Why does volume go to zero? - Mathematics
january 2012 by arsyed
"I suppose you could say that adding a dimension "makes the volume bigger" for the hypersphere, but it does so even more for the unit you measure the volume with, namely the unit cube. So the numerical value of the volume does go towards zero.
Really, of course, it is apples to oranges because volumes of different dimensions are not commensurable -- it makes no sense to compare the area of the unit disk with the volume of the unit sphere.
All we can say is that in higher dimensions, a hypersphere is a successively worse approximation to a hypercube (of side length twice the radius). They coincide in dimension one, and it goes downward from there."
geometry
dimensionality
math
hypercube
hypersphere
curse
Really, of course, it is apples to oranges because volumes of different dimensions are not commensurable -- it makes no sense to compare the area of the unit disk with the volume of the unit sphere.
All we can say is that in higher dimensions, a hypersphere is a successively worse approximation to a hypercube (of side length twice the radius). They coincide in dimension one, and it goes downward from there."
january 2012 by arsyed
Analysis of Boolean Functions (Ryan O'Donnell)
january 2012 by arsyed
"I am writing a book about Fourier analysis of boolean functions. I will serialize it on this blog; I hope to publish two new sections each week.
Besides covering the basic theory, I intend to describe applications in theoretical computer science and other areas of math, including combinatorics, probability, social choice, and geometry."
blogs
books
math
compsci
functions
boolean
Besides covering the basic theory, I intend to describe applications in theoretical computer science and other areas of math, including combinatorics, probability, social choice, and geometry."
january 2012 by arsyed
Radically Elementary Probability and Statistics (Charlie Geyer)
january 2012 by arsyed
"my attempt to fill in some details that Nelson leaves out of his (very short) book"
math
probability
nonstandard-analysis
edward-nelson
january 2012 by arsyed
[1008.4831] Foundations of Inference (Kevin H. Knuth, John Skilling)
december 2011 by arsyed
"We present a foundation for inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying finite lattices of logical statements in a way that satisfies general lattice symmetries. With other applications in mind, our derivations assume minimal symmetries, relying on neither complementarity nor continuity or differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of rules governing quantification of the lattice. These rules form the familiar probability calculus. We also derive a unique quantification of divergence and information. Taken together these results form a simple and clear foundation for the quantification of inference."
math
statistics
probability
logic
information-theory
inference
foundations
december 2011 by arsyed
[math/9903160] The surprise examination or unexpected hanging paradox (Timothy Y. Chow)
december 2011 by arsyed
"The apparently trifling unexpected hanging paradox has generated an enormous philosophical literature. We introduce the mathematician to this literature, paying special attention to aspects that involve nontrivial mathematics. This xxx version of the paper contains an exhaustive bibliography that the editors of the Monthly deemed too lengthy to publish. The bibliography will be continually updated and readers are encouraged to inform the author of any omissions that they discover."
math
logic
paradox
history
game-theory
backward-induction
december 2011 by arsyed
The Matrix Factorization Jungle (Igor Carron)
september 2011 by arsyed
"Matrix Decompositions has a long history and generally centers around a set of known factorizations such as LU, QR, SVD and eigendecompositions. With the advent of new methods based on random projections and convex optimization that started in part in the compressive sensing literature, we are seeing a surge of very diverse algorithms dedicated to many different kinds of matrix factorizations with constraints based on rank, positivity, sparsity,... As a result of this large increase in interest, I have decided to keep a list of them here following the success of the big picture in compressive sensing."
math
matrix
factorization
optimization
machine-learning
compressed-sensing
ref
september 2011 by arsyed
Examples of common false beliefs in mathematics. - MathOverflow
july 2011 by arsyed
"To paraphrase Kai Lai Chung in his book "Green, Brown, and Probability",
"The Markov property means that the past has no after-effect on the future when the present is known; but beware, big mistakes have been made through misunderstanding the exact meaning of the words "when the present is known".""
math
probability
markov
"The Markov property means that the past has no after-effect on the future when the present is known; but beware, big mistakes have been made through misunderstanding the exact meaning of the words "when the present is known".""
july 2011 by arsyed
Symmetry Studies
june 2011 by arsyed
"Promoting applications of symmetry arguments to data analysis"
blogs
math
symmetry
statistics
june 2011 by arsyed
Pedagoguery Software: GrafEq
june 2011 by arsyed
"for producing graphs of implicit equations and inequalities"
software
math
plotting
inequalities
implicit
june 2011 by arsyed
Kill Math (Bret Victor)
may 2011 by arsyed
"Why did I spend my entire career working in a medium where I couldn't see what I was integrating? It seemed to me that analyzing a differential equation without exploring it in phase space was like analyzing a piece of sheet music without actually hearing it.
