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Sequences formed by symbols are found in diverse fields, including genome sequences, written texts and computer codes. An interesting question is whether a sequence of symbols contains correlated structures. Existing methods to characterize correlations require a numerical representation of the sequence. In this regard, mapping a sequence of text into a sequence of numerical values is a key step for assessing correlation analysis. This work proposes a methodology to study correlations in a sequence of symbols. In the first step, the sequence of symbols is mapped in a multivariate numerical sequence formed by unit vectors in a vectorial space. The main feature of such mapping is that symbols are equally weighted, thus avoiding the numerical overrepresentation of symbols. In the second step, a multivariate version of the detrended fluctuation analysis is used to quantify correlations in the numerical sequence. Genome sequences (first COVID-19), written English texts and comovements between Bitcoin and gold markets were used to illustrate the proposed methodology’s performance. The results showed that the balanced numerical mapping of symbolic sequences and the multivariate DFA provides valuable insights into the correlations in a sequence of symbols.

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov — M. Kerr and the author — together with D. Hernández Serrano — can also be deduced from this general expression.

In this paper, we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on one function of one variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal theory of PDEs.

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we give a new proof by relating it to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed form, and discuss nonlinear Laplace transformations and symmetric PDE models.

A critique of some central themes in Pentti Haikonen's recent book, Consciousness and Robot Sentience, is offered. Haikonen maintains that the crucial question concerning consciousness is how the inner workings of the brain or an artificial system can appear, not as inner workings, but as subjective experience. It is argued here that Haikonen's own account fails to answer this question, and that the question is not in fact the right one to ask anyway. It is argued that making the required changes to the question reveals an important lacuna in Haikonen's explanation of consciousness.

Physically, information carriers are encountered in two occurrences, either in native form as physical structures, or in arbitrarily coded, symbolic form such as signal systems or sequences of signs. The symbolic form may rigorously be associated with the existence of life. In contrast, structural information may be present in various physical processes or structures independent of life. The self-organised emergence of symbolic information from structural information may be called ritualisation. A century ago, Julian Huxley had introduced this term in behavioural biology. Subsequently, this evolutionary key process of the emergence of animal and social communication was studied in depth by Konrad Lorenz, Günter Tembrock and other ethologists. Ritualisation exhibits typical features of kinetic phase transitions of the 2^nd kind. From a more general viewpoint, the origin of life, the appearance of human languages and the emergence of human social categories such as money can also be understood as ritualisation transitions. Occurring at some stage of evolutionary history, these transitions have in common that after the crossover, arbitrary symbols are issued and recognised by information-processing devices, by transmitters and receivers in the sense of Shannon’s information theory. In this paper, general properties of the ritualisation transition and the related code symmetry are described. These properties are demonstrated by tutorial examples of very different such transitions in natural, social and technical evolution, reviewed from the perspective of the emergence of symbolic information and its structural historicity.