In this paper we consider the standard Schur problem for a square matrix A, namely the similarity unitary transformation of A into upper Schur form containing the eigenvalues of A on its diagonal. Since the profound work of Issai Schur (1909), this is a fundamental issue in the theory and applications of matrices. Nevertheless, certain details concerning the Schur problem need further clarification especially in connection with the perturbation analysis of the Schur decomposition relative to perturbations in A. In particular, the concept of regular solution to the perturbed Schur form is introduced and illustrated by several examples. We also introduce the concepts of diagonally spectral matrices and of quasi-Schur condensed forms of a matrix A, and show that they may be much less sensitive to perturbations in A.

In inflationary cosmology the form of the potential is still an open problem. In this work, second-order effects of the inflationary potential are evaluated and related to the known formula for the primordial perturbations in a wide range of scales. We found effects that may help to unravel the unknown inflationary potential form and impose new constraints on the parameters that define this potential. In particular, we demonstrate here that even slight deviations in the inflaton potential can lead to significant differences in the calculated spectra if inflation persists for sufficiently long and the normal modes of perturbations are affected by these variations.

The adversarial vulnerability of deep learning models is a critical issue that must be addressed to ensure the safe commercialization of AI technologies. Although numerous studies on adversarial defense methods are actively being conducted from various perspectives, most of them still provide limited robustness, and even the relatively trusted adversarial training is no exception. To develop more reliable defense methods, ongoing research exploring the properties and causes of adversarial vulnerabilities is essential. In this study, we focus on a hypothesis regarding the existence of adversarial examples: The adversarial examples represent low-probability “pockets” in the manifold. Assuming that the hypothesis holds true, we propose a method for generating perturbation: “mixed perturbation (MP)”, which aims at discovering diverse pocket samples in a defensive perspective. The proposed method generates perturbations by leveraging information from both the main task and auxiliary tasks in multi-task learning scenarios, combining them through random weighted summation. The generated mixed perturbation intends to maintain the primary directionality of the main task perturbation to improve the model’s main task recognition performance while introducing variability in the perturbation directions. We then utilize them for adversarial training to form more robust decision boundary. Through experiments and analyses conducted on five benchmark datasets, we validated the effectiveness of our proposed method.

Background: This study aimed to investigate how external perturbations caused by the treadmill belt's acceleration (Acc) and deceleration (Dec) during the Initial Contact (Initial), Mid Stance (Mid) and Pre-Swing (ToeOff) phase affect gait regularity in young adults. Methods: Twenty-one healthy young females walked on a treadmill in a virtual environment (Grail, Motek), where four unexpected perturbations were applied to the left belt at Initial, Mid, and ToeOff. Sample Entropy (SampEn) was calculated for the center of mass (CoM) displacements, for six perturbation scenarios in three directions: anterior-posterior (AP), medial-lateral (ML), and vertical (vert), with SampEn vector lengths (m) ranging from 2 to 10. Results: The CoM displacement exhibited its highest regularity (low SampEn values) in the AP and vert-directions during Dec_ToeOff, across all m values. Similarly, this pattern was observed in the ML-direction, but exclusively for m = 2 and 4. The least regular CoM trajectories (high SampEn values) were for Dec_Mid in the AP-direction, across all m values. This trend persisted in the ML-direction, only for m = 2 and 4. However, the most irregular CoM displacements in the ML-direction occurred during Dec_ToeOff for the remaining m values. Vertical CoM displacements exhibited the highest irregularities during Dec_Initial for m ≥ 4. Conclusions: Evaluating the regularity of CoM displacements using SampEn can be a useful tool for assessing how gait perturbations are handled.

In this research, 117,718 ionospheric perturbations, with a space size (t) of 20–300 s but no limit on amplitude (A) have been automatically searched globally by a software from ion density data measured by the DEMETER satellite for more than 6 years. The influence of the solar activity on the ionosphere has been firstly examined and the results have presented that the solar activities can globally enhance ionospheric irregularities but rarely induce plasma variations more than 100%. A statistical work has been performed on ion PERs to check their dependence on local time and it is shown that there are 24.8% perturbations appeared in daytime (10:30 LT) and 75.2% ones in nighttime (22:30 LT). Ionospheric fluctuations with an absolute amplitude A < 10% tend to be background variations and the percentages of positive perturbations with small A < 20% are 64% in daytime and 26.8% in nighttime, respectively, but this number is revers for mid-large amplitude PERs. There is a demarcation point in space size t = 120 s and occurrence probabilities of day PERs are always higher than that of nighttime ones before this point while this trend is contrary after this point. Distributions of PER according to various amplitudes or space scales have been characterized by typical seasonal variations either in daytime or in nighttime. The EIA only exists in dayside equinox and winter occupying two lowlatitude crests with lower Np. The huge WSA appears in all periods except for dayside summer full of PERs with enhanced amplitude, especially in winter night. The WN-like structure can be obviously figured out in all seasons showing absolutely space large-scales. In the meanwhile, several magnetic anomalous zones of planetary scale non-dipole fields, such as the SAMA, Northern Africa anomaly, and so on, have been also successfully detected by extreme negative ion perturbations in this time.

In this lecture notes, we will introduce Asymptotics, then we will give a short glimpse on Perturbation theory (regular versus singular), which plays a crucial role especially in theoretical physics. Our goal is to find Asymptotic series that approximates the values of integrals depending on some parameter or the solutions of differential equations. These methods that lead to obtain more effective algorithms of numerical evaluation are called: Asymptotic Methods.

This paper explores the interactions of dynamic dark energy with other components of the cosmos, such as dark matter and baryonic matter, and its wide-ranging implications in cosmology. By investigating the interplay between dynamic dark energy and the universe's large-scale structure, we aim to provide a deeper understanding of cosmic evolution and the behavior of dark energy in different cosmological epochs. We also examine the potential of dynamic dark energy to offer novel insights into long-standing cosmological puzzles and their impact on the formation and evolution of cosmic structures.

A system of two nonlinear difference equations under stochastic perturbations is considered. Nonlinearity of the exponential type in each equation of the system under consideration depends on all variables of the system. The stability in probability of a positive equilibrium of the system is studied via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated via examples and figures with numerical simulations of solutions of a considered system of stochastic difference equations. The proposed way of investigation can be applied to nonlinear systems of higher dimension and with other types of nonlinearity, for both difference equations and for delay differential equations.

The lightcone gauge is a set of what are called the observational coordinates adapted to our past lightcone. We develop this gauge by producing a perturbed spacetime metric that describes the geometry of our past lightcone where observations are usually obtained. We connect the produced observational metric to the perturbed Friedmann-Lema\^itre-Robertson-Walker metric in the standard general gauge or what is the so-called 1+3 gauge. We derive the relations between these perturbations of spacetime in the observational coordinates and those perturbations in the standard metric approach, as well as the dynamical equations for the perturbations in observational coordinates. We also calculate the observables in the lightcone gauge and re-derive them in terms of Bardeen potentials to first order. A verification is made of the observables in the perturbed lightcone gauge with those in the standard gauge. The advantage of the method developed is that the observable relations are simpler than in the standard formalism, and they are expressed in terms of the metric components which in principle are measurable. We use the perturbed lightcone gauge in galaxy surveys and the calculations of galaxy number density contrast. The significance of the new gauge is that by considering the null-like light propagations the calculations are much simpler due to the non-consideration of the angular deviations.