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url string | text string | date timestamp[s] | meta dict |
|---|---|---|---|
https://openstax.org/books/college-algebra-corequisite-support-2e/pages/6-review-exercises | College Algebra with Corequisite Support 2e
# Review Exercises
### Review Exercises
##### Exponential Functions
1 .
Determine whether the function $y=156 ( 0.825 ) t y=156 ( 0.825 ) t$ represents exponential growth, exponential decay, or neither. Explain
2 .
The population of a herd of deer is represented by the ... | 2022-01-24T22:06:17 | {
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https://sabbiu.com/p-norms-taxicab-norm-euclidean-norm-and-infinity-norm/ | Sabbiu Shah
p-Norms (Taxicab Norm, Euclidean Norm and infinity-norm)
Norms are a measure of distance. Norm is defined as follows: For $p\geq1$,
$||x||_p \equiv \sqrt[p]{|x_1|^p + |x_2|^p + ... + |x_n|^p}$
Taxicab Norm (1-Norm)
When $p=1$, then the norm is said to be taxicab norm. The distance derived from this nor... | 2021-10-20T00:50:48 | {
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https://encyclopediaofmath.org/wiki/Difference_equation | # Difference equation
An equation containing finite differences of an unknown function. Let $y ( n) = y _ {n}$ be a function depending on an integer argument $n = 0, \pm 1, \pm 2 , . . .$; let
$$\Delta y _ {n} = \ y _ {n + 1 } - y _ {n} ,\ \ \Delta ^ {m + 1 } y _ {n} = \ \Delta ( \Delta ^ {m} y _ {n} ),$$
$$\Delta ^... | 2022-06-27T08:03:10 | {
"domain": "encyclopediaofmath.org",
"url": "https://encyclopediaofmath.org/wiki/Difference_equation",
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https://www.cut-the-knot.org/m/Algebra/MarghidanuWithGeneralization.shtml | # Dorin Marghidanu's Inequality with Generalization
### Problem
Let $a,b\in\mathbb{R},\,$ $a\lt b,\,$ and $x,y,z\in [a,b].\,$ Prove that
$\displaystyle (x+y)^2+(y+z)^2+(z+x)^2+12ab\le 4(a+b)(x+y+z).$
Where does equality hold? Generalize!
### Solution 1
For $u,v\in [a,b],\,$ $(u+v-2a)(u+v-2b)\le 0,\,$ so that
(1)... | 2017-11-17T17:08:41 | {
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"lm_q2_sc... |
https://mattbaker.blog/2013/07/03/quadratic-reciprocity-and-zolotarevs-lemma/ | # Quadratic reciprocity and Zolotarev’s Lemma
I want to explain a beautiful proof of the Law of Quadratic Reciprocity from c. 1874 due to Egor Ivanovich Zolotarev (1847-1878). Some time ago I reformulated Zolotarev’s argument (as presented here) in terms of dealing cards and I posted a little note about it on my web p... | 2022-06-26T02:02:02 | {
"domain": "mattbaker.blog",
"url": "https://mattbaker.blog/2013/07/03/quadratic-reciprocity-and-zolotarevs-lemma/",
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"lm_q1_score": 0.9920620063679783,
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http://zbmath.org/?q=an:1119.53006&format=complete | # zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "strai... | 2013-12-12T19:02:39 | {
"domain": "zbmath.org",
"url": "http://zbmath.org/?q=an:1119.53006&format=complete",
"openwebmath_score": 0.7562640905380249,
"openwebmath_perplexity": 9657.388025247168,
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620061332854,
"lm_q2_score": 0.6584174938590246,
"... |
https://www.mathcounterexamples.net/a-diagonalizable-matrix-over-q-but-not-over-z/ | # A diagonalizable matrix over Q but not over Z
Let us recall that a square matrix $$A$$ is called diagonalizable if it is similar to a diagonal matrix, i.e. if there exists an invertible matrix $$P$$ such that $$P^{-1}AP$$ is a diagonal matrix.
It is well known that a square matrix of dimension $$n \ge 1$$ over a fi... | 2020-09-30T05:09:45 | {
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"url": "https://www.mathcounterexamples.net/a-diagonalizable-matrix-over-q-but-not-over-z/",
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https://online.stat.psu.edu/statprogram/book/export/html/574 | # M.4 Matrix Inverse
M.4 Matrix Inverse
Inverse of a Matrix
The matrix B is the inverse of matrix A if $AB = BA = I$. This is often denoted as $B = A^{-1}$ or $A = B^{-1}$. When taking the inverse of the product of two matrices A and B,
$(AB)^{-1} = B^{-1} A^{-1}$
When taking the determinate of the inverse of the m... | 2020-05-31T17:09:25 | {
"domain": "psu.edu",
"url": "https://online.stat.psu.edu/statprogram/book/export/html/574",
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https://en.wikipedia.org/wiki/257_(number) | 257 (number)
← 256 257 258 →
Cardinal two hundred fifty-seven
Ordinal 257th
(two hundred fifty-seventh)
Factorization prime
Prime yes
Greek numeral ΣΝΖ´
Roman numeral CCLVII
Binary 1000000012
Ternary 1001123
Quaternary 100014
Quinary 20125
Senary 11056
Octal 4018
Duodecimal 19512
Vigesimal CH20
Base 36 7536
257 (two... | 2018-03-19T09:17:26 | {
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"lm_q2_score": 0.6584174938590246,
"lm_q1... |
https://opentextbc.ca/intermediatealgebraberg/chapter/4-5-geometric-word-problems/ | Chapter 4: Inequalities
# 4.5 Geometric Word Problems
It is common to run into geometry-based word problems that look at either the interior angles, perimeter, or area of shapes. When looking at interior angles, the sum of the angles of any polygon can be found by taking the number of sides, subtracting 2, and then m... | 2023-01-27T11:51:08 | {
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"url": "https://opentextbc.ca/intermediatealgebraberg/chapter/4-5-geometric-word-problems/",
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"lm... |
https://phys.libretexts.org/TextBooks_and_TextMaps/Astronomy_and_Cosmology_TextMaps/Map%3A_Celestial_Mechanics_(Tatum)/2%3A_Conic_Sections/2.7%3A_The_General_Conic_Section | $$\require{cancel}$$
# 2.7: The General Conic Section
The equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \label{2.7.1} \tag{2.7.1}$
represents an ellipse whose major axis is along the $$x$$ axis and whose centre is at the origin of coordinates. But what if its centre is not at the origin, and if the major axis is a... | 2018-08-21T04:09:36 | {
"domain": "libretexts.org",
"url": "https://phys.libretexts.org/TextBooks_and_TextMaps/Astronomy_and_Cosmology_TextMaps/Map%3A_Celestial_Mechanics_(Tatum)/2%3A_Conic_Sections/2.7%3A_The_General_Conic_Section",
"openwebmath_score": 0.8829994201660156,
"openwebmath_perplexity": 209.50392004604868,
"lm_name": ... |
https://typeset.io/papers/a-note-on-independent-sets-in-trees-wtn5zzcnv7 | Journal ArticleDOI
# A note on independent sets in trees
01 Jan 1988-SIAM Journal on Discrete Mathematics (Society for Industrial and Applied Mathematics)-Vol. 1, Iss: 1, pp 105-108
TL;DR: A simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given b... | 2022-12-10T05:45:34 | {
"domain": "typeset.io",
"url": "https://typeset.io/papers/a-note-on-independent-sets-in-trees-wtn5zzcnv7",
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http://zbmath.org/?q=an:1117.65077 | # zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "strai... | 2014-04-17T15:40:35 | {
"domain": "zbmath.org",
"url": "http://zbmath.org/?q=an:1117.65077",
"openwebmath_score": 0.8438723087310791,
"openwebmath_perplexity": 4287.674141460762,
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"lm_q1_score": 0.9920620051945144,
"lm_q2_score": 0.6584174938590246,
"lm_q1q2_score": ... |
https://mathstrek.blog/2012/08/04/combinatorial-game-theory-ii/ | ## Lesson 2
In this lesson, we will focus on a special type of game called Nim. Although it’s only one out of infinitely many possible games, understanding it in depth will be very beneficial in analysing a much larger class of games, known as impartial games. (The exact definition of this term is deferred until a lat... | 2019-10-15T14:03:17 | {
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"lm_q1_score": 0.9920620047251287,
"lm_q2_score": 0.658... |
https://www.coursehero.com/sg/college-algebra/properties-of-equality/ | # Properties of Equality
The addition property of equality can be used to undo subtraction when solving equations.
