texonom/texonom-md · Datasets at Hugging Face

title stringlengths 1 544 ⌀	parent stringlengths 0 57 ⌀	created stringlengths 11 12 ⌀	editor stringclasses 1 value	creator stringclasses 4 values	edited stringlengths 11 12 ⌀	refs stringlengths 0 536 ⌀	text stringlengths 1 26k	id stringlengths 32 32
Equinumerosity	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022		일대일 대응이 존재하는 경우	907dd17f4fab4b9b8212011bc680e5bf
Finite Set	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022			63ed2bdb689946429effb738e5422255
Infinite Set	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022			93755bd42acf41e2ac688467ad687885
Infinite Set	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022			971d3736190a4752a1ee3d1e2fea7028
Naive set theory	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022		자연 언어로 정의	67e68d75439f41bb8eb032834706044d
Poset	Set Theory Notion	Feb 25, 2023	Alan Jo	Alan Jo	Feb 25, 2023		> [그럴 것 같은데.. 아니라고?](https://youtube.com/watch?v=DPOYCjgvQ2o&feature=shares)	9a126737d3414bfc9225fe391603b9ad
Power Set	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022		주어진 집합의 모든 부분집합들로 구성된 집합	58038c26c9ef49e1bf59cae859c90394
Russell's *paradox*	null	null	null	null	null	null	### 자기언급의 역설 집함론의 오류 지적 집합은 모든 것을 포함할 수 있다는 공리에서 비롯되었다 집합의 집합은 더이상 집합으로 여기지 않는다는 공리로 해결 > [당신이 수학을 모르는 이유. (feat. 불완전성의 정리)](https://www.youtube.com/watch?v=oippSXvxUlw&t=590s)	7ecbb574edd24c5ab9066b8a5fcc6dea
Symmetric difference	Set Theory Notion	Dec 5, 2022	Alan Jo	Alan Jo	Dec 5, 2022		대칭차 $$A \triangle B$$	07bdea8546c04dabb5cd4ec7fd61b67b
Uncountable Set	Set Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022		자연수의 멱집합이나 무리수의 집합이 비가산 무리수 비가산이니 당연히 실수도 비가산	287e751cba3c466e92f123ca2f04e13f
ZFC	Set Theory Notion	Jul 29, 2023	Alan Jo	Alan Jo	Jul 29, 2023	[Axiom](https://texonom.com/axiom-01147181157e4e68a84a5af85ffdd035)	### ZFC Axioms \|Title\| \|:-:\| \|[Axiom of Choice](https://texonom.com/axiom-of-choice-a872716df36a48dcbb45a7a633c56071)\|	f844451d3f134288a92febc7b8e8257b
Cantor function	Cantor set	null	null	null	null	null	> [칸토어 함수 - 위키백과, 우리 모두의 백과사전](https://ko.wikipedia.org/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4_%ED%95%A8%EC%88%98)	bab4b944c0cb44bdb0089d40ec6b1542
Axiom of Choice	ZFC Axioms	Jul 29, 2023	Alan Jo	Alan Jo	Jul 29, 2023		필요충분 [Well ordering theorem](https://texonom.com/well-ordering-theorem-cd609c76a0f04e3d889ef64d34860b47) [Zorn’s Lemma](https://texonom.com/zorns-lemma-7bbd0f0a383e4800baf7bdfde87612c6) > [충격적인 멘붕 포인트. 이보다 쉬울 순 없다!](https://youtu.be/Y37xTQArewo)	a872716df36a48dcbb45a7a633c56071
Well ordering theorem	Axiom of Choice	null	null	null	null	null	어떤 집합에서도 순서가능	cd609c76a0f04e3d889ef64d34860b47
Zorn’s Lemma	Axiom of Choice	null	null	null	null	null		7bbd0f0a383e4800baf7bdfde87612c6
Lebesgue Theorem	Measure Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022		르베그 미분가능성 정리 단조함수의 미분가능성을 보장해 주는 매우 강력한 정리	50dd4d0d814244799c6b43126d669c5d
Measure Zero	Measure Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022	[Null Set](https://texonom.com/null-set-1f83671ec46848a4b8232daad30419bd)		84f55b9439c743f390eb69ddbd5f4a5b
Null Set	Measure Theory Notion	Apr 5, 2022	Alan Jo	Alan Jo	Apr 5, 2022		매우 작아 무시할 수 있는	1f83671ec46848a4b8232daad30419bd
램지이론	Number Theory Example	Jan 5, 2021	Alan Jo	Seong-lae Cho	Jan 16, 2022		그래이엄 수	742ac0cefcaa43018afea17a47f3ef34
Analytic number theory	Number Theory Example	Feb 22, 2022	Alan Jo	Alan Jo	Feb 22, 2022		### Analytic number theory Topics \|Title\| \|:-:\| \|[Goldbach's conjecture](https://texonom.com/goldbachs-conjecture-4d4a15ceede54e8494dbfc65099f02a4)\| \|[Riemann hypothesis](https://texonom.com/riemann-hypothesis-014b08e38ba14b358aeffb997f2c94af)\|	2ca0555c60b344aba8890dc7741bf191
BCD	Number Theory Example	May 5, 2021	Alan Jo	Alan Jo	Jan 16, 2022		## Binary-coded decimal 이진화 십진법 십진수처럼 코딩된 이진수 8421 code 이진수 네 자리를 묶어 십진수 한 자리로 사용하는 기수 [EBCDIC](https://texonom.com/ebcdic-47243ad2398d47eca866395c6857b675) ### pros and cons > [[BCD,8421코드 총정리]BCD코드는 언제 사용할까, BCD 장점, BCD가산법, 계산하는 법, 왜 6을 더하는가? 이진화 십진법 특징](https://jhnyang.tistory.com/232)	1d73c802caab457486434bb9e661aa95
Collatz conjecture	Number Theory Example	Jan 16, 2022	Alan Jo	Alan Jo	Jan 16, 2022		1937년에 처음으로 이 추측을 제기한 로타르 콜라츠의 이름을 딴 것으로 3n+1 추측, 울람 추측, 혹은 헤일스톤 수열 등 여러 이름 임의의 자연수가 다음 조작을 거쳐 항상 1이 된다는 추측 1. 짝수라면 2로 나눈다. 2. 홀수라면 3을 곱하고 1을 더한다. 3. 1이면 조작을 멈추고, 1이 아니면 첫 번째 단계로 돌아간다.	d6e27f23c9d34778a8f072bcf28edc81
Goldbach's conjecture	Analytic number theory Topics	Dec 29, 2019	Alan Jo	Alan Jo	Feb 22, 2022			4d4a15ceede54e8494dbfc65099f02a4
Riemann hypothesis	Analytic number theory Topics	Feb 22, 2022	Alan Jo	Alan Jo	Mar 28, 2023	[Bernhard Riemann](https://texonom.com/bernhard-riemann-166d61214be5439ca5f7ba1090063a3e)	### Riemann hypothesis Notion \|Title\| \|:-:\| \|[Riemann Zeta Function](https://texonom.com/riemann-zeta-function-0cf40635bf4940a0a7067a73f998ddda)\| > [Mathematician Yitang Zhang Confirms Partial Solution to Riemann Hypothesis](https://pandaily.com/mathematician-yitang-zhang-confirms-partial-solution-to-riemann-hypothesis/)	014b08e38ba14b358aeffb997f2c94af
Riemann Zeta Function	Riemann hypothesis Notion	Feb 22, 2022	Alan Jo	Alan Jo	Feb 22, 2022			0cf40635bf4940a0a7067a73f998ddda
EBCDIC	BCD	null	null	null	null	null	## Extended Binary Coded Decimal Interchange Code 확장 이진화 십진법 교환 부호( > [확장 이진화 십진법 교환 부호 - 위키백과, 우리 모두의 백과사전](https://ko.wikipedia.org/wiki/%ED%99%95%EC%9E%A5_%EC%9D%B4%EC%A7%84%ED%99%94_%EC%8B%AD%EC%A7%84%EB%B2%95_%EA%B5%90%ED%99%98_%EB%B6%80%ED%98%B8)	47243ad2398d47eca866395c6857b675
Factorial	Number Theory Notion	Mar 17, 2023	Alan Jo	Alan Jo	Mar 17, 2023		[Stirling’s Formula](https://texonom.com/stirlings-formula-e41933453d844d278269dc90647d8b96) > [99% 모르는 팩토리얼의 비밀!](https://youtube.com/watch?v=9Ti8EhK4eRw&feature=shares)	5b4059c4407b4c2f9c3be0f8357ae7ae
Infinite	Number Theory Notion	May 5, 2021	Alan Jo	Alan Jo	Jan 16, 2022		무한 앞에서 모든 유한한 값은 0이되기도 하지만 무한 앞에서 모든 가능성이 가능해진다 원숭이가 무한하게 타자를 치는 것 안에 모든 문학이 있는 것처럼 무한은 숫자보다 과정이라고 여겨야	f4d16a6949084f6c9c9763e78492486b
Number System	Number Theory Notion	Feb 22, 2022	Alan Jo	Alan Jo	Feb 22, 2022		### Number Systems \|Title\| \|:-:\| \|[Binary Number](https://texonom.com/binary-number-a28fb645048d4737afb24dcf3f69858d)\| \|[Complex Number](https://texonom.com/complex-number-f3fcb42a92474099b8359581af59bde1)\|	4bfd6d8ac2e34aef9e0c847ccd7c1378
Real Number	Number Theory Notion	Apr 19, 2023	Alan Jo	Alan Jo	Sep 3, 2023		unlike [Complex Number](https://texonom.com/complex-number-f3fcb42a92474099b8359581af59bde1), real number system is not [Algebraically Closed Field](https://texonom.com/algebraically-closed-field-a7ae2a1584f04bcfab20c148c98113cd) ### Real Number Type \|Title\| \|:-:\| \|[Rational Number](https://texonom.com/rational-number-131e8af784874e9a9eb4cd4a39d517b6)\| \|[Irrational Number](https://texonom.com/irrational-number-1da708bf326a4e6cb8938e54179cb6bc)\|	2c5c6fff21ab429da38ba685b40bacf3
Stirling’s Formula	Factorial	null	null	null	null	null	### **Stirling's approximation** $$ln(n!) = nln\;n - n + O(ln\;n)$$ > [Stirling's approximation](https://en.wikipedia.org/wiki/Stirling's_approximation) > [99% 모르는 팩토리얼의 비밀!](https://youtube.com/watch?v=9Ti8EhK4eRw&feature=shares)	e41933453d844d278269dc90647d8b96
Binary Number	Number Systems	Feb 22, 2022	Alan Jo	Alan Jo	Feb 22, 2022			a28fb645048d4737afb24dcf3f69858d
Complex Number	Number Systems	Apr 19, 2023	Alan Jo	Alan Jo	Jun 7, 2023	[Euler's Formula](https://texonom.com/eulers-formula-71f5337221904ba5b26e1398af5de0ab) [Field](https://texonom.com/field-a539c71297824a56a0a7139c871aaa08)	### [Algebraically Closed Field](https://texonom.com/algebraically-closed-field-a7ae2a1584f04bcfab20c148c98113cd) $$z = re^{i\theta}$$ ### Complex Number Notion \|Title\| \|:-:\| \|[Imaginary Number](https://texonom.com/imaginary-number-0c4b551380914570ba48a73039f9fc96)\| \|[Complex Polar Coordinate](https://texonom.com/complex-polar-coordinate-53f5972726ff4d79b59c6f037ccf863a)\| \|[Complex Function](https://texonom.com/complex-function-fa0da3f578d0479495020b48ea3fe290)\| \|[Complex conjugate](https://texonom.com/complex-conjugate-251d154b491f499cae0d769f2529eaf0)\| > [세상에서 가장 아름다운 수식을 이해해보자 (이과용)](https://www.youtube.com/watch?v=kgTSUZjVqas)	f3fcb42a92474099b8359581af59bde1
Complex conjugate	Complex Number Notion	May 13, 2023	Alan Jo	Alan Jo	May 13, 2023		## 켤레 ### Complex conjugate Notion \|Title\| \|:-:\| \|[Conjugate Symmetric](https://texonom.com/conjugate-symmetric-dccf53586f444e49b9bd84efe348bdf3)\|	251d154b491f499cae0d769f2529eaf0
Complex Function	Complex Number Notion	May 13, 2023	Alan Jo	Alan Jo	May 13, 2023		### Complex Functions \|Title\| \|:-:\| \|[Hermitian function](https://texonom.com/hermitian-function-af9c5415235f4d3da67d4e0e15389cff)\|	fa0da3f578d0479495020b48ea3fe290
