Behaviour at points of discontinuity II

This study proposes a method of detecting discontinuous points in physical discontinuities in light intensity fringe pattern (IFP). After changing the experimental conditions, several images were captured with CCD and converted into wrapped phase maps with phase shifting technology (PST). In low-noise wrapped phase maps, discontinuous and jump points were searched directly; and in high-noise wrapped phase maps, these points were searched after filtering. Noise discontinuous points were screened by image comparison to correctly select physical discontinuous points. The location of physical discontinuities was marked according to the assembly principle to prevent diffusion on both sides along the discontinuous line by phase unwrapping from crossing the line. In doing so, the surface appearance of the specimen was phase unwrapped successfully.

Insect flight along river corridors is a fundamental process that facilitates sustainable succession and diversity of aquatic and terrestrial insect communities in highly dynamic fluvial environments. This study examines variations in the thickness of the insect boundary layer (i.e., the pre-surface atmosphere layer in which air velocity does not exceed the sustained speed of flying insects) caused by interactions between diurnal winds and the heterogenous habitat mosaics in the floodplain of a braided river. Based on advective–diffusive theory, we develop and test a semi-empirical model that relates vertical flux of flying insects to vertical profiles of diurnal winds. Our model suggests that, in the logarithmic layer of wind, the density of insect fluxes decreases exponentially with the altitude due to the strong physical forcing. Inside the insect boundary layer, the insect fluxes can increase with the altitude while the winds speed remains nearly constant. We suggest a hypothesis that there is a close correspondence between the height of discontinuity points in the insect profiles (e.g. points with abrupt changes of the insect flux) and the displacement heights of the wind profiles (e.g. points above which the wind profile is logarithmic). Vertical profiles were sampled during three time-intervals at three different habitat locations in the river corridor: a bare gravel bar, a gravel bar with shrubs, and an island with trees and shrubs. Insects and wind speed were sampled and measured simultaneously over each location at 1.5-m intervals up to approximately 17 m elevation. The results support our working hypothesis on close correspondence between discontinuity and displacement points. The thickness of the insect boundary layer matches the height of the discontinuity points and was about 5 m above the bare gravel bar and the gravel bar with shrubs. Above the island, the structure of the insect boundary layer was more complex and consisted of two discontinuity points, one at the mean height of the trees’ crowns (ca. 15 m), and a second, internal boundary layer at the top of the shrubs (ca. 5 m). Our findings improve the understanding of how vegetation can influence longitudinal and lateral dispersal patterns of flying insects in river corridors and floodplain systems. It also highlights the importance of preserving terrestrial habitat diversity in river floodplains as an important driver of both biotic and abiotic (i.e., morphology and airscape) heterogeneity.

Denote by S a topological space. All functions considered are real valued. Convergence means pointwise convergence unless otherwise stated. Suppose f is a function defined on S, xCS, and f is not continuous at x. The statement that (x,f(x)) is a removable point of discontinuity means that there exists a function g which agrees with f on S- {x} and which is continuous at x. The statement that the function u defined on S is upper semicontinuous means that, if xeS and d >u(x), then there exists a neighborhood V of x such that, if yE V, then d>u(y). The function I is lower semicontinuous if -I is upper semicontinuous. 1.2. Statement of Theorems. THEOREM 1. Suppose M is a linear space of real valued functions defined on S which contains a nonzero constant function and which is closed under the operation of absolute value, and U is the set to which u belongs only in case u is the greatest lower bound of a countable subset of M. Then, if the function f defined on S is the limit of a sequence of functions in M, it is the uniform limit of a sequence each term of which is the difference of two members of U, each of which is bounded above. THEOREM 2. Suppose S is perfectly normal and f is a function defined on S. Each two of the following three statements are equivalent: (1) the function f is the limit of a sequence of functions, each of which has at most a finite number of points of discontinuity, each of which is removable; (2) the function f is the limit of a sequence of functions, each of which has at most countably many points of discontinuity; and (3) there exist a function g which is the limit of a sequence of continuous functions and a countable subset T of S such that, if xeS- T, f(x) = g(x).

