What is the function of cognition? On one influential account, cognition evolved to co-ordinate behaviour with environmental change or complexity (Godfrey-Smith 1996). Liberal interpretations of this view ascribe cognition to an extraordinarily broad set of biological systems – even bacteria, which modulate their activity in response to salient external cues, would seem to qualify as cognitive agents. However, equating cognition with adaptive flexibility per se glosses over important distinctions in the way biological organisms deal with environmental complexity. Drawing on contemporary advances in theoretical biology and computational neuroscience, we cash these distinctions out in terms of different kinds of generative models, and the representational and uncertainty-resolving capacities they afford. This analysis leads us to propose a formal criterion for delineating cognition from other, more pervasive forms of adaptive plasticity. On this view, biological cognition is rooted in a particular kind of functional organisation; namely, that which enables the agent to detach from the present and engage in counterfactual (active) inference.

One of creators of quantum mechanics P. Jordan in his work on quantum biology claimed that life's missing laws were the rules of chance and probability of the quantum world. The article presents author’s results of studying frequencies (or probabilities) of nucleotides on so-called epi-chains of long DNA sequences of various eukaryotic and prokaryotic genomes. DNA epi-chains are algorithmically constructed subsequencies of DNA nucleotide sequences. According to the algorithm of construction of any epi-chain of the order n, the epi-chain is such nucleotide subsequence, in which the numerations of adjacent nucleotides differ by natural number n (n = 1, 2, 3, 4,…). Correspondingly each epi-chain of order n ≥ 2 contains n times less nucleotides than the original DNA sequence. The presented results unexpectedly discover that in long single-stranded and double-stranded DNA of any tested genome its DNA epi-chains of different orders n (values n are not too large) have practically identical frequencies (or probabilities) of each kind of nucleotides. These data allow considering DNA as a regular rich set of epi-chains, which can play a certain role in genetic and epigenetic phenomena as the author belives. Appropriate rules of nucleotide frequencies on epi-chains of long DNA sequences are formulated for further their tests on a wider set of genomes. These results testify on existence of long-range coherence in long DNA and remind the Fröhlich's theory of long-range coherence in biological systems. The phenomenological data are discussed from different standpoints: the DNA double helices and helical antennas with circular polarizations of electromagnetic waves; relations with the Fröhlich's theory; numerical analysis of DNA epi-chains under binary representations of nucleotides. Results are useful for developing quantum and algebraic biology.

We propose a non-invasive approach to control the quality of spheroids and their fusion into a complex bioconstruct. The proposed method is based on the union of the nutrient concentration change calculated using Fick's law and the "reaction-diffusion" equations, taking into account absorption coefficient with specifying the exact fusion geometry using implicit Function Repre-sentation (FRep) functions. The proposed approach allows us to analyze the viability of cells within the spheroid, predict the fusion of spheroids, and accurately model complex heteroge-neous biostructures as a future task for our research. These results will significantly accelerate the development of such a promising field of additive biotechnologies as 3D bioprinting.

In this perspective article, we ponder topographical enhancements of artificial neural networks. In our recent paper in JMLR, we proved a quasi-equivalence between the network width and depth and also discussed the power of intra-links, which can be viewed as network height. In 1982, Hopfield published a network to model human associative memory, which contains many loops for dynamic evolution toward fixed points. Based on noising-denoising loops, diffusion models are recently developed to enable Bayesian modeling and inference with big data. Furthermore, we envision development of multi-AI-agent systems through “netware” engineering as a quantum leap of software engineering for emergent behaviors and autonomous AI at individual and population levels. We believe that the novel use of links and loops in space and time via multi-scale coupling would catalyze the next-generation neural networks.

Examining the similarity of event environments or surroundings, more precisely settings, provides additional insight in analyzing event sequences as it provides information about the context and potential common factors that may have influenced them. This article proposes a new similarity measure for event setting sequences, which involve the space and time in which events occur. While similarity measures for spatiotemporal event sequences have been studied, the settings and setting sequences have not yet been studied. While modeling event setting sequences we consider spatial and temporal scales to define the bounds of the setting and incorporates dynamic variables alongside static variables. Using a matrix-based representation and an extended Jaccard index we developed new similarity measures that allow for the use of all variable data types. We successfully used these similarity measures coupled with other multivariate statistical analysis approaches in a case study involving setting sequences and pollution event sequences associated with the same monitoring stations, which validate the hypothesis that more similar spatial-temporal settings or setting sequences may generate more similar events or event sequences. In conclusion, these similarity measures have many potential real-world applications, and offer researchers a powerful tool for understanding different factors and their dynamics corresponding to occurrences of spatiotemporal event sequences.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

Motivated by a common Mathematical Finance topic, this paper surveys several results concerning windings of 2-dimensional processes, including planar Brownian motion, complex-valued Ornstein-Uhlenbeck processes and planar stable processes. In particular, we present Spitzer's asymptotic Theorem for each case. We also relate this study to the pricing of Asian options.

Variational Auto-Encoders (VAEs) are deep generative models used for unsupervised learning, however their standard version is not topology-aware in practice since the data topology may not be taken into consideration. In this paper, we propose two different approaches with the aim to preserve the topological structure between the input space and the latent representation of a VAE. Firstly, we introduce InvMap-VAE as a way to turn any dimensionality reduction technique, given an embedding it produces, into a generative model within a VAE framework providing an inverse mapping into original space. Secondly, we propose the Witness Simplicial VAE as an extension of the Simplicial Auto-Encoder to the variational setup using a witness complex for computing the simplicial regularization, and we motivate this method theoretically using tools from algebraic topology. The Witness Simplicial VAE is independent of any dimensionality reduction technique and together with its extension, Isolandmarks Witness Simplicial VAE, preserves the persistent Betti numbers of a data set better than a standard VAE.

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (eg, Galileo's group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R. F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.

Perhaps the greatest challenge in the graft copolymerization of acrylic acid onto starch initiated by Fenton’s is the selectivity of the grafting reaction. Low selectivity means that too much of unwanted homopolymer polyacrylic acid is produced. In this paper, it has been analyzed why with this reaction system, involving a water soluble monomer and indirect initiation, the graft selectivity is such a challenge. Several other grafting reactions, especially with monomers of low water solubility or when direct initiation is applied, show a better grafting performance. A scheme is presented based on reaction engineering principles, that supports an explanation of the observed differences. A more selective activation of starch is a factor, but perhaps even more important is a low monomer to starch ratio at the starting sites of graft reactions. Based on these insights, several methods to improve the graft selectivity are evaluated, with the help of literature data but mostly by experimental explorations in our laboratory. Most promising appears to be a dedicated dosage method, to add monomer in portions over time, tested in a batch reactor with gelatinized cassava starch. This way, both grafting percentage and graft selectivity could be improved. Such could not be realized with the more conventional dosage, which is to add all monomer before the start of the reaction. This can be considered a principal breaktrough. Still, more research and development would be needed to refine the method and to implement the idea also in a continuous reactor at larger scale.

Let $f\left(1\right)=1$ , and let $f(n+1)=2^{2^{f\left(n\right)}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {x_i · x_j = x_k : i, j, k ∈ {1, . . . , n}} ∪ {x_i + 1 = x_k : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x₁, . . . , x_n, then each such solution (x₁, . . . , x_n) satisfies x₁, . . . , x_n ≤ f (2n). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set $\mathfrak{M}$ ⊆ has a finite-fold Diophantine representation, then $\mathfrak{M}$ is computable.