Complex Polar Coordinate	Complex Number Notion	Apr 19, 2023	Alan Jo	Alan Jo	Apr 19, 2023	[Polar Coordinate](https://texonom.com/polar-coordinate-affa11aeb26241cdabf2a592e73daf91)	### Normal Coordinate $R + Ii$ can be expressed in normal coordinate $(R, I)$ ### Polar Coordinate let $\theta = tan^{-1}\frac{I}{R}, r = \sqrt{R^2+I^2}$, then $R + Ii = r(cos\theta + isin\theta)$ by [Euler's Formula](https://texonom.com/eulers-formula-71f5337221904ba5b26e1398af5de0ab) $r(cos\theta + isin\theta) = re^{j\theta}$	53f5972726ff4d79b59c6f037ccf863a
Imaginary Number	Complex Number Notion	Apr 19, 2023	Alan Jo	Alan Jo	Apr 19, 2023		i 또는 j로 사용한다 전류(I)랑 헷갈리거나 변수명으로 i많이 써서 j쓰는 곳이 실무에서 많다 MatLab이나 python등	0c4b551380914570ba48a73039f9fc96
Conjugate Symmetric	Complex conjugate Notion	May 13, 2023	Alan Jo	Alan Jo	May 13, 2023		## Hermitian Symmetry 공액 대칭	dccf53586f444e49b9bd84efe348bdf3
Hermitian function	Complex Functions	May 13, 2023	Alan Jo	Alan Jo	Jun 25, 2023		[Complex conjugate](https://texonom.com/complex-conjugate-251d154b491f499cae0d769f2529eaf0) is equal to the original function	af9c5415235f4d3da67d4e0e15389cff
Recurrence relation	Sequence Theory Notion	Oct 17, 2022	Alan Jo	Alan Jo	Sep 14, 2023	[Mathematical induction](https://texonom.com/mathematical-induction-428dbeddb59940dca2267474cb23631c)	## 점화식 1. Guess the form of the solution 2. Verify by induction 3. Solve for constants or [The Master Method](https://texonom.com/the-master-method-26345dcafc17491ca0f75b6e137647db) ### Recurrence relation Notion \|Title\| \|:-:\| \|[Recursion-tree method](https://texonom.com/recursion-tree-method-e45153b2672b4ff2ba0abea475c3b898)\|	1c0b90005c30437fa61c5e049782e4be
Sequence	Sequence Theory Notion	Oct 3, 2021	Alan Jo	Alan Jo	Sep 12, 2023		### 수열 ### Sequences \|Title\| \|:-:\| \|[Fibonacci sequence](https://texonom.com/fibonacci-sequence-830af82ec2bf49cc90da16fd1a201b14)\|	638838659a104ce3acc741ff806adf79
Series	Sequence Theory Notion	Oct 3, 2021	Alan Jo	Alan Jo	Aug 29, 2023		### Sum of every sequence ### Famous Series \|Title\| \|:-:\| \|[Taylor Series](https://texonom.com/taylor-series-66da8b67bfb041b2bdff0d16d12cf25a)\| \|[Fourier Series](https://texonom.com/fourier-series-034521e2dae346b8bf6b9095a602c9a7)\| \|[Power Series](https://texonom.com/power-series-c191ba27a1de41d8b2ea7e33b56fe9f6)\| ### Series Notion \|Title\| \|:-:\| \|[Convergent series](https://texonom.com/convergent-series-bd6d5a85531d4b89b7979707fe48b100)\| \|[Divergent series](https://texonom.com/divergent-series-1a5c65d9e0644d53a46f1b416fcabcc6)\|	da206bccff4241e7afb1c73a3aac306d
Fourier Series	Famous Series	Mar 16, 2023	Alan Jo	Alan Jo	Sep 13, 2023	[Fourier Transform](https://texonom.com/fourier-transform-5ee1f9063a9743dd9846ddc600286d5a) [Wave](https://texonom.com/wave-0b508bdcc4ac46759240dbc9b37d7994) [Periodic Function](https://texonom.com/periodic-function-3fa775a544934003bfbf39101551d1e1)	### Fourier Series Notion \|Title\| \|:-:\| \|[CTFS](https://texonom.com/ctfs-bfcd6b43900e44d983ae5c6537338b8b)\| \|[DTFS](https://texonom.com/dtfs-924b5dfa43dd46638de66cc15f61b538)\| \|[Fourier Series Basis](https://texonom.com/fourier-series-basis-64de5af81a8643c3a9f92f8e8e9bb2bb)\| \|[Parseval's theorem](https://texonom.com/parsevals-theorem-d132edfc5b734643934a11b42bcdc075)\| \|[Fourier Series Tilda function](https://texonom.com/fourier-series-tilda-function-b8495f56f23a4f58897df1dadacbce11)\| > [Fourier series](https://en.wikipedia.org/wiki/Fourier_series)	034521e2dae346b8bf6b9095a602c9a7
Taylor Series	Famous Series	Oct 3, 2021	Alan Jo	Alan Jo	Aug 29, 2023	[Calculus](https://texonom.com/calculus-8e37e7dac61e435da6a6970da7296f73)	도함수들의 한 점에서의 값으로 계산된 항의 무한합으로 해석함수를 나타내는 방법 ### Taylor Series Notion \|Title\| \|:-:\| \|[Taylor's Theorem](https://texonom.com/taylors-theorem-ecb251f43b1744a6a0ac5fe53a9d28d4)\| \|[Taylor series expansion](https://texonom.com/taylor-series-expansion-f83425f12a0e4d7e9122395a85e300e3)\|	66da8b67bfb041b2bdff0d16d12cf25a
Square wave	Non-sinusoidal waveforms	Mar 21, 2023	Alan Jo	Alan Jo	Sep 20, 2023		### [Fourier Series](https://texonom.com/fourier-series-034521e2dae346b8bf6b9095a602c9a7) $$A\Sigma_{k=1}^\infty\frac{1}{k}sin(2\pi kx)$$ ### Template Gallery \|Title\| \|:-:\| \|[Boxcar function](https://texonom.com/boxcar-function-fdf2d3b368f540bbbdaab928819f8da4)\| > [Square wave](https://en.wikipedia.org/wiki/Square_wave)	f730f0419f3843c389357eecc59f8d94
Taylor's Theorem	Taylor Series Notion	Oct 3, 2021	Alan Jo	Alan Jo	Aug 29, 2023			ecb251f43b1744a6a0ac5fe53a9d28d4
Taylor series expansion	Taylor Series Notion	Oct 3, 2021	Alan Jo	Alan Jo	Aug 29, 2023		[quadratic approximation](https://texonom.com/quadratic-approximation-3bd053180ba547e6a97ecf6b57f9af2d)	f83425f12a0e4d7e9122395a85e300e3
quadratic approximation	Taylor series expansion	null	null	null	null	null	second-orderTaylor series expansion	3bd053180ba547e6a97ecf6b57f9af2d
Convergent series	Series Notion	Mar 7, 2023	Alan Jo	Alan Jo	Jun 11, 2023			bd6d5a85531d4b89b7979707fe48b100
Divergent series	Series Notion	Mar 9, 2023	Alan Jo	Alan Jo	Jun 11, 2023		swing (oscillate) or go infinite	1a5c65d9e0644d53a46f1b416fcabcc6
Closure	Topology Notion	Jun 4, 2023	Alan Jo	Alan Jo	Jun 4, 2023			f8ea031aff2a42e28ce310accdd3f38a
Compactness	Topology Notion	Jun 4, 2023	Alan Jo	Alan Jo	Jun 4, 2023			4548634a95f645d2bf9f2027de247d5d
Euclidean Topology	Topology Notion	Jun 4, 2023	Alan Jo	Alan Jo	Jun 4, 2023			d6b8e43b16264d8aaa37f576522e7063
Heine Borel Theorem	Topology Notion	Jun 4, 2023	Alan Jo	Alan Jo	Jun 4, 2023			9364af6db0584563850234356131360e
M*anifold*	Topology Notion	Apr 30, 2023	Alan Jo	Alan Jo	Jul 12, 2023		### 다양체 국소적으로 유클리드 공간과 닮은 위상 공간 시계에서 3시와 9시 사이 시간 차이를 거리가 아니라 원의 거리인 각으로 재는 것처럼 2차원이 아니라 1차원 ### M*anifold Notion* \|Title\| \|:-:\| \|[Manifold hypothesis](https://texonom.com/manifold-hypothesis-0cb18ad52c9049eea1cdb57a52e6ef93)\| \|[Latent Manifold](https://texonom.com/latent-manifold-849d5f347d0848808fa05cd904106782)\| \|[Kähler manifold](https://texonom.com/khler-manifold-fca18f34330d4e6b8e74083f19eef2c5)\| > [Manifold](https://en.wikipedia.org/wiki/Manifold)	c6aa008ada9148309693c43c5033ad18
Poincare Conjecture	Topology Notion	Jun 4, 2023	Alan Jo	Alan Jo	Jun 4, 2023			ed72bbeef3774339adc4c3dfc7e83282
Kähler manifold	Manifold Notion	Jul 12, 2023	Alan Jo	Alan Jo	Jul 12, 2023		### Kähler manifold Notion \|Title\| \|:-:\| \|[Hodge Theory](https://texonom.com/hodge-theory-b69cbfba28e74a219b577372752bffae)\|	fca18f34330d4e6b8e74083f19eef2c5
Latent Manifold	null	null	null	null	null	null		849d5f347d0848808fa05cd904106782
M*anifold *hypothesis	Manifold Notion	May 24, 2023	Alan Jo	Alan Jo	Jun 1, 2023	[Distance Algorithm](https://texonom.com/distance-algorithm-9b12bb4903424559a97f32c00f5ec351)	assume that data is concentrated around a low-dimensional manifold 기존 차원보다 manifolded dimension distance가 더 중요 > [Manifold hypothesis](https://en.wikipedia.org/wiki/Manifold_hypothesis)	0cb18ad52c9049eea1cdb57a52e6ef93
Hodge Theory	Kähler manifold Notion	Jul 12, 2023	Alan Jo	Alan Jo	Jul 12, 2023		### Hodge theory Notion \|Title\| \|:-:\| \|[Hodge conjecture](https://texonom.com/hodge-conjecture-9c13dfb6c4fe4aafa9a6faabf24415f0)\|	b69cbfba28e74a219b577372752bffae
Hodge conjecture	Hodge theory Notion	Jul 12, 2023	Alan Jo	Alan Jo	Jul 12, 2023		> [Hodge conjecture](https://en.wikipedia.org/wiki/Hodge_conjecture) > [[난제] 호지추측 Hodge conjecture](https://www.youtube.com/watch?v=_OgxpHAbqAE)	9c13dfb6c4fe4aafa9a6faabf24415f0
archimedes cattle problem	Math Conundrums	Jun 13, 2020	Alan Jo	Alan Jo	Aug 17, 2021			e2e3467d3cba449cb999fa65d3d1020d
Asymptotic analysis	Analysis Types	Sep 7, 2023	Alan Jo	Alan Jo	Sep 7, 2023		### Asymptotic analysis Notion \|Title\| \|:-:\| \|[Asymptotic Notation](https://texonom.com/asymptotic-notation-44f31e0bfbaa455cb26f55423ed292a7)\| \|[The Master Method](https://texonom.com/the-master-method-26345dcafc17491ca0f75b6e137647db)\| > [Asymptotic analysis](https://en.wikipedia.org/wiki/Asymptotic_analysis)	017e73c405534be58baf14dde7824082
Asymptotic Notation	null	null	null	null	null	null	### They are set of functions non-tight bounds lower case notation means less than or greater than (not equal) ### Asymptotic Notations \|Title\| \|:-:\| \|[Big Θ](https://texonom.com/big-06ed557ed5c042c0b3b590320b4cf119)\| \|[Big O](https://texonom.com/big-o-a4d2742321a248b7b13068b1c4eb3215)\| \|[Big Ω](https://texonom.com/big-f4152bdb293042eca409b2e9f4f6b986)\|	44f31e0bfbaa455cb26f55423ed292a7