Linear equations are the simplest types of equations that contain variables. In a linear equation in one variable, each term can be written as a number or a product of the variable and a number. The varia... | 2019-01-19T14:52:40 | {
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"url": "https://www.coursehero.com/sg/college-algebra/properties-of-equality/",
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"openwebmath_perplexity": 357.5886018251146,
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.992062004490436,
"lm_q2_score": 0... |
http://mathonline.wikidot.com/convergence-criterion-for-series-in-hilbert-spaces | Convergence Criterion for Series in Hilbert Spaces
# Convergence Criterion for Series in Hilbert Spaces
Recall from the Bessel's Inequality for Inner Product Spaces page that if $H$ is an inner product space and $(x_n)_{n=1}^{\infty}$ is an orthonormal sequence of points in $H$ then for every $y \in H$ we have that:
... | 2017-07-27T14:16:30 | {
"domain": "wikidot.com",
"url": "http://mathonline.wikidot.com/convergence-criterion-for-series-in-hilbert-spaces",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620043730896,
"lm_q2_... |
https://jeremykun.com/category/linear-algebra/ | The Inner Product as a Decision Rule
The standard inner product of two vectors has some nice geometric properties. Given two vectors $x, y \in \mathbb{R}^n$, where by $x_i$ I mean the $i$-th coordinate of $x$, the standard inner product (which I will interchangeably call the dot product) is defined by the formula
$\d... | 2017-05-29T01:57:44 | {
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"url": "https://jeremykun.com/category/linear-algebra/",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620068373637,
"lm_q2_score": 0.658417487156366,
"lm... |
https://freopen.com/math/2020/03/29/Box-Muller-Algorithm.html | # Box-Muller Algorithm
## Description
$x_1$ and $x_2$ are two independent $uniform(0,1)$ random variables, and let
$r = \sqrt{-2 \log{x_1}}, \quad \theta = 2 \pi x_2$
Then
$z_1 = r \cos{\theta}, \quad z_2 = r \sin{\theta}$
are independent $normal(0,1)$ random variabls.
## Proof
$p(z_1,z_2) = \frac{1}{2\pi} e^{\... | 2023-03-23T23:25:41 | {
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"url": "https://freopen.com/math/2020/03/29/Box-Muller-Algorithm.html",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.992062006602671,
"lm_q2_score": 0.6584174871... |
https://www.tungsteno.io/post/exp-change_of_variable/ | If $F'(x)=f(x)$, then $(F(g(x)))'=f(g(x))g'(x)$. Thus, the integral of a function of the type
is always solved as
It is very useful to denote this trick with a change of variable: let
If we differentiate both sides of the expression and keep track of variable with respect to which we are differentiating with $\mathr... | 2020-02-18T10:26:39 | {
"domain": "tungsteno.io",
"url": "https://www.tungsteno.io/post/exp-change_of_variable/",
"openwebmath_score": 0.9674426317214966,
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620061332854,
"lm_q2_score": 0.658417487156366... |
http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n | # Multiplicative group of integers modulo n
Not to be confused with Integers modulo n.
In modular arithmetic the set of congruence classes relatively prime to the modulus number, say n, form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive res... | 2014-07-25T18:39:59 | {
"domain": "wikipedia.org",
"url": "http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620058985926,
"lm_q2_score":... |
http://educ.jmu.edu/~waltondb/ModelCalculus/calculus-overview.html | # Section5.1An Overview of Calculus¶ permalink
When we studied sequences, we learned that the behavior of the sequence could be described in terms of its increments. (Recall 3.4.) The following summarizes what we learned about sequences.
##### Behavior of a Sequence Using Increments
Every sequence $(x_k)$ has a corr... | 2017-03-27T16:28:29 | {
"domain": "jmu.edu",
"url": "http://educ.jmu.edu/~waltondb/ModelCalculus/calculus-overview.html",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620042557432,
"lm_q2_score": 0.6584174... |
https://stats.stackexchange.com/questions/30858/how-to-calculate-cumulative-distribution-in-r/30860 | # How to calculate cumulative distribution in R?
I need to calculate the cumulative distribution function of a data sample.
Is there something similar to hist() in R that measure the cumulative density function?
I have tries ecdf() but i can't understand the logic.
## locked by whuber♦May 19 '17 at 12:35
This ques... | 2019-11-21T13:09:16 | {
"domain": "stackexchange.com",
"url": "https://stats.stackexchange.com/questions/30858/how-to-calculate-cumulative-distribution-in-r/30860",
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"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.... |
https://mathematica.stackexchange.com/questions/163331/how-would-i-use-mathematica-to-solve-this-equation | # How would I use mathematica to solve this equation?
I don't have much experience with solving equations using mathematica. I have the following equation:
$$A=u\cdot f^{-1}(u)-\int_a^{f^{-1}(u)}f(x)dx$$ For some given constant $A>0$ and $a\in[0,1]$, and given function $f$, with the following properties:
• $f(x)\geq... | 2019-07-18T03:27:47 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/163331/how-would-i-use-mathematica-to-solve-this-equation",
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"openwebmath_perplexity": 1525.1162041938808,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score"... |
https://mathematica.stackexchange.com/questions/51127/how-to-apply-compound-styles-to-curves-in-e-g-plot | # How to apply compound styles to curves in e.g. Plot
Say, I have a colorlist
listColor={Black,Brown,Red,Cyan}
Now, I have some Plot function that I can use this list:
Plot[{Sin[x],Cos[x],x,x^2},{x,1,100},PlotStyle->listColor]
Everything went fine. Now, I wanted to make the plot style "thick"
Plot[{Sin[x],Cos[x... | 2021-04-16T22:17:43 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/51127/how-to-apply-compound-styles-to-curves-in-e-g-plot",
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"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score":... |
https://gmatclub.com/forum/the-cost-price-of-m-articles-is-the-same-as-the-selling-price-of-n-262916.html | GMAT Changed on April 16th - Read about the latest changes here
It is currently 25 Apr 2018, 09:18
GMAT Club Daily Prep
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... | 2018-04-25T16:18:57 | {
"domain": "gmatclub.com",
"url": "https://gmatclub.com/forum/the-cost-price-of-m-articles-is-the-same-as-the-selling-price-of-n-262916.html",
"openwebmath_score": 0.4424494802951813,
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"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.... |
https://mathematica.stackexchange.com/questions/47449/how-do-i-make-the-color-of-arrowheads-in-a-graph-different-from-the-colors-of-th/157629 | # How do I make the color of arrowheads in a graph different from the colors of the arrows?
