New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences

A perfect n-gon is an abstraction of a regular n-gon when regarded in the real projective plane. The vertices of a regular n-gon P lie on n parallel classes of lines. The lines in any parallel class meet at a point at infinity. We call these points the perspective points of P. The vertices of P lie on a circle and the perspective points of P lie on the line at infinity in the projective plane, so we can say that the combined set of vertices and perspective points lie on a (reducible) cubic curve consisting of a line and a circle. In our Main Theorem we show that the combined set of vertices and perspective points of any perfect polygon lie on a cubic curve which may be irreducible. In case the cubic is irreducible, a well-known algebra which we call a geometric triple system can be defined on its points. We show that perfect polygons can be obtained as translates of these algebras.

In this paper we discuss triple systems whose elements come from an abelian group G. Every triple of elements with the same sum in G will correspond to three collinear points in the projective plane. We will show that the set of all points in such a triple system must lie on a cubic curve γ which we call the envelope of the system. We will determine the number and location of equivalent triple systems which have the same cubic envelope. These results will be used to determine exactly which finite abelian groups can be used to construct a geometric triple system.

The binary hypercommutative (BH) variety, also referred to as the variety of thirdpoint groupoids in Harris (Thirdpoint groupoids, Masters thesis, Virginia State University, 2008), is a collection of algebras associated naturally with the nonsingular points of an irreducible cubic curve. The identities which define these algebras have, as consequences, expressions whose value is invariant under any permutation of the variables comprising the expression. We characterize such expressions and Conjecture that they can be used to determine points of intersection between a given cubic curve and an arbitrary algebraic curve. We prove the Conjecture in the case of the intersection of a cubic γ and a conic β: If γ, β meet in the five points a,b,c,d,e in the real projective plane, then they meet also in the point e*{(a*b)*(c*d)} where a*b denotes the point of intersection besides a,b where the line joining a,b meets γ. The expression e*{(a*b)*(c*d)} is invariant in the BH variety. The Conjecture is also proved for two types of singular cubics. We provide illustrations as strong evidence, but cannot prove similar statements pertaining to the intersection of a nonsingular cubic with higher degree algebraic curves.

The Ternary Hypercommutative (TH) variety is an equational class of algebras determined by the identities:

A word in this variety is a properly parenthetized expression consisting of variables and the ternary operation [ , , ]. If T is a TH algebra, and elements from T are substituted for the variables in a word, then the resulting element of T is the value of the word with respect to T. A word is monotone if no variable is repeated, and a monotone word is invariant if its value with respect to any TH algebra remains unchanged under any permutation of its variables. Thus to say [a,b,c] is invariant and is a shorthand way to express the identities [a,b,c] = [a,c,b] = [b,a,c] = [b,c,a] = [c,a,b] = [c,b,a]. The geometric significance of the TH variety is found in the study of intersection properties of circles with certain types of algebraic curves. For example, if α is an ellipse and a, b, c are any three points of α, then the circle (a,b,c) meets α in a unique fourth point, counting multiplicities, which we denote by our ternary operation [a,b,c]. Identities (i), (ii) are clearly satisfied by this operation, and in Fletcher (Group Circle Systems on Conics, New Frontiers of Multidisciplinary Research in STEAM-H, Springer International Publishing, Cham, 2014) it is shown that axiom (iii) also holds. We call algebraic curves whose intersection with circles satisfies the TH axioms supercyclic. We characterize supercyclic curves and determine the invariant words in the TH variety. Evidence is also given for a Conjecture regarding the significance of these words with respect to intersection properties of supercyclic curves and general algebraic curves.

We present the results of numerical simulation for image segmentation based on the chain distance clustering algorithm. The key issue is the use of the p-adic metric, where p > 1 is a prime number, at the scale of levels of brightness (pixel wise). In previous studies the p-adic metric was used mainly in combination with spectral methods. In this paper this metric is explored directly, without preparatory transformations of images. The main distinguishing feature of the p-adic metric is that it reflects the hierarchic structure of information presented in an image. Different classes of images match with in general different prime p (although the choice p = 2 works on average). Therefore the presented image segmentation procedure has to be combined with a kind of learning to select the prime p corresponding to the class of images under consideration.

In this note we present the Prime Number Paradigm: “Nature encodes its laws with the aid of prime numbers.” We claim that prime number skeletons of data-sets contain the basic structures represented them. This paradigm is illustrated by a variety of data: from astrophysics to geophysics (including petroleum data), and reading comprehension. We remark that the ideas about the fundamental (and even mystical) role played by primes in mathematics and in physics were discussed by many scientists during the last 2000 years. The main advantage of our approach is its close connection with real experimental data. To present deeper patterns in data, the prime skeletons can be extended to p-adic skeletons. P-adic numbers play the important role number theory and recently they started to be widely used in theoretical physics, form string theory and theory of complex disordered systems to geophysics.