The Master Method	Asymptotic analysis Notion	Sep 7, 2023	Alan Jo	Alan Jo	Sep 10, 2023	[Divide and Conquer paradigm](https://texonom.com/divide-and-conquer-paradigm-9b10e85e70214b9b8f0bd293f5c297bd) [Recurrence relation](https://texonom.com/recurrence-relation-1c0b90005c30437fa61c5e049782e4be)	### The master method applies to recurrences of the form $$T(n) = aT(n/b) + f(n)$$ where $a\ge 1, b\ge1$, and $f$ is [Asympotically positive](https://texonom.com/asympotically-positive-b0f0899d95154b9a9a5d0620f0786f2b) ### The Master Method Notion \|Title\| \|:-:\| \|[The Master Method Case](https://texonom.com/the-master-method-case-e93bcfbae2cf4ca3a4bd9b297fa45ec6)\| \|[Asympotically positive](https://texonom.com/asympotically-positive-b0f0899d95154b9a9a5d0620f0786f2b)\| > [Master theorem (analysis of algorithms)](https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms))	26345dcafc17491ca0f75b6e137647db
Big Θ	null	null	null	null	null	null	$$\Theta(g(n)) = O(g(n)) \cap \Omega(g(n))$$ ### satisfy both upper bound and lower bound After a certain point the requirement matches, we can say Big θ $\Theta(n^2) = \Theta(1000n^2)$ theoretically but practically, it matters ### Big θ Notion \|Title\| \|:-:\| > [Big-θ (빅 세타) 표기법 (개념 이해하기) \| 알고리즘 \| Khan Academy](https://ko.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation) > [https://velog.io/@wan088/시간-복잡도-big-O-big-big-](https://velog.io/@wan088/시간-복잡도-big-O-big-big-)	06ed557ed5c042c0b3b590320b4cf119
Big Ω	Asymptotic Notations	Sep 7, 2023	Alan Jo	Alan Jo	Sep 10, 2023		### Non-tight Lower bound We write $f(n) = O(g(n))$ if there exist constants $c>0, n_0>0$ such that $0 \le cg(n) \le f(n) $ for all $n \ge n_0$ > [https://velog.io/@wan088/시간-복잡도-big-O-big-big-](https://velog.io/@wan088/시간-복잡도-big-O-big-big-)	f4152bdb293042eca409b2e9f4f6b986
Big O	null	null	null	null	null	null	### Non-tight Upper bound We write $f(n) = O(g(n))$ if there exist constants $c>0, n_0>0$ such that $0 \le f(n) \le cg(n)$ for all $n \ge n_0$ ### Big O Notion \|Title\| \|:-:\| \|[Time Complexity](https://texonom.com/time-complexity-cd2b4611ca52442fa7a77c1af0f91b2f)\| \|[Space Complexity](https://texonom.com/space-complexity-d873465dea934141a0edfd11c6df65c5)\| > [시간 복잡도 (big-O, big-Ω, big-θ)](https://velog.io/@wan088/시간-복잡도-big-O-big-big-)	a4d2742321a248b7b13068b1c4eb3215
Space Complexity	Big O Notion	Apr 29, 2021	Alan Jo	Alan Jo	Sep 7, 2023			d873465dea934141a0edfd11c6df65c5
Time Complexity	Big O Notion	Apr 29, 2021	Alan Jo	Alan Jo	Sep 7, 2023		### Time Complexities \|Title\| \|:-:\| \|[Best case time complexity](https://texonom.com/best-case-time-complexity-b1172d189af5470e85d7c988940dcc81)\| \|[Average case time complexity](https://texonom.com/average-case-time-complexity-92e4cbb0014a4f5689e1e616f42e0fdd)\| \|[Best case time complexity](https://texonom.com/best-case-time-complexity-6667e4f2a30a473991033732aa7d75e4)\| ### Time Complexity Notion \|Title\| \|:-:\| \|[Complexity growth pattern](https://texonom.com/complexity-growth-pattern-c40a6ab140fa46df99db601e07aaa962)\|	cd2b4611ca52442fa7a77c1af0f91b2f
Average case time complexity	Time Complexities	Sep 7, 2023	Alan Jo	Alan Jo	Sep 17, 2023	[Random Variable](https://texonom.com/random-variable-0bc19e0582784ec3a6e0de5b91296212)	## expected complexity > [Average-case complexity](https://en.wikipedia.org/wiki/Average-case_complexity)	92e4cbb0014a4f5689e1e616f42e0fdd
Best case time complexity	Time Complexities	Sep 7, 2023	Alan Jo	Alan Jo	Sep 7, 2023		보통 의미없다	6667e4f2a30a473991033732aa7d75e4
Best case time complexity	Time Complexities	Sep 7, 2023	Alan Jo	Alan Jo	Sep 7, 2023			b1172d189af5470e85d7c988940dcc81
Complexity growth pattern	null	null	null	null	null	null		c40a6ab140fa46df99db601e07aaa962
Asympotically positive	The Master Method Notion	Sep 10, 2023	Alan Jo	Alan Jo	Sep 10, 2023		### A function if there exists an $N$ such that $f(n) \gt 0$ for all $n\ge N$	b0f0899d95154b9a9a5d0620f0786f2b
The Master Method Case	The Master Method Notion	Sep 7, 2023	Alan Jo	Alan Jo	Sep 12, 2023		### Proof is on [Introduction to Algorithm](https://texonom.com/introduction-to-algorithm-e7e86c2968934959b53c2c2160bec4c6) proof is easy to understand ![](https://www.notion.so/image/https%3A%2F%2Fprod-files-secure.s3.us-west-2.amazonaws.com%2F0bf522c6-2468-4c71-99e3-68f5a25d4225%2Faee544bb-0bbc-4756-82ec-888ec58cee19%2FUntitled.png?table=block&id=31d702c1-ce01-45df-9c17-160f8d038702&cache=v2) ![](https://www.notion.so/image/https%3A%2F%2Fprod-files-secure.s3.us-west-2.amazonaws.com%2F0bf522c6-2468-4c71-99e3-68f5a25d4225%2Fa7dc6a24-54b1-4a4a-bfb8-6e272ad7605b%2FUntitled.png?table=block&id=f0847bdd-c435-478d-832f-8fa609b46c02&cache=v2) if $f(n)$ does not meet regularity condition, you can’t use the master theorem in the third case ### Example ![](https://www.notion.so/image/https%3A%2F%2Fprod-files-secure.s3.us-west-2.amazonaws.com%2F0bf522c6-2468-4c71-99e3-68f5a25d4225%2F43a55351-d727-4e4e-9852-a77f7764484c%2FUntitled.png?table=block&id=58c45eb6-fb81-4880-80aa-4847772f8caa&cache=v2) > [Master theorem (analysis of algorithms)](https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms))	e93bcfbae2cf4ca3a4bd9b297fa45ec6
Extrapolation	Regression analysis Notion	Apr 12, 2023	Alan Jo	Alan Jo	Apr 12, 2023		> [Extrapolation](https://en.wikipedia.org/wiki/Extrapolation) > [외삽법 (Extrapolation)](https://m.blog.naver.com/bisanghara/120183969936)	b3788658ec7745a0a41d41b54d22c66b
Loss Function	null	null	null	null	null	null	### Gradient Available 하나의 input data에 대해서 오차를 계산하는 함수 - [KL Divergence](https://texonom.com/kl-divergence-c7964872f5184a7baaf312605405aef6) - [Cross Entropy](https://texonom.com/cross-entropy-fefc71fd930a41a7842f39bccb3abcc9) decision-theoretic object that we are bringing in to quantify the negative consequences of error ### Loss Functions \|Title\| \|:-:\| \|[Mean Squared Error](https://texonom.com/mean-squared-error-18a36a7c349f495e9f960c7b605ad8ed)\| \|[RMSE](https://texonom.com/rmse-f61d17a91b174155bd3562c23e67b804)\| \|[Binary Crossentropy](https://texonom.com/binary-crossentropy-a0d2492e4aee49c2b7926653041a541a)\| \|[Categorical Crossentropy](https://texonom.com/categorical-crossentropy-b883d1401ffc426daea9bac208c8c5df)\| \|[L1 Loss Function](https://texonom.com/l1-loss-function-809f81be29194faaa6adfb9057327bbb)\| \|[Zero-One Loss](https://texonom.com/zero-one-loss-077d869edbfc4c09ac4441d0e547e961)\| \|[Hinge Loss](https://texonom.com/hinge-loss-7ccd3c37b1c64cf5b27a6b1d36067c1a)\| ### Loss Function Notion \|Title\| \|:-:\| \|[Test Loss](https://texonom.com/test-loss-9148d37e324240d19e41f9319fa8a0ab)\| \|[Training Loss](https://texonom.com/training-loss-3316afa9758643118624ad288b809861)\| > [0023 Loss & Metric - Deepest Documentation](https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0023/) ### Difference by Example > [What is the difference between a loss function and an error function?](https://stats.stackexchange.com/questions/359043/what-is-the-difference-between-a-loss-function-and-an-error-function) ### Loss Functions > [Module: tf.keras.losses \| TensorFlow v2.12.0](https://www.tensorflow.org/api_docs/python/tf/keras/losses) > [Objective Function, Loss Function, Cost Function의 차이](https://ganghee-lee.tistory.com/28) [https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0023/](https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0023/)	e8f6343914494828988137987cf459f9
Ordinary Least Squares	Regression analysis Notion	Jun 25, 2023	Alan Jo	Alan Jo	Jun 25, 2023		## OLS > [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares)	d19e8054a63141ad8b53e4efb974f2b3
Regression	Regression analysis Notion	Oct 6, 2021	Alan Jo	Alan Jo	Apr 18, 2023	[Optimization](https://texonom.com/optimization-984100dcbc454057bffd1b97be6ef134) [Data Classification](https://texonom.com/data-classification-0ed6ffa3555b4879be09f0501f0b580c)	### the target variable is continuous ### Regressions \|Title\| \|:-:\| \|[Linear Regression](https://texonom.com/linear-regression-0137f4fba64746f6bc28e059d0aa6dd1)\| \|[Logistic Regression](https://texonom.com/logistic-regression-8ed39aeb672648b1933df8911996d5d7)\| ### Regression Notion \|Title\| \|:-:\| \|[Cost Function](https://texonom.com/cost-function-66c21423cc7347909016b3423f2ada7e)\| \|[IRLS](https://texonom.com/irls-ca318bd9d30e4054b87415503e34c0f3)\| \|[Multi-Class Regression](https://texonom.com/multi-class-regression-b0971c1ace5240abb9d6032522766455)\|	d43885ba8cd044f4942cb591bdf1a612
**Regularization**	Regression analysis Notion	Mar 27, 2023	Alan Jo	Alan Jo	Mar 27, 2023		## simpler [Regularization](https://texonom.com/regularization-c85433c8c0554eba8edf0035b7fd334c) concept is first adapted to [Ridge regression](https://texonom.com/ridge-regression-8520dc9911234f62bc6fa25976af0972) model Norm is 절댓값 ### **Regularization Norms** \|Title\| \|:-:\| \|[L0 Norm](https://texonom.com/l0-norm-47471a6f6dea484fbf30a4c46cde8152)\| \|[L1 Norm](https://texonom.com/l1-norm-d316024c475e4eb691785783756bce57)\| \|[L2 Norm](https://texonom.com/l2-norm-38c15917350a4c82a11003474ac7d280)\|	c85433c8c0554eba8edf0035b7fd334c
Residual Sum of Squares	Regression analysis Notion	Jun 25, 2023	Alan Jo	Alan Jo	Jun 25, 2023		## RSS > [Residual sum of squares](https://en.wikipedia.org/wiki/Residual_sum_of_squares)	343bc935d5264d209c2adb3eea06ca60