I'm graphing a Markov process
mp = DiscreteMarkovProcess[{1, 0, 0}, ({
{0.6, 0.1, 0.3},
{0.2, 0.7, 0.1},
{0.3, 0.3, 0.4}
})];
and would like to have arrows whose thicknesses corresponds to the transition probabilities, with ... | 2019-07-21T17:52:23 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/47449/how-do-i-make-the-color-of-arrowheads-in-a-graph-different-from-the-colors-of-th/157629",
"openwebmath_score": 0.23747175931930542,
"openwebmath_perplexity": 4429.338296535963,
"lm_name": "Qwen/Qwen-72B",
"lm_labe... |
https://gmatclub.com/forum/the-sum-of-the-first-k-positive-integers-is-equal-to-k-k-1-2-what-is-126289.html?fl=similar | It is currently 12 Dec 2017, 14:04
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
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we will pick new questions that match your level based o... | 2017-12-12T22:04:24 | {
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"url": "https://gmatclub.com/forum/the-sum-of-the-first-k-positive-integers-is-equal-to-k-k-1-2-what-is-126289.html?fl=similar",
"openwebmath_score": 0.5512520670890808,
"openwebmath_perplexity": 6500.427905585682,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes\n\n",
"... |
http://math.stackexchange.com/questions/106824/what-are-various-proofs-good-for?answertab=oldest | # What are various proofs good for?
There are plenty of questions around here, which are proven to be right or wrong in various ways. I wonder, what one can learn from these differing ways of how to prove something, despite the fact that: The more proofs, the better.
Let's say a statement is something like a way $A\t... | 2015-07-31T03:47:42 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/106824/what-are-various-proofs-good-for?answertab=oldest",
"openwebmath_score": 0.853007972240448,
"openwebmath_perplexity": 640.8261751897438,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.865224... |
https://forum.allaboutcircuits.com/threads/matlab-code-to-find-all-prime-numbers-between-2-and-5000.80722/ | # Matlab code to find all prime numbers between 2 and 5000
#### Dapnc
Joined Feb 7, 2013
3
Need to write a matlab program that will find all prime numbers between 2 and 5000 but I can't use any built in matlab functions. Please help. It also has to run quickly.
#### tshuck
Joined Oct 18, 2012
3,534
Need to write a ... | 2022-10-05T18:06:12 | {
"domain": "allaboutcircuits.com",
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"lm_q1_scor... |
https://gmatclub.com/forum/a-group-of-8-friends-want-to-play-doubles-tennis-how-many-55369-20.html | It is currently 18 Nov 2017, 07:15
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https://tex.stackexchange.com/questions/480294/ploting-the-roots-of-a-polynomial | # Ploting the roots of a polynomial
I want to plot the roots of any given polynomial on the complex plane.
Example: Let $P(x)=x^4-x^3-1$ be given. I want to plot on the $Oxy$ complex plane all four roots of this polynomial.
I suppose Tikz might be useful tool on this case, but I don't have experience on this package... | 2020-01-20T21:20:06 | {
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https://greprepclub.com/forum/if-x-0-and-two-sides-of-a-certain-triangle-2579.html | It is currently 14 Jul 2020, 10:35
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https://stats.stackexchange.com/questions/231285/dropping-one-of-the-columns-when-using-one-hot-encoding | # Dropping one of the columns when using one-hot encoding
My understanding is that in machine learning it can be a problem if your dataset has highly correlated features, as they effectively encode the same information.
Recently someone pointed out that when you do one-hot encoding on a categorical variable you end u... | 2019-06-16T23:17:57 | {
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https://tex.stackexchange.com/questions/227181/formulas-should-start-at-the-same-position | # Formulas should start at the same position
I am a very beginner of LaTeX and I do not have any kind of experience with it.
How can I make formulas start at the same position such that they are aligned exactly underneath each other.
\documentclass[oneside,12pt]{scrartcl}
\usepackage[ngerman]{babel}
\usepackage[fixa... | 2019-09-15T14:07:46 | {
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https://mathematica.stackexchange.com/questions/95473/fill-a-circle-in-an-image | # Fill a Circle in an Image
Let's suppose I have something like this (the image resulting from this):
ColorNegate[Rasterize[Graphics[Circle[{100, 100}, 50]]]]
Now, what I want to do is fill the inside of the circle (in the image, not transforming Circle->Disk :) ) with White (the color of the circle).
Ideas?
• Bi... | 2021-06-21T14:22:35 | {
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http://www.onemathematicalcat.org/Math/Geometry_obj/GeoGebra_tutorial/GeoGebra_basics.htm | GEOGEBRA TUTORIAL: GEOGEBRA BASICS
This tutorial firms up many of the concepts introduced in GeoGebra Worksheet: Triangles,
and also introduces some new features of GeoGebra.
This tutorial will be most helpful if you're trying everything in GeoGebra while you're reading.
Here's one way to do this:
• click on APPLET ... | 2015-07-28T17:42:30 | {
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"lm_q1_score": 0.86522... |
https://ellejae.com/archive/2caa12-show-that-satisfies-the-differential-equation-with-initial-condition | 0\\ Differential Equation Practice Problems With Solutions. In calculus, the term usually refers to the starting condition for finding the particular solution for a differential equation. How can I make an Android app "forget" that it installed on my phone before? where C is the constant to solve for using the 2nd cond... | 2022-10-02T05:57:44 | {
"domain": "ellejae.com",
"url": "https://ellejae.com/archive/2caa12-show-that-satisfies-the-differential-equation-with-initial-condition",
"openwebmath_score": 0.6854680180549622,
"openwebmath_perplexity": 1036.3032906210562,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0... |
https://tex.stackexchange.com/questions/131125/better-way-to-display-long-division | # Better way to display long division?
I'm currently in the process of trying to create a worksheet for my students with long division problems for them to practice. Unfortunately, the best I've been able to come up with so far in terms of displaying long division like how they write it is:
Which could work if need b... | 2022-05-20T06:37:54 | {
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https://mathematica.stackexchange.com/questions/linked/88312 | 27k views
### $\LaTeX$ and Mathematica
I quite often would like to draw graphics in my $\LaTeX$ documents using Mathematica. I have encountered three problems. I would like to know if there are any workarounds to these problems I would ...
5k views
### Python-style plots in Mathematica
I love making plots in Mathem... | 2019-12-13T00:30:10 | {
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"lm_q1_score": 0.8652240756264639,
"lm_q2_score": 0.754... |
https://brilliant.org/discussions/thread/pencil-logic-puzzles/ | ×
# Pencil logic puzzles
Do you know Sudoku? Or maybe Paint by Numbers (also known as Nonogram)? Sudoku and Nonogram are examples of the many kinds of puzzles also known as "pencil logic puzzles" (for the lack of better terms).
