Life and works of Revaz Grigolia
Department of Mathematics, University of Salerno, Italy
On the theory of epistemic Łukasiewicz logic corresponding to the Chang algebra with application in Immune system
We describe specific fragments of the immune system using relational systems (Kripke frames) that represent the semantics of the novel logic EŁ-Modal Epistemic Łukasiewicz logic. The language of EŁ extends the infinitely valued Łukasiewicz logic Ł by introducing a unary connective, interpreted as a modal epistemic operator that denotes knowledge and quasi-knowledge. This paper demonstrates that theorems within this logic hold for the immune system model. Furthermore, we propose conjectures related to the model that are unresolved by immune scientists, striving to prove or refute these conjectures as theorems. Additionally, we investigate the decidability and unification problems of the corresponding logic and its admissible rules.
Interdisciplinary Department, Tbilisi State University and
Institute of Cybernetics, Georgian Technical University, Tbilisi, Georgia
Fractional and Parametric Dynamical Models
The introduction of different types of exponential functions, which are eigenfunctions of suitable differential operators, has made it possible to extend classical mathematical problems and entities such as linear dynamical systems, Bernoulli and Euler numbers and polynomials, population dynamics models, and the Laplace transform. In particular, the Laguerre-type derivatives, the fractional derivative, and the parametric version of these operators have been used in this context.
In this lecture, we consider the case of fractional and parametric exponential operators to derive the corresponding dynamical models. Numerical examples of some of the models examined using the computational algebra program Mathematica©, are shown throughout the article.
Joint work with Diego Caratelli and Pierpaolo Natalini
Exploring Brownian Motion on Diverse Structures
In this talk, we delve into the captivating interplay between stochastic processes and geometric structures, with a focus on Brownian motion as it unfolds across lines, circles, curves, and planar domains. We investigate the intricate probability laws governing these motions and reveal surprising phenomena that arise when randomness meets geometry.
A highlight of the presentation is the study of the so-called Brownian triangle—a triangle whose vertices are determined by three independent planar Brownian motions. We analyze the distribution of its area, uncovering remarkable stochastic properties and connections to classical geometric probability. In addition, the talk addresses the relationship between curvature and randomness. We explore how the curvature of a curve influences the behavior of a Brownian particle traveling near it over small time intervals, shedding new light on the interplay between deterministic geometry and probabilistic dynamics.
Attendees will gain insight into modern techniques in stochastic geometry and see how these methods open doors to novel applications and theoretical advancements in the understanding of complex random systems.
Department of Mathematics and Statistics, Eastern Michigan University, USA
Tuning Proliferation and Death in Drosophila development and Disease
During development, morphogen gradients encode positional information to pattern morphological structures during organogenesis. Some gradients, like that of Dpp in the fly wing, remain proportional to the size of growing organs—that is, they scale. Gradient scaling keeps morphological patterns proportioned in organs of different sizes. We have found a mechanism of scaling that ensures that, when the gradient is smaller than the organ, cell death trims the developing tissue to match the size of the gradient. Scaling is controlled by molecular associations between Dally and Pentagone, known factors involved in scaling, and a key factor that mediates cell death, Flower. We show that Flower activity in gradient expansion is not dominated by cell death, but by the activity of Dally/Pentagone on scaling. We found a connection between scaling and cell death that may uncover a molecular toolbox hijacked by tumours.
Department of Molecular & Clinical Cancer Medicine, University of Liverpool, United Kingdom
Theoretical and Empirical Conditions for Validating the Montgomery-Koyama-Smith Equation in Estimating Total Leaf Area per Shoot
Leaves are the principal organs of photosynthesis, and their total area (leaf area) is a fundamental biophysical trait that directly governs a plant’s photosynthetic capacity, light interception efficiency, and energy balance. Accurate estimation of individual and total leaf area is therefore essential. The Montgomery equation establishes a proportional relationship between individual leaf area (A) and the product of leaf length and width (LW). Recently, this principle was extended to estimate the total leaf area per shoot or culm (AT) using the equation AT = kKSLKSWKS, where LKS denotes the sum of individual leaf widths, WKS is the maximum leaf length, and kKS is a proportionality coefficient. This formulation is referred to as the Montgomery-Koyama-Smith equation (MKSE). However, empirical evidence indicates that AT scales allometrically with LKSWKS, with a scaling exponent less than one. In this study, we demonstrate that the MKSE holds true under the following conditions: (i) the leaf areas of each shoot, when sorted in ascending order, follow a geometric sequence; (ii) the common ratio of these geometric sequences is consistent across shoots; (iii) the number of leaves per shoot is constant; and (iv) leaf width scales isometrically or allometrically with leaf length according to a power law. Violation of any of conditions (ii) through (iv) results in an allometric relationship AT ∝ (LKSWKS)α with α < 1. This work provides a theoretical foundation and practical guidance for the appropriate application of the MKSE in estimating total leaf area, thereby enhancing its utility in ecological and physiological studies.
