Algebraic Dynamics Research Articles

Generation of genetic codes with 2-adic codon algebra and adaptive dynamics

This is a brief review on modeling genetic codes with the aid of 2-adic dynamical systems. In this model amino acids are encoded by the attractors of such dynamical systems. Each genetic code is coupled to the special class of 2-adic dynamics. We consider the discrete dynamical systems, These are the iterations of a function F:Z2→Z2, where Z2 is the ring of 2-adic numbers (2-adic tree). A genetic code is characterized by the set of attractors of a function belonging to the code generating functional class. The main mathematical problem is to reduce degeneration of dynamic representation and select the optimal generating function. Here optimality can be treated in many ways. One possibility is to consider the Lipschitz functions playing the crucial role in general theory of iterations. Then we minimize the Lip-constant. The main issue is to find the proper biological interpretation of code-functions. One can speculate that the evolution of the genetic codes can be described in information space of the nucleotide-strings endowed with ultrametric (treelike) geometry. A code-function is a fitness function; the solutions of the genetic code optimization problem are attractors of the code-function. We illustrate this approach by generation of the standard nuclear and (vertebrate) mitochondrial genetics codes.

Algebraic Dynamical Systems in Machine Learning

We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learning problems on structured or non-numerical data, including ‘hybrid symbolic-numeric’ models.

Birational maps with transcendental dynamical degree

AbstractWe give examples of birational selfmaps of , whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (B) in particular. We demonstrate that any B-differentiable function gives rise to a null shear-free congruence (NSFC) on the B-vector space CM and on its Minkowski subspace M. Making use of the Kerr–Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of B-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in CM. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the “Unique Worldline” and discuss the connections of the theory with the Feynman–Wheeler concept of “One-Electron Universe”.

Remarks on algebraic dynamics in positive characteristic

AbstractIn this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. Applying the arithmetic degree and canonical height in positive characteristic, we prove the Dynamical Mordell–Lang Conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit Conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell–Lang and prove a counting result for backward orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.

New families satisfying the dynamical uniform boundedness principle over function fields

We extend a technique, originally due to the first author and Poonen, for proving cases of the Strong Uniform Boundedness Principle (SUBP) in algebraic dynamics over function fields of positive characteristic. The original method applied to unicritical polynomials for which the characteristic does not divide the degree. We show that many new 1-parameter families of polynomials satisfy the SUBP, including the family of all quadratic polynomials in even characteristic. We also give a new family of non-polynomial, non-Lattès rational functions that satisfies the SUBP.

Partial differential equations of entropy analysis on ternary hybridity nanofluid flow model via rotating disk with hall current and electromagnetic radiative influences

The flow of a fluid across a revolving disc has several technical and industrial uses. Examples of rotating disc flows include centrifugal pumps, viscometers, rotors, fans, turbines, and spinning discs. An important technology with implications for numerous treatments utilized in numerous sectors is the use of hybrid nanofluids (HNFs) to accelerate current advancements. Through investigation of ternary nanoparticle impacts on heat transfer (HT) and liquid movement, the thermal properties of tri-HNFs were to be ascertained in this study. Hall current, thermal radiation, and heat dissipation have all been studied in relation to the use of flow-describing equations. The ternary HNFs under research are composed of the nanomolecules aluminum oxide (Al2O3), copper oxide (CuO), silver (Ag), and water (H2O). For a number of significant physical characteristics, the physical situation is represented utilizing the boundary layer investigation, which produces partial differential equations (PDEs). The rheology of the movement is extended and computed in a revolving setting under the assumption that the movement is caused by a rotatingfloppy. Before the solution was found using the finite difference method, complicated generated PDEs were transformed into corresponding ODEs (Keller Box method). A rise in the implicated influencing factors has numerous notable physical impacts that have been seen and recorded. The Keller Box method (KBM) approach is also delivered for simulating the determination of nonlinear system problems faced in developing liquid and supplementary algebraic dynamics domains. The rate of entropy formation rises as the magnetic field parameter and radiation parameter increase. Entropy production rate decreases as the Brinkman number and Hall current parameter become more enriched. The thermal efficiency of ternary HNFs compared to conventional HNFs losses to a low of 4.8% and peaks to 5.2%.

