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Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions
No full textBernstein-Schwarzman conjectured that the quotient of a complex affine spaceby an irreducible complex crystallographic group generated by reflections is aweighted projective space. The conjecture was proved by Schwarzman andTokunaga-Yoshida in dimension 2 for almost all such groups, and for allcrystallographic reflection groups of Coxeter type by Looijenga,Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that theconjecture is true for the crystallographic reflection group in dimension 3 forwhich the associated collineation group is Klein's simple group of order 168.In this case the quotient is the 3-dimensional weighted projective space withweights 1, 2, 4, 7. The main ingredient in the proof is the computation of thealgebra of invariant theta functions. Unlike the Coxeter case, the invariantalgebra is not free polynomial, and this was the major stumbling block.Comment: 21 pages, 1 figure. Final version typeset in the EPIGA styl
Twin-width and permutations
No full textInspired by a width invariant on permutations defined by Guillemot and Marx,Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs,which is a parameter describing its structural complexity. This invariant hasbeen further extended to binary structures, in several (basically equivalent)ways. We prove that a class of binary relational structures (that is:edge-colored partially directed graphs) has bounded twin-width if and only ifit is a first-order transduction of a~proper permutation class. As aby-product, we show that every class with bounded twin-width contains at most pairwise non-isomorphic -vertex graphs
Diagonal F-splitting and Symbolic Powers of Ideals
No full textLet be any ideal in a strongly -regular, diagonally -split ring essentially of finite type over an -finite field. We show that for all for which the formula makes sense. We use this to show a number of novelcontainments between symbolic and ordinary powers of prime ideals in thissetting, which includes all determinantal rings and a large class of toricrings in positive characteristic. In particular, we show that for all prime ideals of height in such rings.Comment: Copy edited and formatted in the EpiGA journal's styleshee
Plato, Aristotle, and Locke on the accumulation of wealth and natural law
No full textThe possibility of a growing accumulation of wealth, what we now refer to as economic growth, was something already considered by Plato, Aristotle and Locke, under the concept of chrematistics. In this paper we show how the economic thinking of these authors cannot be fully understood without considering the intimate relationship they establish between politics and property accumulation. In addition to continuities and ruptures in the arguments, there can be seen a growing understanding of the phenomenon of economic growth in such a way that, when we arrive at Locke, an evident paradigm shift can be appreciated. This change is rooted in the contributions of scholastic thinking for which the acquisition of property through human labour or industry enjoys legitimacy according to natural law
A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct
No full textWe study the expressiveness and succinctness of history-deterministicpushdown automata (HD-PDA) over finite words, that is, pushdown automata whosenondeterminism can be resolved based on the run constructed so far, butindependently of the remainder of the input word. These are also known asgood-for-games pushdown automata. We prove that HD-PDA recognise more languagesthan deterministic PDA (DPDA) but not all context-free languages (CFL). Thisclass is orthogonal to unambiguous CFL. We further show that HD-PDA can beexponentially more succinct than DPDA, while PDA can be double-exponentiallymore succinct than HD-PDA. We also study HDness in visibly pushdown automata(VPA), which enjoy better closure properties than PDA, and for which we showthat deciding HDness is ExpTime-complete. HD-VPA can be exponentially moresuccinct than deterministic VPA, while VPA can be exponentially more succinctthan HD-VPA. Both of these lower bounds are tight. We then compare HD-PDA withPDA for which composition with games is well-behaved, i.e. good-for-gamesautomata. We show that these two notions coincide, but only if we considerpotentially infinitely branching games. Finally, we study the complexity ofresolving nondeterminism in HD-PDA. Every HDPDA has a positional resolver, afunction that resolves nondeterminism and that is only dependant on the currentconfiguration. Pushdown transducers are sufficient to implement the resolversof HD-VPA, but not those of HD-PDA. HD-PDA with finite-state resolvers aredeterminisable
The Volterra Integrable case. Novel analytical and numerical results
