A first step toward higher order chain rules in abelian functor calculus

One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when $n=2$. We describe how to obtain the second higher order directional derivative chain rule for abelian functors. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors

Green's Relations in Finite Transformation Semigroups

We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1

Towards run-time monitoring of web services conformance to business-level agreements

Web service behaviour is currently specified in a mixture of ways, often using methods that are only partially complete. These range from static functional specifications, based on interfaces in WSDL and preconditions in RIF, to business process simulations using executable process-based models such as BPEL, to detailed quality of service (QoS) agreements laid down in a service level agreement (SLA). This paper recognises that something similar to a SLA is required at the higher business level to govern the contract between service producers, brokers and consumers. We call this a business level agreement (BLA) and within this framework, seek to unify disparate aspects of functional specification, QoS and run-time verification. We propose that the method for validating a web service with respect to its advertised BLA should be based on run-time service monitoring. This is a position paper towards defining these goals

Conjugacy and Equivalence of Weighted Automata and Functional Transducers

International audienceWe show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, or any (skew) ¯eld and that the same holds true for functional tranducers as well

Effective Theories for Circuits and Automata

Abstracting an effective theory from a complicated process is central to the study of complexity. Even when the underlying mechanisms are understood, or at least measurable, the presence of dissipation and irreversibility in biological, computational and social systems makes the problem harder. Here we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback. We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory. We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.Comment: 11 pages, 9 figure

Reducing Computational Costs in the Basic Perturbation Lemma

Homological Perturbation Theory [11, 13] is a well-known general method for computing homology, but its main algorithm, the Basic Perturbation Lemma, presents, in general, high computational costs. In this paper, we propose a general strategy in order to reduce the complexity in some important formulas (those following a specific pattern) obtained by this algorithm. Then, we show two examples of application of this methodology.

Comparative RNAseq Analysis of the Insect-Pathogenic Fungus <i>Metarhizium anisopliae</i> Reveals Specific Transcriptome Signatures of Filamentous and Yeast-Like Development

The fungus Metarhizium anisopliae is a facultative insect pathogen used as biological control agent of several agricultural pests worldwide. It is a dimorphic fungus that is able to display two growth morphologies, a filamentous phase with formation of hyphae and a yeast-like phase with formation of single-celled blastospores. Blastospores play an important role for M. anisopliae pathogenicity during disease development. They are formed solely in the hemolymph of infected insects as a fungal strategy to quickly multiply and colonize the insect’s body. Here, we use comparative genome-wide transcriptome analyses to determine changes in gene expression between the filamentous and blastospore growth phases in vitro to characterize physiological changes and metabolic signatures associated with M. anisopliae dimorphism. Our results show a clear molecular distinction between the blastospore and mycelial phases. In total 6.4% (n = 696) out of 10,981 predicted genes in M. anisopliae were differentially expressed between the two phases with a fold-change > 4. The main physiological processes associated with up-regulated gene content in the single-celled yeast-like blastospores during liquid fermentation were oxidative stress, amino acid metabolism (catabolism and anabolism), respiration processes, transmembrane transport and production of secondary metabolites. In contrast, the up-regulated gene content in hyphae were associated with increased growth, metabolism and cell wall re-organization, which underlines the specific functions and altered growth morphology of M. anisopliae blastospores and hyphae, respectively. Our study revealed significant transcriptomic differences between the metabolism of blastospores and hyphae. These findings illustrate important aspects of fungal morphogenesis in M. anisopliae and highlight the main metabolic activities of each propagule under in vitro growth conditions

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201