The algebraic dynamics of the pentagram map

AbstractThe pentagram map, introduced by Schwartz [The pentagram map. Exp. Math.1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J.162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

Fully algebraic and self-consistent effective dynamics in a static quantum embedding

Quantum embedding approaches involve the self-consistent optimization of a local fragment of a strongly correlated system, entangled with the wider environment. The `energy-weighted' density matrix embedding theory (EwDMET) was established recently as a way to systematically control the resolution of the fragment-environment coupling, and allow for true quantum fluctuations over this boundary to be self-consistently optimized within a fully static framework. In this work, we reformulate the algorithm to ensure that EwDMET can be considered equivalent to an optimal and rigorous truncation of the self-consistent dynamics of dynamical mean-field theory (DMFT). A practical limitation of these quantum embedding approaches is often a numerical fitting of a self-consistent object defining the quantum effects. However, we show here that in this formulation, all numerical fitting steps can be entirely circumvented, via an effective Dyson equation in the space of truncated dynamics. This provides a robust and analytic self-consistency for the method, and an ability to systematically and rigorously converge to DMFT from a static, wave function perspective. We demonstrate that this improved approach can solve the correlated dynamics and phase transitions of the Bethe lattice Hubbard model in infinite dimensions, as well as one- and two-dimensional Hubbard models where we clearly show the benefits of this rapidly convergent basis for correlation-driven fluctuations. This systematically truncated description of the effective dynamics of the problem also allows access to quantities such as Fermi liquid parameters and renormalized dynamics, and demonstrates a numerically efficient, systematic convergence to the zero-temperature dynamical mean-field theory limit.

Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita–Takesaki theory in our context.

Algebraic dynamics of k-valued fuzzy cognitive maps and its stabilization

Fuzzy Cognitive Maps (FCMs) as knowledge-based tools are widely implemented for knowledge representation and reasoning. To reach final equilibrium states is essential for the results of causal reasoning in practical applications by using FCMs. However, the dynamics and the stabilization problem of FCMs are still not adequately studied so far. In this article, the algebraic dynamics of k-valued fuzzy cognitive maps and its stabilization is analytically and quantitatively investigated. Based on the technique of semi-tensor product (STP), k-valued FCMs are firstly converted into discrete-time linear representation. General formulas are obtained to calculate the number of fixed points and limit cycles of finite-state FCMs, and a necessary and sufficient criterion to determine attractors is proposed, by means of which an algorithm for calculating all attractors and their basins are achieved. Furthermore, the stabilization problem of k-valued FCMs is discussed. By introducing controls to certain concepts in FCMs, the system can be globally stabilized in equilibrium states. The related necessary and sufficient criterion is proposed and the corresponding method to achieve the global stabilization of FCMs under constant controls is given. Examples are shown to demonstrate the effectiveness and feasibility of the proposed scheme.

Exploration of the algebraic traveling wave solutions of a higher order model

PurposeThe purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena.Design/methodology/approachThe authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions.FindingsAs a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections.Research limitations/implicationsThe paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure.Practical implicationsBy considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections.Originality/valueTo the best of the authors’ knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.

Tropical Limits of Decimated Polynomials

Motivated by some problems that originate in Statistical Physics and Algebraic Dynamics, we discuss a particular renormalization mechanism of multivariate Laurent polynomials which is called a decimation, and the corresponding tropical limiting shape result obtained in Arzhakova et al. (Decimation of principal actions. Preprint, 2018).

On Solutions of the Fuji-Suzuki-Tsuda System

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlev\'e VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.

Algebraic dynamics of the lifts of Frobenius

We study the algbraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic is the Frobenius. In particular, we prove a version of the dynamical Manin-Mumford conjecture and the dynamical Mordell-Lang conjecture for the coherent backward orbits for such endomorphisms. We also give a new proof of a dynamical version of the Tate-Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedding into a dynamical system for some endomorphism of a projective space.

Quantitative and Neurovascular Anatomy of the Growing Gracilis Muscle in the Human Fetuses.

