One-parameter Family Of Transformations Research Articles

On the motion of a magnetized rigid body

A method introduced by Yehia (J Phys A Math Gen 32:7565–7580, 1999 , for example) for generalizing known results of the problem of the motion of a rigid body is extended here to take magnetization by rotation (Barnett–London effect) into account. In the general case of anisotropic ferromagnets, the equations of motion are proved to be covariant under a one-parameter family of transformations which generalize the problem by inserting a parameter, with a definite physical meaning, into the dynamical equations. As an example, we generalize a particular solution of the problem of the motion of a gyrostat in a central field of attraction. It turns out that the generalized solution represents a new particular solution of the problem of the motion by inertia of a rigid body in an ideal incompressible fluid.

On Dynamical Properties of a One-Parameter Family of Transformations Arising in Averaging of Semigroups

To describe the dynamics of quantum systems with degenerate symmetric but not self-adjoint Hamiltonian, we consider the Naimark extension of the Hamiltonian to a self-adjoint operator in an extended Hilbert space. We relate to the symmetric Hamiltonian a one-parameter family of averaged dynamical transformations of the set of quantum states obtained from a unitary group of transformations of the extended Hilbert space by using a conditional expected value to an algebra of bounded operators acting in the original space. We establish the absence of the semigroup property and injectivity of the family of averaged dynamical transformations. We obtain a representation of trajectories of the averaged family of dynamical transformations by maximum points of functionals on the space of mappings of the time interval into the set of quantum states.

On the inverse Noether's theorem in nonlinear micropolar continua

In this article, the modified Noether's theorem is established in general form. Then, the inverse theorem is used in nonlinear micropolar continua in order to derive one-parameter family of transformations under which the corresponding functional is invariant. Next, the conservation laws are written. They include, as a special case, the conservation laws of micropolar elastostatics, the balance laws of elastodynamics and elastostatics. We do not analyse any of these special cases, because they may be obtained very easily.

Transformations between surfaces in $\mathbb{R}^4$ with flat normal and/or tangent bundles

We exhibit several transformations of surfaces M in \mathbb{R}^4 : a transformation of flat surfaces that gives surfaces with flat normal bundle (semiumbilical surfaces); and its inverse that from a semiumbilical surface obtains a flat surface; then a one-parameter family of transformations f on flat semiumbilical immersed surfaces (FSIS), such that df(T_pM) is totally orthogonal to T_pM, and that give FSIS. This family satisfies a Bianchi type of permutability property.

Boundary layer flows with Dufour and Soret effects: I: Forced and natural convection

Similarity solutions of laminar boundary layer flows in forced and natural convection are examined aiming at the evaluation of the coupled effects of heat and mass transport. Momentum, energy and mass balance equations are simultaneously solved under the assumption of the validity of constitutive equations for the heat and mass fluxes of dilute solutes obeying linear forms in accordance with the thermodynamics of irreversible processes. Forced and natural convection cases depend on different similarity variables, with different exponents for the axial variable and different significant physical properties compose the pertinent dimensionless groups. In the first part of this paper thermal diffusion and diffusion thermo effects are studied for the boundary layer flows about a vertical flat plate in forced and in natural convection, separately. The second part deals with the intermediate cases of non-similar of mixed convection, where it is determined by a one-parameter family of transformations ( 0 ⩽ ϒ ⩽ 1 ) , which reduce to both extreme cases. The mixed convection will be presented in a forthcoming continuation. A modification of the method of solution based on Adomian's polynomials for the resulting boundary value problem is employed.

Geometric phase for cyclic motions and the quantum state space metric

The geometric phase is generalized to a cyclic motion resulting from any one-parameter family of transformations in the Hilbert space. This is applied to the translation in one spatial dimension of the Bloch wave function. Physical meanings are given to the Fubini-Study metric on the projective Hilbert space, and a metric in the classical phase space is deduced as the classical limit of this quantum metric.

Scaling properties of a transformation defined on cellular automaton rules

A one-parameter family of transformations defined on the set of all one-dimensional cellular automata is studied. The class of a given cellular automaton is unchanged under the set of these transformations. For class 3 cellular automata, transformed statistical quantities satisfy simple scaling properties.