692 research outputs found
Permutations of Massive Vacua
We discuss the permutation group G of massive vacua of four-dimensional gauge
theories with N=1 supersymmetry that arises upon tracing loops in the space of
couplings. We concentrate on superconformal N=4 and N=2 theories with N=1
supersymmetry preserving mass deformations. The permutation group G of massive
vacua is the Galois group of characteristic polynomials for the vacuum
expectation values of chiral observables. We provide various techniques to
effectively compute characteristic polynomials in given theories, and we deduce
the existence of varying symmetry breaking patterns of the duality group
depending on the gauge algebra and matter content of the theory. Our examples
give rise to interesting field extensions of spaces of modular forms.Comment: 44 pages, 1 figur
The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems
Since its inception as a student project in 2001, initially just for the
handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library
has been continuously improved and extended by joining scrupulous research on
the theoretical foundations of (possibly non-convex) numerical abstractions to
a total adherence to the best available practices in software development. Even
though it is still not fully mature and functionally complete, the Parma
Polyhedra Library already offers a combination of functionality, reliability,
usability and performance that is not matched by similar, freely available
libraries. In this paper, we present the main features of the current version
of the library, emphasizing those that distinguish it from other similar
libraries and those that are important for applications in the field of
analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table
Interpretable Graph Networks Formulate Universal Algebra Conjectures
The rise of Artificial Intelligence (AI) recently empowered researchers to
investigate hard mathematical problems which eluded traditional approaches for
decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields
laying the foundations of modern mathematics -- is still completely unexplored.
This work proposes the first use of AI to investigate UA's conjectures with an
equivalent equational and topological characterization. While topological
representations would enable the analysis of such properties using graph neural
networks, the limited transparency and brittle explainability of these models
hinder their straightforward use to empirically validate existing conjectures
or to formulate new ones. To bridge these gaps, we propose a general algorithm
generating AI-ready datasets based on UA's conjectures, and introduce a novel
neural layer to build fully interpretable graph networks. The results of our
experiments demonstrate that interpretable graph networks: (i) enhance
interpretability without sacrificing task accuracy, (ii) strongly generalize
when predicting universal algebra's properties, (iii) generate simple
explanations that empirically validate existing conjectures, and (iv) identify
subgraphs suggesting the formulation of novel conjectures
Combinatorics of the Permutahedra, Associahedra, and Friends
I present an overview of the research I have conducted for the past ten years
in algebraic, bijective, enumerative, and geometric combinatorics. The two main
objects I have studied are the permutahedron and the associahedron as well as
the two partial orders they are related to: the weak order on permutations and
the Tamari lattice. This document contains a general introduction (Chapters 1
and 2) on those objects which requires very little previous knowledge and
should be accessible to non-specialist such as master students. Chapters 3 to 8
present the research I have conducted and its general context. You will find:
* a presentation of the current knowledge on Tamari interval and a precise
description of the family of Tamari interval-posets which I have introduced
along with the rise-contact involution to prove the symmetry of the rises and
the contacts in Tamari intervals;
* my most recent results concerning q, t-enumeration of Catalan objects and
Tamari intervals in relation with triangular partitions;
* the descriptions of the integer poset lattice and integer poset Hopf
algebra and their relations to well known structures in algebraic
combinatorics;
* the construction of the permutree lattice, the permutree Hopf algebra and
permutreehedron;
* the construction of the s-weak order and s-permutahedron along with the
s-Tamari lattice and s-associahedron.
Chapter 9 is dedicated to the experimental method in combinatorics research
especially related to the SageMath software. Chapter 10 describes the outreach
efforts I have participated in and some of my approach towards mathematical
knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche
Structural and universal completeness in algebra and logic
In this work we study the notions of structural and universal completeness
both from the algebraic and logical point of view. In particular, we provide
new algebraic characterizations of quasivarieties that are actively and
passively universally complete, and passively structurally complete. We apply
these general results to varieties of bounded lattices and to quasivarieties
related to substructural logics. In particular we show that a substructural
logic satisfying weakening is passively structurally complete if and only if
every classical contradiction is explosive in it. Moreover, we fully
characterize the passively structurally complete varieties of MTL-algebras,
i.e., bounded commutative integral residuated lattices generated by chains.Comment: This is a preprin
An algebraic investigation of Linear Logic
In this paper we investigate two logics from an algebraic point of view. The
two logics are: MALL (multiplicative-additive Linear Logic) and LL (classical
Linear Logic). Both logics turn out to be strongly algebraizable in the sense
of Blok and Pigozzi and their equivalent algebraic semantics are, respectively,
the variety of Girard algebras and the variety of girales. We show that any
variety of girales has equationally definable principale congruences and we
classify all varieties of Girard algebras having this property. Also we
investigate the structure of the algebras in question, thus obtaining a
representation theorem for Girard algebras and girales. We also prove that
congruence lattices of girales are really congruence lattices of Heyting
algebras and we construct examples in order to show that the variety of girales
contains infinitely many nonisomorphic finite simple algebras
Checking NFA equivalence with bisimulations up to congruence
16pInternational audienceWe introduce bisimulation up to congruence as a technique for proving language equivalence of non-deterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp. We compare our algorithm to the recently introduced antichain algorithms, by analysing and relating the two underlying coinductive proof methods. We give concrete examples where we exponentially improve over antichains; experimental results moreover show non negligible improvements on random automata
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
Non-Redundant Implicational Base of Many-Valued Context Using SAT
Some attribute implications in an implicational base of a derived context of many-valued context can be inferred from some other attribute implications together with its scales. The scales are interpretation of some values in the many-valued context therefore they are a prior or an existing knowledge. In knowledge discovery, the such attribute implications are redundant and cannot be considered as new knowledge. Therefore the attribute implicational should be eliminated. This paper shows that the redundancy problem exists and formalizes a model to check the redundancy
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