Université Libre de Bruxelles
MATH-F-420: Differential geometry
Monday 16:00-18:00, P.OF.2058
Announcement for this course.
Miscellanea: test problems, exam, etc.
Here is the materials of Prof. Vakil ‘s course on Foundations of Algebraic Geometry 07-08 in Stanford University:
From on Warner’s book.
Smooth curve on manifold :
A mapping . Let , we define the tangent vector of the curve at is the vector
we apply the formula
where is an any function on .
Put and , the above formula implies
This is directional derivative.
This paragraph we follow the book of Rudin . On linear functionals, there is special relationship between integration and linear functionals. In , a vector space, for any positive measure , the mapping
example, let be the set of all continuous functions on the unit interval . Then
has two properties:
so it is a linear functional on . Moreover, this is a positive linear functional: if then .
Consider a segment and the class of all with and or support of is . So we get . It is important that the length of related to the values of the functional .
There is an important theorem of F. Riesz, this illustrates to above event
Theorem 1 (F. Riesz) Let be a closed subset of . Then every positive linear functional on there corresponds a finite positive Borel measure on such that
 W. Rudin, Real and Complex analysis, Mc.Graw-Hill, 1970.
Let be a real analytic function with . Let and be a compact subset in . Then the (classical) \L ojasiewicz inequality asserts that:
- There exist such that
\noindent Let be a real analytic function with and . The \L ojasiewicz gradient inequality asserts that:
As a consequence, in (1), the order of zero of an analytic function is finite, and if is close to then is close to the zero set of . However, if is not compact, the latter is not always true and the inequality (1) does not always hold. The inequality (2) is similar to (1), it is not true in the case of is non-compact.
Two inequalities (1) and (2) have some special cases. For example, in the inequality (1), if has only isolated zero, i.e. , this implies . Hence, we have
On the other hand, being different from (2), we have another inequality:
There are some relations between and in complex case and real cases…
We introduce to the classical moment problem, a short history. We refer to Akhiezer’s book:
N. I, Akhiezer, The Classical Moment Problem and Some Related Questions in
Analysis, Oliver & Boyd, Edinburgh/London, 1965.
and Christiansen’s notes: Moment (on Steven Miller’s page ).
The moment problem is a classical problem in analysis. This problem occurs for the first time in the work of Chebychev in 1873. After that, T.Stieltjes (1894-1895) and A.Markov consider more general case. Chebychev and A.Markov took the moment problem in the relationship with probability theory. The first solution and discussion of extended moment problem is due to Hamburger, he studied Classical moment problem (one-dimensional).
Classical moment problem (one-dimensional) Given an infinite sequence of real numbers (). Does there exist a positive Borel measure such that:
In general, we have Classical moment problem (multidimensional)
Given a function . Does there exist a positive Borel measure such that:
In the case one-dimensional moments, the sequence is a function and we have .
Two measure and are called equivalent if they satisfy:
In other words, we say they have same moments.
The measure is called determinate if there only exists such that and indeterminate otherwise.
The aims of the multidimensional moment problem are:
- To find necessary and sufficient conditions for existence of measure satisfying (*).
- To be able to decide determinacy.
- In the indeterminate case to give a complete description of all measures satisfying (*).