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The double-chain deoxyribonucleic acid model, which is important to the retention and transfer of genetic material in biological domains, is examined in this study. It is important because, it bridges the gap between theoretical physics and molecular biology by offering a more thorough and precise explanation of DNA behavior. In this model, the bottom combination represents hydrogen bonds between base pairs in the two long, evenly elastic filaments that represent the two polynucleotide chains of the deoxyribonucleic acid molecule. The Lie symmetry analysis is used to explain the Lie invariance criteria. This leads to a four-dimensional Lie algebra where the translation point symmetries in space and time correlate with the conservation of mass and energy, respectively, and the remaining point symmetries are dilation and scaling. The double-chain deoxyribonucleic acid partial differential model is reduced to an ordinary differential equation, and built Lie subalgebras are found first, along with invariant closed-form solutions. The Cauchy problem for the double-chain deoxyribonucleic acid model cannot be solved by the inverse scattering transform method; therefore, the analytical Riccati-Bernoulli suboptimal differential equation approach technique is used to build the exact solution. The appropriate parametric values are taken in contour, two, and three dimensions to graphically illustrate the solution. A physically meaningful and intuitive interpretation of the system dynamics is required in order to make the Hamiltonian function under consideration easier to comprehend and analyze. One of the numerous conservation principles commonly seen in systems defined by a Hamiltonian function is energy conservation. The conservation laws are determined for the model under consideration, which are essential for deciphering and solving complex problems and are used to illustrate deep understandings of how physical systems behave. Understanding the stability and long-term behavior of the system depends on these preserved quantities. To assess the governing system’s sensitivity, a sensitive analysis is offered.

The secondary regulation hydrostatic transmission technology and H ∞ control are applied to the mobile platform of the robot in this paper. The H ∞ control system and the secondary regulation electro‐hydraulic drive system of the mobile platform are designed. In view of the nonlinear characteristics such as dead zone, hysteresis loop, and Coulomb friction of the secondary regulation hydrostatic transmission system, the Hamiltonian form of the system was constructed by applying the Hamiltonian functional method. Based on the Hamiltonian function, the robust controller was designed, and simulation and experimental studies were carried out. Good control performance was achieved, and the dynamic characteristics of the system were significantly improved, such as faster response, minimal overshoot, and reduced static error. It has strong anti‐interference ability and good robustness. The designed mobile platform is suitable for working in the field, high‐speed and heavy‐load conditions, with large load capacity and strong traction ability. It can realize energy recovery and reuse, greatly reducing the installed power of the mobile platform.

The paper studies the geometry of Liouville foliation generated by integrable Hamiltonian system. It is shown that regular leaves are two-dimensional surface of zero Gaussian curvature and zero Gaussian torsion.

Nonlinear random vibration is a common phenomenon, and predicting its probability density is an essential component of vibration engineering. This paper proposes a data-driven method for identifying explicit expressions of response probability densities in random vibrating systems with implicit Hamiltonian functions. The process concludes with two steps, identifying the Hamiltonian function and calculating the probability density of the stationary response. The former uses the differential equations of the motion of the quasi-Hamiltonian systems to identify Hamiltonian functions from the simulated data, while the latter estimates the logarithm of the probability density from the identified Hamiltonian functions and acquires an explicit expression. Their unknown coefficients can be attributed to the solution of a set of undetermined equations. The proposed method is applied to systems in which the Hamiltonian functions cannot be simply derived, such as those with complicated stiffness. Two examples are presented to demonstrate the applicability and effectiveness of our method, i.e., the nonlinear vibration energy harvester (VEH) and the Duffing oscillator with LuGre friction. The proposed technique outperforms Monte Carlo simulations (MCSs) in efficiency. The results show that our method is insensitive to parameters and can be used for identifying transient probability density. Its application scope is wider than the stochastic averaging method.

A one-parameter family of trans-series asymptotics as$\\tau \\to \\pm \\infty$and$\\tau \\to \\pm {\\rm i} \\infty$for solutions of the degenerate Painlevé III equation (DP3E),$$ u^{\\prime \\prime}(\\tau) = \\frac{(u^{\\prime} (\\tau))^{2}}{u(\\tau)} - \\frac{u^{\\prime}(\\tau)}{\\tau} + \\frac{1}{\\tau}\\bigl(-8 \\varepsilon (u(\\tau))^{2} + 2ab\\bigr) + \\frac{b^{2}}{u(\\tau)},$$where$\\varepsilon \\in \\lbrace \\pm 1 \\rbrace$ ,$a \\in \\mathbb{C}$ , and$b \\in \\mathbb{R} \\setminus \\lbrace 0 \\rbrace$ , are parametrised in terms of the monodromy data of an associated first-order$2 \\times 2$matrix linear ODE via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the$\\sigma$ -forms of the DP3E are also obtained. The actions of various Lie-point symmetries for the DP3E are derived.

By using the classical Lie algebra, the stationary zero curvature equation, and the Lenard recursion equations, we obtain the non-isospectral TD hierarchy. Two kinds of expanding higher-dimensional Lie algebras are presented by extending the classical Lie algebra. By solving the expanded non-isospectral zero curvature equations, the multi-component non-isospectral TD hierarchies are derived. The Hamiltonian structure for one of them is obtained by using the trace identity.

A family of multibosonic complex-symmetric Hamiltonians possessing both the real and complex spectra is studied, with emphasis upon the properties of the latter subfamily. In it one treats resonances as eigenstates of a non-Hermitian effective quantum Hamiltonian. As long as the search for their complex energy eigenvalues is not easy, a reduced task is considered in which one only evaluates the auxiliary real quantities called singular values. Several forms of representation of Greens functions in terms of (possibly, matrix) continued fractions are shown to offer an efficient approach to this task.

This paper obtains new results on optimal control problems with fuzzy parameters. First, we transform the problem into an interval optimal control problem. After writing the interval Hamiltonian function, we present two methods with a joint application of generalized Hukuhara difference for eliminating the uncertainty and writing the necessary optimality conditions. In the first method, we use the flexibility function and in the second one, we apply the convex combination of the bounds of the Hamiltonian function to state these conditions. The main advantage of these methods is to overcome the previous shortcomings in this context. The effectiveness of the suggested methods is demonstrated through a numerical experiment.

For the chiral oscillator described by a second order and degenerate Lagrangian with special Euclidean group of symmetries, we show, by cotangent bundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of oscillator algebra, the central extension of special Euclidean algebra in two dimensions.

We describe a model of three-dimensional quantum space. Starting from a well-studied model of the fuzzy sphere, we extend it by connecting a large — potentially infinite — collection of such spheres to represent a three-dimensional volume. We verify that the chosen Hamiltonian is appropriate; in the continuum limit, the spectrum converges to that of standard quantum mechanics. Specifically, we analyse the behaviour of the spectrum of the Coulomb problem in the finite case.