No full textIn the present paper we reconsider the integrable case of the Hamiltonian-species Volterra system, as it has been introduced by Vito Volterra in 1937and significantly enrich the results already published in the ArXiv in 2019 bytwo of the present authors (M. Scalia and O. Ragnisco). In fact, we present anew approach to the construction of conserved quantities and comment about thesolutions of the equations of motion; we display mostly new analytical andnumerical results, starting from the classical predator-prey model and arrivingat the general -species modelComment: 24 pages, 9 figures. This paper is a natural prosecution of the work arXiv:1903.03595 and presents different enrichments and enhancements of the results there appeared. In this second version several typos have been correcte
A higher-order transformation approach to the formalization and analysis of BPMN using graph transformation systems
No full textThe Business Process Modeling Notation (BPMN) is a widely used standardnotation for defining intra- and inter-organizational workflows. However, theinformal description of the BPMN execution semantics leads to differentinterpretations of BPMN elements and difficulties in checking behavioralproperties. In this article, we propose a formalization of the executionsemantics of BPMN that, compared to existing approaches, covers more BPMNelements while also facilitating property checking. Our approach is based on ahigher-order transformation from BPMN models to graph transformation systems.To show the capabilities of our approach, we implemented it as an open-sourceweb-based tool
Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory
No full textBishop's measure theory (BMT) is an abstraction of the measure theory of alocally compact metric space , and the use of an informal notion of aset-indexed family of complemented subsets is crucial to its predicativecharacter. The more general Bishop-Cheng measure theory (BCMT) is aconstructive version of the classical Daniell approach to measure andintegration, and highly impredicative, as many of its fundamental notions, suchas the integration space of -integrable functions , rely onquantification over proper classes (from the constructive point of view). Inthis paper we introduce the notions of a pre-measure and pre-integration space,a predicative variation of the Bishop-Cheng notion of a measure space and of anintegration space, respectively. Working within Bishop Set Theory (BST), andusing the theory of set-indexed families of complemented subsets andset-indexed families of real-valued partial functions within BST, we apply theimplicit, predicative spirit of BMT to BCMT. As a first example, we present thepre-measure space of complemented detachable subsets of a set with theDirac-measure, concentrated at a single point. Furthermore, we translate in ourpredicative framework the non-trivial, Bishop-Cheng construction of anintegration space from a given measure space, showing that a pre-measure spaceinduces the pre-integration space of simple functions associated to it.Finally, a predicative construction of the canonically integrable functions, as the completion of an integration space, is included
Temporal Sequencing of Documents
No full textWe outline an unsupervised method for temporal rank ordering of sets ofhistorical documents, namely American State of the Union Addresses and DEEDS, acorpus of medieval English property transfer documents. Our method relies uponeffectively capturing the gradual change in word usage via a bandwidth estimatefor the non-parametric Generalized Linear Models (Fan, Heckman, and Wand,1995). The number of possible rank orders needed to search through for costfunctions related to the bandwidth can be quite large, even for a small set ofdocuments. We tackle this problem of combinatorial optimization using theSimulated Annealing algorithm, which allows us to obtain the optimal documenttemporal orders. Our rank ordering method significantly improved the temporalsequencing of both corpora compared to a randomly sequenced baseline. Thisunsupervised approach should enable the temporal ordering of undated documentsets
Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation
No full textIn this paper, Lie symmetry analysis method is applied to the(2+1)-dimensional time fractional Kadomtsev-Petviashvili (KP) equation with themixed derivative of Riemann-Liouville time-fractional derivative andinteger-order -derivative. We obtained all the Lie symmetries admitted bythe KP equation and used them to reduce the (2+1)-dimensional fractionalpartial differential equation with Riemann-Liouville fractional derivative tosome (1+1)-dimensional fractional partial differential equations withErd\'{e}lyi-Kober fractional derivative or Riemann-Liouville fractionalderivative, thereby getting some exact solutions of the reduced equations. Inaddition, the new conservation theorem and the generalization of Noetheroperators are developed to construct the conservation laws for the equationstudied