We study gravitationally collapsing models of pressureless dust, fluids with pressure, and the generalized Chaplygin gas (GCG) shell in $(2+1)$-dimensional spacetimes. Various collapse scenarios are investigated under a variety of the background configurations such as anti-de Sitter (AdS) black hole, de Sitter (dS) space, flat and AdS space with a conical deficit. As with the case of a disk of dust, we find that the collapse of a dust shell coincides with the Oppenheimer-Snyder type collapse to a black hole provided the initial density is sufficiently large. We also find—for all types of shell—that collapse to a naked singularity is possible under a broad variety of initial conditions. For shells with pressure this singularity can occur for a finite radius of the shell. We also find that GCG shells exhibit diverse collapse scenarios, which can be easily demonstrated by an effective potential analysis.

In this paper we will give a new proof by using group action to prove the uniqueness of maximal Sperner families Fmaxn of [n]. We will also prove the uniqueness of Sperner families F of [n] with |F|=n⌊n2⌋-1 by using a combinatorial approach. Furthermore, by using the uniqueness of Sperner family, we will classify all the structures of (1,2) superimposed codes of size 9×10 and 9×11.

In this paper we deal with equitable partitions of association schemes. We try to generalize a result in group theory and show examples that a generalization of a certain property conjectured for permutation groups does not hold for association schemes.

We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed.

The classical equations of motion for the dissipative RLC coupled two-dimensional circuit with a power source are solved using two-step canonical transformation. We applied our investigation to the system whose power source is the form of sinusoidally oscillating with time.

By means of a two-step unitary transformation, the exact quantum mechanical solution of two-dimensional mesoscopic circuit coupled via inductance is derived. The wavefunction, propagator and expectation value of Hamiltonian are evaluated. The fluctuations of charges and currents are calculated and uncertainty products between charges and their conjugate currents are obtained. We apply our results to the system with special cases of the power source such as the sinusoidal and the sawtooth power source and show that the probability densities oscillate with time due to the power source.