Plane Wave Reflection in a Compressible Half Space with Initial Stress

In this paper, the problem of wave propagation in a compressible half-space with an initial stress is considered. General discussion on the speed of wave in the presence of an initial stress is presented. Furthermore, reflection of a homogeneous plane P−wave is also studied. A special strain energy function dependent on this initial stress is used to understand the response of the materials. Explicit formulas for the reflection coefficients are also presented. General nonlinear theory and the theory of invariants are used to derive theoretical results. Graphical illustration of theoretical results for various numerical values of parameters show that initial stress has considerable bearing on the behavior of a plane wave.

TIME-DEPENDENT WIGNER DISTRIBUTION FUNCTION EMPLOYED IN SQUEEZED SCHRÖDINGER CAT STATES: |Ψ(t)〉 = N-1/2(|β〉+eiϕ|-β〉)

The Wigner distribution function (WDF) for the time-dependent quadratic Hamiltonian system is investigated in the squeezed Schrödinger cat states with the use of Lewis–Riesenfeld theory of invariants. The nonclassical aspects of the system produced by superposition of two distinct squeezed states are analyzed with emphasis on their application into special systems beyond simple harmonic oscillator. An application of our development to the measurement of quantum state by reconstructing the WDF via Autler–Townes spectroscopy is addressed. In addition, we considered particular models such as Cadirola–Kanai oscillator, frequency stable damped harmonic oscillator, and harmonic oscillator with time-variable frequency as practical applications with the object of promoting the understanding of nonclassical effects associated with the WDF.

NEW INVARIANT OF 4-MOVES

Study of equivalence classes of links up to n-moves plays an important role in the theory of invariants based on the skein relation and, in particular, skein modules. In this paper, we consider Nakanishi's 4-move conjecture [12]. The modification of the conjecture to 2-component link (homotopically trivial links) is a question proposed by Kawauchi [10]. We define a new invariant of links which is preserved by 4-moves and analyze its potential strength. In particular, we show that our invariant allows us to obtain results of [8, 9, 13] concerning 4-moves.

ARRANGEMENTS OF NESTED CURVES

Motivated by Arnold's theory of invariants of plane curves, we introduce the semi-group of equivalence classes of arrangements of nested curves. There exists a natural invariant of plane curves without inverse self-tangencies with values in this semi-group. We show that the associated Grothendieck group is ℤ × ℤ. These two factors correspond to previously known invariants of plane curves without inverse self-tangencies, namely Whitney's index and Arnold's J- invariant. We show that arrangements of nested curves are not classified by their finite type invariants.