Lecture Note A localized MAPS

Lecture Notes on Construction of Genetic Linkage Maps

A note on blow-up in parabolic equations with local and localized sources

‎This note deals with the systems of parabolic equations with local and localized sources involving $n$ components‎. ‎We obtained the exponent regions‎, ‎where $kin {1,2,cdots,n}$ components may blow up simultaneously while the other $(n-k)$ ones still remain bounded under suitable initial data‎. ‎It is proved that different initial data can lead to different blow-up phenomena even in the same ...

A Note on Spectrum Preserving Additive Maps on C*-Algebras

Mathieu and Ruddy proved that if be a unital spectral isometry from a unital C*-algebra Aonto a unital type I C*-algebra B whose primitive ideal space is Hausdorff and totallydisconnected, then is Jordan isomorphism. The aim of this note is to show that if be asurjective spectrum preserving additive map, then is a Jordan isomorphism without the extraassumption totally disconnected.

18.782 Arithmetic Geometry Lecture Note 16

Our goal for this lecture is to prove that morphisms of projective varieties are closed maps. In fact we will prove something stronger, that projective varieties are complete, a property that plays a role comparable to compactness in topology. For varieties, compactness as a topological space does not mean much because the Zariski topology is so coarse. Indeed, every subset of An (and hence of ...

Combinatorics lecture notes (I). Combinatorial maps and compact data structures

These notes roughly correspond to the material of my 3 first lectures. The aim of these lectures is to present different directions of research in combinatorics, with enumeration and planarity as a common flavor. Pictures are missing in this version of the notes... too bad for those who missed the lectures. There are probably also many inaccuracies (let’s say errors...), please indicate them to...