Parametrizations of the Poisson–Schrödinger equations in spherical symmetry

We consider the asymptotically flat standing wave solutions to the Poisson–Schrödinger system of equations. These equations are also known as the Schrödinger–Newton equations and are the Newtonian limit of the Einstein–Klein–Gordon equations. The asymptotically flat standing wave solutions to the Poisson–Schrödinger equations are known as static states. These solutions can be parametrized using a variety of choices of two continuous parameters and one discrete parameter, each having a useful physical-geometrical interpretation. The values of the discrete variable determines the number of nodes (zeros) in the solution. We use numerical inversion techniques to analyze transformations between various informative choices of parametrization by relating each of them to a standard set of three parameters. Based on our computations, we propose explicit formulas for these relationships. Our computations also show that for the standard choice of continuous variables, the zero-node ground state yields a minimum value of a geometrically natural discrete variable. We give an explicit formula for this minimum value. We use these results to confirm two related observations from previous work by the author and others, and suggest additional applications and approaches to understand these phenomena analytically.

Deep Gene Networks and Response to Stress

We consider systems of differential equations with polynomial and rational nonlinearities and with a dependence on a discrete parameter. Such systems arise in biological and ecological applications, where the discrete parameter can be interpreted as a genetic code. The genetic code defines system responses to external perturbations. We suppose that these responses are defined by deep networks. We investigate the stability of attractors of our systems under sequences of perturbations (for example, stresses induced by environmental changes), and we introduce a new concept of biosystem stability via gene regulation. We show that if the gene regulation is absent, then biosystems sooner or later collapse under fluctuations. By a genetic regulation, one can provide attractor stability for large times. Therefore, in the framework of our model, we prove the Gromov–Carbone hypothesis that evolution by replication makes biosystems robust against random fluctuations. We apply these results to a model of cancer immune therapy.

Vector autoregression process. Stationarity and simulation

For vector discrete-parameter random autoregressive processes and for a mixed autoregression/moving-average model, we obtain conditions which should be satisfied by the correlation functions or the model coefficients in order that the process be weakly stationary. Fairly simple tests are used. Algorithms for modeling such vector stationary processes are given. Examples are presented clarifying testing criteria for stationarity of models defned in terms of the coefficients or the correlation functions of the process.

Torsors for difference algebraic groups

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also present an application to the Galois theory of differential equations depending on a discrete parameter.

Towards improving edge quality using combinatorial optimization and a novel skeletonize algorithm

Background Object detection and image segmentation of regions of interest provide the foundation for numerous pipelines across disciplines. Robust and accurate computer vision methods are needed to properly solve image-based tasks. Multiple algorithms have been developed to solely detect edges in images. Constrained to the problem of creating a thin, one-pixel wide, edge from a predicted object boundary, we require an algorithm that removes pixels while preserving the topology. Thanks to skeletonize algorithms, an object boundary is transformed into an edge; contrasting uncertainty with exact positions. Methods To extract edges from boundaries generated from different algorithms, we present a computational pipeline that relies on: a novel skeletonize algorithm, a non-exhaustive discrete parameter search to find the optimal parameter combination of a specific post-processing pipeline, and an extensive evaluation using three data sets from the medical and natural image domains (kidney boundaries, NYU-Depth V2, BSDS 500). While the skeletonize algorithm was compared to classical topological skeletons, the validity of our post-processing algorithm was evaluated by integrating the original post-processing methods from six different works. Results Using the state of the art metrics, precision and recall based Signed Distance Error (SDE) and the Intersection over Union bounding box (IOU-box), our results indicate that the SDE metric for these edges is improved up to 2.3 times. Conclusions Our work provides guidance for parameter tuning and algorithm selection in the post-processing of predicted object boundaries.

MATHEMATICAL MODEL OF A TWELVE-PULSE CONVERTER WITH SIXTEEN ZONE CONTROL OF THE OUTPUT VOLTAGE AND ACTIVE-INDUCTIVE LOADS

Analysis of the electromagnetic processes is organized in this article in electric circuit with semiconductor commutator. Mathematical model twelve pulses of the converter is created for analysis of the electromagnetic processes in semiconductor converter with width-pulse regulation of the output voltage. The broughted graphs, which reflect the electromagnetic processes in electric circuit. Method multivariable function was used when performing calculation. The mathematical model of the converter is created for five zoned regulations of the output voltage. Using method multivariable function was found current and voltage of the load, as well as input currents of the converter. The load of the converter has actively inductive nature. Article is devoted to the development of a method of multi-parametric modulating functions by means of working out of new mathematical models and definition of functions and the algorithmic equations for the analysis on sub-system components of electromagnetic processes in electric circuits of variable structure with sinusoidal, direct and pulsing voltage. Introduction of functions with discrete parameters in the algorithmic equations for analysis of processes in circuits with semiconductor commutators simplifies modeling on subsystem components. The mathematical model of steady-state processes and transients in electric circuits of semiconductor converters of modulation type with multi-channel zonal use of phase and line voltages of a three-phase network of power supplies is developed. The mathematical model of electric circuits of thyristor shapers of electro-discharge pulses for the analysis and the matching of capacitors charging modes with decrease several times of electric resistance of technological load is also created. The obtained results have a great value for development theoretical electrical engineering in a direction of simplification of calculations of electromagnetic processes in electric circuits with semi-conductor converters of the electric power. The electromagnetic processes in electric circuit under width-pulse regulation possible to analyse with use the algorithmic equations multivariable function, which argument are a system parameters semiconductor commutator, signal of control, phases to network of the power supply and time. Introduction multivariable function with discrete parameter in algorithmic equations of the analysis formed and connecting processes in electric circuit of the variable structure allows to reflect change of this structure under system components, simplifying modeling and analysis of such processes to account of the generalization of the got equations. Except specified correlations and diagrams designed model allows to analyse forms of the output voltages and current of the separate power modules.

