Eccentric and Concentric Motion Motion of Hamstring during the Leg Curl

The force experienced by the hamstring during the leg curl has been numerically investigated using the conservation theorem. The center of the meniscus is assumed to be the pivot point along with the uniform distribution of forces in the frictionless environment. The variation of force experienced by the hamstring during the concentric motion of the leg curl has been derived and graphically illustrated. It is found that the force experienced by the hamstring increases with the increase in length of the lower leg and its weight as well. The magnitude of force decreases with the increase in distance from the pivot to insertion. However, the magnitude of force increases from about 3.60 to 4.79 kN in the practically valid region 3 to 4 cm distance from the pivot to insertion with the increase in weight of lower leg from 5 to 15 kg. On the other hand, the magnitude of force increases from about 3.75 to 9.80 kN with the increase in weight suspended on the machine from 10 to 40 kg. In addition, the force decreases with the increase in upper leg dimension, but it linearly increases with the increase in the angle of suspension.

Exact Solutions and Conservation Laws of the Time-Fractional Gardner Equation with Time-Dependent Coefficients

In this paper, we employ the certain theory of Lie symmetry analysis to discuss the time-fractional Gardner equation with time-dependent coefficients. The Lie point symmetry is applied to realize the symmetry reduction of the equation, and then the power series solutions in some specific cases are obtained. By virtue of the fractional conservation theorem, the conservation laws are constructed.

Violation of magnetic flux conservation by superconducting nanorings

The behavior of magnetic flux in the ring-shaped finite-gap superconductors is explored from the view-point of the flux-conservation theorem which states that under the variation of external magnetic field "the magnetic flux through the ring remains constant" (see, e.g., [L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuos Media, vol. 8 (New York, Pergamon Press, 1960), Section 42]). Our results, based on the time-dependent Ginzburg-Landau equations and COMSOL modeling, made it clear that in the general case, this theorem is incorrect. While for rings of macroscopic sizes the corrections are small, for micro and nanorings they become rather substantial. The physical reasons behind the effect are discussed. The dependence of flux deviation on ring sizes, bias temperature, and the speed of external flux evolution are explored. The detailed structure of flux distribution inside of the ring opening, as well as the electric field distribution inside the ring's wire cross section are revealed. Our results and the developed finite element modeling approach can assist in elucidating various fundamental topics in superconducting nanophysics and in the advancement of nanosize superconducting circuits prior to time-consuming and costly experiments.

The (3 + 1)-dimensional Wazwaz–KdV equations: the conservation laws and exact solutions

In this study, we dealt with the new conservation theorem and the auxiliary method. We have applied the theorem and method to (3 + 1)-dimensional modified Wazwaz–KdV equations. When we applied a new conservation theorem to given equations, the obtained conservation laws did not satisfy the divergence condition. So, we modified the obtained conservation laws. These conservation laws contain extra terms. Finally, we applied the auxiliary method to given equations. We obtained two solution families with six exact solutions. All the obtained solutions are different from each other. For a suitable value of the solutions, the 3D and 2D surfaces have been plotted by Maple.

An Arithmetically Complete Predicate Modal Logic

This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.

The analysis of conservation laws, symmetries and solitary wave solutions of Burgers–Fisher equation

In this paper, the conservation laws, significant symmetries’ application, and traveling wave solutions are obtained for Burger–Fisher equation (BFE). Conservation laws have a great importance for partial and fractional differential equations and their solutions, especially in physics implementations. The conservation theorem and partial Noether approach are implemented for conservation laws for this equation, and the extended sinh-Gordon expansion method (esGEM) is presented for new solitary wave solutions. All obtained conservation laws are trivial conservation laws. The new and comprehensive solitary wave solutions of the equation by the esGEM are also obtained.

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

In this paper, we consider conservation laws and exact solutions of the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov equation. Firstly, we construct conservation laws of the given equation with the help of the conservation theorem; the developed conservation laws are modified conservation laws. Then, we obtain exact solutions of the given equation via the (G′/G,1/G)-expansion method. The obtained solutions are classified as trigonometric solutions, hyperbolic solutions and rational solutions. Furthermore, graphical representations of the obtained solutions are given.

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3 + 1)-dimensional extended quantum Zakharov–Kuznetsov equation in quantum physics

A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for obtaining the exact solution of some reduced transform equations. Finally, by invoking the new conservation theorem developed by Nail H. Ibragimov, the conservation laws of QZK equation have been derived.