Numerical integration of DAE's seminar

Scientific Computing Group Consistent Initial Values x (t) =sin(t) x 0 (t) +y (t) =0 x 0 is a consistent initial value, if there existsa smooth solution thatfullfillsx (t 0) =x 0 and this solution is definedforallt.


On SomeCriteria for the Balanced Projectivity of Modules over ...

Consider the following commutative diagram with exact rows, and torsion-free modules Mand M *: 0 L N α μ M τ 0 0 L * N * β M * 0 If the top row is balanced-exact and there existsa homomorphismρ : M * →M suchthatτρ =1 , then the bottom row is likewise balanced-exact.


Sample Problems for Exam II

Recall that a space Xis locally path-connected if for eachx∈Xandeach neighborhood Uofx, there existsa neighborhood Vofxsuchthat V⊆Uand Vis path-connected.


Real valued functions in Pointfree Topology

(2) For everyf 2 USC (X) and everyg 2 LSC (X) withf g, there existsa continuoush 2 C (X) such thatf h g. M. Katˇ etov, On real-valued functions in topological spaces, Fund. Math. 38 (1951) 85-91; correction 40 (1953) 203-205.


Undecidability everywhere

There existsa nite group! 2. Thef.p. group Zcannotbe embedded in any nite group. Other Markovproperties: trivial, abelian, nilpotent, solvable, free, torsion-free.


Separating Hyperplanes

Then there existsa hyperplane separating b from K . THEOREM: K R n convex, nonempty. b 62 K . Then K can be separated from b by a hyperplane. COROLLARY: SUPPORTING HYPERPLANE THEOREM: b a boundary point of a convex set K R n .


The Deduction Theorem, Optimal Proof Systems, and Complete ...

There existsa polynomial psuch that for all printable sets of tautologies * the proof system P ∪* is closed under substitutions of variables with respect top.


Chapter1. Limits and Con tinuity

We say that f ( x )has right-hand limit Lat x 0,and write lim x→x + 0 f ( x )= L if for every number *> 0 there existsa corresponding number δ> 0such that for all x x 0 <x<x 0 + δ⇒|f ( x ) −L|<*. 6


Math 231BPartial Differential Equations

B [ u;v ] is bounded and hence by the Riesz Representation Theorem (RRT) there existsa unique element Au such that B [ u;v ]= hAu;vi; for all v2H: However the map u!



Conversely, by Feldman-Moore[10], if E is an arbitrary count-ableBorel equivalence relation on the standard Borelspace X , then there existsa countable group G an daB or el action of G on X such that E = E X G.


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