Towards Improving Edge Quality using Combinatorial Optimization and a Novel Skeletonize Algorithm

Background: Object detection and image segmentation of regions of interest provide the foundation for numerous pipelines across disciplines. Robust and accurate computer vision methods are needed to properly solve image-based tasks. Multiple algorithms have been developed to solely detect edges in images. Constrained to the problem of creating a thin, one-pixel wide, edge from a predicted object boundary, we require an algorithm that removes pixels while preserving the topology. Thanks to skeletonize algorithms, an object boundary is transformed into an edge; contrasting uncertainty with exact positions.Methods: To extract edges from boundaries generated from different algorithms, we present a computational pipeline that relies on: a novel skeletonize algorithm, a non-exhaustive discrete parameter search to find the optimal parameter combination of a specific post-processing pipeline, and an extensive evaluation using three data sets from the medical and natural image domains (kidney boundaries, NYU-Depth V2, BSDS 500). While the skeletonize algorithm was compared to classical topological skeletons, the validity of our post-processing algorithm was evaluated by integrating the original post-processing methods from six different works.Results: Using the state of the art metrics, precision and recall based Signed Distance Error (SDE) and the Intersection over Union bounding box (IOU-box), our results indicate that the SDE metric for these edges is improved up to 2.3 times.Conclusions: Our work provides guidance for parameter tuning and algorithm selection in the post-processing of predicted object boundaries.

Generation of synchronized configuration space trajectories with workspace path constraints for an ensemble of robots

We present an approach to generate path-constrained synchronous motion for the coupled ensemble of robots. In this article, we refer to serial-link manipulators and mobile bases as robots. We assume that the relative motion constraints among the objects in the environment are given. We represent the motion constraints as path constraints and pose the problem of path-constrained synchronous trajectory generation as a non-linear optimization problem. Our approach generates configuration space trajectories for the robots to manipulate the objects such that the given motion constraints among the objects are satisfied. We present a method that formulates the problem as a discrete parameter optimization problem and solves it using successive constraint refinement techniques. The method adaptively selects the parametric representation of the configuration variables for a given scenario. It also generates an approximate solution as the starting point for the successive constraint refinement stages to reduce the computation time. We discuss in detail why successive constraint refinement strategies are useful for solving this class of problems. We demonstrate the effectiveness of the proposed method on challenging test cases in simulation and physical environments with high-degree-of-freedom robotic systems.

Optimal algorithm for distinguishing radio navigation signals by a set of “spaced” correlators

The classical algorithm for signal distinction, signal detecting and estimating signal parameters consists in analyzing discrete parameter values using a correlator. The value of the parameter with the maximum absolute value of the correlator is taken as an estimate. Obviously, this is accompanied by losses in sensitivity and noise immunity, since the specified discrete parameter values do not accurately correspond to the true parameter values of the real signal. In this case, the accuracy of the parameter estimation, even at large signal-to-noise ratios, is limited by the value of the correlators placement interval. Therefore, it is of interest to optimally use the entire set of correlators for parameter estimation and signal detection. The article presents the derivation of algorithm for distinguishing signals by a given parameter by a set of "spaced" correlators. Unlike the classical algorithm, it uses decisive statistics not by one, but by a pair of neighboring correlators, detuned by the correlation interval. In this case, at first, the number of the interval between correlators is estimated according to the maximum of the decisive statistics, and then the value of the parameter is refined within this interval. Additionally, the algorithm allows you to estimate the signal amplitude. The proposed algorithm is compared with the classical one. By means of simulation, the dependences on the energy potential of the average probability of signal distinction for both algorithms are plotted. It is shown that the proposed algorithm has a higher probability of correct distinction than the classical algorithm. It is also shown that the maximum and average energy losses of the distinction algorithm based on a set of "spaced" correlators are less than the losses of the classical algorithm. Thus, the proposed algorithm for distinction signals by a set of "spaced" correlators has greater noise immunity and accuracy of estimating the desired parameter than the classical distinction algorithm.