |
## Context
I am delving into the study of probability distributions and their entropy, with a particular focus on identifying a distribution that maximizes entropy under specific constraints. An area of interest is the application of the sigmoid function, widely recognized in logistic regression and neural networks,... |
Question:
>Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any eventually non-constant sequence?
**Motivation.**
Suppose $X$ is a topological vector space over $\mathbb{R}$ or $\mathbb{C}$. If we define a *sequentially separated*... |
## Context
I am delving into the study of probability distributions and their entropy, with a particular focus on identifying a distribution that maximizes entropy under specific constraints. An area of interest is the application of the sigmoid function, widely recognized in logistic regression and neural networks,... |
I'm currently looking at an old exam and I have encountered the following task:
$ A=\left( \begin{array}{rrr}
2 & -1 \\
-1 & 2 \\
\end{array}\right)$ $b=\left( \begin{array}{rrr}
1 \\
1 \\
\end{array}\right) $
Alongside this there's an [image][1] with two coordinate systems and contour lines. The exact ... |
Let $G$ be a finite group acting freely on a smooth projective complex variety $X$. This induces an action on $\mathrm{Coh}(X)$ by pullback. Let $\pi\colon X \rightarrow Y=X/G$ denote the quotient.
How do I prove the following claim?
**Claim** $\pi_\ast \mathcal{O}_X$ splits as a direct sum according to the irred... |
i need to prove that there is no sequence $(a_n)_{n=1}^\infty$ that the set of its partial limits is [3,4) $\cup$ [5,6], now i think i should be doing it by contradiction where i assume such sequence does exist, but i don't know how to tackle that, like where do i start?
i tried assuming for two subsequences $a_{n_k}$... |
Let's start with question 2, because it's easier.
In one sense, an angle between two line segments in higher dimensions does make sense, because the two line segments (typically) form a two-dimensional plane. We can always embed a plane in $\Bbb{R}^n$ (or any real inner product space) into $\Bbb{R}^2$ isometrically,... |
I need to prove that there is no sequence $(a_n)_{n=1}^\infty$ such that the set of its partial limits is $[3, 4) \cup [5, 6]$, now I think I should be doing it by contradiction where I assume such sequence does exist, but I don't know how to tackle that, like where do I start?
I tried assuming for two subsequences $a... |
Question:
>Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any eventually non-constant sequence?
**Motivation.**
Suppose $X$ is a topological vector space over $\mathbb{R}$ or $\mathbb{C}$. If we define a *sequentially separated*... |
Conjectures About Primes: Things We *Don't Know*
===
In general, a conjecture is a statement that is unproven, but believed by many mathematicians, perhaps even a vast majority of them, to be true--though this is not always the case. In some cases, disproving one of these conjectures would disrupt a lot of other ma... |
I'm currently looking at an old exam and I have encountered the following task:
$ A=\left( \begin{array}{rrr}
2 & -1 \\
-1 & 2 \\
\end{array}\right)$ $b=\left( \begin{array}{rrr}
1 \\
1 \\
\end{array}\right) $
Alongside this there's an [image][1] with a coordinate system and contour lines. The exact sol... |
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