Pencil logic puzzles are puzzles that usually have a grid (so you usually don't need to s... | 2017-09-20T00:31:02 | {
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"url": "https://brilliant.org/discussions/thread/pencil-logic-puzzles/",
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"lm_q1_score": 0.8652240686758842,
"lm_q2_score": 0.754914... |
https://gmatclub.com/forum/the-arithmetic-mean-of-a-collection-of-5-positive-integers-not-necess-245617.html | GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 18 Nov 2018, 00:07
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https://gmatclub.com/forum/at-an-upscale-fast-food-restaurant-shin-can-buy-3-burgers-7-shakes-49205.html | GMAT Question of the Day - Daily to your Mailbox; hard ones only
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https://shapero.xyz/blog/index-1.html | # Conservation laws
A conservation law is a type of PDE that describes the transport of extensive quantities like mass, momentum, and energy. The most general form of a hyperbolic conservation law for a field $q$ is
$$\frac{\partial q}{\partial t} + \nabla\cdot f(q) = s,$$
where $f$ is the flux function and $s$ the ... | 2023-03-27T09:27:36 | {
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"lm_q1_score": 0.986777180969715,
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https://www.ask-math.com/greatest-integer-function.html | # Greatest Integer Function
The greatest integer function is also known as floor function or step function. It is written as f(x) = $[\![x ]\!]$
The value of $[\![x ]\!]$ is the largest integer that is less than or equal to x.
Example 1 : greatest integer of 2.7 = $[\![2.7 ]\!]$ So the largest integer is 2
So, $[\![2.... | 2021-09-26T00:28:04 | {
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https://www.gradesaver.com/textbooks/math/other-math/thinking-mathematically-6th-edition/chapter-5-number-theory-and-the-real-number-system-chapter-summary-review-and-test-review-exercises-page-336/136 | ## Thinking Mathematically (6th Edition)
$\frac{1}{2},\,\,\frac{1}{4},\,\,\frac{1}{8},\,\,\frac{1}{16},\,\,\frac{1}{32}\ \text{and}\ \frac{1}{64}$.
For the second term put $n=2$ in the general formula stated above. \begin{align} & {{a}_{2}}={{a}_{1}}{{r}^{2-1}} \\ & =\frac{1}{2}\cdot {{\left( \frac{1}{2} \right)}^{1}}... | 2019-12-05T22:22:44 | {
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"openwebmath_score": 1.0000100135803223,
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https://forum.azimuthproject.org/plugin/ViewComment/16682 | Boolean algebras are an important kind of poset. The power set functor defines an (contravariant) equivalence \$$P: \text{Set}^{\text{op}} \to \text{Bool}\$$: any function \$$f: X \to Y\$$ corresponds to the *preimage* map \$$f*: PY \to PX\$$, which is a monotone map, and also a boolean algebra homomorphism, meaning it... | 2020-01-20T14:16:14 | {
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771801919951,
"lm_q2_score": 0.66192... |
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-5-trigonometric-identities-section-5-2-verifying-trigonometric-identities-5-2-exercises-page-209/67 | ## Trigonometry (11th Edition) Clone
$$\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}=\sec^2\theta-\tan^2\theta$$ We simplify the left side and find that the expression is an identity.
$$\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}=\sec^2\theta-\tan^2\theta$$ The left side is more complicat... | 2019-10-14T02:14:01 | {
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"openwebmath_score": 0.9635627269744873,
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https://drexel28.wordpress.com/2010/06/09/munkres-chapter-2-section-19-part-ii/ | # Abstract Nonsense
## Munkres Chapter 2 Section 19 (Part II)
9.
Problem: Show that the axiom of choice (AOC) is equivalent to the statement that for any indexed family $\left\{U_{\alpha}\right\}_{\alpha\in\mathcal{A}}$ of non-empty sets, with $\mathcal{A}\ne\varnothing$, the product $\displaystyle \prod_{\alpha\in\... | 2017-05-24T15:42:16 | {
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"lm_q1_score": 0.986777180191995,
"lm_q2_... |
http://jde27.uk/lgla/30_adjoint_accessible.html | Before we classify the representations of SU(3), I want to introduce a representation which makes sense for any Lie group, the adjoint representation, and study it for SU(3).
Definition:
Given a matrix group G with Lie algebra little g, the adjoint representation is the homomorphism capital Ad from big G to big G L o... | 2022-10-06T08:06:07 | {
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"url": "http://jde27.uk/lgla/30_adjoint_accessible.html",
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"lm_q2_score": 0.66192288918838,
"lm_q1q2... |
http://mathhelpforum.com/calculus/47975-approximation-partial-derivative.html | # Math Help - An Approximation to a partial Derivative
1. ## An Approximation to a partial Derivative
An Approximation to a partial Derivative
If a function is known to have
fx(30,24) = -.4
fy(30,24) = 1.2
f(30,24) = 50
Estimate this value of F.
f(30.2,23.9)
Thank you for helping me. I am stuck on the last part
2... | 2015-07-06T15:42:09 | {
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https://jaketae.github.io/study/stieltjes/ | # On Expectations and Integrals
Expectation is a core concept in statistics, and it is no surprise that any student interested in probability and statistics may have seen some expression like this:
$\mathbb{E}[X] = \sum_{x \in X} x f(x) \tag{1}$
In the continuous case, the expression is most commonly presented in te... | 2022-06-26T01:22:02 | {
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"lm_q2_score": 0.66192288918838,
"lm_q1q2_sco... |
https://www.gmatmath.online/fractions-percentages/ | # Fractions / Percentages
In doing problems that involve percentages, whether or not they also involve fractions, it’s important to remember that percentages are just fractions. For example, 87% is the same as 87/100, or, written as a decimal, it’s .87. When we do a problem that includes both fractions and percentages... | 2022-05-25T07:12:33 | {
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"lm_q1_score": 0.986777179025415,
"lm_q2_score": 0.66192288918838,... |
https://gmachine1729.livejournal.com/2021/06/06/ | ## June 6th, 2021
### On the chain rule and change of variables of integrals
Originally published at 狗和留美者不得入内. You can comment here or there.
Theorem 1 (Chain rule) Let , , where and are open in , such that are differentiable on their respective domains. Then is also differentiable on , with for all .
Proof: We fi... | 2021-10-28T05:18:26 | {
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https://en.m.wikibooks.org/wiki/Real_Analysis/Darboux_Integral | # Real Analysis/Darboux Integral
Real AnalysisDarboux Integral
Another popular definition of "integration" was provided by Jean Gaston Darboux and is often used in more advanced texts, such as this wikibook, due to its introductory ease. In this chapter, we will define the Darboux integral, and demonstrate the equiv... | 2016-05-04T00:02:35 | {
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https://quasirandomideas.wordpress.com/2010/04/22/math2111-chapter-1-fourier-series-additional-material-l_2-convergence-of-fourier-series/ | # Math2111: Chapter 1: Fourier series. Additional Material: L_2 convergence of Fourier series.
In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.
In this post I present some ideas whic... | 2018-02-21T16:57:52 | {
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"openwebmath_perplexity": 98.20205888890122,
"lm_name": "Qwen/Qwen-72B",
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https://proofwiki.org/wiki/Planes_are_Subspaces_of_Space | # Planes are Subspaces of Space
## Theorem
The two-dimensional subspaces of $\R^3$ are precisely the homogeneous planes of solid analytic geometry.
## Proof
Follows directly from Equivalent Statements for Vector Subspace Dimension One Less.
$\blacksquare$ | 2022-06-25T20:09:43 | {
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http://math.stackexchange.com/questions/422598/average-number-of-heads-in-filtered-coin-toss?answertab=votes | Average number of heads in filtered coin toss
I have a coin that, when tossed, produces heads with probability $p \geq 0.5$ and tails with probability $1-p$.