Test Loss	Loss Function Notion	May 9, 2023	Alan Jo	Alan Jo	May 9, 2023			9148d37e324240d19e41f9319fa8a0ab
Training Loss	Loss Function Notion	May 9, 2023	Alan Jo	Alan Jo	May 9, 2023			3316afa9758643118624ad288b809861
**Binary Crossentropy**	Loss Functions	Mar 27, 2023	Alan Jo	Alan Jo	May 9, 2023			a0d2492e4aee49c2b7926653041a541a
**Categorical Crossentropy**	Loss Functions	Mar 27, 2023	Alan Jo	Alan Jo	May 9, 2023			b883d1401ffc426daea9bac208c8c5df
Hinge Loss	Loss Functions	Jun 7, 2023	Alan Jo	Alan Jo	Jun 7, 2023			7ccd3c37b1c64cf5b27a6b1d36067c1a
L1 Loss Function	Loss Functions	Mar 27, 2023	Alan Jo	Alan Jo	May 9, 2023		![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F13e50fc7-c832-42d3-89ab-9a7bf6888ead%2FUntitled.png?table=block&id=467fbe92-c25d-4b74-87dc-7b3b71daa445&cache=v2) > [L1, L2 Norm, Loss, Regularization?](https://junklee.tistory.com/29)	809f81be29194faaa6adfb9057327bbb
Mean Squared Error	Loss Functions	Mar 27, 2023	Alan Jo	Alan Jo	Jun 13, 2023	[L2 Norm](https://texonom.com/l2-norm-38c15917350a4c82a11003474ac7d280)	## L2 ![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F764b1ec6-99b5-4466-9f8b-0dd793d8f300%2FUntitled.png?table=block&id=acbfe6df-50bf-4ab0-ad00-192b80d9b686&cache=v2) > [Mean squared error](https://en.wikipedia.org/wiki/Mean_squared_error) > [[딥러닝] 목적/손실 함수(Loss Function) 이해 및 종류](https://needjarvis.tistory.com/567)	18a36a7c349f495e9f960c7b605ad8ed
RMSE	Loss Functions	Mar 27, 2023	Alan Jo	Alan Jo	May 16, 2023	[L2 Norm](https://texonom.com/l2-norm-38c15917350a4c82a11003474ac7d280)	### Root mean squared Error	f61d17a91b174155bd3562c23e67b804
Zero-One Loss	Loss Functions	Jun 7, 2023	Alan Jo	Alan Jo	Jun 7, 2023			077d869edbfc4c09ac4441d0e547e961
Cost Function	null	null	null	null	null	null	## Error Function, J function 모든 input dataset에 대해서 오차를 계산하는 함수 each parameter $$J(\theta) = \frac{1}{2}\Sigma_{i=1}^n(h_\theta(x^{(i)}) - y^{(i)})^2$$ ### Matrix Vector notation $$J(\theta) = \frac{1}{2}(X\theta - \vec{y})^T(X\theta - \vec{y})$$ ### Lest squares cost function > [Objective Function, Loss Function, Cost Function의 차이](https://ganghee-lee.tistory.com/28) [Least Squares Error](https://texonom.com/least-squares-error-93b101745714452aa1656b71320e08b6)	66c21423cc7347909016b3423f2ada7e
IRLS	Regression Notion	Mar 16, 2023	Alan Jo	Alan Jo	Apr 18, 2023	[Newton–Raphson method](https://texonom.com/newtonraphson-method-0bc249bfb0fe4e498829110bf785eb6c)	### Iterative Re-weighted Least Squares Iterative re-weighted least squares (IRLS) is an optimization algorithm used to solve nonlinear least squares problems. It is an iterative algorithm that updates the solution in each step until convergence is achieved. In each iteration, the algorithm re-weights the observations based on the current estimate of the solution. This makes the algorithm more robust to outliers and can help it converge more quickly. update rule is similar with	ca318bd9d30e4054b87415503e34c0f3
Multi-Class Regression	Regression Notion	Mar 21, 2023	Alan Jo	Alan Jo	Mar 21, 2023			b0971c1ace5240abb9d6032522766455
Least Squares Error	Cost Function	null	null	null	null	null	$$E_{LS}$$ 최적화를 수행하는 도구로 사용하는 것이 Least Squares 관찰값들은 매번 Noise가 존재할 수 있다. 이런 경우 정확한 parameter값을 구하는데 error가 최소화되도록 구하는 방법 error는 보통 관찰값과 추정값의 제곱으로 표현 어떤 식에서 구해야할 parameter가 4개일 때, 이 때 실험을 5번 진행되어 관찰되는 값이 parameter보다 많을 때 정확한 parameter 값을 구하기 위하여 사용하는 최적화 방법이 바로 Least Squares > [Untitled](https://taeyoung96.github.io/slam/SLAM_02/)	93b101745714452aa1656b71320e08b6
Linear Regression	Regressions	Mar 9, 2023	Alan Jo	Alan Jo	Jun 7, 2023	[Objective function](https://texonom.com/objective-function-5bf6c1fc7c854b5aaaacd17d5e094464)	### Linear Predictor (minimal [Perceptron](https://texonom.com/perceptron-1deb66f486d54c93bb928d8afba1864c)) $$h(x) = \Sigma_{i=0}^{d}\theta_ix_i = \theta^Tx$$ in Linear Regression there is always global optimum - update $\theta$ - [Gradient Descent](https://texonom.com/gradient-descent-c1342b13182f4fb6959023a75b5e2ff8) like algorithms > - add gradient of derivative of cost function J $$J(\theta) = \frac{1}{2}\Sigma_{i=1}^n(h_\theta(x^{(i)}) - y^{(i)})^2 = \frac{1}{2}(X\theta - \vec{y})^T(X\theta - \vec{y})$$ $$\theta := \theta + \alpha \Sigma(y_i - h_\theta(x_i))x_i$$ - Closed form - solve equation - assume $X^TX$ is invertible $$\theta = (X^TX)^{-1}X^T\vec{y}$$ ### Linear Regression Notion \|Title\| \|:-:\| \|[Basis Function](https://texonom.com/basis-function-994de24f84dd437d9d563ca2a87652d6)\| \|[Locally weighted Linear Regression](https://texonom.com/locally-weighted-linear-regression-6f405fcc12f64e0b9ec061c87d81e1e2)\| \|[Ridge regression](https://texonom.com/ridge-regression-8520dc9911234f62bc6fa25976af0972)\| \|[Lasso Regression](https://texonom.com/lassoregression-01bff5049e0e40faa3c1ce3fb68f1c76)\| \|[LMS](https://texonom.com/lms-3646b634f8974a51b8184a23f3defa1d)\| > [선형회귀 - 공돌이의 수학정리노트](https://angeloyeo.github.io/2020/08/24/linear_regression.html)	0137f4fba64746f6bc28e059d0aa6dd1
Logistic Regression	Regressions	Mar 16, 2023	Alan Jo	Alan Jo	May 21, 2023	[Sigmoid Function](https://texonom.com/sigmoid-function-417e17b995cf40c996cec78f699ea59b) [Logit Function](https://texonom.com/logit-function-104e55a12b234813a020cb09ec51c754) [Gaussian Discriminant Analysis](https://texonom.com/gaussian-discriminant-analysis-67ec78f9fd6944e68671c610aeadb079) [Data Classification](https://texonom.com/data-classification-0ed6ffa3555b4879be09f0501f0b580c)	Sigmoid 함수를 사용하여 0과 1 사이의 값을 출력, log-likelihood를 최대화 ![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ff587cf29-0b05-4917-a3f4-6d1add80bbb5%2FUntitled.png?table=block&id=11da7a73-d114-412e-80a4-98503f07fc58&cache=v2) to make convex cost function ![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3cd79553-1597-4710-8b7a-e4cf3761cdb6%2FUntitled.png?table=block&id=403a0496-8b25-4ad8-9e9f-40e88eedabea&cache=v2) ### Logistic Regression Notion \|Title\| \|:-:\| \|[Multi-Class Logistic Regression](https://texonom.com/multi-class-logistic-regression-545d3a376a4d49dd8845111405909327)\| $$h_\theta(x) = g(\theta^Tx) = \frac{1}{1 + e^{-\theta^Tx}}$$ This leads to $p(u\|x;\theta)$ described by [Bernoulli Distribution](https://texonom.com/bernoulli-distribution-bddaf7e81c9a4df8ab389521fc56e676) $$p(y\|x;\theta) = (h_\theta(x))^y(1 - h_\theta(x))^{1 - y}$$ the [Likelihood function](https://texonom.com/likelihood-function-0750726af35a45bb86036dbc7507f725) is $$l(\theta) = logL(\theta) = \Sigma_{i=1}^ny^{(i)}logh(x^{(i)}) + (1 - y^{(i)})log(1 - h(x^{(i)}))$$ But 로지스틱 회귀의 y값은 you can use [Newton–Raphson method](https://texonom.com/newtonraphson-method-0bc249bfb0fe4e498829110bf785eb6c) or [Stochastic Gradient Descent](https://texonom.com/stochastic-gradient-descent-d8b8d008e0a34f4bb55175ffba21db44) and prior one achieves faster convergence or [IRLS](https://texonom.com/irls-ca318bd9d30e4054b87415503e34c0f3) is much faster difference between [Linear Regression](https://texonom.com/linear-regression-0137f4fba64746f6bc28e059d0aa6dd1) is that usage of sigmoid function or logistic function	8ed39aeb672648b1933df8911996d5d7
Basis Function	Linear Regression Notion	Mar 14, 2023	Alan Jo	Alan Jo	Apr 6, 2023	[Overfitting](https://texonom.com/overfitting-24c3b183372845e8999ad7f7a0ba5035)	## Feature Map $\phi(x)$, Basis Function $h(x)$ $$h_\theta(x) = \Sigma_{l=1}^L\theta_l\phi_l(x) = \theta^T\phi(x)$$ we can extend the linear regression into a linear combination of basis functions $${R^d \rightarrow R} \;\; (x \in R^d)$$ - polynomial basis $x^l$ - gaussian basis $exp(-\frac{\|\|x - \mu_l\|\|^2}{2\sigma^2})$ ### Using nonlinear basis functions, we allow the function to be nonlinear	994de24f84dd437d9d563ca2a87652d6
**Lasso Regression**	Linear Regression Notion	Mar 27, 2023	Alan Jo	Alan Jo	Mar 27, 2023	[L1 Norm](https://texonom.com/l1-norm-d316024c475e4eb691785783756bce57)	> [Regularization(정규화): Ridge regression/LASSO](https://sanghyu.tistory.com/13) > [Ridge regression(릿지 회귀)와 Lasso regression(라쏘 회귀) 쉽게 이해하기](https://rk1993.tistory.com/entry/Ridge-regression와-Lasso-regression-쉽게-이해하기)	01bff5049e0e40faa3c1ce3fb68f1c76
LMS	Linear Regression Notion	Apr 6, 2023	Alan Jo	Alan Jo	Apr 6, 2023		## Least Mean Square $\alpha$ is hyperparameter for fastness of learning ### LMS Notion \|Title\| \|:-:\| \|[LMS with Kernel Trick](https://texonom.com/lms-with-kernel-trick-b2130bc9ef1d41bbb779435512612157)\|	3646b634f8974a51b8184a23f3defa1d