I thought about my instructor's question from years ago: "Why does the solution have two arbitrary constants?" And it was immediately obvious: you have to choose a starting point for the trajectory. The two constants correspond to the x and y where you drop your turtle and start integrating. I had solved "initial value problems" for years. But I had never been able to, literally, point my finger at the initial value."
math
visualization
education
ode
integration
dynamical
geometry
I thought about my instructor's question from years ago: "Why does the solution have two arbitrary constants?" And it was immediately obvious: you have to choose a starting point for the trajectory. The two constants correspond to the x and y where you drop your turtle and start integrating. I had solved "initial value problems" for years. But I had never been able to, literally, point my finger at the initial value."
may 2011 by arsyed
[1105.3943] Fun With Very Large Numbers
may 2011 by arsyed
"We give an example of a formula involving the sinc function that holds for every N = 0, 1, 2, ..., up to about 10^102832732165, then fails for all larger N. We give another example that begins to fail after about N ~ exp(exp(exp(exp(exp(exp(e)))))). This number is larger than the Skewes numbers."
math
numbers
almost-identity
papers
may 2011 by arsyed
Mathematical book reviews make for good reading (sometimes) (ChapterZero)
may 2011 by arsyed
"The author’s preoccupation with convexity occasionally produces grotesque results. His "geometric" proof of the Stone-Weierstrass approximation theorem employs the Hahn-Banach theorem, the Alaoglu-Bourbaki theorem, the Kreĭn-Milman theorem, and the Riesz-Kakutani theorem (whose proof in turn uses the Hahn-Banach theorem and the extremal disconnectedness of the Stone-Čech compactification of a discrete space); as technique runs amok, some may say that they really understand the Stone-Weierstrass theorem for the first time, but others will weep silently for Stone’s beautiful proof."
math
reviews
funny
may 2011 by arsyed
Sailing in state space (Rod Carvalho)
may 2011 by arsyed
"Lastly, we arrive at a most childish idea: in some cases, controller design can be viewed as shaping the streamlines differently in different parts of the state space. This is childish because it is purely conceptual. One obviously cannot design optimal controllers by doodling with crayons."
math
dynamical
control
may 2011 by arsyed
Top Facts - Gauss Facts
april 2011 by arsyed
"Gauss never runs out of room in the margin."
math
gauss
humor
april 2011 by arsyed
Theano v0.3.1 documentation
march 2011 by arsyed
"Theano is a Python library that allows you to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficiently." [...] "If numpy is to be compared to MATLAB and sympy to Mathematica, Theano is a sort of hybrid of the two which tries to combine the best of both worlds."
python
libs
scicomp
gpu
math
matrix
expression
compiler
march 2011 by arsyed
#31: Flavors of Geometry
march 2011 by arsyed
"An Elementary Introduction to Modern Convex Geometry, by Keith M. Ball, 1-58
Hyperbolic Geometry, by James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry, 59-115
Complex Dynamics in Several Variables, by John Smillie and Gregery T. Buzzard, 117-150
Volume Estimates and Rapid Mixing, by Béla Bollobás, 151-194"
books
math
geometry
convex
Hyperbolic Geometry, by James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry, 59-115
Complex Dynamics in Several Variables, by John Smillie and Gregery T. Buzzard, 117-150
Volume Estimates and Rapid Mixing, by Béla Bollobás, 151-194"
march 2011 by arsyed
PyDX
february 2011 by arsyed
"PyDX is a package for working with calculus (differential geometry), arbitrary precision arithmetic (using gmpy), and interval arithmetic."
python
libs
math
scicomp
automatic-differentiation
ode
differential-geometry
via:chl
february 2011 by arsyed
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