719 research outputs found
The geometry of blueprints. Part I: Algebraic background and scheme theory
In this paper, we introduce the category of blueprints, which is a category
of algebraic objects that include both commutative (semi)rings and commutative
monoids. This generalization allows a simultaneous treatment of ideals resp.\
congruences for rings and monoids and leads to a common scheme theory. In
particular, it bridges the gap between usual schemes and -schemes
(after Kato, Deitmar and Connes-Consani). Beside this unification, the category
of blueprints contains new interesting objects as "improved" cyclotomic field
extensions of and "archimedean valuation
rings". It also yields a notion of semiring schemes.
This first paper lays the foundation for subsequent projects, which are
devoted to the following problems: Tits' idea of Chevalley groups over
, congruence schemes, sheaf cohomology, -theory and a unified
view on analytic geometry over , adic spaces (after Huber),
analytic spaces (after Berkovich) and tropical geometry.Comment: Slightly revised and extended version as in print. 51 page
The AKNS-hierarchy
We present here an overview for the Encyclopaedia of Mathematics of the various forms and properties of this system of equations together with its geometric and Lie algebraic background
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Injectivity radius of representations of triangle groups and planar width of regular hypermaps
We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type with a given planar width
Flows and stochastic Taylor series in Ito calculus
For stochastic systems driven by continuous semimartingales an explicit
formula for the logarithm of the Ito flow map is given. A similar formula is
also obtained for solutions of linear matrix-valued SDEs driven by arbitrary
semimartingales. The computation relies on the lift to quasi-shuffle algebras
of formulas involving products of Ito integrals of semimartingales. Whereas the
Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is
classically expanded as a formal sum indexed by permutations, the analogous
formula in Ito calculus is naturally indexed by surjections. This reflects the
change of algebraic background involved in the transition between the two
integration theories
Algebraic Structures in the Coupling of Gravity to Gauge Theories
This article is an extension of the author's second master thesis [1]. It
aims to introduce to the theory of perturbatively quantized General Relativity
coupled to Spinor Electrodynamics, provide the results thereof and set the
notation to serve as a starting point for further research in this direction.
It includes the differential geometric and Hopf algebraic background, as well
as the corresponding Lagrange density and some renormalization theory. Then, a
particular problem in the renormalization of Quantum General Relativity coupled
to Quantum Electrodynamics is addressed and solved by a generalization of
Furry's Theorem. Next, the restricted combinatorial Green's functions for all
two-loop propagators and all one-loop divergent subgraphs thereof are
presented. Finally, relations between these one-loop restricted combinatorial
Green's functions necessary for multiplicative renormalization are discussed.
Keywords: Quantum Field Theory; Quantum Gravity; Quantum General Relativity;
Quantum Electrodynamics; Perturbative Quantization; Hopf Algebraic
RenormalizationComment: 57 pages, 259 Feynman diagrams, article; minor revisions; version to
appear in Annals of Physic
Supersymmetric extension of Moyal algebra and its application to the matrix model
We construct operator representation of Moyal algebra in the presence of
fermionic fields. The result is used to describe the matrix model in Moyal
formalism, that treat gauge degrees of freedom and outer degrees of freedom
equally.Comment: to appear in Mod.Phys.Let
Algorithms for Determining the Order of the Group of Points on an EllipticCurve with Application in Cryptography
Eliptické křivky jsou rovinné křivky, jejíž body vyhovují Weierstrassově rovnici. Jejich hlavní využití je v kryptografii, kde představují důležitý nástroj k tvorbě těžko rozluštitelných kódů bez znalosti klíče, který je v porovnání s ostatními šifrovacími systémy krátký. Díky těmto přednostem jsou hojně využívány. Abychom mohli kódovat a dekódovat zprávy v systému eliptických křivek, musíme znát řád dané eliptické křivky. K jeho získání se mimo jiné používá Shanksův algoritmus a jeho vylepšená varianta, Mestreho algoritmus.The elliptic curves are plane curves whose points satisfy the Weierstrass equation. Their main application is in the cryptography, where they represent an important device for creating code which is hard to break without knowing the key and which is short in comparison with other encoding methods. The elliptic curves are widely used thanks to these advantages. To be able to code and decode in the elliptic curve cryptography we must know the order of the given elliptic curve. The Shank's algorithm and its improved version, the Mestre's algorithm, are used for its determining.
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