[en] Universal generalized Helmholtz equations were tested for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with arbitrary magnetization and general Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with longitudinal magnetization, which are obtained from generalized Helmholtz equations. From the general Helmholtz equations for gyrotropic waveguides are derived the partial Helmholtz equations for gyrotropic waveguides with certain orthogonal cross-sectional shapes (in particular: rectangular, round, elliptical) with a specific (longitudinal, normal, tangent) magnetization. General Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes in longitudinal magnetization and partial Helmholtz equations for gyrotropic elliptic waveguides in longitudinal magnetization are used during testing. (paper)

[en] Variational principles, Helmholtz's variational self-adjointness conditions and Santilli's method of the indirect analytic representation factors are extended to the case if dealing with fermionic systems described by Grassmann variables

[en] We extend the Cauchy– Riemann (or Wirtinger) operators and the Laplacian on ${\mathbb{C}}^m$ to zero-degree currents on a (possibly singular) Riemann subdomain $D$ of a complex space (without recourse to resolution of singularities). The former extension gives rise to an adjoint operator ${\stackrel{¯<!--\; ??\; -->}{\mathrm{\partial <!--\; ???\; -->}}}^{\ast <!--\; ???\; -->}$ for the $\stackrel{¯<!--\; ??\; -->}{\mathrm{\partial <!--\; ???\; -->}}$-operator on extendable test forms on $D$ (the components of ${\stackrel{¯<!--\; ??\; -->}{\mathrm{\partial <!--\; ???\; -->}}}^{\ast <!--\; ???\; -->}$ are the Wirtinger derivations). By means of the Wirtinger derivations, we generalize Gunning's theorem on the Cauchy– Riemann criterion (in the weak sense) for locally integrable functions to zero-degree currents on a complex space. To prove this result, we first give a generalization of Weyl's lemma to a Helmholtz operator. In the case of a continuous (resp. Lipschitzian) zero-degree current, we give a characterization of 'weak holomorphy' in terms of a local mean-value property (resp. an Euler operation). Wirtinger derivations also enable us to give explicit representations of the Green operator for the modified Laplacian ${\mathcal{S}}_{p,1,0}:=-<!--\; ???\; -->{\mathrm{△<!--\; ???\; -->}}_p+\mathrm{I}\mathrm{d}$ (acting weakly on the Sobolev space $H^{-<!--\; ???\; -->1}(D)$) and of Riesz's isomorphism between the Sobolev spaces $H_c^1(D{)}^{\ast <!--\; ???\; -->}$ and $H_c^1(D)$. (paper)

[en] The starting flows in regular polygonal ducts of S = 3, 4, 5, 6, 8 sides are determined by the method of eigenfunction superposition. The necessary S-fold symmetric eigenfunctions and eigenvalues of the Helmholtz equation are found either exactly or by boundary point match. The results show the starting time is governed by the first eigenvalue. (paper)

No abstract available

[en] Two procedures to introduce the familiar retarded potentials of Maxwell’s equations are reviewed. The first well-known procedure makes use of the Lorenz-gauge potentials of Maxwell’s equations. The second less-known procedure applies the retarded Helmholtz theorem to Maxwell’s equations. Both procedures are compared in the context of an undergraduate presentation of electrodynamics. The covariant form of both procedures is discussed for completeness. As a related discussion, two procedures to introduce the unfamiliar instantaneous potentials of Maxwell’s equations are also reviewed. The first procedure applies the standard Helmholtz theorem to Maxwell’s equations and the second one uses the Coulomb-gauge potentials of Maxwell’s equations. The retarded and instantaneous forms of the potentials of Maxwell’s equations are briefly commented upon. The retarded Helmholtz theorem is used to introduce the retarded potentials of Maxwell’s equations with magnetic monopoles. (paper)

[en] Conformal invisibility devices are only supposed to work within the valid range of geometrical optics. Here, we show by numerical simulations and analytical arguments that for certain quantized frequencies, they are nearly perfect even in a regime that clearly violates geometrical optics. The quantization condition follows from the analogy between the Helmholtz equation and the stationary Schroedinger equation.

[en] The self-inductance of a current-carrying circular coil and the mutual inductances of the Helmholtz coils and coil-sole-noid systems have been measured and calculated theoretically. The experiments and the required equipment are suited to an undergraduate laboratory. The theoretical calculation involve the use of simple numerical integration methods for evaluating the magnetic field of the circular coil and the inductances. The calculated values agree with the measurements within the experimental error. The material presented can be proposed to the students as a laboratory project. (Author) 7 refs

[en] We revisit the Ornstein–Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic ‘equation of state’ of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa. (paper)

No abstract available