108 research outputs found
Numerical Integrators Development Environment
Tato práce se zabývá transformací soustav diferenciálních rovnic do polynomiálního tvaru. Takto transformované soustavy diferenciálních rovnic je poté možno řešit pomocí Taylorova rozvoje. Tato metoda umožňuje počítat numerické řešení počáteční úlohy dynamickou volbou řádu tak, aby byla splněna požadovaná přesnost. Práce matematicky dokazuje, že transformované soustavy diferenciálních rovnic mají stejné řešení, jako soustavy původních rovnic. Tato transformace je využitelná pro všechny matematické funkce běžně používané v technických aplikacích. Práce se dále zabývá optimalizací dané problematiky a implementuje ji v přiloženém programu taylor. Program umožňuje matematické a grafické zpracování řešení zadaných diferenciálních rovnic podle zvolených parametrů.This term project describes transformation of system of diferential equations into polynomial form. Such transformed systems of diferential equations can be subsequently solved using Taylor series. This method enables computing of initial problem's numeric solution using dynamical order selection in order to achieve required accuracy. The work mathematically proves, that transformed systems of diferential equations have the same solution as the original systems. This transformation can be used for all mathematic functions commonly used in technical applications. The work also focuses on optimization of given problem and implements it in programme taylor. This progamme enables user to solve given diferential equations with chosen parameters.
On the Chabauty space of locally compact abelian groups
This paper contains several results about the Chabauty space of a general
locally compact abelian group. Notably, we determine its topological dimension,
we characterize when it is totally disconnected or connected; we characterize
isolated points.Comment: 24 pages, 0 figur
An "almost" full embedding of the category of graphs into the category of groups
We construct a functor from the category of graphs to the category of groups
which is faithful and "almost" full, in the sense that it induces bijections of
the Hom sets up to trivial homomorphisms and conjugation in the category of
groups.
We provide several applications of this construction to localizations (i.e.
idempotent functors) in the category of groups and the homotopy category.Comment: 24 pages; to appear in Adv. Math
An almost full embedding of the category of graphs into the category of abelian groups
We construct an embedding G of the category of graphs into the category of
abelian groups such that for graphs X and Y we have Hom(GX,GY)=Z[Hom(X,Y)], the
free abelian group whose basis is the set Hom(X,Y). The isomorphism is
functorial in X and Y. The existence of such an embedding implies that,
contrary to a common belief, the category of abelian groups is as complex and
comprehensive as any other concrete category. We use this embedding to settle
an old problem of Isbell whether every full subcategory of the category of
abelian groups, which is closed under limits, is reflective. A positive answer
turns out to be equivalent to weak Vopenka's principle, a large cardinal axiom
which is not provable but believed to be consistent with standard set theory.
Several known constructions in the category of abelian groups are obtained as
quick applications of the embedding. In the revised version we add some
consequences to the Hovey-Palmieri-Stricland problem about existence of
arbitrary localizations in a stable homotopy categoryComment: 20 page
Indestructibility of Vopenka's Principle
We show that Vopenka's Principle and Vopenka cardinals are indestructible
under reverse Easton forcing iterations of increasingly directed-closed partial
orders, without the need for any preparatory forcing. As a consequence, we are
able to prove the relative consistency of these large cardinal axioms with a
variety of statements known to be independent of ZFC, such as the generalised
continuum hypothesis, the existence of a definable well-order of the universe,
and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic
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