Bamboo Research Institute, Nanjing Forestry University, Nanjing, China
Padè Approximants and theory of Appéll polynomials
The use of approximants of Padè type are employed to develop a method aimed at opening new perspectives in the theory of Appell polynomials. In this talk, the expansion of amplitude A(t) of the Appell polynomials family in terms of rational approximants yields the possibility of determining the approximation of the an(x) in terms of other special polynomials. Application of this approach to Hermite polynomials yields highly accurate approximations in terms of truncated exponential polynomials. Further, monomiality conditions are explored and formalism is extended to consider the Padè approximants within the context of umbral notation.
Superelliptical Kinematics of the Nature according to Gielis Super Formula
Superellipses are geometric tools that offer a very useful way to model a variety of complex three-dimensional shapes, including objects, buildings, seeds, flowers, human and animal limbs, and aircraft fuselages. Because of the diversity of nature and the way the world is constructed based on the principle of rotation, defining rotational motion, which is never a single shape, is crucial. In this study, we give a new form of rotational motion that combines rotational motion with superellipses, the best way to express nature. Using the Rodrigues and Cayley methods for superelliptic rotational motion, we inroduce the corresponding rotation matrix forms.
However, the simplest and most important method for describing rotational motion around any axis is the use of quaternions. Due to the useful applications of quaternions, the definition of superelliptic rotation and superelliptic quaternions will be very helpful in understanding nature. Therefore, the Gielis function is used to define superelliptic quaternions using the superelliptic inner product and the superelliptic vector product. Unlike other quaternions, this quaternion is defined by a positive-valued function that represents a wide variety of superelliptic rotational motions, in addition to circular and elliptic motions, using the Gielis function. This quaternionic structure expresses motion on the superelliptic relative to the Gielis function. In other words, this quaternion structure describes superelliptic rotational motion about any axis and is a general quaternion structure. We use mathematical programs to visualize and characterize various applications of the theory to demonstrate how superelliptic quaternions can be used.
Department of Mathematics, Arts and Science Faculty, Amasya University, Amasya, Turkey
Extrinsic characteristics of immersions
We outline the history of the idea of deformation of space, which lead to the concept of curvature invariants, as we understand them today, including contributions of E. Bacaloglu and F. Casorati, among others. We pursue the following question: what is the best way to quantify the deformation of space? This important question could be viewed in a new paradigm after 1956, when John F. Nash, Jr. proved that a Riemannian manifold can be immersed isometrically into an Euclidean ambient space of dimension sufficiently large. This important theorem allowed to view the representation of space from its exterior, from an outside perspective. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash’s Theorem effectively is finding useful relationships between intrinsic and extrinsic quantities characterising immersions.
A turning point in the history of the question we pursue was an enlightening paper written by B.-Y. Chen in 1993, which paved the way for a deeper understanding of the meaning of the Riemannian inequalities between intrinsic and extrinsic quantities. Our present discussion invites a reflection on whether we could hope to characterise submanifolds by using mainly extrinsic quantities.
The following Problem is raised: Are there any other extrinsic relations that determine the topology or the geometry of an embedded geometric object? How do we define them? How much insight do they provide, when we look at the geometric object “from the outside”?
Department of Mathematics, California State University Fullerton, CA, USA
Anisotropic Compact Star Model in Generalized Tolman-Kuchowicz Background with a Quadratic Equation of State.
This study investigates a set of solutions to the Einstein field equations for the uncharged, static, and spherically symmetric compact star PSR J0952–0607 by applying the generalized Tolman-Kuchowicz space-time metric along with a quadratic equation of state. Through graphical analysis, the model parameter n is established, and a stable stellar structure for the compact star model is developed. The stability of the proposed model is analyzed using the Tolman-Oppenheimer-Volkoff equation and the Harrison-Zeldovich-Novikov criterion. The anisotropic compact star model meets all necessary stability standards, including the adiabatic index, the causality requirement, Herrera’s cracking condition, Buchdahl’s condition, and is free from central singularities.
Department of Mathematics, Pandit Deendayal Energy University, Gandhinagar, India
Can we build commutative R-functions using Geometry?
We present recent progress on the extension of binary R-functions to commutative N-ary R-functions. Starting from a geometric reinterpretation of the binary case, which yields a simple and direct parameterization, we extend and further simplify this concept in order to propose a new family of R-functions with N arguments. This approach opens the way to a unified and geometrically intuitive framework for designing commutative logical constructions in higher dimensions.