Pointwise limit distribution results are given for the isotonic regression estimator at a point of discontinuity. The cases treated are independent data, - and f -mixing data and subordinated Gaussian long range dependent data. Pointwise limit results for the nonparametric maximum likelihood estimator of a monotone density are given at a point of discontinuity, for independent data. The limit distributions are non-standard and differ from the ones obtained for differentiable regression and density functions.

According to a well-known result, the collection of all ω-limit sets of a continuous map of the interval equipped with the Hausdorff metric is a compact metric space. In this paper, a similar result is proved for piecewise continuous maps with finitely many points of discontinuity, if the points of discontinuity are not periodic for any variant of the map. A variant of f is a map g coinciding with f at any point of continuity and being continuous from one side at any point of discontinuity. It is also shown that ω-limit sets of these maps are locally saturating, another property known for continuous maps. However, contrary to the situation for continuous maps, there are piecewise continuous maps having locally saturating sets which are not ω-limit sets. A condition implying that a locally saturating set is an ω-limit set is presented.

In the present paper, our boundary value problem is Sturm–Liouville problem with retarded argument and transmission conditions at the finitely many points of discontinuity. This is the first work containing several points of discontinuity in the theory of differential equations with retarded argument. The main purpose of this paper is to obtain asymptotic formulas for the eigenvalues and corresponding eigenfunctions of this boundary value problem. The simplicity of the eigenvalues is proved. In special case when our problem with retarded argument and transmission conditions at one point of discontinuity, the obtained results coincide with the corresponding results in Bayramov et al. (Appl Math Comput 191, 592–600, 2007).

In spectral-like resolution-WENO hybrid schemes, if the switch function takes more grid points as discontinuity points, the WENO scheme is often turned on, and the numerical solutions may be too dissipative. Conversely, if the switch function takes less grid points as discontinuity points, the hybrid schemes usually are found to produce oscillatory solutions or just to be unstable. Even if the switch function takes less grid points as discontinuity points, the final hybrid scheme is inclined to be more stable, provided the spectral-like resolution scheme in the hybrid scheme has moderate shock-capturing capability. Following this idea, we propose nonlinear spectral-like schemes named weighted group velocity control (WGVC) schemes. These schemes show not only high-resolution for short waves but also moderate shock capturing capability. Then a new class of hybrid schemes is designed in which the WGVC scheme is used in smooth regions and the WENO scheme is used to capture discontinuities. These hybrid schemes show good resolution for small-scales structures and fine shock-capturing capabilities while the switch function takes less grid points as discontinuity points. The seven-order WGVC-WENO scheme has also been applied successfully to the direct numerical simulation of oblique shock wave-turbulent boundary layer interaction.

This paper provides an analytical framework to study the performance of linear companding techniques proposed in the OFDM literature, thus settling the numerous controversial claims that are based solely on simulation results. Linear companding transforms are widely employed to reduce the peak-to- average-power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems. Two main linear companding classes have been considered in the literature: linear symmetrical transform (LST) and linear asymmetrical transform (LAST). In the literature, the bit error rate (BER) performance superiority of the basic LAST (with one discontinuity point) over the LST is claimed based on computer simulations. Also, it has been claimed that a LAST with two discontinuity points outperforms the basic LAST with one discontinuity point. These claims are however not substantiated with analytical results. Our analysis shows that these claims are, in general, not always true. We derive a sufficient condition, in terms of the companding parameters, under which the BER performance of a general LAST with M-1 discontinuity points is superior to that of LST. The derived condition explains the contradictions between different reported results in the literature and validates some other reported simulation results. It also serves as a guideline in the process of choosing proper values for companding parameters to obtain a specific trade-off between PAPR reduction capability and BER performance. In particular, the derived sufficient condition shows that the BER performance for LAST depends on the slopes of the LAST rather than on the number of discontinuity points as has been indicated so far. Moreover, we derive conditions for the companding parameters in order to keep the average transmitted power unchanged after companding. Our theoretical derivations are supported by simulation results.

