Computing and Optimization in the Physical and Social Sciences – Slides of Ahmadi

Computing and Optimization in the Physical and Social Sciences:

http://www.princeton.edu/~amirali/Public/Teaching/ORF363_COS323/F14/

Index of /~amirali/Public/Teaching/ORF363_COS323/F14

[ICO] Name Last modified Size Description

[PARENTDIR] Parent Directory
[   ] ORF363_COS323_F14_Lec1.pdf 2014-09-13 21:11 2.2M
[   ] ORF363_COS323_F14_Lec1.ppt 2014-09-13 21:15 8.0M
[   ] ORF363_COS323_F14_Lec2.pdf 2014-09-24 19:25 2.2M
[   ] ORF363_COS323_F14_Lec3.pdf 2014-09-25 23:54 5.7M
[   ] ORF363_COS323_F14_Lec4.pdf 2014-10-03 01:42 3.4M
[   ] ORF363_COS323_F14_Lec5.pdf 2014-10-04 12:03 3.1M
[   ] ORF363_COS323_F14_Lec6.pdf 2014-10-19 11:04 2.4M
[   ] ORF363_COS323_F14_Lec7.pdf 2014-10-18 10:18 2.6M
[   ] ORF363_COS323_F14_Lec8.pdf 2014-10-18 12:31 2.6M
[   ] ORF363_COS323_F14_Lec9.pdf 2014-11-03 02:02 4.4M
[   ] ORF363_COS323_F14_Lec10.pdf 2014-11-14 19:59 5.4M
[   ] ORF363_COS323_F14_Lec11.pdf 2015-01-04 16:51 1.5M
[   ] ORF363_COS323_F14_Lec12.pdf 2015-01-04 18:10 1.4M
[   ] ORF363_COS323_F14_Lec13.pdf 2015-01-04 18:28 4.4M
[   ] ORF363_COS323_F14_Lec14.pdf 2014-12-05 18:50 3.4M
[   ] ORF363_COS323_F14_Lec15.pdf 2014-12-11 15:25 4.3M
[   ] ORF363_COS323_F14_Lec16.pdf 2015-01-04 18:29 1.7M
[   ] cvx_examples.m 2014-10-04 01:47 1.1K

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Some illustrations of dynamical systems

Blog of Gabriel Peyré: https://twitter.com/gabrielpeyre

For example: Gradient flows: https://twitter.com/gabrielpeyre/status/1007865434320850944

The gradient field defines the steepest descent direction. The gradient flow dynamic defines a segmentation of the space into attraction bassins of the local minimizers.  pic.twitter.com/wW0flPEWor

— Gabriel Peyré (@gabrielpeyre) 16/6/2018.

 

MATH-F-420: Differential geometry of Verbitsky

Misha Verbitsky

Université Libre de Bruxelles

MATH-F-420: Differential geometry

Monday 16:00-18:00, P.OF.2058

Announcement for this course.

Slides:

Handouts:

Miscellanea: test problems, exam, etc.

 

Source: http://verbit.ru/ULB/GEOM-2015/

On the diffetential of a mapping 2

In the case {\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m} case, there is a linear map, which is “linear approximation” of {\psi}. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If {M} and {N} are smooth manifolds and {\psi \colon M \rightarrow N} is a smooth map then for each {m \in M}, the map

\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N

is defined by

\displaystyle d\psi(v)(f) = v(f \circ \psi)

is called the pushforward. Actually,

\displaystyle d\psi \colon TM \rightarrow TN.

Suppose that {\dim{M} \ge \dim{N}} and {f \colon M \rightarrow N} is a differentiable mapping. We have

Definition 1 The mapping {f} is called a trivial fibration (differentiable) on {N} if there exists a differential manifold {F}, is called fibre of {f}, and a diffeomorphism

\displaystyle \phi \colon M \rightarrow N \times F

such that the following diagram is commutative

sodo

Representation of a linear functional

We review results of Haviland and Riesz on the reperesentation of a linear functional.

Definition 1 Let {X} be a subset of {\mathbb{R}^n} and {C(X)} be algebra of continuous functions on {X}. A positive linear functional on {C(X)} is a linear functional {L} with {L(f) \ge 0} for all {f \in C(X)} such that {f(a) \ge 0, \forall a \in X}.

We recall Haviland’s result in \cite{Marshall2} (also see \cite{Ha1, Ha2}), with {\mathbb{R}[x_1, \dots, x_n]} denotes the ring of real multivariable polynomials:

Theorem 2 (Haviland) For a linear functional {L: \mathbb{R}[x_1, \dots, x_n]} and closed set {K} in {\mathbb{R}^n}, the following are equivalent:

  1. {L} comes from a Borel measure on {K}, i.e., {\exists} a Borel measure {\mu} on {K} such that, {\forall f \in \mathbb{R}[x_1, \dots, x_n], L(f) = \int f d\mu.}
  2. {L(f) \ge 0} holds for all {f \in \mathbb{R}[x_1, \dots, x_n]} such that {f \ge 0} on {K}.

In Haviland’s theorem, a positive linear functional extended from ring of real multivariable polynomials to larger subalgebra and this theorem can be derived as a consequence of the following Riesz Representation Theorem (see \cite[p. 77]{KS}):

Theorem 3 (Riesz Representation Theorem) Let {X} be a locally compact Hausdorff space and let {L: C_c(X) \rightarrow \mathbb{R}} be a positive linear functional. Then there exists a unique Borel measure {\mu} on {X} such that

\displaystyle L(f) = \int f d\mu, \forall f \in C_c(X).

{C_c(X)} is the algebra of continuous functions with compact support.