Université Bourgogne Europe, IUT Le Creusot, France
Nature of the order parameters in disordered systems: meaning of spatial patterns in thermodynamics
The notion of order is somewhat abstract one, so different fields use it differently. In solid state physics, the order means unambiguously the perfect periodicity, while defects present disorder. In this manner, traditional physicists used to judge the order by regularities or patterns observed in the spatial arrangement of atoms. However, we soon encounter difficulties with this definition, when wider classes of material, such as quasicrystals, are examined. The extreme case is living systems. Although there is no periodicity, we can intuitively understand that living systems must be highly ordered materials. Schrodinger described living systems as "aperiodic crystal". This inconsistency between order and periodicity requires us to revisit the definition of the order in thermodynamics. We need to make our mind free from geometrical regulations. Careful analysis of the order parameters observed in various phenomena leads us to conclude that the essential requirement of the order parameter is the existence of a definite value <x> for the time average of dynamical variable x(t). For solids, an atom position fluctuates around the equilibrium position. Therefore, the equilibrium atom position can generally be regarded as the order parameter for solids. This view resolves the self-contradictory concept "aperiodic crystal" for living systems.
Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
爻 (yáo): The Lattice of Change — A Symbolic Geometry and Philosophy of Interconnected Reality
This paper explores the Chinese character 爻 (yáo) not merely as a linguistic element within the I Ching but as a symbolic unit of dynamic transformation, spatial logic, and metaphysical structure. By interpreting 爻 through interdisciplinary lenses—including symbolic geometry, philosophical dualism, topology, and cross-cultural analogs such as the quincunx from Sir Thomas Browne's "The Garden of Cyrus"—this research proposes that 爻 encodes a lattice of change: a woven structure representing the recursive, interconnected, and transformative nature of reality.
Nova Southeastern University, Florida USA
Statistical universals of genomic DNAs and a quantum-logical approach to the analysis of genetic structures
Mechanical Engineering Research Institute of Russian Academy of Sciences, Moscow, Russia
Möbius phenomenon in Generalised Möbius-Listing surfaces and bodies with different radial cross-section
The report describes the Generalised Möbius-Listing surfaces and bodies with different radial cross sections. These geometric objects have one common property: each of them, at a certain number (m,n) and diametrical cut, exhibits the Mobius phenomenon. Among the considered flat figures (or lines) radial cross sections are such objects as: Euler spiral -“clothoid”, Lame curves, Grandi Roses, Gielis curves, Epicycloid, Hypocycloid and so on.
Department of Mathematics, Tbilisi State University, Georgia
Playing Graphs. Music through Algorithms
This study applies scientific and computational methods to the musical context, proposing a quantitative and replicable framework for the analysis of musical scores. Starting from digital encodings in MusicXML format, automatic data parsing enables the extraction and structured representation of melodic, textual, and harmonic information through data structures and directed graphs. Each musical element is modeled as a node enriched with metadata, allowing for precise and measurable representations of compositional features. By combining techniques from data mining, Graph Theory, and Graph Neural Networks, this approach aims to transform musical manuscripts into analyzable datasets, enabling automatic information retrieval and pattern recognition. Finally, this methodology lays the groundwork for developing generative models capable of exploring and simulating creative processes in composition
Department of Engineering, University of Palermo, Italy
Finsler geometry, spacetime and gravity
Finsler geometry is a wide generalization of the Lorentzian one, appearing when one wishes to allow for anisotropies between different spacetime directions even at an infinitesimal scale. The interest in Finsler geometry has quickly grown in recent years due: in addition to its mathematical naturalness, it has been linked to phenomena in the frontier of knowledge in physics, ranging from quantum gravity phenomenology to the accelerated expansion of the universe.
After covering its foundations, this talk will review the proposals that have been made for extending the Einstein field equations to Finsler spacetimes. The focus will be on those admitting a variational formulation, and the different not-so-classical geometric objects that appear in them. We will see how the solutions change depending on whether there is a (nonlinear) connection independent of the metric, and what some exact solutions look like.
A Point-Theory of Morphogenesis
Building on earlier work with generalised conic sections, we use the superformula to introduce ultra-flexibility instead of rigidity as encoded in the geometry of Euclid and Descartes. By considering Points as ultra-extensible primitives, we define Points endowed with shape, size, and historical continuity. This Point-Theory of Morphogenesis addresses multiple challenges for a mathematical theory of morphogenesis for both natural and abstract shapes. The theory is formalised by a minimal set of one definition, two axioms, and two postulates.
Geniaal BV, Antwerp
Department of Biosciences Engineering, University of Antwerp, Belgium