Identification of approximate symmetries in biological development

Virtually all forms of life, from single-cell eukaryotes to complex, highly differentiated multicellular organisms, exhibit a property referred to as symmetry. However, precise measures of symmetry are often difficult to formulate and apply in a meaningful way to biological systems, where symmetries and asymmetries can be dynamic and transient, or be visually apparent but not reliably quantifiable using standard measures from mathematics and physics. Here, we present and illustrate a novel measure that draws on concepts from information theory to quantify the degree of symmetry, enabling the identification of approximate symmetries that may be present in a pattern or a biological image. We apply the measure to rotation, reflection and translation symmetries in patterns produced by a Turing model, as well as natural objects (algae, flowers and leaves). This method of symmetry quantification is unbiased and rigorous, and requires minimal manual processing compared to alternative measures. The proposed method is therefore a useful tool for comparison and identification of symmetries in biological systems, with potential future applications to symmetries that arise during development, as observed in vivo or as produced by mathematical models. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter

The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. For algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE and can be systematically computed. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq equation reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.

Multi-track displaced vertices at B-factories

We propose a program at B-factories of inclusive, multi-track displaced vertex searches, which are expected to be low background and give excellent sensitivity to non-minimal hidden sectors. Multi-particle hidden sectors often include long-lived particles (LLPs) which result from approximate symmetries, and we classify the possible decays of GeV-scale LLPs in an effective field theory framework. Considering several LLP production modes, including dark photons and dark Higgs bosons, we study the sensitivity of LLP searches with different number of displaced vertices per event and track requirements per displaced vertex, showing that inclusive searches can have sensitivity to a large range of hidden sector models that are otherwise unconstrained by current or planned searches.

Group Theory: Mathematical Expression of Symmetry in Physics

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

Theoretical and experimental status of rare charm decays

Rare charm decays offer the unique possibility to explore flavor-changing neutral-currents in the up-sector within the Standard Model and beyond. Due to the lack of effective methods to reliably describe its low energy dynamics, rare charm decays have been considered as less promising for long. However, this lack does not exclude the possibility to perform promising searches for New Physics per se, but a different philosophy of work is required. Exact or approximate symmetries of the Standard Model allow to construct clean null-test observables, yielding an excellent road to the discovery of New Physics, complementing the existing studies in the down-sector. In this review, we summarize the theoretical and experimental status of rare charm [Formula: see text] transitions, as well as opportunities for current and future experiments such as LHCb, Belle II, BES III, the FCC-ee and proposed tau-charm factories. We also use the most recent experimental results to report updated limits on lepton-flavor conserving and lepton-flavor violating Wilson coefficients.

Particles dynamics around time conformal quintessential Schwarzschild black hole

In this paper, the effect of time is explored on the dynamics of neutral and charged particles around the Schwarzschild (Sch) black hole (BH) environed by quintessence. We introduce a general time conformal factor [Formula: see text] to the quintessential Sch spacetime, where [Formula: see text] is a time ([Formula: see text])-dependent function and [Formula: see text], a small parameter, that causes perturbation in the corresponding spacetime. The Noether symmetry equation is obtained with the help of Lagrangian corresponding to time conformal Sch spacetime environed by quintessence. Consequently, we get a 19 partial differential equations (PDEs) system. The solution of this system gives Noether symmetries, Noether approximate symmetries, and the corresponding conservation laws. The dynamics of neutral/charged particles are studied as well as graphically analyzed at the foot of these invariants.

FOURTH-ORDER PATTERN FORMING PDES: PARTIAL AND APPROXIMATE SYMMETRIES

This paper considers pattern forming nonlinear models arising in the study of thermal convection and continuous media. A primary method for the derivation of symmetries and conservation laws is Noether’s theorem. However, in the absence of a Lagrangian for the equations investigated, we propose the use of partial Lagrangians within the framework of calculating conservation laws. Additionally, a nonlinear Kuramoto-Sivashinsky equation is recast into an equation possessing a perturbation term. To achieve this, the knowledge of approximate transformations on the admissible coefficient parameters is required. A perturbation parameter is suitably chosen to allow for the construction of nontrivial approximate symmetries. It is demonstrated that this selection provides approximate solutions.