Minimum mean square error estimation and approximation of the Bayesian updateAlexander Litvinenko
We develop a Bayesian update surrogate. Our formula allows us to update polynomial chaos coefficients. In contrast to classical Bayesian approach, we suggest to update PCE coefficients. We show that classical Kalman filter is a particular case of our update.
Lattice rules are one of the two main classes of methods for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) integration. In this tutorial, we recall the definition and summarize the key properties of lattice rules. We discuss what classes of functions these rules are good to integrate, and how their parameters can be chosen in terms of variance bounds for these classes of functions. We consider integration lattices in the real space as well as in a polynomial space over the finite field F2. We provide various numerical examples of how these rules perform compared with standard Monte Carlo. Some examples involve high-dimensional integrals, others involve Markov chains. We also discuss software design for RQMC and what software is available.
Minimum mean square error estimation and approximation of the Bayesian updateAlexander Litvinenko
We develop a Bayesian update surrogate. Our formula allows us to update polynomial chaos coefficients. In contrast to classical Bayesian approach, we suggest to update PCE coefficients. We show that classical Kalman filter is a particular case of our update.
Lattice rules are one of the two main classes of methods for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) integration. In this tutorial, we recall the definition and summarize the key properties of lattice rules. We discuss what classes of functions these rules are good to integrate, and how their parameters can be chosen in terms of variance bounds for these classes of functions. We consider integration lattices in the real space as well as in a polynomial space over the finite field F2. We provide various numerical examples of how these rules perform compared with standard Monte Carlo. Some examples involve high-dimensional integrals, others involve Markov chains. We also discuss software design for RQMC and what software is available.
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxjeremylockett77
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w.
2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the second part which is discussing basis and dimension.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
I am George P. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Perth, Australia. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
5. Recommendation 5
Easy to start, easy to finish :
Calculus : James Stewart, Calculus
Linear algebra : Gilbert Strang, Introduction to Linear
Algebra
Mathematical Statistics : Hogg, Introduction to
Mathematical Statistics
Warning! - easy to start, hard to finish :
Calculus : 김홍종, 미적분학
Linear Algebra : 이인석 - 선형대수와 군
Mathematical Statistics : 김우철 - 수리통계학
11. Vector, Vector space 11
Vector is element of Vector space.
Then, what is vector space?
12. Vector, Vector space 12
Let V be a vector space. Then for v, w, z ∈ V , r, s ∈ F
1. Vector space is abelian group. That is,
1.1 v + w = w + v
1.2 v + (w + z) = (v + w) + z
1.3 ∃0 ∈ V s.t. 0 + v = v + 0 = v for ∀v ∈ V
1.4 For ∀v ∈ V , ∃(−v) ∈ V s.t. v + (−v) = (−v) + v = 0
2. Vector space is F-module. That is,
2.1 r · (s · v) = (r · s) · v for ∀v ∈ V
2.2 For identity 1 ∈ R, 1 · v = v · 1 = v for ∀v ∈ V .
2.3 (r + s) · (v + w) = av + bv + aw + bw.
If F = R, we call V as real vector space, and if F = C, we
call V as complex vector space.
14. Vector, Vector space 14
1. Vector space is a set, where addition and scalar
multiplication is well-defined.
2. Vector is element of vector space
15. Linear combination 15
Let v1, v2, · · · , vn be vectors in vector space V . And let
a1, a2, · · · , an be real numbers. Then a linear combination of
v1, v2, · · · , vn is defined as:
a1v1 + a2v2 + · · · + anvn
16. Linearly independent 16
Let v1, v2, · · · , vn be vectors of a vector space V . Then, if
solution of equation with variables a1, a2, · · · , an expressed as
0 = a1v1 + a2v2 + · · · + anvn
has unique solution a1 = a2 = · · · = an = 0, then we say
v1, v2, · · · , vn is linearly independent.
17. Examples of linearly independent set 17
Let S = {(1, 0), (0, 1)}. Then, equation
0 = a · (1, 0) + b · (0, 1) = (a, b)
have unique solution a = b = 0. Thus, S = {(1, 0), (0, 1)} is
linearly independent.
18. Basis 18
Let V be a vector space and S = {v1, v2, · · · , vn} be linearly
independent vectors of V . Then if
Span(S) =< S >= {a1v1 +a2v2 +· · ·+anvn|ai ∈ R, i = 1, · · · , n}
becomes same as V , that is, if Span(S)=V , we call S as the
basis of V .
19. Dimension of vector space 19
Let V be a vector space. Then dimension of the vector space
is defined as:
dim V = max{|S| : S ⊂ V, S is linearly independent set}
That is, dimension is maximum number of number of elements
of linearly independent subset of given vector space
20. Linear map 20
Let V, W be two vector space. Then a linear map between
vector spaces L : V → W satisfies:
1. L(v + w) = L(v) + L(w) for ∀v, w ∈ V
2. L(rv) = r · L(v) for ∀v ∈ V, r ∈ R
21. Fundamental Theorem of Linear Algebra 21
Theorem (Fundamental Theorem of Linear Algebra)
Let V , W be two vector spaces with dimension n, m,
respectively. And let L : V → W be a linear map between these
two vector spaces. Then, there is a matrix ML ∈ Mm,n(R) s.t.
L(v) = ML · v
for ∀v ∈ V . That is, the set of all linear map and the set
of all matrices is same. Or, equivalently, matrix and linear
map has 1-1 correspondence(same).