I start a coin-tossing experiment. Whenever I get more than one tail in a row, I discard the second tail and toss again, so that my results look like one long c... | 2016-02-07T17:46:27 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/422598/average-number-of-heads-in-filtered-coin-toss?answertab=votes",
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"lm_label": "1. YES\n2. YES",
"lm_q1_sco... |
http://math.stackexchange.com/questions/267419/problem-on-two-ideals-and-their-quotient | # Problem on two ideals and their quotient
For two ideals $I$ and $J$ in a commutative ring $R$, define $I : J = \{a\in R : aJ \subset I\}$. In the ring $\mathbb{Z}$ of all integers, if $I = 12\mathbb {Z}$ and $J = 8\mathbb {Z}$, find $I : J$.
How should I solve this problem? Can anyone help me please? Thanks for you... | 2015-07-30T18:12:54 | {
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"url": "http://math.stackexchange.com/questions/267419/problem-on-two-ideals-and-their-quotient",
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"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.986777177... |
https://mathoverflow.net/questions/260322/the-mathfrak-l-functor-on-textsfprof | # The $\mathfrak L$ functor on $\textsf{Prof}$
$\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}$ Recall that Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ allows us to define the functor $$\L : \Prof(A,B) \to \Prof(B,A)$$ ($\cal V$ is a cosmos in which $A,B$ are enri... | 2020-10-01T20:23:33 | {
"domain": "mathoverflow.net",
"url": "https://mathoverflow.net/questions/260322/the-mathfrak-l-functor-on-textsfprof",
"openwebmath_score": 0.9708478450775146,
"openwebmath_perplexity": 231.6987091853297,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771774699747,
"lm_... |
https://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-11-infinite-sequences-and-series-11-1-sequences-11-1-exercises-page-745/80 | ## Calculus 8th Edition
a) The sequence is increasing and has an upper bound. b) $\lim\limits_{n \to \infty}a_{n} = 2$
a) $a_{1}= \sqrt 2$ $a_{n+1} = \sqrt (2+a_{n})$ For $n=1$ $a_{1+1} = \sqrt (2+a_{1})$ $a_{2}= \sqrt (2+\sqrt 2)$ since $\sqrt (2+\sqrt 2) \gt \sqrt 2$ Then $a_{2} \gt a_{1}$ which suggests that the se... | 2020-04-08T16:56:55 | {
"domain": "gradesaver.com",
"url": "https://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-11-infinite-sequences-and-series-11-1-sequences-11-1-exercises-page-745/80",
"openwebmath_score": 0.9934211373329163,
"openwebmath_perplexity": 59.17286879753058,
"lm_name": "Qwen/Qwen-72B",
... |
https://en.wikipedia.org/wiki/Stolz-Ces%C3%A0ro_theorem | # Stolz–Cesàro theorem
(Redirected from Stolz-Cesàro theorem)
In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time.
The Stolz–Cesàro theorem can be viewed... | 2017-03-29T02:04:46 | {
"domain": "wikipedia.org",
"url": "https://en.wikipedia.org/wiki/Stolz-Ces%C3%A0ro_theorem",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771770811146,
"lm_q2_score": 0.66192288918838... |
http://www.martinorr.name/blog/2012/04/25/the-masser-wustholz-isogeny-theorem/ | # Martin's Blog
## The Masser-Wüstholz isogeny theorem
Posted by Martin Orr on Wednesday, 25 April 2012 at 14:09
Let and be two isogenous abelian varieties over a number field . Can we be sure that there is an isogeny between them of small degree, where "small" is an explicit function of and ? In particular, our bou... | 2018-07-19T01:26:40 | {
"domain": "martinorr.name",
"url": "http://www.martinorr.name/blog/2012/04/25/the-masser-wustholz-isogeny-theorem/",
"openwebmath_score": 0.9774165153503418,
"openwebmath_perplexity": 325.1440369673869,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771766922545,
"lm_q2... |
https://math.stackexchange.com/questions/1654123/understanding-the-definition-of-the-direct-sum-of-subspaces-of-a-vector-space | # Understanding the definition of the direct sum of subspaces of a vector space
I have a question regarding the definition of direct sum of a vector space in relation to subspaces.
Definition: A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1\cap W_2... | 2020-02-21T19:20:26 | {
"domain": "stackexchange.com",
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"openwebmath_perplexity": 25.175489080794133,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. ... |
https://mathstodon.xyz/@0foldcv/100630473390402761 | Consider two random complex numbers
z = u₁ + iv₁ and
w = u₂ + iv₂,
where u₁, v₁, u₂, v₂ are independent standard normal random variables (N(0,1)).
Then what is the probability distribution of the absolute value of the product |zw|?
Some empirical investigation (simulation) shows that the distribution looks like this:
... | 2020-09-26T20:58:33 | {
"domain": "mathstodon.xyz",
"url": "https://mathstodon.xyz/@0foldcv/100630473390402761",
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"lm_q1_score": 0.9867771763033945,
"lm_q2_score": 0.6619228891883799,... |
https://stacks.math.columbia.edu/tag/0EHL | Lemma 10.15.3. Let $R$ be a ring. Let $x \in R$, $I \subset R$ an ideal, and $\mathfrak p_ i$, $i = 1, \ldots , r$ be prime ideals. Suppose that $x + I \not\subset \mathfrak p_ i$ for $i = 1, \ldots , r$. Then there exists an $y \in I$ such that $x + y \not\in \mathfrak p_ i$ for all $i$.
Proof. We may assume there ar... | 2021-09-23T10:29:38 | {
"domain": "columbia.edu",
"url": "https://stacks.math.columbia.edu/tag/0EHL",
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"lm_q1_score": 0.9867771763033943,
"lm_q2_score": 0.66192288918838,
"lm_q1q2_... |
https://enacademic.com/dic.nsf/enwiki/15538/Quadrilateral | Edges and vertices 4
Schläfli symbol {4} (for square)
Area various methods;
see below
Internal angle (degrees) 90° (for square)
In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and some... | 2020-08-12T15:37:25 | {
"domain": "enacademic.com",
"url": "https://enacademic.com/dic.nsf/enwiki/15538/Quadrilateral",
"openwebmath_score": 0.8638280630111694,
"openwebmath_perplexity": 1118.806624041225,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771759145343,
"lm_q2_score": 0.6619228891... |
https://socratic.org/questions/what-is-the-unit-vector-that-is-orthogonal-to-the-plane-containing-3i-2j-3k-and--1 | # What is the unit vector that is orthogonal to the plane containing (3i + 2j - 3k) and (i -2j + 3k) ?