**Equinumerosity**

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

일대일 대응이 존재하는 경우

907dd17f4fab4b9b8212011bc680e5bf

Finite Set

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

63ed2bdb689946429effb738e5422255

Infinite Set

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

93755bd42acf41e2ac688467ad687885

Infinite Set

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

971d3736190a4752a1ee3d1e2fea7028

Naive set theory

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

자연 언어로 정의

67e68d75439f41bb8eb032834706044d

Poset

Set Theory Notion

Feb 25, 2023

Alan Jo

Feb 25, 2023

> [그럴 것 같은데.. 아니라고?](https://youtube.com/watch?v=DPOYCjgvQ2o&feature=shares)

9a126737d3414bfc9225fe391603b9ad

Power Set

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

주어진 집합의 모든 부분집합들로 구성된 집합

58038c26c9ef49e1bf59cae859c90394

Russell's ***paradox***

null

### **자기언급**의 역설 집함론의 오류 지적 집합은 모든 것을 포함할 수 있다는 공리에서 비롯되었다 집합의 집합은 더이상 집합으로 여기지 않는다는 공리로 해결 > [당신이 수학을 모르는 이유. (feat. 불완전성의 정리)](https://www.youtube.com/watch?v=oippSXvxUlw&t=590s)

7ecbb574edd24c5ab9066b8a5fcc6dea

Symmetric difference

Set Theory Notion

Dec 5, 2022

Alan Jo

Dec 5, 2022

대칭차 $$A \triangle B$$

07bdea8546c04dabb5cd4ec7fd61b67b

Uncountable Set

Set Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

자연수의 멱집합이나 무리수의 집합이 비가산 무리수 비가산이니 당연히 실수도 비가산

287e751cba3c466e92f123ca2f04e13f

ZFC

Set Theory Notion

Jul 29, 2023

Alan Jo

Jul 29, 2023

[Axiom](https://texonom.com/axiom-01147181157e4e68a84a5af85ffdd035)

f844451d3f134288a92febc7b8e8257b

Cantor function

Cantor set

null

> [칸토어 함수 - 위키백과, 우리 모두의 백과사전](https://ko.wikipedia.org/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4_%ED%95%A8%EC%88%98)

bab4b944c0cb44bdb0089d40ec6b1542

Axiom of Choice

ZFC Axioms

Jul 29, 2023

Alan Jo

Jul 29, 2023

필요충분 [Well ordering theorem](https://texonom.com/well-ordering-theorem-cd609c76a0f04e3d889ef64d34860b47) [Zorn’s Lemma](https://texonom.com/zorns-lemma-7bbd0f0a383e4800baf7bdfde87612c6) > [충격적인 멘붕 포인트. 이보다 쉬울 순 없다!](https://youtu.be/Y37xTQArewo)

a872716df36a48dcbb45a7a633c56071

Well ordering theorem

Axiom of Choice

null

어떤 집합에서도 순서가능

cd609c76a0f04e3d889ef64d34860b47

Zorn’s Lemma

Axiom of Choice

null

7bbd0f0a383e4800baf7bdfde87612c6

Lebesgue Theorem

Measure Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

**르베그 미분가능성 정리** 단조함수의 미분가능성을 보장해 주는 매우 강력한 정리

50dd4d0d814244799c6b43126d669c5d

Measure Zero

Measure Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

[Null Set](https://texonom.com/null-set-1f83671ec46848a4b8232daad30419bd)

84f55b9439c743f390eb69ddbd5f4a5b

Null Set

Measure Theory Notion

Apr 5, 2022

Alan Jo

Apr 5, 2022

매우 작아 무시할 수 있는

1f83671ec46848a4b8232daad30419bd

램지이론

Number Theory Example

Jan 5, 2021

Alan Jo

Seong-lae Cho

Jan 16, 2022

그래이엄 수

742ac0cefcaa43018afea17a47f3ef34

Analytic number theory

Number Theory Example

Feb 22, 2022

Alan Jo

Feb 22, 2022

### Analytic number theory Topics |Title| |:-:| |[Goldbach's conjecture](https://texonom.com/goldbachs-conjecture-4d4a15ceede54e8494dbfc65099f02a4)| |[Riemann hypothesis](https://texonom.com/riemann-hypothesis-014b08e38ba14b358aeffb997f2c94af)|

2ca0555c60b344aba8890dc7741bf191

BCD

Number Theory Example

May 5, 2021

Alan Jo

Jan 16, 2022

## Binary-coded decimal 이진화 십진법 십진수처럼 코딩된 이진수 8421 code 이진수 네 자리를 묶어 십진수 한 자리로 사용하는 기수 [EBCDIC](https://texonom.com/ebcdic-47243ad2398d47eca866395c6857b675) ### pros and cons > [[BCD,8421코드 총정리]BCD코드는 언제 사용할까, BCD 장점, BCD가산법, 계산하는 법, 왜 6을 더하는가? 이진화 십진법 특징](https://jhnyang.tistory.com/232)

1d73c802caab457486434bb9e661aa95

Collatz conjecture

Number Theory Example

Jan 16, 2022

Alan Jo

Jan 16, 2022

1937년에 처음으로 이 추측을 제기한 로타르 콜라츠의 이름을 딴 것으로 3n+1 추측, 울람 추측, 혹은 헤일스톤 수열 등 여러 이름 임의의 자연수가 다음 조작을 거쳐 항상 1이 된다는 추측 1. 짝수라면 2로 나눈다. 2. 홀수라면 3을 곱하고 1을 더한다. 3. 1이면 조작을 멈추고, 1이 아니면 첫 번째 단계로 돌아간다.

d6e27f23c9d34778a8f072bcf28edc81

Goldbach's conjecture

Analytic number theory Topics

Dec 29, 2019

Alan Jo

Feb 22, 2022

4d4a15ceede54e8494dbfc65099f02a4

Riemann hypothesis

Analytic number theory Topics

Feb 22, 2022

Alan Jo

Mar 28, 2023

[Bernhard Riemann](https://texonom.com/bernhard-riemann-166d61214be5439ca5f7ba1090063a3e)

### Riemann hypothesis Notion |Title| |:-:| |[Riemann Zeta Function](https://texonom.com/riemann-zeta-function-0cf40635bf4940a0a7067a73f998ddda)| > [Mathematician Yitang Zhang Confirms Partial Solution to Riemann Hypothesis](https://pandaily.com/mathematician-yitang-zhang-confirms-partial-solution-to-riemann-hypothesis/)

014b08e38ba14b358aeffb997f2c94af

Riemann Zeta Function

Riemann hypothesis Notion

Feb 22, 2022

Alan Jo

Feb 22, 2022

0cf40635bf4940a0a7067a73f998ddda

EBCDIC

BCD

null

## Extended Binary Coded Decimal Interchange Code 확장 이진화 십진법 교환 부호( > [확장 이진화 십진법 교환 부호 - 위키백과, 우리 모두의 백과사전](https://ko.wikipedia.org/wiki/%ED%99%95%EC%9E%A5_%EC%9D%B4%EC%A7%84%ED%99%94_%EC%8B%AD%EC%A7%84%EB%B2%95_%EA%B5%90%ED%99%98_%EB%B6%80%ED%98%B8)

47243ad2398d47eca866395c6857b675

Factorial

Number Theory Notion

Mar 17, 2023

Alan Jo

Mar 17, 2023

[Stirling’s Formula](https://texonom.com/stirlings-formula-e41933453d844d278269dc90647d8b96) > [99% 모르는 팩토리얼의 비밀!](https://youtube.com/watch?v=9Ti8EhK4eRw&feature=shares)

5b4059c4407b4c2f9c3be0f8357ae7ae

Infinite

Number Theory Notion

May 5, 2021

Alan Jo

Jan 16, 2022

무한 앞에서 모든 유한한 값은 0이되기도 하지만 무한 앞에서 모든 가능성이 가능해진다 원숭이가 무한하게 타자를 치는 것 안에 모든 문학이 있는 것처럼 무한은 숫자보다 과정이라고 여겨야

f4d16a6949084f6c9c9763e78492486b

Number System

Number Theory Notion

Feb 22, 2022

Alan Jo

Feb 22, 2022

4bfd6d8ac2e34aef9e0c847ccd7c1378

Real Number

Number Theory Notion

Apr 19, 2023

Alan Jo

Sep 3, 2023

unlike [Complex Number](https://texonom.com/complex-number-f3fcb42a92474099b8359581af59bde1), real number system is not [Algebraically Closed Field](https://texonom.com/algebraically-closed-field-a7ae2a1584f04bcfab20c148c98113cd) ### Real Number Type |Title| |:-:| |[Rational Number](https://texonom.com/rational-number-131e8af784874e9a9eb4cd4a39d517b6)| |[Irrational Number](https://texonom.com/irrational-number-1da708bf326a4e6cb8938e54179cb6bc)|