The logical combinatorial approach of the pattern recognition theory works with the description of objects in terms of a combination of quantitative and qualitative variables, giving the possibility to consider absent information for the values of some variables in the object description. This approach uses supervised classification algorithms, which are based on the concept of partial precedence, that is, partial analogies (an object can be alike to another object, not in its totality). These characteristics are suitable to model classification problems in Medicine. The objective of this work is to show the usefulness of the logical combinatorial approach to solve problems of pattern recognition in Medicine by illustrating three case studies: the differential diagnosis of Glaucoma, a method for comparing somatotypes (human body types in terms of physical structure), and the prognosis of rehabilitation of patients with cleft lip and palate.

The infinite Darcy–Prandtl number model is an effective reduced model for describing convection in a fluid-saturated porous medium. It is well known that the deterministic model does not possess a unique invariant measure. In this work, we study the dynamics of the infinite Darcy–Prandtl number model, under an additive stochastic forcing of its low modes. This is the so-called stochastic infinite Darcy–Prandtl number model. We prove that the stochastically forced system, does indeed possess a unique invariant measure.

Barrier options are a class of exotic options that are traded in over-the-counter markets worldwide. These options are particularly attractive for their lower cost compared to vanilla options. However, the closed form analytical solutions for the partial differential equations modeling these options are not easy to obtain and therefore one usually seeks numerical approaches to find them. In this paper, we consider two types of exotic options, namely a single barrier European down-and-out call and a double barrier European knock-out call options. Like some other standard and nonstandard options, these barrier options also have non-smooth payoffs at the exercise price. This non-smooth payoff is the main cause of the reduction in accuracy when the classical numerical methods, for example, lattice method, Monte Carlo method, or other methods based on finite difference and finite elements are used to solve such problems. In fact, the same happens when one uses the spectral method which is known to preserve the exponential accuracy. In order to retain this high-order accuracy, in this paper we propose a spectral decomposition method which approximates the unknown solution by rational interpolants on each sub-domain. The resulting semi-discrete problem is solved by a contour integral method. Our numerical results affirm that the proposed approach is very robust and gives very reliable results.

The p-adic aspects of Topological Geometrodynamics (TGD) will be discussed. Introduction gives a short summary about classical and quantum TGD. This is needed since the p-adic ideas are inspired by TGD based view about physics.

p-Adic mass calculations relying on p-adic generalization of thermodynamics and super-symplectic and super-conformal symmetries are summarized. Number theoretical existence constrains lead to highly non-trivial and successful physical predictions. The notion of canonical identification mapping p-adic mass squared to real mass squared emerges, and is expected to be a key player of adelic physics allowing to map various invariants from p-adics to reals and vice versa.

A view about p-adicization and adelization of real number based physics is proposed. The proposal is a fusion of real physics and various p-adic physics to single coherent whole achieved by a generalization of number concept by fusing reals and extensions of p-adic numbers induced by given extension of rationals to a larger structure and having the extension of rationals as their intersection.

Also the understanding of the correspondence between real and p-adic physics at various levels—space-time level, imbedding space level, and level of “world of classical worlds” (WCW)—is a challenge. The gigantic isometry group of WCW and the maximal isometry group of imbedding space give hopes about a resolution of the problems. Strong form of holography (SH) allows a non-local correspondence between real and p-adic space-time surfaces induced by algebraic continuation from common string world sheets and partonic 2-surfaces. Also local correspondence seems intuitively plausible and is based on number theoretic discretization as intersection of real and p-adic surfaces providing automatically finite “cognitive” resolution. The existence of p-adic variants of Kähler geometry of WCW is a challenge, and NTU might allow to realize it.

I will also sum up the role of p-adic physics in TGD inspired theory of consciousness. Negentropic entanglement (NE) characterized by number theoretical entanglement negentropy (NEN) plays a key role. Negentropy Maximization Principle (NMP) forces the generation of NE. The interpretation is in terms of evolution as increase of negentropy resources.

To be and to become an American: past, present, and future. We analyze the dynamics of the socio-cultural evolution of America as a player in a 1-agent game, integrating multiple immigrant-based cultural identities: American, in continuous random cross-cultural interactions with one another, assign a probability to each of the American ten core values which we recall, thereby defining at every instant a population state as a vector of probabilities. Adapting a methodology from Evolutionary Game Theory (replicator-like in human context), we uncover all possible dynamical game scenarios, to include Nash Equilibria, Eventual Nash Equilibria, Nash Limit Cycles, and Isochrons, a state of self-sustained oscillations in decision-making and where individual preferences evolve with the same constant phase. (Social Isochrons). We conjecture a game scenario for America stability and prosperity as the co-existence of asymptotically stable community Nash Equilibria based on the respective cultural identities, around a Nash Limit Cycle inside-attractive and outside-repulsive, together with a national Nash Equilibrium, asymptotically stable, around a Nash Limit Cycle inside-repulsive while outside-attractive, all in the interior of a 9-simplex, convex compact subset of the R ¹⁰ Euclidean space. However, socio-cultural structures being highly hierarchical, such an analysis should be extended to the much richer non-Archimedean/p-adic simplex.