Dec 19, 2016
The answer is =〈0,-3/sqrt13,-2/sqrt13〉
#### Explanation:
We do a cross product to find the vector orthogonal to the plane
The vector is given by the determinant
$| \left(\hat{i} , \hat{j} , \hat{... | 2019-10-17T22:38:19 | {
"domain": "socratic.org",
"url": "https://socratic.org/questions/what-is-the-unit-vector-that-is-orthogonal-to-the-plane-containing-3i-2j-3k-and--1",
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"openwebmath_perplexity": 732.8212764301463,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_sco... |
https://jimmyjhickey.com/ST702-Estimators-Cramer-Rao | ## Example Problems on Estimators
ST702 Homework 5 on Estimators
# 7.45
Let $X_1, X_2, \dots X_n$ be iid from a distribution with mean $\mu$ and variance $\sigma^2$, and let $S^2$ be the usual unbiased estimator of $\sigma^2$. In Example 7.3.4 we saw that, under normality, the MLE has smaller MSE than $S^2$. In this... | 2021-05-06T03:16:41 | {
"domain": "jimmyjhickey.com",
"url": "https://jimmyjhickey.com/ST702-Estimators-Cramer-Rao",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771809697151,
"lm_q2_score": 0.661922882519... |
https://web2.0calc.com/questions/the-graph-of-the-parametric-equations-begin-align | +0
# The graph of the parametric equations \begin{align*} x&=\cos t,\\ y&=\sin t,
0
218
1
+474
The graph of the parametric equations
\begin{align*} x&=\cos t,\\ y&=\sin t, \end{align*}
meets the graph of the parametric equations
\begin{align*} x &= 2+ 4\cos s,\\ y &= 3+4\sin s, \end{align*}
at two points. Find the s... | 2019-04-26T14:52:32 | {
"domain": "0calc.com",
"url": "https://web2.0calc.com/questions/the-graph-of-the-parametric-equations-begin-align",
"openwebmath_score": 1.000008463859558,
"openwebmath_perplexity": 3971.3694577416563,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9867771809697151,
"lm... |
https://yannisparissis.wordpress.com/2011/03/10/dmat0101-notes-3-the-fourier-transform-on-l1/ | ## DMat0101, Notes 3: The Fourier transform on L^1
1. Definition and main properties.
For ${f\in L^1({\mathbb R}^n)}$, the Fourier transform of ${f}$ is the function
$\displaystyle \mathcal{F}(f)(\xi)=\hat{f}(\xi)=\int_{{\mathbb R}^n}f(x)e^{-2\pi i x\cdot \xi}dx,\quad \xi\in{\mathbb R}^n.$
Here ${x\cdot y}$ denotes... | 2017-10-20T12:32:13 | {
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"url": "https://yannisparissis.wordpress.com/2011/03/10/dmat0101-notes-3-the-fourier-transform-on-l1/",
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"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.98677718058... |
http://appliedclassicalanalysis.net/2016/08/02/integrate-int_-11sqrtfrac1x1-x-mathrmd-x-2/ | # Integrate $$\int_{-1}^{1}\sqrt{\frac{1+x}{1-x}} \mathrm{d} x$$
This integral appeared in Inside Interesting Integrals by Paul Nahin in the problem set of chapter 3. Using Wolfram Alpha, we get
\int\limits_{-1}^{1}\sqrt{\frac{1+x}{1-x}} \mathrm{d} x = \pi
\label{eq:1}
\tag{1}
Nahin suggests the following trig subst... | 2018-11-15T14:42:17 | {
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"url": "http://appliedclassicalanalysis.net/2016/08/02/integrate-int_-11sqrtfrac1x1-x-mathrmd-x-2/",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0... |
https://www.delftstack.com/howto/python/rmse-python/ | # Using RMSE in Python
Fariba Laiq Jan 03, 2023 Jan 07, 2022 Python Python Math
RMS (root mean square), also known as the quadratic mean, is the square root of the arithmetic mean of the squares of a series of numbers.
RMSE (root mean square error) gives us the difference between actual results and our calculated re... | 2023-01-29T06:21:47 | {
"domain": "delftstack.com",
"url": "https://www.delftstack.com/howto/python/rmse-python/",
"openwebmath_score": 0.20868198573589325,
"openwebmath_perplexity": 2164.4807291833217,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771801919951,
"lm_q2_score": 0.6619228825191... |
http://www.mathematicalfoodforthought.com/2006/01/finished-product-topic-number-theory_22.html | ## Sunday, January 22, 2006
### Finished Product. Topic: Number Theory. Level: Olympiad.
Problem: (1970 IMO - #1) Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
Solution: We claim that no suc... | 2020-01-18T11:30:48 | {
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"url": "http://www.mathematicalfoodforthought.com/2006/01/finished-product-topic-number-theory_22.html",
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"openwebmath_perplexity": 128.3106356257361,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_scor... |
http://makeyourownmandelbrot.blogspot.com/ | ## Friday, 29 December 2017
### Escape Condition for Julia and Mandelbrot Fractals
This post is reproduced from the blog for my next book, Make Your Own Algorithmic Art.
This post isn't a tutorial on creating the Julia or Mandelbrot fractals - you can learn about that here. Here we'll focussed on a specific bit of m... | 2018-04-21T02:14:31 | {
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"url": "http://makeyourownmandelbrot.blogspot.com/",
"openwebmath_score": 0.37631216645240784,
"openwebmath_perplexity": 1132.2566989987056,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771798031351,
"lm_q2_score": 0.6619228825191872,
"lm_q... |
https://proofwiki.org/wiki/Direct_Product_of_Sylow_p-Subgroups_is_Sylow_p-Subgroup | # Direct Product of Sylow p-Subgroups is Sylow p-Subgroup
## Theorem
Let $G_1$ and $G_2$ be groups.
Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively.
Let $H_1$ be a Sylow $p$-subgroup of $G_1$.
Let $H_2$ be a Sylow $p$-subgroup of $G_2$.
Then $H_1 \times H_2$ is a Sylow $p$-subgroup of $G_1 \times... | 2021-01-25T04:04:34 | {
"domain": "proofwiki.org",
"url": "https://proofwiki.org/wiki/Direct_Product_of_Sylow_p-Subgroups_is_Sylow_p-Subgroup",
"openwebmath_score": 0.9561023712158203,
"openwebmath_perplexity": 52.53486798737504,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771798031351,
"lm... |
https://ja.mathigon.org/course/intro-probability/random-variables | # ProbabilityRandom Variables
An event may be regarded as function of the outcome of an experiment: based on the outcome, we can say that the event occurred or didn't occur. We will often be interested in specifying richer information about the outcome of an experiment than a simple yes or no. Specifically, we will of... | 2021-11-27T16:45:38 | {
"domain": "mathigon.org",
"url": "https://ja.mathigon.org/course/intro-probability/random-variables",
"openwebmath_score": 0.8250868916511536,
"openwebmath_perplexity": 265.6448386587733,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771798031351,
"lm_q2_score": 0.6619... |
https://www.physicsforums.com/threads/maxima-and-minima-in-calculus.1044462/ | # Maxima and Minima in calculus
• MHB
WMDhamnekar
MHB
Question: Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
Let r and h be the radius and height of the right circular cylinder inscribed in a given cone of radi... | 2023-03-21T02:33:12 | {
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"url": "https://www.physicsforums.com/threads/maxima-and-minima-in-calculus.1044462/",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771798031351,
"lm_q2... |
https://proofwiki.org/wiki/Carmichael_Number_has_3_Odd_Prime_Factors | # Carmichael Number has 3 Odd Prime Factors
## Theorem
Let $n$ be a Carmichael number.
Then $n$ has at least $3$ distinct odd prime factors.
## Proof
By Korselt's Theorem, $n$ is odd.
Therefore $n$ has at least $1$ odd prime factor.
