Newest Questions
1,695,436 questions
-2
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14
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How do I find the missing values? [closed]
I encountered this question on Khan Academy link: [https://www.khanacademy.org/math/statistics-probability/analyzing-categorical-data/two-way-tables-for-categorical-data/v/analyzing-trends-categorical-...
1
vote
0
answers
10
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Straightedge and compass construction: $B,C$ from $A,I,G$
I have been using this question as a benchmark for AIs: so far, they've been awful.
The position of the vertex $A$, the incenter $I$ and the centroid $G$ of a triangle are known. Prove that it is ...
- 370k
0
votes
0
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13
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Justify whether this complex simultaneous equation is correct
Given:
$z$ and $w$ are complex numbers
$$z+iw=3+4i$$
$$\bar z-w=2+6i$$
I substitute $z=a+bi$, $w=x+yi$, $\bar z =a-bi$
So,
$$a-y=3 (1)$$
$$b+x=4(2)$$
$$a-x=2(3)$$
$$-b-y=6(4)$$
After eliminate $(2)+(3)...
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1
vote
0
answers
10
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Are simplicial abelian groups a coalgebra?
Given a topological space $X$, then $AW:C_*(X)\to C_*(X)\otimes C_*(X)$ makes $C_*(X)$ into a coalgebra. Given $\sigma\in C_n(X)$Where $AW$ is defined as
\begin{align*}
AW(\sigma)&=\sum_{k=0}^{n}(\...
- 91
0
votes
0
answers
9
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Definable subsets of a random graph are random graphs?
I have to prove this statement:
Let $A\subseteq N\vDash T_{rg}$ where $T_{rg}$ is the theory of random graphs in the language $L=\{r\}$.
For every $\varphi(x) \in L(A)$, if $\varphi$ is consistent ...
- 133
0
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0
answers
10
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Prove that the zig-zag product of $G$ and $H$ (where $H$ is the smaller of the two) lifts $H^2$
Prove that the zig-zag product of $G$ and $H$ (where $H$ is the smaller of the two) lifts $H^2$.
I was reading Expander Graphs and their Applications (Lecture notes for a course by Nati Linial and ...
- 1,801
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answers
30
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0
votes
0
answers
16
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Is the Mandelbrot set a projection of a 3D object?
Is the Mandelbrot set a projection of a 3D object? Some of the images I have seen look like there is depth if you allow yourself to look at it that way. See attached image for example. Some views look ...
- 382
0
votes
1
answer
17
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How to apply Frobenius method to solve differential equation with non constant exponential coefficients?
I have a differential equation with non constant exponential coefficients say:
$$e^{-x^2}\frac{\partial^2 X(x)}{\partial x^2}+(9x^2-7)e^{-x^2}X(x)-9X(x)=0$$
I am using Frobenius method of solving ...
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0
votes
0
answers
6
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Meaning of the discriminant of a general Jacobian
Let's say I have two functions: $y(a,b)$ and $z(a,b)$, and I use the following Jacobian:
$J=\left(\begin{array}{cc}\frac{\partial y}{\partial a} & \frac{\partial y}{\partial b} \\ { \frac{\partial ...
- 277
0
votes
1
answer
23
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Radially outward tangent of polar curve
The slope of a polar function $f$ is generally given by the formula $f'(x) = \frac{f'(\theta)\sin(\theta) + f(\theta)\cos(\theta)}{-f(\theta)\sin(\theta) + f'(\theta)\cos(\theta)}$ However, the rate ...
- 393
0
votes
0
answers
6
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Image of the transgression map and Hochschild-Serre sequence
Let $1\longrightarrow M\longrightarrow H\longrightarrow G\longrightarrow 1$ be a short exact sequence of finite groups, where $M$ is cyclic normal subgroup of $H$. Let $\mathbb{F}$ be a field of ...
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-1
votes
1
answer
35
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What is an example of an infinite player game in game theory? [closed]
I've heard about infinite player games in a book I'm reading and also there's a heading in the zbmath classification scheme about them. I'm not talking about infinite games, i.e, games that never end. ...