22. Linear map with Neural Network 22
Let X = {x1, x2, · · · , xn} be given dataset. Then neural
network N with L hidden layers with each activation function
σ1, σ2, · · · , σL+1 is expressed as follows:
N(xi) = σL+1(ML+1(· · · (M2(σ1(M1xi))) · · · ))
where Mj are matrices.(j = 1, · · · , L + 1)
23. Norm of Vector 23
Let V be a vector space with dimension n. And let
v = (v1, v2, · · · , vn) be vector. Then, we call Lp norm of V as:
||v||p =
p
|v1|p + |v2|p + · · · + |vn|p
=
n
i=1
|vi|p
1
p
Conventionally, if we say norm or Euclidean norm, we mean
L2 norm. Furthermore, if we say Manhattan norm or
Taxicab norm, we mean L1 norm.
24. Distance 24
Given a vector space V and set of positive real numbers
including 0, denoted as R∗ = R+ ∪ {0}, a distance d is a
function from V × V → R∗ which satisfies following properties :
1. d(v, v) = 0 for ∀v ∈ V
2. d(v, w) ≥ 0 for , w ∈ V
3. d(v, w) = d(w, v) for ∀v, w ∈ V
4. d(v, u) ≤ d(v, w) + d(w, u) for ∀v, w, u ∈ V
26. Inner Product 26
Let V be a vector space. Then, for a function ·, · : V × V → R,
if ·, · satisfies
1. v, v ≥ 0 for ∀v ∈ V
2. v, v = 0 if and only if v = 0
3. v, w = w, v for ∀v, w ∈ V
4. av, w = a v, w for ∀v, w ∈ V and a ∈ R
5. v + w, u = v, u + w, u for ∀v, w, u ∈ V
we call ·, · a inner product
27. Eigenvector, Eigenvalue 27
Let V be vector space and let A : V → V be a linear map. Then
if λ ∈ R and 0 = v ∈ V satisfies
Av = λv
we say v is eigenvector of A, λ is eigenvalue of A.
28. Eigenvector, Eigenvalue 28
How to find eigenvectors, eigenvalues?
Av = λv
⇔Av = λInv
⇔(A − λIn)v = 0
We said v = 0. Therefore, if (A − λIn) is invertible,
(A − λIn)−1
(A − λIn)v = 0
⇔Inv = 0
⇔v = 0
Contradiction. Therefore, (A − λIn) should not be invertible.
This means, eigenvalues of A should be solution of the equation
det(A − tIn) = 0
29. Eigenvector, Eigenvalue 29
Characteristic polynomial : φA(t) = det(tIn − A)
Eigenvalues : Solutions of φA(t) = 0 −→ We get n
eigenvalues if n × n matrix is given(including multiplicity).
31. Topology, Topological space 31
Let X be a set. Then topology TX ⊆ 2X defined on X
satisfies:
1. ∅, X ∈ TX
2. Let Λ be a nonempty set. For all α ∈ Λ, if Uα ∈ TX, then
α∈Λ
Uα ∈ TX
3. If U1, · · · , Un ∈ TX, then U1 ∩ · · · ∩ Un ∈ TX
If TX is a topology of X, we say that (X, TX) as topological
space. we just abbreviate (X, TX) as X. We say elements of
topology as open set
32. Homeomorphism 32
Let X, Y be two topological spaces. Then if there exists a
function f : X → Y s.t
1. f is continuous.
2. f is bijective. This means f−1 exists.
3. f−1 is continuous.
Then, we say that f is homeomorphism. And we say X and
Y are homeomorphic.
33. Examples of homeomorphic objects 33
Cup with one handle and donut(torus)1.
Two dimensional triangle and rectangle, and circle.
R ∪ {∞} and circle.
1
https://en.wikipedia.org/wiki/Homeomorphism
34. Topological manifold 34
Let M be a topological space. Then, if M satisfies:
1. For each p ∈ M, there exists an open set Up ∈ TX s.t.
p ∈ Up is homeomorphic to Rn.
2. M is Hausdorff space.
3. M is second countable.
36. Dimension of manifold 36
Let M be a manifold. Then, for every p ∈ M, there exists an
open set Up ∈ TX s.t. Up is homeomorphic to RN . Then, we say
this n as dimension of manifold.
37. Embedding of manifold in euclidean space 37
Let M be a manifold. Then, there exists a euclidean space RN
s.t. M is embdded in RN . We say this N as dimension of
embedding space.
38. Manifold Hypothesis 38
Let X = {x1, · · · , xn} be a set of n data points. Then, Manifold
hypothesis is:
X consists a manifold that is embedded in high dimension,
which is in fact low dimensional manifold.
39. Manifold conjecture 39
Let X = {x1, · · · , xn} be a set of n data points. Then, Manifold
conjecture is:
What is the exact expression of manifold hypothesis?
How does the manifold look like?
41. Universal Approximation Theorem - prerequisite 41
1. Dense subset
Let X be a topological space and let S ⊆ X. Then, S is dense
in X means:
For every element p ∈ S, and for every open set Up ∈ TX
containing p, if Up ∩ X = 0, we say that S is dense in X.
One important example of dense subset of R is Q. We say that
Q is dense in R. We denote this as Q = R
42. Universal Approximation Theorem - prerequisite 42
2. Sigmoidal function
A sigmoidal function is a monotonically increasing continuous
function σ : R → [0, 1] s.t.
σ(x) =
1 x → +∞
0 x → −∞
43. Universal Approximation Theorem - prerequisite 43
3. Neural network with one hidden layer
A neural network with one hidden layer is expressed as:
N(x) =
N
j=1
αjσ(yT
j x + θj)
44. Universal Approximation Theorem 44
Theorem (Universal Approximation Theorem)
Let N be a neural network with one hidden layer and sigmoidal
activation function. And let C0 be set of continuous function.
Then, collection of N is dense in C0.