This study is intended to obtain the algebraic growth dynamics of the gracilis in fetuses and determine the variations of neurovascular pedicle(s) of the gracilis, to aid infant surgeries. Forty fetuses (19 males and 21 females) were included in the study. Gestational mean age of the fetuses was 22.40 ± 2.67 (range, 18-28) weeks. Numerical values were obtained using a digital caliper and a digital image analysis software. Linear functions for the surface area, width, anterior, and posterior margin lengths of the gracilis were calculated, respectively, as: y = -289.307 + 20.501 × age (weeks), y = -7.113 + 0.622 × age (weeks), y = -24.421 + 3.434 × age (weeks), and y = -24.397 + 3.314 × age (weeks). In addition, length and width of the gracilis tendon were calculated as y = -6.676 + 0.934 × age (weeks) and y = -0.732 + 0.074 × age (weeks), respectively. Parameters of the gracilis had no statistically significant difference regarding side and sex (P > 0.05). In all the specimens, the gracilis was innervated by the anterior branch of the obturator nerve. Blood supply of the gracilis was identified to be derived from 1 single artery in 38 sides of total 80 (47.5%), from 2 arteries in 36 (45%) and from 3 arteries in 6 (7.5%). In 74 sides (92.5%), the nerve was superficial to the main artery, whereas in 6 sides (7.5%), it was deeper. The data of the present study could be beneficial for surgeons in infant surgeries to treat conditions such as obstetrical brachial plexus paralysis, facial palsy, or anal incontinence. Linear functions can be utilized to better evaluate the growth course of the gracilis in fetuses and to predict the dimensions thereof. Additionally, comprehending the structure and recognizing the variations of the gracilis nerves and arteries can help to protect the neurovascular pedicle(s) of the gracilis during the operations.

Dynamics of algebras in quantum unstable systems

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is nonunitary. This allows to describe observables that are initially noncommutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.

Algebraic dynamics of skew-linear self-maps

Let X X be a variety defined over an algebraically closed field k k of characteristic 0 0 , let N ∈ N N\in \mathbb {N} , let g : X ⇢ X g:X\dashrightarrow X be a dominant rational self-map, and let A : A N ⟶ A N A:\mathbb {A}^N\longrightarrow \mathbb {A}^N be a linear transformation defined over k ( X ) k(X) , i.e., for a Zariski open dense subset U ⊂ X U\subset X , we have that for x ∈ U ( k ) x\in U(k) , the specialization A ( x ) A(x) is an N N -by- N N matrix with entries in k k . We let f : X × A N ⇢ X × A N f:X\times \mathbb {A}^N\dashrightarrow X\times \mathbb {A}^N be the rational endomorphism given by ( x , y ) ↦ ( g ( x ) , A ( x ) y ) (x,y)\mapsto (g(x), A(x)y) . We prove that if the determinant of A A is nonzero and if there exists x ∈ X ( k ) x\in X(k) such that its orbit O g ( x ) \mathcal {O}_g(x) is Zariski dense in X X , then either there exists a point z ∈ ( X × A N ) ( k ) z\in (X\times \mathbb {A}^N)(k) such that its orbit O f ( z ) \mathcal {O}_f(z) is Zariski dense in X × A N X\times \mathbb {A}^N or there exists a nonconstant rational function ψ ∈ k ( X × A N ) \psi \in k(X\times \mathbb {A}^N) such that ψ ∘ f = ψ \psi \circ f=\psi . Our result provides additional evidence to a conjecture of Medvedev and Scanlon.

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Abstract Let 𝕂 {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over 𝕂 {\mathbb{K}} . Let f : X ⊢ X {f:X\vdash X} and g : Y ⊢ Y {g:Y\vdash Y} be dominant correspondences, and π : X ⇢ Y {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that π ∘ f = g ∘ π {\pi\circ f=g\circ\pi} . We define relative dynamical degrees λ p ( f | π ) ≥ 1 {\lambda_{p}(f|\pi)\geq 1} for any p = 0 , … , dim ⁡ ( X ) - dim ⁡ ( Y ) {p=0,\dots,\dim(X)-\dim(Y)} . These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy ( φ , ψ ) {(\varphi,\psi)} from π 2 : ( X 2 , f 2 ) → ( Y 2 , g 2 ) {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to π 1 : ( X 1 , f 1 ) → ( Y 1 , g 1 ) {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have λ p ( f 1 | π 1 ) ≥ λ p ( f 2 | π 2 ) {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when 𝕂 = ℂ {\mathbb{K}=\mathbb{C}} . Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

Self-dual connections and the equations of fundamental fields in a Weyl–Cartan space

Spaces with a Weyl-type connection and torsion of a special type induced by the structure of the differentiability conditions in the algebra of complex quaternions are considered. The consistency of these conditions implies the self-duality of curvature. The Maxwell and SL(2, C) Yang-Mills fields associated with the irreducible components of the connection also turn out to be self-dual, so that the corresponding equations are fulfilled on the solutions of the generating system. Using the twistor structure of the latter, its general solution is obtained. The singular locus has a string-like (particle-like) structure and generates the self-consistent algebraic dynamics of the string system.