2c5c6fff21ab429da38ba685b40bacf3

Stirling’s Formula

Factorial

null

### ****Stirling's approximation**** $$ln(n!) = nln\;n - n + O(ln\;n)$$ > [Stirling's approximation](https://en.wikipedia.org/wiki/Stirling's_approximation) > [99% 모르는 팩토리얼의 비밀!](https://youtube.com/watch?v=9Ti8EhK4eRw&feature=shares)

e41933453d844d278269dc90647d8b96

Binary Number

Number Systems

Feb 22, 2022

Alan Jo

Feb 22, 2022

a28fb645048d4737afb24dcf3f69858d

Complex Number

Number Systems

Apr 19, 2023

Alan Jo

Jun 7, 2023

[Euler's Formula](https://texonom.com/eulers-formula-71f5337221904ba5b26e1398af5de0ab) [Field](https://texonom.com/field-a539c71297824a56a0a7139c871aaa08)

### [Algebraically Closed Field](https://texonom.com/algebraically-closed-field-a7ae2a1584f04bcfab20c148c98113cd) $$z = re^{i\theta}$$ ### Complex Number Notion |Title| |:-:| |[Imaginary Number](https://texonom.com/imaginary-number-0c4b551380914570ba48a73039f9fc96)| |[Complex Polar Coordinate](https://texonom.com/complex-polar-coordinate-53f5972726ff4d79b59c6f037ccf863a)| |[Complex Function](https://texonom.com/complex-function-fa0da3f578d0479495020b48ea3fe290)| |[Complex conjugate](https://texonom.com/complex-conjugate-251d154b491f499cae0d769f2529eaf0)| > [세상에서 가장 아름다운 수식을 이해해보자 (이과용)](https://www.youtube.com/watch?v=kgTSUZjVqas)

f3fcb42a92474099b8359581af59bde1

**Complex conjugate**

Complex Number Notion

May 13, 2023

Alan Jo

May 13, 2023

251d154b491f499cae0d769f2529eaf0

Complex Function

Complex Number Notion

May 13, 2023

Alan Jo

May 13, 2023

fa0da3f578d0479495020b48ea3fe290

Complex Polar Coordinate

Complex Number Notion

Apr 19, 2023

Alan Jo

Apr 19, 2023

[Polar Coordinate](https://texonom.com/polar-coordinate-affa11aeb26241cdabf2a592e73daf91)

### Normal Coordinate $R + Ii$ can be expressed in normal coordinate $(R, I)$ ### Polar Coordinate let $\theta = tan^{-1}\frac{I}{R}, r = \sqrt{R^2+I^2}$, then $R + Ii = r(cos\theta + isin\theta)$ by [Euler's Formula](https://texonom.com/eulers-formula-71f5337221904ba5b26e1398af5de0ab) $r(cos\theta + isin\theta) = re^{j\theta}$

53f5972726ff4d79b59c6f037ccf863a

Imaginary Number

Complex Number Notion

Apr 19, 2023

Alan Jo

Apr 19, 2023

i 또는 j로 사용한다 전류(I)랑 헷갈리거나 변수명으로 i많이 써서 j쓰는 곳이 실무에서 많다 MatLab이나 python등

0c4b551380914570ba48a73039f9fc96

Conjugate Symmetric

Complex conjugate Notion

May 13, 2023

Alan Jo

May 13, 2023

## Hermitian Symmetry 공액 대칭

dccf53586f444e49b9bd84efe348bdf3

**Hermitian function**

Complex Functions

May 13, 2023

Alan Jo

Jun 25, 2023

[Complex conjugate](https://texonom.com/complex-conjugate-251d154b491f499cae0d769f2529eaf0) is equal to the original function

af9c5415235f4d3da67d4e0e15389cff

**Recurrence relation**

Sequence Theory Notion

Oct 17, 2022

Alan Jo

Sep 14, 2023

[Mathematical induction](https://texonom.com/mathematical-induction-428dbeddb59940dca2267474cb23631c)

## 점화식 1. Guess the form of the solution 2. Verify by induction 3. Solve for constants or [The Master Method](https://texonom.com/the-master-method-26345dcafc17491ca0f75b6e137647db) ### **Recurrence relation Notion** |Title| |:-:| |[Recursion-tree method](https://texonom.com/recursion-tree-method-e45153b2672b4ff2ba0abea475c3b898)|

1c0b90005c30437fa61c5e049782e4be

Sequence

Sequence Theory Notion

Oct 3, 2021

Alan Jo

Sep 12, 2023

638838659a104ce3acc741ff806adf79

Series

Sequence Theory Notion

Oct 3, 2021

Alan Jo

Aug 29, 2023

### Sum of every sequence ### Famous Series |Title| |:-:| |[Taylor Series](https://texonom.com/taylor-series-66da8b67bfb041b2bdff0d16d12cf25a)| |[Fourier Series](https://texonom.com/fourier-series-034521e2dae346b8bf6b9095a602c9a7)| |[Power Series](https://texonom.com/power-series-c191ba27a1de41d8b2ea7e33b56fe9f6)| ### Series Notion |Title| |:-:| |[Convergent series](https://texonom.com/convergent-series-bd6d5a85531d4b89b7979707fe48b100)| |[Divergent series](https://texonom.com/divergent-series-1a5c65d9e0644d53a46f1b416fcabcc6)|

da206bccff4241e7afb1c73a3aac306d

Fourier Series

Famous Series

Mar 16, 2023

Alan Jo

Sep 13, 2023

[Fourier Transform](https://texonom.com/fourier-transform-5ee1f9063a9743dd9846ddc600286d5a) [Wave](https://texonom.com/wave-0b508bdcc4ac46759240dbc9b37d7994) [Periodic Function](https://texonom.com/periodic-function-3fa775a544934003bfbf39101551d1e1)

### Fourier Series Notion |Title| |:-:| |[CTFS](https://texonom.com/ctfs-bfcd6b43900e44d983ae5c6537338b8b)| |[DTFS](https://texonom.com/dtfs-924b5dfa43dd46638de66cc15f61b538)| |[Fourier Series Basis](https://texonom.com/fourier-series-basis-64de5af81a8643c3a9f92f8e8e9bb2bb)| |[Parseval's theorem](https://texonom.com/parsevals-theorem-d132edfc5b734643934a11b42bcdc075)| |[Fourier Series Tilda function](https://texonom.com/fourier-series-tilda-function-b8495f56f23a4f58897df1dadacbce11)| > [Fourier series](https://en.wikipedia.org/wiki/Fourier_series)

034521e2dae346b8bf6b9095a602c9a7

Taylor Series

Famous Series

Oct 3, 2021

Alan Jo

Aug 29, 2023

[Calculus](https://texonom.com/calculus-8e37e7dac61e435da6a6970da7296f73)

도함수들의 한 점에서의 값으로 계산된 항의 무한합으로 해석함수를 나타내는 방법 ### Taylor Series Notion |Title| |:-:| |[Taylor's Theorem](https://texonom.com/taylors-theorem-ecb251f43b1744a6a0ac5fe53a9d28d4)| |[Taylor series expansion](https://texonom.com/taylor-series-expansion-f83425f12a0e4d7e9122395a85e300e3)|

66da8b67bfb041b2bdff0d16d12cf25a

Square wave

Non-sinusoidal waveforms

Mar 21, 2023

Alan Jo

Sep 20, 2023

### [Fourier Series](https://texonom.com/fourier-series-034521e2dae346b8bf6b9095a602c9a7) $$A\Sigma_{k=1}^\infty\frac{1}{k}sin(2\pi kx)$$ ### Template Gallery |Title| |:-:| |[Boxcar function](https://texonom.com/boxcar-function-fdf2d3b368f540bbbdaab928819f8da4)| > [Square wave](https://en.wikipedia.org/wiki/Square_wave)

f730f0419f3843c389357eecc59f8d94

Taylor's Theorem

Taylor Series Notion

Oct 3, 2021

Alan Jo

Aug 29, 2023

ecb251f43b1744a6a0ac5fe53a9d28d4

Taylor series expansion

Taylor Series Notion

Oct 3, 2021

Alan Jo

Aug 29, 2023

[quadratic approximation](https://texonom.com/quadratic-approximation-3bd053180ba547e6a97ecf6b57f9af2d)

f83425f12a0e4d7e9122395a85e300e3

quadratic approximation

Taylor series expansion

null

second-orderTaylor series expansion

3bd053180ba547e6a97ecf6b57f9af2d

Convergent series

Series Notion

Mar 7, 2023

Alan Jo

Jun 11, 2023

bd6d5a85531d4b89b7979707fe48b100

Divergent series

Series Notion

Mar 9, 2023

Alan Jo

Jun 11, 2023

swing (oscillate) or go infinite

1a5c65d9e0644d53a46f1b416fcabcc6

Closure

Topology Notion

Jun 4, 2023

Alan Jo

Jun 4, 2023

f8ea031aff2a42e28ce310accdd3f38a

Compactness

Topology Notion

Jun 4, 2023

Alan Jo

Jun 4, 2023

4548634a95f645d2bf9f2027de247d5d

Euclidean Topology

Topology Notion

Jun 4, 2023

Alan Jo

Jun 4, 2023

d6b8e43b16264d8aaa37f576522e7063

Heine Borel Theorem

Topology Notion

Jun 4, 2023

Alan Jo

Jun 4, 2023

9364af6db0584563850234356131360e

M***anifold***

Topology Notion

Apr 30, 2023

Alan Jo

Jul 12, 2023

### 다양체 **국소적으로 유클리드 공간과 닮은 위상 공간** 시계에서 3시와 9시 사이 시간 차이를 거리가 아니라 원의 거리인 각으로 재는 것처럼 2차원이 아니라 1차원 ### M***anifold Notion*** |Title| |:-:| |[Manifold hypothesis](https://texonom.com/manifold-hypothesis-0cb18ad52c9049eea1cdb57a52e6ef93)| |[Latent Manifold](https://texonom.com/latent-manifold-849d5f347d0848808fa05cd904106782)| |[Kähler manifold](https://texonom.com/khler-manifold-fca18f34330d4e6b8e74083f19eef2c5)| > [Manifold](https://en.wikipedia.org/wiki/Manifold)

c6aa008ada9148309693c43c5033ad18

Poincare Conjecture

Topology Notion

Jun 4, 2023

Alan Jo

Jun 4, 2023

ed72bbeef3774339adc4c3dfc7e83282

Kähler manifold

Manifold Notion

Jul 12, 2023

Alan Jo

Jul 12, 2023

fca18f34330d4e6b8e74083f19eef2c5

Latent Manifold

null

849d5f347d0848808fa05cd904106782

M***anifold *****hypothesis**

Manifold Notion

May 24, 2023

Alan Jo

Jun 1, 2023

[Distance Algorithm](https://texonom.com/distance-algorithm-9b12bb4903424559a97f32c00f5ec351)

assume that data is concentrated around a low-dimensional manifold 기존 차원보다 manifolded dimension distance가 더 중요 > [Manifold hypothesis](https://en.wikipedia.org/wiki/Manifold_hypothesis)

