A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Level -1/2 Realization of Quantum N-Toroidal Algebra in Type Cn

We construct a level -1/2 vertex representation of the quantum [Formula: see text]-toroidal algebra of type [Formula: see text], which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type [Formula: see text] as a by-product.

Simulating of X-states and the two-qubit XYZ Heisenberg system on IBM quantum computer

Two qubit density matrices which are of X-shape, are a natural generalization of Bell Diagonal States (BDSs) recently simulated on the IBM quantum device. We generalize the previous results and propose a quantum circuit for simulation of a general two qubit X-state, implement it on the same quantum device, and study its entanglement for several values of the extended parameter space. We also show that their X-shape is approximately robust against noisy quantum gates. To further physically motivate this study, we invoke the two-spin Heisenberg XYZ system and show that for a wide class of initial states, it leads to dynamical density matrices which are X-states. Due to the symmetries of this Hamiltonian, we show that by only two qubits, one can simulate the dynamics of this system on the IBM quantum computer.

Rigid impacts of three-dimensional rocking structures

Studies of rocking motion aim to explain the remarkable earthquake resistance of rocking structures. State-of-the-art assessment methods are mostly based on planar models, despite ongoing efforts to understand the significance of three-dimensionality. Impacts are essential components of rocking motion. We present experimental measurements of free-rocking blocks on a rigid surface, focusing on extreme sensitivity of impacts to geometric imperfections, unpredictability, and the emergence of three-dimensional motion via spontaneous symmetry breaking. These results inspire the development of new impact models of three-dimensional facet and edge impacts of polyhedral objects. Our model is a natural generalization of existing planar models based on the seminal work of George W. Housner. Model parameters are estimated empirically for rectangular blocks. Finally, new perspectives in earthquake assessment of rocking structures are discussed.

Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

Competitive Algorithms for Online Multidimensional Knapsack Problems

In this paper, we study the online multidimensional knapsack problem (called OMdKP) in which there is a knapsack whose capacity is represented in m dimensions, each dimension could have a different capacity. Then, n items with different scalar profit values and m-dimensional weights arrive in an online manner and the goal is to admit or decline items upon their arrival such that the total profit obtained by admitted items is maximized and the capacity of knapsack across all dimensions is respected. This is a natural generalization of the classic single-dimension knapsack problem and finds several relevant applications such as in virtual machine allocation, job scheduling, and all-or-nothing flow maximization over a graph. We develop two algorithms for OMdKP that use linear and exponential reservation functions to make online admission decisions. Our competitive analysis shows that the linear and exponential algorithms achieve the competitive ratios of O(θα ) and O(łogł(θα)), respectively, where α is the ratio between the aggregate knapsack capacity and the minimum capacity over a single dimension and θ is the ratio between the maximum and minimum item unit values. We also characterize a lower bound for the competitive ratio of any online algorithm solving OMdKP and show that the competitive ratio of our algorithm with exponential reservation function matches the lower bound up to a constant factor.

Fragmentations with self-similar branching speeds

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

The role of colour flows in matrix element computations and Monte Carlo simulations

We discuss how colour flows can be used to simplify the computation of matrix elements, and in the context of parton shower Monte Carlos with accuracy beyond leading-colour. We show that, by systematically employing them, the results for tree-level matrix elements and their soft limits can be given in a closed form that does not require any colour algebra. The colour flows that we define are a natural generalization of those exploited by existing Monte Carlos; we construct their representations in terms of different but conceptually equivalent quantities, namely colour loops and dipole graphs, and examine how these objects may help to extend the accuracy of Monte Carlos through the inclusion of subleading-colour effects. We show how the results that we obtain can be used, with trivial modifications, in the context of QCD+QED simulations, since we are able to put the gluon and photon soft-radiation patterns on the same footing. We also comment on some peculiar properties of gluon-only colour flows, and their relationships with established results in the mathematics of permutations.

On geodesically reversible Finsler manifolds

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.