By Korselt's Theorem, for each prime factor of $n$:
$p^2 \nmid n$
$\paren {p - ... | 2020-09-30T06:36:15 | {
"domain": "proofwiki.org",
"url": "https://proofwiki.org/wiki/Carmichael_Number_has_3_Odd_Prime_Factors",
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"openwebmath_perplexity": 238.8129658963121,
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986777179414275,
"lm_q2_score": 0.6... |
https://docs.aptech.com/gauss/beta.html | beta¶
Purpose¶
Computes the standard Beta function, also called the Euler integral. The beta function is defined as:
$B(x, y) = \int_{0}^{1} t^{x−1}(1−t)^{y−1}dt$
Format¶
f = beta(x, y)
Parameters
• x (scalar or NxK matrix) – may be real or complex
• y (LxM matrix) – ExE conformable with x.
Returns
f (NxK matri... | 2022-05-22T20:30:39 | {
"domain": "aptech.com",
"url": "https://docs.aptech.com/gauss/beta.html",
"openwebmath_score": 0.8688353896141052,
"openwebmath_perplexity": 8418.280773871347,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.986777179414275,
"lm_q2_score": 0.6619228825191871,
"lm_q1q2_... |
http://mathhelpforum.com/pre-calculus/103828-problem-regarding-trig-identities.html | # Math Help - Problem regarding Trig Identities
1. ## Problem regarding Trig Identities
I need to use trig identities to find the indicated trig functions. The problem that I'm stuck on gives me that tan(theta)=5 ; with this im supposed to find cot(theta) cos(theta) tan(90- theta) and csc(theta). I am not really sure... | 2015-04-21T05:13:05 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/pre-calculus/103828-problem-regarding-trig-identities.html",
"openwebmath_score": 0.9357950091362,
"openwebmath_perplexity": 811.6443197826328,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9867771790254151,... |
https://brainmass.com/math/complex-analysis/computing-integrals-via-analytic-continuation-589748 | Explore BrainMass
# Computing integrals via analytic continuation
Not what you're looking for? Search our solutions OR ask your own Custom question.
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
Using the identity:
Integral from 0 to infinity of x^(-q)... | 2021-05-10T01:54:45 | {
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"url": "https://brainmass.com/math/complex-analysis/computing-integrals-via-analytic-continuation-589748",
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"openwebmath_perplexity": 2309.9834013608906,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9867... |
https://www.shaalaa.com/question-bank-solutions/graph-maxima-minima-the-minimum-value-x-2-250-x-a-75-b-50-c-25-d-55_46681 | Share
Books Shortlist
# Solution for The Minimum Value of ( X 2 + 250 X ) is (A) 75 (B) 50 (C) 25 (D) 55 - CBSE (Science) Class 12 - Mathematics
#### Question
The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is
(a) 75
(b) 50
(c) 25
(d) 55
#### Solution
(a) 75
$\text { Given }: f\left( x \right) = x^2 +... | 2019-05-26T09:59:49 | {
"domain": "shaalaa.com",
"url": "https://www.shaalaa.com/question-bank-solutions/graph-maxima-minima-the-minimum-value-x-2-250-x-a-75-b-50-c-25-d-55_46681",
"openwebmath_score": 0.24300260841846466,
"openwebmath_perplexity": 853.3120647393165,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"l... |
https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/Thebault0.shtml | # In the Spirit of Thébault I
Dr. Ray Viglione from Kean University and his undergraduate student Purna Patel found a variation of Victor Thébault's first problem that dealt with the configuration of squares built on the sides of a parallelogram.
Given a parallelogram, the centers of the squares drawn on both sides o... | 2018-12-14T00:02:01 | {
"domain": "cut-the-knot.org",
"url": "https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/Thebault0.shtml",
"openwebmath_score": 0.48492857813835144,
"openwebmath_perplexity": 1939.4636334229044,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986777178636555,
"lm_q2_s... |
https://www.effortlessmath.com/math-topics/how-to-find-the-number-of-solutions-in-a-system-of-equations/ | # How to Find the Number of Solutions in a System of Equations?
Depending on how the linear equations in a system touch each other, there will be a different number of solutions to the system. Here you get familiarized with how to find the number of solutions in a system of equations.
A system of linear equations usu... | 2023-03-21T10:29:37 | {
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"lm_label": "1. YES\n2. YES",
"lm_q1_score"... |
https://plainmath.net/6638/to-determine-the-correct-graph-for-the-function-g-x-equal-frac-1-2-f-plus-is | To determine.The correct graph for the function[g(x)=-frac{1}{2}f(x)+1 is B
To determine.
The correct graph for the function $g\left(x\right)=-\frac{1}{2}f\left(x\right)+1$ is B
You can still ask an expert for help
• Questions are typically answered in as fast as 30 minutes
Solve your problem for the price of one c... | 2022-06-28T13:01:30 | {
"domain": "plainmath.net",
"url": "https://plainmath.net/6638/to-determine-the-correct-graph-for-the-function-g-x-equal-frac-1-2-f-plus-is",
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"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9... |
https://www.cheenta.com/medians-of-triangle-prmo-2018-problem-10/ | What is the NO-SHORTCUT approach for learning great Mathematics?
# How to Pursue Mathematics after High School?
For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.
Try this beautiful problem from Geometry based on medians of triangle
## Medians of triangle |... | 2021-08-06T00:00:22 | {
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"url": "https://www.cheenta.com/medians-of-triangle-prmo-2018-problem-10/",
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"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771782476948,
"lm_q2_score": 0.6619... |
http://clay6.com/qa/7087/the-money-to-be-spent-for-the-welfare-of-the-employees-of-a-firm-is-proport | # The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue).If the total revenue(in rupees) received from the sale of Xunits of a product is given by R(x)=$3x^2+36x+5$,find the marginal revenue,when x=5,and write which value does the q... | 2017-11-24T01:57:24 | {
"domain": "clay6.com",
"url": "http://clay6.com/qa/7087/the-money-to-be-spent-for-the-welfare-of-the-employees-of-a-firm-is-proport",
"openwebmath_score": 0.8562685251235962,
"openwebmath_perplexity": 437.02433110930195,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.98677717... |
https://encyclopediaofmath.org/wiki/Conformal_radius_of_a_domain | # Conformal radius of a domain
A characteristic of a conformal mapping of a simply-connected domain, defined as follows: Let $D$ be a simply-connected domain with more than one boundary point in the $z$- plane. Let $z _ {0}$ be a point of $D$. If $z _ {0} \neq \infty$, then there exists a unique function $w = f ( z)$,... | 2023-02-05T04:29:54 | {
"domain": "encyclopediaofmath.org",
"url": "https://encyclopediaofmath.org/wiki/Conformal_radius_of_a_domain",
"openwebmath_score": 0.971750020980835,
"openwebmath_perplexity": 113.56213662043339,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771778588347,
"lm_q2_score... |
https://abrahamphysics.com/mechanics/massDistributions/momentOfInertia/momentOfInertia.htm | ## Moment of Inertia
The moment of inertia describes how mass is distributed around a point. How we calculate it depends on the type of system. We call a system discrete if it is composed of only point like particles. We call a system continuous if you cannot describe your system that way. Essentially, can you invoke ... | 2021-06-23T21:16:02 | {
"domain": "abrahamphysics.com",
"url": "https://abrahamphysics.com/mechanics/massDistributions/momentOfInertia/momentOfInertia.htm",
"openwebmath_score": 0.6263046264648438,
"openwebmath_perplexity": 196.07558195499598,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986777177... |
https://meangreenmath.com/2015/09/22/different-ways-of-solving-a-contest-problem-part-3/ | # Different ways of solving a contest problem (Part 3)
The following problem appeared on the American High School Mathematics Examination (now called the AMC 12) in 1988:
If $3 \sin \theta = \cos \theta$, what is $\sin \theta \cos \theta$?
