335,399 research outputs found
Implicit quantification made explicit : how to interpret blank nodes and universal variables in Notation3 Logic
Since the invention of Notation3 Logic, several years have passed in which the theory has been refined and applied in different reasoning engines like Cwm, EYE, and FuXi. But despite these developments, a clear formal definition of Notation3’s semantics is still missing. This does not only form an obstacle for the formal investigation of that logic and its relations to other formalisms, it has also practical consequences: in many cases the interpretations of the same formula differ between reasoning engines. In this paper we tackle one of the main sources of that problem, namely the uncertainty about implicit quantification. This refers to Notation3’s ability to use bound variables for which the universal or existential quantifiers are not explicitly stated, but implicitly assumed. We provide a tool for clarification through the definition of a core logic for Notation3 that only supports explicit quantification. We specify an attribute grammar which maps Notation3 formulas to that logic according to the different interpretations and thereby define
the semantics of Notation3. This grammar is then implemented and used to test the impact of the differences between interpretations on practical cases. Our dataset includes Notation3 implementations from former research projects and test cases developed for the reasoner EYE. We find that 31% of these files are understood differently by different reasoners. We further analyse these cases and categorize them in different classes of which we consider one most harmful: if a file is manually written by a user and no specific built-in predicates are used (13% of our critical files), it is unlikely that this user is aware of possible differences. We therefore argue the need to come to an agreement on implicit quantification, and discuss the different possibilities
New interpretations of the higher Stasheff--Tamari orders
In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on
triangulations of cyclic polytopes and conjectured them to coincide. We open up
an algebraic angle for approaching this conjecture by showing how these orders
arise naturally in the representation theory of the higher Auslander algebras
of type , denoted . For this we give new combinatorial
interpretations of the orders, making them comparable. We then translate these
combinatorial interpretations into the algebraic framework. We also show how
triangulations of odd-dimensional cyclic polytopes arise in the representation
theory of , namely as equivalence classes of maximal green
sequences. We furthermore give the odd-dimensional counterpart to the known
description of -dimensional triangulations as sets of non-intersecting
-simplices of a maximal size. This consists in a definition of two new
properties which imply that a set of -simplices produces a
-dimensional triangulation.Comment: 41 pages, 10 figures; v2: fixed typos and added references; v3: fixed
typos, added references, other minor revisions; v4: added references, changed
convention for multiplying arrows in path algebr
On Relativistic Generalization of Perelman's W-entropy and Statistical Thermodynamic Description of Gravitational Fields
Using double 2+2 and 3+1 nonholonomic fibrations on Lorentz manifolds, we
extend the concept of W-entropy for gravitational fields in the general
relativity, GR, theory. Such F- and W-functionals were introduced in the Ricci
flow theory of three dimensional, 3-d, Riemannian metrics by G. Perelman,
arXiv: math.DG/0211159. Nonrelativistic 3-d Ricci flows are characterized by
associated statistical thermodynamical values determined by W--entropy.
Generalizations for geometric flows of 4-d pseudo-Riemannian metrics are
considered for models with local thermodynamical equilibrium and separation of
dissipative and non-dissipative processes in relativistic hydrodynamics. The
approach is elaborated in the framework of classical filed theories
(relativistic continuum and hydrodynamic models) without an underlying kinetic
description which will be elaborated in other works. The 3+1 splitting allows
us to provide a general relativistic definition of gravitational entropy in the
Lyapunov-Perelman sense. It increases monotonically as structure forms in the
Universe. We can formulate a thermodynamic description of exact solutions in GR
depending, in general, on all spacetime coordinates. A corresponding 2+2
splitting with nonholonomic deformation of linear connection and frame
structures is necessary for generating in very general form various classes of
exact solutions of the Einstein and general relativistic geometric flow
equations. Finally, we speculate on physical macrostates and microstate
interpretations of the W-entropy in GR, geometric flow theories and possible
connections to string theory (a second unsolved problem also contained in
Perelman's works) in the Polyakov's approach.Comment: latex2e, v4 is an accepted to EPJC substantial extension of a former
letter type paper on 10 pages to a research article on 41 pages; a new author
added, the paper's title and permanent and visiting affiliations were
correspondingly modified; and new results, conclusions and references are
provide
Paradoxical Interpretations of Urban Scaling Laws
Scaling laws are powerful summaries of the variations of urban attributes
with city size. However, the validity of their universal meaning for cities is
hampered by the observation that different scaling regimes can be encountered
for the same territory, time and attribute, depending on the criteria used to
delineate cities. The aim of this paper is to present new insights concerning
this variation, coupled with a sensitivity analysis of urban scaling in France,
for several socio-economic and infrastructural attributes from data collected
exhaustively at the local level. The sensitivity analysis considers different
aggregations of local units for which data are given by the Population Census.
We produce a large variety of definitions of cities (approximatively 5000) by
aggregating local Census units corresponding to the systematic combination of
three definitional criteria: density, commuting flows and population cutoffs.
We then measure the magnitude of scaling estimations and their sensitivity to
city definitions for several urban indicators, showing for example that simple
population cutoffs impact dramatically on the results obtained for a given
system and attribute. Variations are interpreted with respect to the meaning of
the attributes (socio-economic descriptors as well as infrastructure) and the
urban definitions used (understood as the combination of the three criteria).
Because of the Modifiable Areal Unit Problem and of the heterogeneous
morphologies and social landscapes in the cities internal space, scaling
estimations are subject to large variations, distorting many of the conclusions
on which generative models are based. We conclude that examining scaling
variations might be an opportunity to understand better the inner composition
of cities with regard to their size, i.e. to link the scales of the city-system
with the system of cities
Finitely Generated Groups Are Universal
Universality has been an important concept in computable structure theory. A
class of structures is universal if, informally, for any
structure, of any kind, there is a structure in with the same
computability-theoretic properties as the given structure. Many classes such as
graphs, groups, and fields are known to be universal.
This paper is about the class of finitely generated groups. Because finitely
generated structures are relatively simple, the class of finitely generated
groups has no hope of being universal. We show that finitely generated groups
are as universal as possible, given that they are finitely generated: for every
finitely generated structure, there is a finitely generated group which has the
same computability-theoretic properties. The same is not true for finitely
generated fields. We apply the results of this investigation to quasi Scott
sentences
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