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INDECOMPOSABLE SOERGEL BIMODULES FOR UNIVERSAL COXETER GROUPS

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2017年第6期369卷 3883-3910页

作者： Elias, Ben Libedinsky, Nicolas Univ Oregon Dept Math Eugene OR 97403 USA Univ Chile Dept Math Santiago Chile Univ Virginia Dept Math Charlottesville VA 22904 USA

We produce an explicit recursive formula which computes the idempotent projecting to any indecomposable Soergel bimodule for a universal Coxeter system. This gives the exact set of primes for which the positive characteristic analogue of Soergel's conjecture holds. Along the way, we introduce the multicolored Temperley-Lieb algebra.

关键词： COXETER groups IDEMPOTENTS INDECOMPOSABLE modules COXETER complexes recursive functions

The Diversity of Categoricity Without Delay

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ALGEBRA AND LOGIC 2017年第2期56卷 171-177页

作者： Kalimullin, I. S. Melnikov, A. G. Ng, K. M. Kazan Volga Reg Fed Univ Ul Kremlevskaya 18 Kazan 420008 Russia Massey Univ Auckland Private Bag 102904 Auckland 0745 New Zealand Nanyang Technol Univ Nanyang Ave 50 Singapore 639798 Singapore

Boundary-induced pattern formation from uniform temporal oscillation

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CHAOS 2018年第4期28卷 045110-045110页

作者： Kohsokabe, Takahiro Kaneko, Kunihiko Univ Tokyo Grad Sch Arts & Sci Dept Basic Sci Meguro Ku 3-8-1 Komaba Tokyo 1538902 Japan Univ Tokyo Grad Sch Arts & Sci Res Ctr Complex Syst Biol Meguro Ku 3-8-1 Komaba Tokyo 1538902 Japan

Pattern dynamics triggered by fixing a boundary is investigated. By considering a reactiondiffusion equation that has a unique spatially uniform and limit cycle attractor under a periodic or Neumann boundary condition, and then by choosing a fixed boundary condition, we found three novel phases depending on the ratio of diffusion constants of activator to inhibitor: transformation of temporally periodic oscillation into a spatially periodic fixed pattern, travelling wave emitted from the boundary, and aperiodic spatiotemporal dynamics. The transformation into a fixed, periodic pattern is analyzed by crossing of local nullclines at each spatial point, shifted by diffusion terms, as is analyzed by using recursive equations, to obtain the spatial pattern as an attractor. The generality of the boundary-induced pattern formation as well as its relevance to biological morphogenesis is discussed. (C) 2018 Author(s).

关键词： REACTION-diffusion equations ATTRACTORS (Mathematics) BOUNDARY value problems recursive functions SPATIOTEMPORAL processes APERIODICITY DYNAMICS

Generic programming in OCAML

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arXiv 2018年

作者： Balestrieri, Florent Mauny, Michel ENSTA-ParisTech Université Paris-Saclay Inria Paris France

We present a library for generic programming in OCAML, adapting some techniques borrowed from other functional languages. The library makes use of three recent additions to OCAML: generalised abstract datatypes are essential to reflect types, extensible variants allow this reflection to be open for new additions, and extension points provide syntactic sugar and generate boiler plate code that simplify the use of the library. The building blocks of the library can be used to support many approaches to generic programming through the concept of view. Generic traversals are implemented on top of the library and provide powerful combinators to write concise definitions of recursive functions over complex tree types. Our case study is a type-safe deserialisation function that respects type abstraction. Copyright © 2018, The Authors. All rights reserved.

关键词： recursive functions

A predicative variant of hyland's effective topos

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arXiv 2018年

作者： Maietti, Maria Emilia Maschio, Samuele Dipartimento di Matematica Universit`a di Padova

Here, we present a subcategory pEff of Hyland's Effective Topos Eff which can be considered a predicative variant of Eff itself. The construction of pEff is motivated by the desire of providing a "predicative"categorical universe of realizers to model the Minimalist Foundation for constructive mathematics which was ideated by the first author with G. Sambin in 2005 and completed into a two-level formal system by the first author in 2009. pEff is a "predicative"categorical universe because its objects and morphisms can be formalized in Feferman's predicative weak theory of inductive definitions dID1. Moreover, it is a predicative variant of the Effective Topos for the following reasons. First, pEff is a list-arithmetic locally cartesian closed pretopos of de- finable objects in dID1with a fibred category of small objects over pEff and a (non-small) classifier of small subobjects. Second, it happens to coincide with the exact completion on the lex category defined as a predicative rendering in dID1of the subcategory of Eff of recursive functions. As a consequence it validates the Formal Church's thesis and it embeds in Eff by preserving the list-arithmetic locally cartesian closed pretopos *** Codes 03F50, 18D30, 18C99, 03D70 Copyright © 2018, The Authors. All rights reserved.

