Symmetry Conservation Research Articles

The Nature of the Chemical Bonds of High-Valent Transition-Metal Oxo (M=O) and Peroxo (MOO) Compounds: A Historical Perspective of the Metal Oxyl-Radical Character by the Classical to Quantum Computations.

This review article describes a historical perspective of elucidation of the nature of the chemical bonds of the high-valent transition metal oxo (M=O) and peroxo (M-O-O) compounds in chemistry and biology. The basic concepts and theoretical backgrounds of the broken-symmetry (BS) method are revisited to explain orbital symmetry conservation and orbital symmetry breaking for the theoretical characterization of four different mechanisms of chemical reactions. Beyond BS methods using the natural orbitals (UNO) of the BS solutions, such as UNO CI (CC), are also revisited for the elucidation of the scope and applicability of the BS methods. Several chemical indices have been derived as the conceptual bridges between the BS and beyond BS methods. The BS molecular orbital models have been employed to explain the metal oxyl-radical character of the M=O and M-O-O bonds, which respond to their radical reactivity. The isolobal and isospin analogy between carbonyl oxide R2C-O-O and metal peroxide LFe-O-O has been applied to understand and explain the chameleonic chemical reactivity of these compounds. The isolobal and isospin analogy among Fe=O, O=O, and O have also provided the triplet atomic oxygen (3O) model for non-heme Fe(IV)=O species with strong radical reactivity. The chameleonic reactivity of the compounds I (Cpd I) and II (Cpd II) is also explained by this analogy. The early proposals obtained by these theoretical models have been examined based on recent computational results by hybrid DFT (UHDFT), DLPNO CCSD(T0), CASPT2, and UNO CI (CC) methods and quantum computing (QC).

Higher-Order Electrocyclizations in Biological and Synthetic Processes.

In general, electrocyclizations follow the Woodward-Hoffmann's rules of conservation of orbital symmetry. These rules have been extensively verified in low-order processes, both in thermal and photochemical reactions, up to eight π-electrons. However, when the number of π-electrons in the system increases, some deviations of that general rules can be found. This focused review highlights the main features of reported higher-order electrocyclizations involving 10, 12, 14, 16 and 18 π-electrons. Some of these examples constitute useful intermediates in the synthesis of biologically active compounds. When computational studies were not included in the reported examples, DFT calculations have been performed to be included in this review. Analysis of the respective pericyclic topologies shows the importance of computational tools for understanding the selectivity observed experimentally.

Infinite Dimensional Symmetry Group Reductions and Conservation Laws of Lin–Reissner–Tsien Equation

Infinite Dimensional Symmetry Group Reductions and Conservation Laws of Lin–Reissner–Tsien Equation

Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u). We carried out a complete Lie symmetry classification by finding forms for α(u) and f(u). This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach.

Lie symmetry analysis and conservation laws of axially uniform strings

Lie symmetry analysis and conservation laws of axially uniform strings

Lie symmetry analysis, exact solutions, and conservation laws to multi-component nonlinear Schrödinger equations

Lie symmetry analysis, exact solutions, and conservation laws to multi-component nonlinear Schrödinger equations

Lie symmetry analysis, conservation laws and exact solutions for variable-coefficients (2+1)-dimensional dissipative long-wave system

The Lie symmetry analysis, optimal system, exact solutions and conservation laws of the (2+1)-dimensional variable-coefficients dissipative long-wave (vcDLW) system are introduced in this paper. Lie symmetry analysis method is used to obtain the vector field firstly. And the Olver method is used to obtain the optimal system of one-dimensional subalgebras of the equations. Based on the optimal system, the equations are similarly reduced. Some representative equations are selected from the equations obtained after similarly reduction, and then the expansion method is applied to these equations, so that some new exact solutions of the (2+1)-dimensional vcDLW equations can be obtained. Finally, we study the conservation laws of the system.

Lie symmetry analysis and conservation laws with soliton solutions to a nonlinear model related to chains of atoms

Lie symmetry analysis and conservation laws with soliton solutions to a nonlinear model related to chains of atoms

Time-fractional generalized fifth-order KdV equation: Lie symmetry analysis and conservation laws

The purpose of this study is to apply the Lie group analysis method to the time-fractional order generalized fifth-order KdV (TFF-KdV) equation. We examine applying symmetry analysis to the TFF-KdV equation with the Riemann–Liouville (R–L) derivative, employing the G′/G-expansion approach to yield trigonometric, hyperbolic, and rational function solutions with arbitrary constants. The discovered solutions are unique and have never been studied previously. For solving non-linear fractional partial differential equations, we find that the G′/G-expansion approach is highly effective. Finally, conservation laws for the equation are well-built with a full derivation based on the Noether theorem.

Seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one.

We consider two-dimensional cellular automata with the von Neumann neighborhood that satisfy two properties of interest from a modeling viewpoint: rotation symmetry (i.e., the local rule is invariant under rotation of the neighborhood by 90^{∘}) and number conservation (i.e., the sum of all the cell states is conserved upon every update). It is known that if the number of states k is smaller than or equal to six, then each rotation-symmetric number-conserving cellular automaton is isomorphic to some k-ary one, i.e., one with state set {0,1,...,k-1}. In this paper, we exhibit an example of a seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one. This example strongly supports our plea that research into multistate cellular automata should not only focus on those that have {0,1,...,k-1} as a state set.

Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen–Lee–Liu equation

Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen–Lee–Liu equation

Tests of Charge–Parity Symmetry and Lepton Flavor Conservation in the Top Quark Sector

The Standard Model (SM) of particle physics is the most general renormalizable theory which is built on a few general principles and fundamental symmetries with the given particle content. However, multiple symmetries are not built into the model and are simply consequences of renormalizabilty, gauge invariance, and particle content of the theory. It is crucial to test the validity of these types of symmetries and related conservation laws experimentally. The CERN LHC provides the highest sensitivity for testing the SM symmetries at high energy scales involving heavy particles such as the top quark. In this article, we are going to review the recent experimental searches of charge–parity and charged-lepton flavor violation in the top quark sector.

Lie symmetry analysis and conservation laws for the time fractional generalized advection–diffusion equation

Lie symmetry analysis and conservation laws for the time fractional generalized advection–diffusion equation

Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers

We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansätze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estimation and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purification of the state employing the linear combination of unitaries (LCU) approach.

Discussion on couple of nonlinear models for lie symmetry analysis, self adjointees, conservation laws and soliton solutions

Discussion on couple of nonlinear models for lie symmetry analysis, self adjointees, conservation laws and soliton solutions

Lie symmetry analysis, particular solutions and conservation laws of a (2+1)-dimensional KdV4 equation.

In this paper, a (2+1)-dimensional KdV4 equation is considered. We obtain Lie symmetries of this equation by utilizing Lie point symmetry analysis method, then use them to perform symmetry reductions. By using translation symmetries, two fourth-order ordinary differential equations are obtained. Solutions of one fourth order ordinary differential equation are presented by using direct integration method and $ (G'/G) $-expansion method respectively. Furthermore, the corresponding solutions are depicted with appropriate graphical representations. The other fourth-order ordinary differential equation is solved by using power series technique. Finally, two kinds of conserved vectors of this equation are presented by invoking the multiplier method and Noether's theorem respectively.

Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation

<abstract><p>The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.</p></abstract>

Lie symmetry analysis, conservation laws and diverse solutions of a new extended (2+1)-dimensional Ito equation

<abstract><p>In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation theorem.</p></abstract>

Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

<abstract><p>In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem.</p></abstract>

Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations

<abstract><p>We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.</p></abstract>