Life and works of Revaz Grigolia
Department of Mathematics, University of Salerno, Italy
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
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.
Department of Mathematics, Arts and Science Faculty, Amasya University, Amasya, Turkey
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
Anisotropic Compact Star Model in Generalized Tolman-Kuchowicz Background with a Quadratic Equation of State.
Abstract: 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 encounters 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
Abstract: 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
Department of Molecular & Clinical Cancer Medicine, University of Liverpool, United Kingdom
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
Department of Mathematics, Tbilisi State University, Georgia
University of Palermo, Italy