In recent years there has been significant interest in understanding magnetohydrodynamic (MHD) Jeffery–Hamel problems involving two-dimensional flow. Such dynamics involve an incompressible, conducting, viscous fluid flowing between two inclined planar walls, forming a converging or diverging channel. In reading the associated literature, we discovered a repeated issue with the reported form of the Hartmann number, linked with a frequent inconsistency in the form of the ordinary differential equation under consideration. The purpose of this note is to bring this fundamental error to light and to suggest some corrections.
Let G be a finite group and let γ(G)=∫01SP(G,x)dx=∑p∈π(G)np(G)sp(G)+1 be the Sylow-integral invariant associated with the Sylow polynomial of G. We show that small values of γ(G) do not force supersolvability. More precisely, if H is any finite group and Nt=∏p∈π(H)Cpt, then γ(H×Nt)→0 as t→∞. Consequently, for every ɛ>0 there exists a finite solvable group G which is not supersolvable and satisfies γ(G)<ɛ. We also compute γ(Ck×A4) explicitly and record infinite subfamilies with γ(G)<6/5.
An intuitionistic fuzzy differential equation (IFDE) with uncertainty in its initial value is studied through membership and non-membership level intervals. Scalar endpoint propagation and reconstruction of an admissible intuitionistic fuzzy number are treated separately. Sufficient reconstruction conditions are established, together with a counterexample showing that compact nested two-parameter intervals need not define an intuitionistic fuzzy number. For a continuous real field that is globally Lipschitz in the state variable, the scalar flow propagates both cut families and preserves compactness, nesting, level continuity, and Atanassov compatibility. This propagation produces a continuous intuitionistic fuzzy trajectory without requiring monotonicity of the field. If the field is also nondecreasing, the trajectory is type-I level differentiable, satisfies the IFDE, and obeys a Gronwall estimate in the cut metric. Nonincreasing and non-monotone fields require different endpoint treatments. A bounded nonlinear equation with a known flow illustrates reconstruction, endpoint convergence, nesting, compatibility, and continuous dependence. The fourth-order Runge–Kutta method attains its expected convergence order.
We probe recent literature on magnetohydrodynamic flow of a Casson fluid in a converging/diverging channel with velocity slip, and the claim that a similarity transformation reduces the governing Navier–Stokes equations to a class of nonlinear ordinary differential equations.We demonstrate that the proposed similarity reduction is mathematically inconsistent and we derive the correct forms. We further compute numerical solutions of both the previously published and corrected boundary value problems. We show that they differ significantly near the channel walls, leading to substantial discrepancies in predicted wall shear and related physical quantities.
We establish a variational formulation for Bailey’s susceptible–infected–removed epidemic model using the Jacobi last multiplier method. An explicit first-order Lagrangian is derived whose Euler–Lagrange equations reproduce the susceptible–infected subsystem, while the removed compartment is recovered through conservation of the total population. The autonomous character of the Lagrangian yields a nontrivial first integral, which provides a description of the epidemic phase curves. We also show that the reduced dynamics admits a noncanonical Hamiltonian formulation. These results reveal an underlying variational and geometric structure in a classical epidemiological model usually regarded as nonconservative.