0cb18ad52c9049eea1cdb57a52e6ef93

Hodge Theory

Kähler manifold Notion

Jul 12, 2023

Alan Jo

Jul 12, 2023

b69cbfba28e74a219b577372752bffae

**Hodge conjecture**

Hodge theory Notion

Jul 12, 2023

Alan Jo

Jul 12, 2023

> [Hodge conjecture](https://en.wikipedia.org/wiki/Hodge_conjecture) > [[난제] 호지추측 Hodge conjecture](https://www.youtube.com/watch?v=_OgxpHAbqAE)

9c13dfb6c4fe4aafa9a6faabf24415f0

archimedes cattle problem

Math Conundrums

Jun 13, 2020

Alan Jo

Aug 17, 2021

e2e3467d3cba449cb999fa65d3d1020d

**Asymptotic analysis**

Analysis Types

Sep 7, 2023

Alan Jo

Sep 7, 2023

### **Asymptotic analysis Notion** |Title| |:-:| |[Asymptotic Notation](https://texonom.com/asymptotic-notation-44f31e0bfbaa455cb26f55423ed292a7)| |[The Master Method](https://texonom.com/the-master-method-26345dcafc17491ca0f75b6e137647db)| > [Asymptotic analysis](https://en.wikipedia.org/wiki/Asymptotic_analysis)

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**Asymptotic Notation**

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### They are set of functions non-tight bounds lower case notation means less than or greater than (not equal) ### **Asymptotic Notations** |Title| |:-:| |[Big Θ](https://texonom.com/big-06ed557ed5c042c0b3b590320b4cf119)| |[Big O](https://texonom.com/big-o-a4d2742321a248b7b13068b1c4eb3215)| |[Big Ω](https://texonom.com/big-f4152bdb293042eca409b2e9f4f6b986)|

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The Master Method

Asymptotic analysis Notion

Sep 7, 2023

Alan Jo

Sep 10, 2023

[Divide and Conquer paradigm](https://texonom.com/divide-and-conquer-paradigm-9b10e85e70214b9b8f0bd293f5c297bd) [Recurrence relation](https://texonom.com/recurrence-relation-1c0b90005c30437fa61c5e049782e4be)

### The master method applies to recurrences of the form $$T(n) = aT(n/b) + f(n)$$ where $a\ge 1, b\ge1$, and $f$ is [Asympotically positive](https://texonom.com/asympotically-positive-b0f0899d95154b9a9a5d0620f0786f2b) ### The Master Method Notion |Title| |:-:| |[The Master Method Case](https://texonom.com/the-master-method-case-e93bcfbae2cf4ca3a4bd9b297fa45ec6)| |[Asympotically positive](https://texonom.com/asympotically-positive-b0f0899d95154b9a9a5d0620f0786f2b)| > [Master theorem (analysis of algorithms)](https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms))

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Big Θ

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$$\Theta(g(n)) = O(g(n)) \cap \Omega(g(n))$$ ### satisfy both upper bound and lower bound After a certain point the requirement matches, we can say Big θ $\Theta(n^2) = \Theta(1000n^2)$ theoretically but practically, it matters ### Big θ Notion |Title| |:-:| > [Big-θ (빅 세타) 표기법 (개념 이해하기) | 알고리즘 | Khan Academy](https://ko.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation) > [https://velog.io/@wan088/시간-복잡도-big-O-big-big-](https://velog.io/@wan088/시간-복잡도-big-O-big-big-)

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Big Ω

Asymptotic Notations

Sep 7, 2023

Alan Jo

Sep 10, 2023

### Non-tight Lower bound We write $f(n) = O(g(n))$ if there exist constants $c>0, n_0>0$ such that $0 \le cg(n) \le f(n) $ for all $n \ge n_0$ > [https://velog.io/@wan088/시간-복잡도-big-O-big-big-](https://velog.io/@wan088/시간-복잡도-big-O-big-big-)

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Big O

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### Non-tight Upper bound We write $f(n) = O(g(n))$ if there exist constants $c>0, n_0>0$ such that $0 \le f(n) \le cg(n)$ for all $n \ge n_0$ ### Big O Notion |Title| |:-:| |[Time Complexity](https://texonom.com/time-complexity-cd2b4611ca52442fa7a77c1af0f91b2f)| |[Space Complexity](https://texonom.com/space-complexity-d873465dea934141a0edfd11c6df65c5)| > [시간 복잡도 (big-O, big-Ω, big-θ)](https://velog.io/@wan088/시간-복잡도-big-O-big-big-)

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Space Complexity

Big O Notion

Apr 29, 2021

Alan Jo

Sep 7, 2023

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Time Complexity

Big O Notion

Apr 29, 2021

Alan Jo

Sep 7, 2023

### Time Complexities |Title| |:-:| |[Best case time complexity](https://texonom.com/best-case-time-complexity-b1172d189af5470e85d7c988940dcc81)| |[Average case time complexity](https://texonom.com/average-case-time-complexity-92e4cbb0014a4f5689e1e616f42e0fdd)| |[Best case time complexity](https://texonom.com/best-case-time-complexity-6667e4f2a30a473991033732aa7d75e4)| ### Time Complexity Notion |Title| |:-:| |[Complexity growth pattern](https://texonom.com/complexity-growth-pattern-c40a6ab140fa46df99db601e07aaa962)|

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Average case time complexity

Time Complexities

Sep 7, 2023

Alan Jo

Sep 17, 2023

[Random Variable](https://texonom.com/random-variable-0bc19e0582784ec3a6e0de5b91296212)

## **expected complexity** > [Average-case complexity](https://en.wikipedia.org/wiki/Average-case_complexity)

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Best case time complexity

Time Complexities

Sep 7, 2023

Alan Jo

Sep 7, 2023

보통 의미없다

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Best case time complexity

Time Complexities

Sep 7, 2023

Alan Jo

Sep 7, 2023

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Complexity growth pattern

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Asympotically positive

The Master Method Notion

Sep 10, 2023

Alan Jo

Sep 10, 2023

### A function if there exists an $N$ such that $f(n) \gt 0$ for all $n\ge N$

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The Master Method Case

The Master Method Notion

Sep 7, 2023

Alan Jo

Sep 12, 2023

### Proof is on [Introduction to Algorithm](https://texonom.com/introduction-to-algorithm-e7e86c2968934959b53c2c2160bec4c6) proof is easy to understand ![](https://www.notion.so/image/https%3A%2F%2Fprod-files-secure.s3.us-west-2.amazonaws.com%2F0bf522c6-2468-4c71-99e3-68f5a25d4225%2Faee544bb-0bbc-4756-82ec-888ec58cee19%2FUntitled.png?table=block&id=31d702c1-ce01-45df-9c17-160f8d038702&cache=v2) ![](https://www.notion.so/image/https%3A%2F%2Fprod-files-secure.s3.us-west-2.amazonaws.com%2F0bf522c6-2468-4c71-99e3-68f5a25d4225%2Fa7dc6a24-54b1-4a4a-bfb8-6e272ad7605b%2FUntitled.png?table=block&id=f0847bdd-c435-478d-832f-8fa609b46c02&cache=v2) if $f(n)$ does not meet regularity condition, you can’t use the master theorem in the third case ### Example ![](https://www.notion.so/image/https%3A%2F%2Fprod-files-secure.s3.us-west-2.amazonaws.com%2F0bf522c6-2468-4c71-99e3-68f5a25d4225%2F43a55351-d727-4e4e-9852-a77f7764484c%2FUntitled.png?table=block&id=58c45eb6-fb81-4880-80aa-4847772f8caa&cache=v2) > [Master theorem (analysis of algorithms)](https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms))

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Extrapolation

Regression analysis Notion

Apr 12, 2023

Alan Jo

Apr 12, 2023

> [Extrapolation](https://en.wikipedia.org/wiki/Extrapolation) > [외삽법 (Extrapolation)](https://m.blog.naver.com/bisanghara/120183969936)

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Loss Function

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### Gradient Available 하나의 input data에 대해서 오차를 계산하는 함수 - [KL Divergence](https://texonom.com/kl-divergence-c7964872f5184a7baaf312605405aef6) - [Cross Entropy](https://texonom.com/cross-entropy-fefc71fd930a41a7842f39bccb3abcc9) decision-theoretic object that we are bringing in to quantify the negative consequences of error ### Loss Functions |Title| |:-:| |[Mean Squared Error](https://texonom.com/mean-squared-error-18a36a7c349f495e9f960c7b605ad8ed)| |[RMSE](https://texonom.com/rmse-f61d17a91b174155bd3562c23e67b804)| |[Binary Crossentropy](https://texonom.com/binary-crossentropy-a0d2492e4aee49c2b7926653041a541a)| |[Categorical Crossentropy](https://texonom.com/categorical-crossentropy-b883d1401ffc426daea9bac208c8c5df)| |[L1 Loss Function](https://texonom.com/l1-loss-function-809f81be29194faaa6adfb9057327bbb)| |[Zero-One Loss](https://texonom.com/zero-one-loss-077d869edbfc4c09ac4441d0e547e961)| |[Hinge Loss](https://texonom.com/hinge-loss-7ccd3c37b1c64cf5b27a6b1d36067c1a)| ### Loss Function Notion |Title| |:-:| |[Test Loss](https://texonom.com/test-loss-9148d37e324240d19e41f9319fa8a0ab)| |[Training Loss](https://texonom.com/training-loss-3316afa9758643118624ad288b809861)| > [0023 Loss & Metric - Deepest Documentation](https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0023/) ### Difference by Example > [What is the difference between a loss function and an error function?](https://stats.stackexchange.com/questions/359043/what-is-the-difference-between-a-loss-function-and-an-error-function) ### Loss Functions > [Module: tf.keras.losses | TensorFlow v2.12.0](https://www.tensorflow.org/api_docs/python/tf/keras/losses) > [Objective Function, Loss Function, Cost Function의 차이](https://ganghee-lee.tistory.com/28) [https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0023/](https://deepestdocs.readthedocs.io/en/latest/002_deep_learning_part_1/0023/)

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Ordinary Least Squares

Regression analysis Notion

Jun 25, 2023

Alan Jo

Jun 25, 2023

## OLS > [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares)

d19e8054a63141ad8b53e4efb974f2b3

Regression

Regression analysis Notion

Oct 6, 2021

Alan Jo

Apr 18, 2023

[Optimization](https://texonom.com/optimization-984100dcbc454057bffd1b97be6ef134) [Data Classification](https://texonom.com/data-classification-0ed6ffa3555b4879be09f0501f0b580c)

### the target variable is continuous ### Regressions |Title| |:-:| |[Linear Regression](https://texonom.com/linear-regression-0137f4fba64746f6bc28e059d0aa6dd1)| |[Logistic Regression](https://texonom.com/logistic-regression-8ed39aeb672648b1933df8911996d5d7)| ### Regression Notion |Title| |:-:| |[Cost Function](https://texonom.com/cost-function-66c21423cc7347909016b3423f2ada7e)| |[IRLS](https://texonom.com/irls-ca318bd9d30e4054b87415503e34c0f3)| |[Multi-Class Regression](https://texonom.com/multi-class-regression-b0971c1ace5240abb9d6032522766455)|

