Numerical upper bounds on growth of automaton groups

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .

Keystroke Dynamics-Based Authentication System Using Empirical Thresholding Algorithm

In a password-based authentication technique, whenever the typed password and username matches the system database, the secure login page allows the client to access it. Despite the password matching, the proposed method checks the similarity between the typing rhythm of entered password and the rhythm of password samples in client's database. In this paper, a novel algorithmic procedure is presented to authenticate the legal client based on empirical threshold values obtained from the timing information of the client's keystroke dynamics. The exploratory outcomes demonstrate an impressive diminish in both false rejection rate and false acceptance rate. Equal error rate and authentication accuracy are also assessed to show the superiority and robustness of the method. Therefore, the proposed keystroke dynamics-based authentication method can be valuable in securing the system protection as a correlative or substitute form of client validation and as a useful resource for identifying the illegal invasion.

REDUCTION OF THE CLASSIFIER IN THE METHOD OF LOGICAL ANALYSIS OF DATA BASED ON THE Ε-, Δ-CRITERION FOR SELECTING PATTERN

Предлагается алгоритмическая процедура редукции классификатора в методе логического анализа данных, основанная на отборе закономерностей с помощью ε-, δ-критерия. Реализация подхода заключается в формировании исходного классификатора как набора закономерностей на базе наблюдений обучающей выборки, применения к полученным правилам процедуры наращивания и последующего их отбора в новый классификатор на базе ε-, δ-критерия. Приводится эмпирическое подтверждение целесообразности данной алгоритмической процедуры. An algorithmic procedure for the reduction of the classifier in the method of logical analysis of data, based on the selection of patterns using the ε-, δ-criterion is proposed. The implementation of the approach consists in the formation of the initial classifier as a set of patterns based on observations of the training sample, application of the increasing procedure to the obtained patterns and their subsequent selection into a new classifier based on the ε-, δ-criterion. An empirical confirmation of the expediency of this algorithmic procedure is given.

Supersymmetric spectroscopy on AdS4 × S7 and AdS4 × S6

New techniques based on Exceptional Field Theory have recently allowed for the calculation of the Kaluza-Klein spectra of certain AdS4 solutions of D = 11 and massive IIA supergravity. These are the solutions that consistently uplift on S7 and S6 from vacua of maximal four-dimensional supergravity with SO(8) and ISO(7) gaugings. In this paper, we provide an algorithmic procedure to compute the complete Kaluza-Klein spectrum of five such AdS4 solutions, all of them $$ \mathcal{N} $$ N = 1, and give the first few Kaluza-Klein levels. These solutions preserve SO(3) and U(1) × U(1) internal symmetry in D = 11, and U(1) (two of them) and no continuous symmetry in type IIA. Together with previously discussed cases, our results exhaust the Kaluza-Klein spectra of known supersymmetric AdS4 solutions in D = 11 and type IIA in the relevant class.

Gauge invariant perturbation theory via homotopy transfer

We show that the perturbative expansion of general gauge theories can be expressed in terms of gauge invariant variables to all orders in perturbations. In this we generalize techniques developed in gauge invariant cosmological perturbation theory, using Bardeen variables, by interpreting the passing over to gauge invariant fields as a homotopy transfer of the strongly homotopy Lie algebras encoding the gauge theory. This is illustrated for Yang-Mills theory, gravity on flat and cosmological backgrounds and for the massless sector of closed string theory. The perturbation lemma yields an algorithmic procedure to determine the higher corrections of the gauge invariant variables and the action in terms of these.

A Systematic Approach for Evaluating the Adoption of Additive Manufacturing in the Product Design Process

Additive Manufacturing (AM) technologies have expanded the possibility of producing unconventional geometries, also increasing the freedom of design. However, in the designer’s everyday work, the decision regarding the adoption of AM for the production of a component is not straightforward. In fact, it is necessary to process much information regarding multiple fields to exploit the maximum potential of additive production. For example, there is a need to evaluate the properties of the printable materials, their compatibility with the specific application, redesign shapes accordingly to AM limits, and conceive unique and complex products. Additionally, procurement and logistics evaluations, as well as overall costs possibly extending to the entire life cycle, are necessary to come to a decision for a new and radical solution. In this context, this paper investigates the complex set of information involved in this process. Indeed, it proposes a framework to support and guide a designer by means of a structured and algorithmic procedure to evaluate the opportunity for the adoption of AM and come to an optimal design. A case study related to an ultralight aircraft part is reported to demonstrate the proposed decision process.

Prediction of Interpolants in Zero Diluted Images

This paper provides an algorithmic procedure to predict interpolants of zero diluted images. Given a digital image, one can zero dilute it by right adjoining a column consisting of ‘0s’ to every column except the last column and inserting a row consisting of ‘0s’ below every row except the last row. This yields a new image with a size (2W-1)×(2H-1), where W is the width and H is the height of the original image. Another way of zero diluting an image is by right adjoining a column consisting of ‘0s’ to every column and inserting a row consisting of ‘0s’ below every row. This yields a new image with a size (2W)×(2H), where W is the width and H is the height of the original image. Alternatively, subsampling of an image is carried out by forcing pixel values in the alternate columns and rows to zero. Thus, the size of the subsampled image is reduced to half of the size of the original image. This means 75% of the information in the original image is lost in the subsampled image. On the other hand, zero dilution of an image does not cause loss of information but increases the possibility of predicting more information. The question that arises here is whether it is possible to predict more pixel values, which are called interpolants so that the reconstructed image is an enhanced version of the original image in resolution. In this paper, two novel interpolant prediction techniques, which are reliable and computationally efficient, are discussed. They are (i) interpolant prediction using neighborhood pixel value averaging and (ii) interpolant prediction using extended morphological filtering. These techniques can be applied to predict interpolants in a subsampled image also.