-1
votes
1
answer
78
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Why do we get natural log on $\int \frac {1}{x} dx $
I was wondering why we always get natural log in general in maths, especially calculus.
The best example is obviously $\int \frac{1}{x} dx = ln(x) + c$
Why is the base e? Is it just an arbitrary value ...
0
votes
0
answers
8
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Are finite extensions of number rings always finite after localization at primes?
Let $K/L$ be a finite extension of number fields and $A/B$ the corresponding extension of number rings. Let $\mathfrak{p} \subset A$ be a (nonzero) prime ideal and $\mathfrak{q} \subset B$ a prime ...
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1
vote
1
answer
33
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Generalized eigenspace of a derivation in an associative algebra
As an associative algebra I just mean a vector space $V$ with a bilinear form $a\cdot b$ for $a,b \in V$, and by a derivation I mean a linear function $\delta: V \rightarrow V$ such that $\delta (a \...
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0
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11
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I have a question about Hartshorne Chapter 2 Proposition 7.2.
Proposition7.2.
Let $\phi : X \rightarrow \mathbb P_A^n$ be a morphism of scheme over $A$, corresponding to an invertible sheaf $\mathcal L$ on $X$ and sections $s_0,...,s_n \in \Gamma(X,\mathcal L)$ ...
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1
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0
answers
10
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transcendence of logarithms and exponentials in differential fields
Let $(K,D)$ be a differential field and $t\notin K$ either a logarithm ($t'=a'/a$) for $a\in K$ with $a\neq0$ or a exponential ($t'/t=a'$) for $a\in K^*$.
Why is $t$ transcendental over $K$?
I started ...
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-1
votes
0
answers
12
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finding a compact connected orientable $2$-manifold of arbitrary Euler characteristic
I was reading the following question here Finding compact connected orientable 3-manifolds of arbitrary Euler characteristic and I was wondering, what will be the difference in the solution of this ...
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-6
votes
1
answer
67
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Conjecture: $\sum_{k=1}^n\frac{1}{k^p} \neq \pi$, for any finite natural $n$ and any complex $p$ [closed]
Conjecture about finite sums of $\frac1{k^p}$ and $\pi$
Let $S_n(p) = \sum_{k=1}^n\frac{1}{k^p}$, where $n \in \mathbb{N}$ and $p \in \mathbb{C}$.
Conjecture (Frenzy, 2025):
For any finite $n$ and ...
0
votes
1
answer
34
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Does the absolute convergence of the Fourier series imply continuity? [duplicate]
Suppose that $f$ is a function on the torus $\mathbb T$ such that its Fourier series is absolutely convergent, i.e. $\sum_{n \in \mathbb Z} |\hat{f}(n)| < \infty$, where $\hat{f}(n)$ is the $n$-th ...
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-4
votes
0
answers
44
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Find flaws in my logic as to why the collatz conjecture is unsolvable [closed]
Solving the collatz conjecture would require being given an iterative piecewise function and determining if it ever returns the same value, otherwise known as looping. Adding a piecewise to check if ...
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votes
0
answers
15
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$T:=\{x\in S: f(x)\leq a_i\text{ for all }i\in\{1,\dots,m\}\}$. $\{x\in S: f(x)<a_i\text{ for all }i\in\{1,\dots,m\}\}=\operatorname{Int}T$ hold? [closed]
I am reading "Analysis on Manifolds" by James R. Munkres.
Let $d$ denote the sup metric $d(x,y)=|x-y|$ on $\mathbb{R}^n$.
Let $D_N:=\left\{x: d(x,B)\geq\frac{1}{N}\,\,\text{ and }\,\,d(x,0)...
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1
vote
0
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18
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Unorthodox implementation for a recursive identification method
Recently I realized that I've coded a custom-made function for recursive output error method which differs a bit from the traditional algorithm and would like to know the perception of my peers. ...
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0
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0
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23
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When will $\chi(X) = 0$?
