324 research outputs found
The tame-wild principle for discriminant relations for number fields
Consider tuples of separable algebras over a common local or global number
field, related to each other by specified resolvent constructions. Under the
assumption that all ramification is tame, simple group-theoretic calculations
give best possible divisibility relations among the discriminants. We show that
for many resolvent constructions, these divisibility relations continue to hold
even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is
corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section
6.
On Jordan's measurements
The Jordan measure, the Jordan curve theorem, as well as the other generic
references to Camille Jordan's (1838-1922) achievements highlight that the
latter can hardly be reduced to the "great algebraist" whose masterpiece, the
Trait\'e des substitutions et des equations alg\'ebriques, unfolded the
group-theoretical content of \'Evariste Galois's work. The present paper
appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte
der Mathematik (1868-1942) for providing an overview of Jordan's works. On the
one hand, we shall especially investigate the collective dimensions in which
Jordan himself inscribed his works (1860-1922). On the other hand, we shall
address the issue of the collectives in which Jordan's works have circulated
(1860-1940). Moreover, the time-period during which Jordan has been publishing
his works, i.e., 1860-1922, provides an opportunity to investigate some
collective organizations of knowledge that pre-existed the development of
object-oriented disciplines such as group theory (Jordan-H\"older theorem),
linear algebra (Jordan's canonical form), topology (Jordan's curve), integral
theory (Jordan's measure), etc. At the time when Jordan was defending his
thesis in 1860, it was common to appeal to transversal organizations of
knowledge, such as what the latter designated as the "theory of order." When
Jordan died in 1922, it was however more and more common to point to
object-oriented disciplines as identifying both a corpus of specialized
knowledge and the institutionalized practices of transmissions of a group of
professional specialists
Solving a Class of Higher-Order Equations over a Group Structure
In recent years, symbolic and constraint-solving techniques have been making major advances and are continually being deployed in new business and engineering applications. A major push behind this trend has been the development and deployment of sophisticated methods that are able to comprehend and evaluate important sub-classes of symbolic problems (such as those in polynomial, linear inequality and finite domains). However, relatively little has been explored in higher-order domains, such as equations with unknown functions. This paper proposes a new symbolic method for solving a class of higher-order equations with an unknown function over the complex domain. Our method exploits the closure property of group structure (for functions) in order to allow an equivalent system of equations to be expressed and solved in the first-order setting. Our work is an initial step towards the relatively unexplored realm of higher-order constraint-solving, in general; and higher-order equational solving, in particular. We shall provide some theoretical background for the proposed method, and also prototype an implementation under Mathematica. We hope that our foray will help open up more sophisticated applications, as well as encourage work towards new methods for solving higher-order constraints.Singapore-MIT Alliance (SMA
The Tame-Wild Principle for Discriminant Relations for Number Fields
Consider tuples ( K1 , … , Kr ) of separable algebras over a common local or global number field F1, with the Ki related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of Ki ∕ F . We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics
Tensor network and (-adic) AdS/CFT
We use the tensor network living on the Bruhat-Tits tree to give a concrete
realization of the recently proposed -adic AdS/CFT correspondence (a
holographic duality based on the -adic number field ). Instead
of assuming the -adic AdS/CFT correspondence, we show how important features
of AdS/CFT such as the bulk operator reconstruction and the holographic
computation of boundary correlators are automatically implemented in this
tensor network.Comment: 59 pages, 18 figures; v3: improved presentation, added figures and
reference
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