Proton-neutron symplectic model description of $^{20}$Ne

A microscopic description of the low-lying positive-parity rotational bands in $^{20}$Ne is given within the framework of the symplectic-based proton-neutron shell-model approach provided by the proton-neutron symplectic model (PNSM). For this purpose a model Hamiltonian is used which includes an algebraic interaction, lying in the enveloping algebra of the $Sp(12,R)$ dynamical group of the PNSM, that introduces both horizontal and vertical mixings of different $SU(3)$ irreducible representations within the $Sp(12,R)$ irreducible collective space of $^{20}$Ne. A good overall description is obtained for the excitation energies of the ground and first two excited $\beta$ bands, as well as for the ground state intraband $B(E2)$ quadrupole collectivity and the known interband $B(E2)$ transition probabilities between the low-lying collective states without the use of an effective charge.

How dynamics constrains probabilities in general probabilistic theories

We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.

Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.

Four-dimensional quantum oscillator and magnetic monopole with U(1) dynamical group

By using an appropriate transformation, it was shown that the quantum system of four-dimensional (4D) simple harmonic oscillator can describe the motion of a charged particle in the presence of a magnetic monopole field. It was shown that the Dirac magnetic monopole has the hidden algebra of U(1) symmetry and by reducing the dimensions of space, the U(1) × U(1) dynamical group for 4D harmonic oscillator quantum system was obtained. Using the group representation and based on explicit solution of the obtained differential equation, the spectrum of system was calculated.

Group consensus of heterogeneous multi-agent systems with fixed topologies

Purpose – The purpose of this paper is to study the dynamical group consensus of heterogeneous multi-agent systems with fixed topologies. Design/methodology/approach – The tool used in this paper to model the topologies of multi-agent systems is algebraic graph theory. The matrix theory and stability theory are applied to research the group consensus of heterogeneous multi-agent systems with fixed topologies. The Laplace transform and Routh criterion are utilized to analyze the convergence properties of heterogeneous multi-agent systems. Findings – It is discovered that the dynamical group consensus for heterogeneous multi-agent systems with first-order and second-order agents can be achieved under the reasonable hypothesizes. The group consensus condition is only relied on the nonzero eigenvalues of the graph Laplacian matrix. Originality/value – The novelty of this paper is to investigate the dynamical group consensus of heterogeneous multi-agent systems with first-order and second-order agents and fixed topologies and obtain a sufficient group consensus condition.

U(6)-Phonon model of nuclear collective motion

The U(6)-phonon model of nuclear collective motion with the semi-direct product structure [HW(21)]U(6) is obtained as a hydrodynamic (macroscopic) limit of the fully microscopic proton–neutron symplectic model (PNSM) with Sp(12, R) dynamical group. The phonon structure of the [HW(21)]U(6) model enables it to simultaneously include the giant monopole and quadrupole, as well as dipole resonances and their coupling to the low-lying collective states. The U(6) intrinsic structure of the [HW(21)]U(6) model, from the other side, gives a framework for the simultaneous shell-model interpretation of the ground state band and the other excited low-lying collective bands. It follows then that the states of the whole nuclear Hilbert space which can be put into one-to-one correspondence with those of a 21-dimensional oscillator with an intrinsic (base) U(6) structure. The latter can be determined in such a way that it is compatible with the proton–neutron structure of the nucleus. The macroscopic limit of the Sp(12, R) algebra, therefore, provides a rigorous mechanism for implementing the unified model ideas of coupling the valence particles to the core collective degrees of freedom within a fully microscopic framework without introducing redundant variables or violating the Pauli principle.

Induced topological pressure for countable state Markov shifts

We generalise Savchenko's definition of topological entropy for special flows over countable Markov shifts by considering the corresponding notion of topological pressure. For a large class of Hölder continuous height functions not necessarily bounded away from zero, this pressure can be expressed by our new notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words, and we are able to set up a variational principle in this context. Investigating the dependence of induced pressure on the subset of words, we give interesting new results connecting the Gurevič and the classical pressure with exhaustion principles for a large class of Markov shifts. In this context we consider dynamical group extensions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.