In this paper, a convolutional neural network for the detection and characterization of impedance discontinuity points in cables is presented. The neural network analyzes time-domain reflectometry signals and produces a set of estimated discontinuity points, each of them characterized by a class describing the type of discontinuity, a position, and a value quantifying the entity of the impedance discontinuity. The neural network was trained using a great number of simulated signals, obtained with a transmission line simulator. The transmission line model used in simulations was calibrated using data obtained from stepped-frequency waveform reflectometry measurements, following a novel procedure presented in the paper. After the training process, the neural network model was tested on both simulated signals and measured signals, and its detection and accuracy performances were assessed. In experimental tests, where the discontinuity points were capacitive faults, the proposed method was able to correctly identify 100% of the discontinuity points, and to estimate their position and entity with a root-mean-squared error of 13 cm and 14 pF, respectively.

Starting from a family of discontinuous piece-wise linear one-dimensional maps, recently introduced as a dynamic model in social sciences, we propose a geometric method for finding the analytic expression of the bifurcation curves, in the space of the parameters, that bound the regions characterized by the existence of stable periodic cycles of any period. The conditions for the creation and the destruction of periodic cycles, as well as the analytic expressions of the bifurcation conditions, are obtained by studying the occurrence of border-collision bifurcations. In this paper we consider the case of maps formed by three linear portions separated by two discontinuity points. After summarizing the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point. Finally we show how the considered map can be obtained as the limit case of a family of continuous maps as a parameter is increased without bounds, and we show how the low period cycles, which are typical of the discontinuous map we consider, emerge from the more complex (i.e. chaotic) behaviors observed in the continuous maps when a parameter value is large enough. From the point of view of the social application the increasing values of the parameter can be interpreted as higher degrees of impulsivity of the agents involved in binary decisions.

This paper extends Regression discontinuity designs with unknown discontinuity points developed by (Porter, J., and P. Yu. 2015. “Regression Discontinuity Designs with Unknown Discontinuity Points: Testing and Estimation.” Journal of Econometrics 189: 132–147.) to allow for state-dependent discontinuity points. We discuss the estimation of the model, and propose test statistics for treatment effect and state dependency in the discontinuity points. We conduct Monte Carlo simulations to compare the proposed estimator with these based on the constant discontinuity RDD and the classic fuzzy RDD, and find that overlooking the state dependency can lead to biased estimates of treatment effects, while the proposed estimator works well and is robust when applied to constant discontinuity RDDs. Monte Carlo experiments also point out that the sizes and powers of the proposed test statistics are generally satisfactory. The model is illustrated with an empirical application.

This study addresses the problem of minimizing functions whose values are obtained by running an expensive computer simulation and where the simulation may fail to converge for some points, i.e. the objective function is discontinuous. An algorithm is proposed which is comprised of three major steps: generating a global surrogate model of the objective function; using an evolutionary algorithm to identify regions where minimizers are likely to exist and then using an efficient local search to converge to a minimizer. In order to handle discontinuity points we implement two mechanisms: a) a global penalty function is generated by interpolation and is used to modify the global surrogate model such that the function value predicted by the global surrogate model are increased in the vicinity of discontinuity points b) in case a discontinuity point is encountered during the local search the search region is temporarily reduced so as to seek a valid point in a smaller region around the current valid point. Performance analysis from minimization of mathematical test functions and a real-world application of airfoil shape optimization show the proposed algorithm efficiently minimizes expensive black-box functions with discontinuity points.

When estimating the grain size, the processing effect for complex grain boundaries is often poor. To solve this problem, the recovery techniques for complex grain boundary are studied. After analysis of steps to estimate the grain size, we know that it is critical to get more accurate boundary before quantitative determination, and the discontinuous points among grain boundaries should be connected. Therefore, two methods to connect the discontinuous points were established, and it showed that watershed method is more efficient; however over-segmentation and under-segmentation exist. Accordingly, recovery techniques of complex grain boundary were put forward, that is to edit the regional minimum manually or to connect the discontinuous points in the gray image before watershed transform. By this technique, the grain boundary can be got more accurately, conveniently and continuously closed, independently of interfere of noise, scratches, illumination etc.

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.

Regression discontinuity designs with unknown discontinuity points: Testing and estimation