关键词： recursive functions

Fusion graph convolutional networks

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arXiv 2018年

作者： Vijayan, Priyesh Yash Chandak Khapra, Mitesh M. Parthasarathy, Srinivasan Ravindran, Balaraman Dept. of CSE Robert Bosch Centre for Data Science AI Indian Institute of Technology Madras Chennai India Dept. of Computer Science University of Massachusetts Amherst United States Dept. of CSE Dept. of Biomedical Informatics Ohio State University Ohio United States

Semi-supervised node classification in attributed graphs, i.e., graphs with node features, involves learning to classify unlabeled nodes given a partially labeled graph. Label predictions are made by jointly modeling the node and its' neighborhood features. State-of-the-art models for node classification on such attributed graphs use differentiable recursive functions that enable aggregation and filtering of neighborhood information from multiple hops. In this work, we analyze the representation capacity of these models to regulate information from multiple hops independently. From our analysis, we conclude that these models despite being powerful, have limited representation capacity to capture multi-hop neighborhood information effectively. Further, we also propose a mathematically motivated, yet simple extension to existing graph convolutional networks (GCNs) which has improved representation capacity. We extensively evaluate the proposed model, F-GCN on eight popular datasets from different domains. F-GCN outperforms the state-of-the-art models for semi-supervised learning on six datasets while being extremely competitive on the other two. Copyright © 2018, The Authors. All rights reserved.

关键词： recursive functions

A Simple Functional Presentation and an Inductive Correctness Proof of the Horn Algorithm

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arXiv 2018年

作者： Ravara, António NOVA-LINCS and Dep. de Informática FCT Universidade NOVA de Lisboa Portugal

We present a recursive formulation of the Horn algorithm for deciding the satisfiability of propositional clauses. The usual presentations in imperative pseudo-code are informal and not suitable for simple proofs of its main properties. By defining the algorithm as a recursive function (computing a least fixed-point), we achieve: 1) a concise, yet rigorous, formalisation;2) a clear form of visualising executions of the algorithm, step-by-step;3) precise results, simple to state and with clean inductive proofs. Copyright © 2018, The Authors. All rights reserved.

关键词： recursive functions

Computing with P Systems

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arXiv 2018年

作者： Syropoulos, Apostolos Doumanis, Stratos Sotiriades, Konstantinos T. Greek Molecular Computing Group Xanthi Greece

P systems are computing conceptual computing devices that are at least as powerful as Turing machines. However, until recently it was not known how one can encode any recursive function as a P system. Here we propose a new encoding of recursive as P systems with graph-like structure, which is the main difference with previous documented attempts. The consequence of this and other such efforts is that they provide a solid ground for the implementation of real programming languages in existing hardware. Copyright © 2018, The Authors. All rights reserved.

关键词： recursive functions

Static and dynamic verification of relational properties on self-composed C code

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arXiv 2018年

作者： Blatter, Lionel Kosmatov, Nikolai Le Gall, Pascale Prevosto, Virgile Petiot, Guillaume CEA List Software Reliability and Security Lab PC 174 Gif-sur-Yvette91191 France CentraleSupelec Université Paris-Saclay Gif-sur-Yvette91190 France

Function contracts are a well-established way of formally specifying the intended behavior of a function. However, they usually only describe what should happen during a single call. Relational properties, on the other hand, link several function calls. They include such properties as non-interference, continuity and monotonicity. Other examples relate sequences of function calls, for instance, to show that decrypting an encrypted message with the appropriate key gives back the original message. Such properties cannot be expressed directly in the traditional setting of modular deductive verification, but are amenable to verification through self-composition. This paper presents a verification technique dedicated to relational properties in C programs and its implementation in the form of a FRAMA-C plugin called RPP and based on self-composition. It supports functions with side effects and recursive functions. The proposed approach makes it possible to prove a relational property, to check it at runtime, to generate a counterexample using testing and to use it as a hypothesis in the subsequent verification. Our initial experiments on existing benchmarks confirm that the proposed technique is helpful for static and dynamic analysis of relational properties. Copyright © 2018, The Authors. All rights reserved.

关键词： recursive functions