Singularly perturbed rank one linear operators

The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

Controlling the adhesive pull-off force via the change of contact geometry

A first-order asymptotic analysis of the Griffith energy balance in the Johnson–Kendall–Roberts model of adhesive contact under non-symmetric perturbation of the contact geometry is presented. The pull-off force is evaluated in explicit form. A particular case of adhesive contact between a relatively stiff sphere and an elastic half-space is considered under the assumption that the sphere geometry is changed by the application of an arbitrary lateral normal surface loading. The effect of the sphere Poisson’s ratio on controlling the adhesive pull-off force is considered. This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.

Improved Salp Swarm Algorithm with Simulated Annealing for Solving Engineering Optimization Problems

Swarm-based algorithm can successfully avoid the local optimal constraints, thus achieving a smooth balance between exploration and exploitation. Salp swarm algorithm(SSA), as a swarm-based algorithm on account of the predation behavior of the salp, can solve complex daily life optimization problems in nature. SSA also has the problems of local stagnation and slow convergence rate. This paper introduces an improved salp swarm algorithm, which improve the SSA by using the chaotic sequence initialization strategy and symmetric adaptive population division. Moreover, a simulated annealing mechanism based on symmetric perturbation is introduced to enhance the local jumping ability of the algorithm. The improved algorithm is referred to SASSA. The CEC standard benchmark functions are used to evaluate the efficiency of the SASSA and the results demonstrate that the SASSA has better global search capability. SASSA is also applied to solve engineering optimization problems. The experimental results demonstrate that the exploratory and exploitative proclivities of the proposed algorithm and its convergence patterns are vividly improved.

Relative contribution of Trend and Interannually-varying SST Anomalies to the 2018 Heat Waves in the Extratropical Northern Hemisphere

The northern extratropics—including regions in northern Europe, northeast Asia, and North America—experienced extremely prolonged heat waves during May–August 2018. Record-breaking surface temperatures, which caused numerous deaths, were observed in several cities. The 2018 heat waves exhibited a circumglobal characteristic owing to a circumpolar perturbation (CCP) in the middle–upper troposphere of the Northern Hemisphere (NH). The CCP had two parts: a wave-like perturbation and a hemispheric perturbation that was almost zonally symmetric. Singular value decomposition analysis revealed that the zonally symmetric perturbation was coupled to the SST warming trend, whereas the wave-like perturbation was primarily coupled to the interannually-varying SST anomaly (SSTA), particularly in the tropical North Pacific, which reached an extreme in 2018. Numerical experiments confirmed that the zonally symmetric component was primarily resulted from the SSTA associated with the warming trend, whereas the interannually-varying SSTAs in the NH contributed mostly to the wave-like perturbation. The warming trend component of SSTA, especially that in the tropics, compounded by the unusually large SSTAs in 2018, was hypothesized to have contributed to inducing the circumpolar circulation anomaly that caused the record-breaking heat waves in the extratropical NH in 2018.

Shape of a sound wave in a weakly-perturbed Bose gas

We employ the Gross-Pitaevskii equation to study acoustic emission generated in a uniform Bose gas by a static impurity. The impurity excites a sound-wave packet, which propagates through the gas. We calculate the shape of this wave packet in the limit of long wavelengths, and argue that it is possible to extract properties of the impurity by observing this shape. We illustrate here this possibility for a Bose gas with a trapped impurity atom – an example of a relevant experimental setup. Presented results are general for all one-dimensional systems described by the nonlinear Schr"{o}dinger equation and can also be used in nonatomic systems, e.g., to analyze light propagation in nonlinear optical media. Finally, we calculate the shape of the sound-wave packet for a three-dimensional Bose gas assuming a spherically symmetric perturbation.

Coexistence of a new type of bound state in the continuum and a lasing threshold mode induced by PT symmetry

Some photonic systems support bound states in the continuum (BICs) that have infinite lifetimes, although their frequencies and momenta are matched to vacuum modes. Using a prototypical system that can be treated analytically, we show that each of these BICs always splits into a pair of new type BIC and lasing threshold mode when a parity-time (PT)–symmetric perturbation is introduced. The radiation loss at the lasing threshold is exactly balanced by the net gain of the particles. These PT symmetry-induced BICs are different from ordinary BICs, as they can be excited by an external source but do not radiate, and they carry a different quality factor divergence rate from that of the ordinary BICs. While most of the attention of PT-symmetric systems is captured by the coalescence of modes at exceptional points, the splitting of ordinary BICs is a new phenomenon that illustrates the rich physics embedded in PT-symmetric systems.

On a Quantum Waveguide with a Small PT-symmetric Perturbation

We consider a quantum waveguide with a small PT-symmetric perturbation described by a potential. We study the spectrum of such a system and show that the perturbation can produce eigenvalues near the threshold of the continuous spectrum.

Influence of Surface Processes over Africa on the Atlantic Marine ITCZ and South American Precipitation

Previous studies show that the climatological precipitation over South America, particularly the Nordeste region, is influenced by the presence of the African continent. Here the influence of African topography and surface wetness on the Atlantic marine ITCZ (AMI) and South American precipitation are investigated. Cross-equatorial flow over the Atlantic Ocean introduced by north–south asymmetry in surface conditions over Africa shifts the AMI in the direction of the flow. African topography, for example, introduces an anomalous high over the southern Atlantic Ocean and a low to the north. This results in a northward migration of the AMI and dry conditions over the Nordeste region. The implications of this process on variability are then studied by analyzing the response of the AMI to soil moisture anomalies over tropical Africa. Northerly flow induced by equatorially asymmetric perturbations in soil moisture over northern tropical Africa shifts the AMI southward, increasing the climatological precipitation over northeastern South America. Flow associated with an equatorially symmetric perturbation in soil moisture, however, has a very weak cross-equatorial component and very weak influence on the AMI and South American precipitation. The sensitivity of the AMI to soil moisture perturbations over certain regions of Africa can possibly improve the skill of prediction.