When I presented this problem to a group of students, I was pleasantly surpri... | 2017-12-13T18:57:20 | {
"domain": "meangreenmath.com",
"url": "https://meangreenmath.com/2015/09/22/different-ways-of-solving-a-contest-problem-part-3/",
"openwebmath_score": 0.8019065856933594,
"openwebmath_perplexity": 288.4872670848113,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771778588... |
https://math.stackexchange.com/questions/2814735/tensor-product-identity-proof-without-explicit-construction | # Tensor product identity: proof without explicit construction
Let $R$ be a commutative ring with identity, and $M$ an $R$-module. Then $$R\otimes_R M \simeq M$$ This can be easily shown directly by construction of the isomorphism, namely $r\otimes m \mapsto rm$ for $r \in R$ and $m \in M$. Indeed, let $rm=0$. Then $r... | 2020-09-25T04:09:58 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2814735/tensor-product-identity-proof-without-explicit-construction",
"openwebmath_score": 0.9897511005401611,
"openwebmath_perplexity": 85.50428538567193,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_sco... |
https://math.stackexchange.com/questions/2420304/variance-in-central-limit-theorem-for-dependent-random-variables | # Variance in central limit theorem for dependent random variables.
I am reading some notes and struggling to show something which should be basic. I will write out the problem below. We have a measure preserving dynamical system $(X,T,\mu)$, and a measurable map $f:X\to \mathbb{R}$. We consider the sequence of identi... | 2019-07-21T15:34:44 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2420304/variance-in-central-limit-theorem-for-dependent-random-variables",
"openwebmath_score": 0.9874310493469238,
"openwebmath_perplexity": 679.976418714891,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1... |
http://zbmath.org/?q=an:1205.35099 | # zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "strai... | 2014-04-18T05:56:09 | {
"domain": "zbmath.org",
"url": "http://zbmath.org/?q=an:1205.35099",
"openwebmath_score": 0.8184961676597595,
"openwebmath_perplexity": 6885.258666866859,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771774699746,
"lm_q2_score": 0.6619228825191872,
"lm_q1q2_score": ... |
http://mathhelpforum.com/algebra/280860-more-come-me-s-been-while.html | # Thread: More to come from me !! It’s been a while
1. ## More to come from me !! It’s been a while
Solve a^2 - 7a - 30 = 0
2. ## Re: More to come from me !! It’s been a while
I would factor here...can you name two factors of -30 whose sum is -7?
3. ## Re: More to come from me !! It’s been a while
Oh I see, it is... | 2018-12-12T01:20:39 | {
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"url": "http://mathhelpforum.com/algebra/280860-more-come-me-s-been-while.html",
"openwebmath_score": 0.8104525804519653,
"openwebmath_perplexity": 1081.9253046340684,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771774699747,
"lm_q2_sco... |
https://testbook.com/question-answer/the-following-balances-are-extracted-at-the-end-of--6156d02786a3de6005a9ca8c | # The following balances are extracted at the end of the accounting period from the books of Radhey Shyam as follows:Plant & Machinery Rs. 2,00,000Furniture Rs. 50,000Building Rs. 5,00,000Depreciation is to be charged:20% on plant & machinery, 10% on furniture and 5% on Building. Calculate the amount of depreciation to... | 2022-01-28T08:02:24 | {
"domain": "testbook.com",
"url": "https://testbook.com/question-answer/the-following-balances-are-extracted-at-the-end-of--6156d02786a3de6005a9ca8c",
"openwebmath_score": 0.8471425771713257,
"openwebmath_perplexity": 13285.621478757124,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_sc... |
https://analysisofmath.wordpress.com/tag/liebniz-series/ | # A Beautiful Convergent Series
In this post I want to find the value of the sum
$\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \zeta \left(n+1 \right)$
Note that for $s>1$, $\zeta(s)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^s}$.
I use following formula that is called Gregory-Leibniz-Madhava’s series
$\displ... | 2021-07-27T19:31:23 | {
"domain": "wordpress.com",
"url": "https://analysisofmath.wordpress.com/tag/liebniz-series/",
"openwebmath_score": 0.9943063855171204,
"openwebmath_perplexity": 798.0821168680992,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771770811146,
"lm_q2_score": 0.661922882519... |
https://aquazorcarson.wordpress.com/2013/03/09/product-of-gaussian-probability-densities/ | product of Gaussian probability densities
So lately I have been following Udacity’s wonderful course on robotics and in particular was fascinated by the procedure of Kalman filter. My attempt here will be to expound on certain mathematical subtleties and missing deductive steps in the videos.
The first one has to do ... | 2018-01-17T11:08:15 | {
"domain": "wordpress.com",
"url": "https://aquazorcarson.wordpress.com/2013/03/09/product-of-gaussian-probability-densities/",
"openwebmath_score": 0.9224775433540344,
"openwebmath_perplexity": 270.01644361246974,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986777177081114... |
https://www.vedantu.com/maths/factors-of-120 | # Factors of 120
## Prime Factorization of 120
The factors of 120 are all the numbers that give the result as the number 120 when they are multiplied together in a pair of two. There are many different factors that are commonly used in the mathematical calculations, for example, the factors of the numbers 56, 90, etc... | 2020-09-23T14:03:41 | {
"domain": "vedantu.com",
"url": "https://www.vedantu.com/maths/factors-of-120",
"openwebmath_score": 0.5938757061958313,
"openwebmath_perplexity": 312.6446849400106,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9867771770811146,
"lm_q2_score": 0.6619228825191872,
"lm_q1... |
https://math.stackexchange.com/questions/2656819/can-any-large-number-n-in-mathbbz-be-written-as-a-sum-of-a-square-and-a-squ | # Can any large number $N\in \mathbb{Z}$ be written as a sum of a square and a square-free?
I'm reading an article by Dan Carmon on square free values of large polynomials over the rational function field. In this article he states the following question:
Does every sufficiently large $N\in \mathbb{Z}$ admit a repres... | 2021-03-05T01:19:19 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2656819/can-any-large-number-n-in-mathbbz-be-written-as-a-sum-of-a-square-and-a-squ",
"openwebmath_score": 0.9405708312988281,
"openwebmath_perplexity": 80.66669689746232,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YE... |
https://www.shaalaa.com/question-bank-solutions/the-mean-following-numbers-68-find-value-x-45-52-60-x-69-70-26-81-94-hence-estimate-median-median-grouped-data_19015 | Share
# The Mean of Following Numbers is 68. Find the Value of ‘X’. 45, 52, 60, X, 69, 70, 26, 81 and 94 Hence Estimate the Median. - Mathematics
Course
#### Question
The mean of following numbers is 68. Find the value of ‘x’.
45, 52, 60, x, 69, 70, 26, 81 and 94
Hence estimate the median.
#### Solution
Mean = ... | 2020-07-08T22:53:21 | {
"domain": "shaalaa.com",
"url": "https://www.shaalaa.com/question-bank-solutions/the-mean-following-numbers-68-find-value-x-45-52-60-x-69-70-26-81-94-hence-estimate-median-median-grouped-data_19015",
"openwebmath_score": 0.3550638258457184,
"openwebmath_perplexity": 2677.94881643834,
"lm_name": "Qwen/Qwen-7... |
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