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Mathematical Aspects of Data Analysis
Christoph B
orgers
Mathematics Department
Tufts University
Spring 2017
Part I
Dimension Reduction Via PCA
1. The principal direction of a cloud of
points in the plane
Think about n points in R2 .
1
0
-1
-1
0
1
CLOUD O
In this document:
1) covering points with disks
2) Ramsey number lower bound
3) Tournaments
4) Large bipartite subgraphs
5) Large independent sets (weak Turan theorem)
6) Large dominating sets
TO DO: hypergraph coloring
Treat this as a zero
0
0
0
0
0
0
0
1
From Euler formula on planar graph E 3V 6, claim that at least 80% of vertices have degree less than 30.
Suppose by contradiction, if more than 20% of vertices have degree 30, then E (30 V 0.2)/2 = 3V (We have to divide
by 2 because in the smallest case
Translation
Fabula Cincinnati
1. 1.
Fabula Cincinnati est fabula virorum Romanorum multorum.
The tale of Cincinnatus is the story of many Roman men.
1. 2.
Pro Roma Cincinnatus viros multos duxit. Etiam erat agricola.
For Rome, and brought him a man of gre
Math 70
Linear Algebra
TUFTS UNIVERSITY
Department of Mathematics
Exam II
November 16, 2015
Sections 1 and 2
Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devices
must be turned off and put away during the
Math 70
Linear Algebra
TUFTS UNIVERSITY
Department of Mathematics
Exam II
March 30, 2015
Sections 1 and 2
Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devices
must be turned off and put away during the exa
Math 70
Linear Algebra
TUFTS UNIVERSITY
Department of Mathematics
Exam 1
Feb 25, 2013
All sections
Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devices
must be turned off and put away during the exam. Unle
Math 70
Linear Algebra
TUFTS UNIVERSITY
Department of Mathematics
Exam I
October 12, 2015
Sections 1 and 2
Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devices
must be turned off and put away during the ex
Math 70
Linear Algebra
TUFTS UNIVERSITY
Department of Mathematics
Exam 2
Nov 5, 2012
All sections
Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devices
must be turned off and put away during the exam. Unles
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Email: [email protected]
Cell (617-388-4956)
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Accurate, thorough, and persistent individual seeking a position
as an Administrative Assistant with Beth Israel Boston , Ma
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borgers
2009/11/17
page 211
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Appendix D
Solutions to Selected Exercises
1.3. When there are 4 candidates, each voter
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2009/11/17
page 75
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Chapter 11
Winner Selection versus
Ranking
We have considered winner selection methods so
Lecture 7
Psychiatric Medications:
Psychiatric medications are the third most prescribed
Mental Illness:
Defined by the culture and society that you live in and the current times; view of mental
illness determined the treatment
o Ancient Greeks and Roma
Lecture 5_Oct 26
Amphetamines:
History of Amphetamines:
o First used in the late 1800s
o There are 2 forms: dextro-amphetamines & meth-amphetamines
o Beginning in the 1920s, they have been used to treat obesity, alcoholism, bedwetting, depression,
schizo
Lecture 2_Sep 21
Neuron:
The building block of human brains
Can make new connections but NEVER grow back
Purpose: to communicate into the nervous system and to create a response within the system
Components of a neuron:
o Dendrites pick up information fro
Opiates
Have been around for thousands of years because of opium poppy plant; first in the
1600s in China
Opiate Addiction:
o Laborers from China were brought in to work on the railroad and they brought
with them their custom of smoking opium
o Common a
Over the Counter Medications (OTCs):
Lecture 4_Oct 19
They are dangerous because:
o Unclear label instructions
o Contain several ingredients
o Can be used for multiple symptoms (not specific)
o Public thinks they are safe
o In English (non-English speaker
Hallucinogens:
Common natural hallucinogens:
o Mushrooms containing psilocybin
o Nutmeg containing myristica oil
o Morning glory seeds
o Peyote cactus containing mescaline: can cause loss of sense of time (2 hours)
o Ergot: fungus that grows on rye plant
Lecture 3_Sep 28
Caffeine:
A member of Xanthines class oldest known stimulants (including theophyllinetea,
theobrominechocolate)
Antagonist drug blocking adenosine neurotransmitter, which creates a calming or sedative effect
How does it work?
o Rapid abso