I was reading the solution of the following question here:
How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?
Here is the solution:
Lets suppose that ...
- 59
0
votes
0
answers
13
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Can an XRF machine reproduce results by a certified laboratory which uses a different method entirely [closed]
I have a project. The aim of the project is determine as to whether a portable xrf machine can reproduce laboratory assay results. The laboratory uses a different method called four acid digestion ...
1
vote
1
answer
98
views
Is there a name for the operator $⊞$ defined as $a ⊞ b = \sqrt{a^2 + b^2}$?
It's not uncommon to have a meaningful relationship c² = a² + b². This happens in both geometry (Pythagorean theorem) and statistics (sources of variance) at least, and I think I've seen it elsewhere. ...
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3
votes
0
answers
29
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Representation preserves center if irreducible and primitive
Suppose $\rho:G \to \operatorname{GL}(V)$ is a continuous representation for $G$ a topological group and $V$ a finite dimensional complex vector space. $G$ is not necessarily finite, and I don't ...
- 474
1
vote
0
answers
17
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Signs of de Rham cycle class maps
For a general smooth complex variety $X$, I know two ways to define de Rham cycle classes with supports for subvarieties $Y\subseteq X$.
The theory of currents.
Currents define a quasi-isomorphic ...
- 425
0
votes
0
answers
13
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Angle between inner normals of a convex domain and its gradient image
Problem:
Let $u: U \to \mathbb{R}$ be a $C^1$ convex function defined on an convex set
$U \subset \mathbb{R}^n$, satisfying
$$
x \in \partial U, \quad Du(x) = y.
$$
Assume that the gradient image $V :=...
- 1
3
votes
1
answer
27
views
A purely periodic sequence must satisfy a homogenous linear recurrence?
Assume that $(a_n)_{n \in \mathbb{N}}$ is an integer sequence such that for any $k \in \mathbb{N}$ there exists an $m(k)$ such that $a_n \equiv a_{n+m(k)} \pmod{k}$ for all $n \in \mathbb{N}$. Is it ...
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0
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0
answers
24
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Dirichlet Characters modulo 15
I'm trying to determine all Dirichlet Characters modulo 15. I know how to determine Dirichlet Characters mod 3 and mod 5, and CRT tells us that $\mathbb Z_{15}^* \cong \mathbb Z_{3}^* \times \mathbb ...
1
vote
0
answers
50
views
What is the $\lim_{k \to \infty} \sum_{i=a^k}^{a^k + a^{k-1} - 1} \frac{1}{i}$ in terms of $a$?
I am exploring the limit of a specific sum involving segments of the harmonic series. The expression is defined for any natural number $a \geq 1$.
What is the value of the following limit in terms of $...
0
votes
0
answers
32
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Accelerating reference frame in a traveling wave
I am working with a traveling wave model common in population genetics (see, for example, Neher and Hallatschek 2012). In these we model the evolution of a 1D fitness ‘wave’ as a population ...
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1
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0
answers
27
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Is there a mistake in the Mayer-Vietoris sequence in the following solution?
I was reading the solution of the following question here:
How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?
Here is the solution:
Lets suppose that ...
- 59
2
votes
0
answers
28
views
Compare a weak topology with the Gelfand topology for the algebra of bounded weakly continuous functions [closed]
Let (for simplicity) $H$ be a separable Hilbert space and $A$ be the Banach algebra of bounded weakly continuous functions of $H$. Then
The Gelfand space of $A$, the space of maximal ideals, can be ...
- 21
1
vote
2
answers
73
views
Generalized associativity law in groups [closed]
I really am sorry because I think I am going to ask something too obvious but I cannot wrap my head around it.
I was trying to prove the general associativity rule in groups for any $n$ elements of a ...
- 11
0
votes
1
answer
55
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Combination of five balls [closed]
Combination of five balls
A basket contains five identical balls except for their colors.
2 White balls and 3 Black balls.
How many different ways are there to get 3 balls at once?
How many different ...