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****Regularization****

Regression analysis Notion

Mar 27, 2023

Alan Jo

Mar 27, 2023

## simpler [Regularization](https://texonom.com/regularization-c85433c8c0554eba8edf0035b7fd334c) concept is first adapted to [Ridge regression](https://texonom.com/ridge-regression-8520dc9911234f62bc6fa25976af0972) model Norm is 절댓값 ### ****Regularization Norms**** |Title| |:-:| |[L0 Norm](https://texonom.com/l0-norm-47471a6f6dea484fbf30a4c46cde8152)| |[L1 Norm](https://texonom.com/l1-norm-d316024c475e4eb691785783756bce57)| |[L2 Norm](https://texonom.com/l2-norm-38c15917350a4c82a11003474ac7d280)|

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**Residual Sum of Squares**

Regression analysis Notion

Jun 25, 2023

Alan Jo

Jun 25, 2023

## RSS > [Residual sum of squares](https://en.wikipedia.org/wiki/Residual_sum_of_squares)

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Test Loss

Loss Function Notion

May 9, 2023

Alan Jo

May 9, 2023

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Training Loss

Loss Function Notion

May 9, 2023

Alan Jo

May 9, 2023

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****Binary Crossentropy****

Loss Functions

Mar 27, 2023

Alan Jo

May 9, 2023

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****Categorical Crossentropy****

Loss Functions

Mar 27, 2023

Alan Jo

May 9, 2023

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Hinge Loss

Loss Functions

Jun 7, 2023

Alan Jo

Jun 7, 2023

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L1 Loss Function

Loss Functions

Mar 27, 2023

Alan Jo

May 9, 2023

![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F13e50fc7-c832-42d3-89ab-9a7bf6888ead%2FUntitled.png?table=block&id=467fbe92-c25d-4b74-87dc-7b3b71daa445&cache=v2) > [L1, L2 Norm, Loss, Regularization?](https://junklee.tistory.com/29)

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Mean Squared Error

Loss Functions

Mar 27, 2023

Alan Jo

Jun 13, 2023

[L2 Norm](https://texonom.com/l2-norm-38c15917350a4c82a11003474ac7d280)

## L2 ![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F764b1ec6-99b5-4466-9f8b-0dd793d8f300%2FUntitled.png?table=block&id=acbfe6df-50bf-4ab0-ad00-192b80d9b686&cache=v2) > [Mean squared error](https://en.wikipedia.org/wiki/Mean_squared_error) > [[딥러닝] 목적/손실 함수(Loss Function) 이해 및 종류](https://needjarvis.tistory.com/567)

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RMSE

Loss Functions

Mar 27, 2023

Alan Jo

May 16, 2023

[L2 Norm](https://texonom.com/l2-norm-38c15917350a4c82a11003474ac7d280)

### Root mean squared Error

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Zero-One Loss

Loss Functions

Jun 7, 2023

Alan Jo

Jun 7, 2023

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Cost Function

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## Error Function, J function 모든 input dataset에 대해서 오차를 계산하는 함수 each parameter $$J(\theta) = \frac{1}{2}\Sigma_{i=1}^n(h_\theta(x^{(i)}) - y^{(i)})^2$$ ### Matrix Vector notation $$J(\theta) = \frac{1}{2}(X\theta - \vec{y})^T(X\theta - \vec{y})$$ ### Lest squares cost function > [Objective Function, Loss Function, Cost Function의 차이](https://ganghee-lee.tistory.com/28) [Least Squares Error](https://texonom.com/least-squares-error-93b101745714452aa1656b71320e08b6)

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IRLS

Regression Notion

Mar 16, 2023

Alan Jo

Apr 18, 2023

[Newton–Raphson method](https://texonom.com/newtonraphson-method-0bc249bfb0fe4e498829110bf785eb6c)

### **Iterative Re-weighted Least Squares** Iterative re-weighted least squares (IRLS) is an optimization algorithm used to solve nonlinear least squares problems. It is an iterative algorithm that updates the solution in each step until convergence is achieved. In each iteration, the algorithm re-weights the observations based on the current estimate of the solution. This makes the algorithm more robust to outliers and can help it converge more quickly. update rule is similar with

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Multi-Class Regression

Regression Notion

Mar 21, 2023

Alan Jo

Mar 21, 2023

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Least Squares Error

Cost Function

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$$E_{LS}$$ 최적화를 수행하는 도구로 사용하는 것이 Least Squares 관찰값들은 매번 Noise가 존재할 수 있다. 이런 경우 정확한 parameter값을 구하는데 error가 최소화되도록 구하는 방법 error는 보통 관찰값과 추정값의 제곱으로 표현 어떤 식에서 구해야할 parameter가 4개일 때, 이 때 실험을 5번 진행되어 관찰되는 값이 parameter보다 많을 때 정확한 parameter 값을 구하기 위하여 사용하는 최적화 방법이 바로 Least Squares > [Untitled](https://taeyoung96.github.io/slam/SLAM_02/)

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Linear Regression

Regressions

Mar 9, 2023

Alan Jo

Jun 7, 2023

[Objective function](https://texonom.com/objective-function-5bf6c1fc7c854b5aaaacd17d5e094464)

### Linear Predictor (minimal [Perceptron](https://texonom.com/perceptron-1deb66f486d54c93bb928d8afba1864c)) $$h(x) = \Sigma_{i=0}^{d}\theta_ix_i = \theta^Tx$$ in Linear Regression there is always global optimum - update $\theta$ - [Gradient Descent](https://texonom.com/gradient-descent-c1342b13182f4fb6959023a75b5e2ff8) like algorithms > - add gradient of derivative of cost function J $$J(\theta) = \frac{1}{2}\Sigma_{i=1}^n(h_\theta(x^{(i)}) - y^{(i)})^2 = \frac{1}{2}(X\theta - \vec{y})^T(X\theta - \vec{y})$$ $$\theta := \theta + \alpha \Sigma(y_i - h_\theta(x_i))x_i$$ - Closed form - solve equation - assume $X^TX$ is invertible $$\theta = (X^TX)^{-1}X^T\vec{y}$$ ### Linear Regression Notion |Title| |:-:| |[Basis Function](https://texonom.com/basis-function-994de24f84dd437d9d563ca2a87652d6)| |[Locally weighted Linear Regression](https://texonom.com/locally-weighted-linear-regression-6f405fcc12f64e0b9ec061c87d81e1e2)| |[Ridge regression](https://texonom.com/ridge-regression-8520dc9911234f62bc6fa25976af0972)| |[Lasso Regression](https://texonom.com/lassoregression-01bff5049e0e40faa3c1ce3fb68f1c76)| |[LMS](https://texonom.com/lms-3646b634f8974a51b8184a23f3defa1d)| > [선형회귀 - 공돌이의 수학정리노트](https://angeloyeo.github.io/2020/08/24/linear_regression.html)

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Logistic Regression

Regressions

Mar 16, 2023

Alan Jo

May 21, 2023

[Sigmoid Function](https://texonom.com/sigmoid-function-417e17b995cf40c996cec78f699ea59b) [Logit Function](https://texonom.com/logit-function-104e55a12b234813a020cb09ec51c754) [Gaussian Discriminant Analysis](https://texonom.com/gaussian-discriminant-analysis-67ec78f9fd6944e68671c610aeadb079) [Data Classification](https://texonom.com/data-classification-0ed6ffa3555b4879be09f0501f0b580c)

Sigmoid 함수를 사용하여 0과 1 사이의 값을 출력, log-likelihood를 최대화 ![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Ff587cf29-0b05-4917-a3f4-6d1add80bbb5%2FUntitled.png?table=block&id=11da7a73-d114-412e-80a4-98503f07fc58&cache=v2) to make convex cost function ![](https://www.notion.so/image/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F3cd79553-1597-4710-8b7a-e4cf3761cdb6%2FUntitled.png?table=block&id=403a0496-8b25-4ad8-9e9f-40e88eedabea&cache=v2) ### Logistic Regression Notion |Title| |:-:| |[Multi-Class Logistic Regression](https://texonom.com/multi-class-logistic-regression-545d3a376a4d49dd8845111405909327)| $$h_\theta(x) = g(\theta^Tx) = \frac{1}{1 + e^{-\theta^Tx}}$$ This leads to $p(u|x;\theta)$ described by [Bernoulli Distribution](https://texonom.com/bernoulli-distribution-bddaf7e81c9a4df8ab389521fc56e676) $$p(y|x;\theta) = (h_\theta(x))^y(1 - h_\theta(x))^{1 - y}$$ the [Likelihood function](https://texonom.com/likelihood-function-0750726af35a45bb86036dbc7507f725) is $$l(\theta) = logL(\theta) = \Sigma_{i=1}^ny^{(i)}logh(x^{(i)}) + (1 - y^{(i)})log(1 - h(x^{(i)}))$$ But 로지스틱 회귀의 y값은 you can use [Newton–Raphson method](https://texonom.com/newtonraphson-method-0bc249bfb0fe4e498829110bf785eb6c) or [Stochastic Gradient Descent](https://texonom.com/stochastic-gradient-descent-d8b8d008e0a34f4bb55175ffba21db44) and prior one achieves **faster convergence** or [IRLS](https://texonom.com/irls-ca318bd9d30e4054b87415503e34c0f3) is much faster difference between [Linear Regression](https://texonom.com/linear-regression-0137f4fba64746f6bc28e059d0aa6dd1) is that usage of sigmoid function or logistic function

8ed39aeb672648b1933df8911996d5d7

Basis Function

Linear Regression Notion

Mar 14, 2023

Alan Jo

Apr 6, 2023

[Overfitting](https://texonom.com/overfitting-24c3b183372845e8999ad7f7a0ba5035)

## Feature Map $\phi(x)$, Basis Function $h(x)$ $$h_\theta(x) = \Sigma_{l=1}^L\theta_l\phi_l(x) = \theta^T\phi(x)$$ we can extend the linear regression into a linear combination of basis functions $${R^d \rightarrow R} \;\; (x \in R^d)$$ - polynomial basis $x^l$ - gaussian basis $exp(-\frac{||x - \mu_l||^2}{2\sigma^2})$ ### Using nonlinear basis functions, we allow the function to be nonlinear

994de24f84dd437d9d563ca2a87652d6

****Lasso Regression****

Linear Regression Notion

Mar 27, 2023

Alan Jo

Mar 27, 2023

[L1 Norm](https://texonom.com/l1-norm-d316024c475e4eb691785783756bce57)

> [Regularization(정규화): Ridge regression/LASSO](https://sanghyu.tistory.com/13) > [Ridge regression(릿지 회귀)와 Lasso regression(라쏘 회귀) 쉽게 이해하기](https://rk1993.tistory.com/entry/Ridge-regression와-Lasso-regression-쉽게-이해하기)

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LMS

Linear Regression Notion

Apr 6, 2023

Alan Jo

Apr 6, 2023

## Least Mean Square $\alpha$ is hyperparameter for fastness of learning ### LMS Notion |Title| |:-:| |[LMS with Kernel Trick](https://texonom.com/lms-with-kernel-trick-b2130bc9ef1d41bbb779435512612157)|

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