- 6,274
0
votes
0
answers
19
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Determinant-like identities for noncommuting operator matrices
Let $V$ be a finite dimensional vector space and $M_{i,j}\in \text{End}(V), \ i,j\in\{1,...,n\}$ are not necessarily commuting operators. Then are the following equality holds?
$$
\frac{1}{n!}\sum_{\...
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4
votes
6
answers
117
views
Maximizing $\sqrt{x^2+4}+\sqrt{(3-x)^2+4}$ for $0\leq x \leq 3$
What is the maximum of $y$ if $0\leq x \leq 3$? $$y=\sqrt{x^2+4}+\sqrt{(3-x)^2+4}$$
MY ATTEMPT
Since $$0\leq x \leq 3$$
$$0\leq x^2 \leq 9$$
$$4\leq x^2+4 \leq 13$$
$$2\leq \sqrt{x^2+4} \leq \sqrt{13}...
- 6,274
1
vote
0
answers
18
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Finding asymptotic lines
I'm studying a problem with negatively curved surfaces ($K<0$) and in a reference (ch. 10) it is claimed that, since the asymptotic lines are defined everywhere and non-intersecting, it's natural ...
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0
votes
1
answer
23
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Hewitt Savage zero-one law, what is event invariant under any finite permutation of the $(X_i)_{i\ge 1}$?
I found two versions of the HS zero-one law. (First in Kallenberg's book, second in the lecture notes of Nicolas Curien here, page 23 )
Let $X_i$ be a sequence of iid rvs on $(\Omega, F,P)$ with law $...
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-1
votes
0
answers
55
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Wait, are gaps between primes bounded or are they not? [closed]
Wikipedia says it's been proven that the gap after a prime $p$ is at most $p^{0.525}$. But Ford et al write: "In 1931, Westzynthius [46] proved that infinitely often, the gap between consecutive ...
- 151
0
votes
0
answers
20
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Polarizations and totally positive endomorphisms
Let $(A, \lambda)$ be a polarized abelian variety (over some field $k$), and write $\alpha \mapsto \alpha^{\dagger}$ for the associated Rosati involution on $\operatorname{End}^0(A) := \operatorname{...
- 584
0
votes
0
answers
12
views
Examples of Related Categorical Variables Without Causality [closed]
I want to find a pair of categorical variables that are related but do not have a cause-and-effect relationship. Are there any 'well-known' examples of this?
- 11
0
votes
1
answer
47
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for a family of functions f, can we argue about the maximum of their derivatives if they are not necessarily differentiable?
I was working through the question below and my solution ended up as such:
Question:
Let f be a continuous function defined on $[0, 1]$ such that
$f (0) =f(1) = 1$ and $ | f (a) - f (b)| < | a - b|,...
3
votes
0
answers
83
views
A challenging integral involving an algebraic function
Inspired by this question, I've come up with an exercise that I hope you will find amusing:
Show that
$$\begin{eqnarray*} &&\int_{0}^{\sqrt[3]{2}}\left[u^3\sqrt[3]{27+u^6+\sqrt{27}\sqrt{27+2u^...
- 370k
1
vote
0
answers
24
views
Placing hexagonal experimental plots (how to include all pattern combinations using a minimum number of tiles) [closed]
I'm trying to design an intercropping experiment testing how crops perform when surrounded by crops of the same or different species. I have three crops I'm testing in various combinations.
The ...
- 19
-1
votes
0
answers
20
views
Can this approach of numerical simulation can be used as possible proof of differential equation solution's qualitative view? [closed]
I found the possible differential equation's solution view using hypothesis and numerical simulation for check. Does this approach can be used as possible proof?
Short conspect of research:
Page 1
...
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2
votes
2
answers
58
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How to determine the universal $C^*$algebra generated by $x^*=x^2=x$?
First, we recall the following lemma which helps us to determine the universal $C^*$algebras.
Lemma Suppose $\{a_g\}$ is a representation of $R$ in a $C^*$algebra $A$ such that :
the set $\{a_g, g\in ...
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