The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ n ≥ d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.

Discrete Hypergeometric Legendre Polynomials

A discrete analog of the Legendre polynomials defined by discrete hypergeometric series is investigated. The resulting polynomials have qualitatively similar properties to classical Legendre polynomials. We derive their difference equations, recurrence relations, and generating function.

The Exponentiated Burr–Hatke Distribution and Its Discrete Version: Reliability Properties with CSALT Model, Inference and Applications

Continuous and discrete distributions are essential to model both continuous and discrete lifetime data in several applied sciences. This article introduces two extended versions of the Burr–Hatke model to improve its applicability. The first continuous version is called the exponentiated Burr–Hatke (EBuH) distribution. We also propose a new discrete analog, namely the discrete exponentiated Burr–Hatke (DEBuH) distribution. The probability density and the hazard rate functions exhibit decreasing or upside-down shapes, whereas the reversed hazard rate function. Some statistical and reliability properties of the EBuH distribution are calculated. The EBuH parameters are estimated using some classical estimation techniques. The simulation results are conducted to explore the behavior of the proposed estimators for small and large samples. The applicability of the EBuH and DEBuH models is studied using two real-life data sets. Moreover, the maximum likelihood approach is adopted to estimate the parameters of the EBuH distribution under constant-stress accelerated life-tests (CSALTs). Furthermore, a real data set is analyzed to validate our results under the CSALT model.

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a "periodized stationary phase method" to discrete Wigner functions of systems with odd prime dimension and show that the π8 gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on t qutrit π8 gates that are followed by Clifford evolution, and show that this only requires 3t2+1 quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼30.8t for 10−2 precision) for any number of π8 gates to full precision.

A new approach to the inverse discrete transmission eigenvalue problem

<p style='text-indent:20px;'>A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.</p>

SIMULATION OF AN UNSTEADY INCOMPRESSIBLE FLUID FLOW THROUGH A PERFORATED PIPELINE

A mathematical model of the unsteady flow of an incompressible viscous fluid through a perforated pipeline is proposed, which is described by a system of nonlinear partial differential equations. In the framework of the model, the purpose is to determine the pressure and the flow rate of the fluid at the pipeline inlet, providing the flow rate and the pressure required at the pipeline outlet. By combining the system of the equations, the original problem is reduced to a boundary-value inverse problem for a nonlinear parabolic equation with respect to fluid flow rate. To solve the boundary inverse problem, the method of nonlocal perturbation of boundary conditions is proposed. A discrete analog of the inverse problem is obtained using the finitedifference approximation, and a special approach is suggested for solving the resulting system of difference equations. As a result, the difference problem for each discrete value of the time variable splits into two second-order difference problems and a linear equation with respect to an approximate value of the desired flow rate at the pipeline inlet. The absolutely stable Thomas method is used to numerically solve the obtained difference problems. After determining the flow rate distribution along the entire pipeline, the pressure at the pipeline inlet is also calculated using an explicit formula. Based on the proposed computational algorithm, the numerical experiments are performed for benchmark problems.

MULTI-ELEMENT SCALE INDICATOR DEVICES IN BUILT-IN SYSTEMS

The work is devoted to investigation of functional principles of data display means building in embedded systems and definition of ways of reliability increasing of information transfer at interaction in user interface. The importance of a visual communication channel with the operator to ensure the protection of information in complex systems and responsible applications is shown. The principles of implementation of the data output subsystem in embedded systems are analyzed and it is found that the required level of information is provided only by multi-element indicator devices. The element base of indicators is investigated and determined that the most effective display elements from a reliable and ergonomic point of view for built-in applications are LEDs. Analysis of the principles of visual presentation of information showed that the analog (discrete-analog) method of data transmission to the operator provides the highest level of ergonomic parameters of indicators. In this case, the best results have a scale indication based on the additive information model. The use of color speeds up the reading of information from the scale. The control schemes of indicator elements for construction of reliable devices are analyzed. It has been found that the use of microcontrollers significantly increases the level of reliability and provides flexibility of such control schemes. In this case, the software used has a significant impact on the reliability and efficiency of solutions. The matrix connection of LEDs, which are switched in a dynamic mode, allows to build effective means of communication with the operator. It is determined that the best set of technical, reliability and ergonomic characteristics will be obtained when implementing data output in embedded systems using LED bar graph display with microcontroller means in bicyclical dynamic mode. However, very little attention has been paid to investigation of the principles of construction and software optimization support for scale information using control schemes based on microcontrollers.

Methodology of non-linear robust estimation for the solutions synthesis of inverse and direct multidisciplinary problems in engineering dimensional chains calculation based on discrete analog data

This paper analyses the definition of inverse and direct problems in engineering dimensional chains calculation based on discrete analogue data and the methodologies for solving these problems. It is shown that the direct dimensional chains calculation, which belongs to the class of inverse boundary value problems in a stochastic formulation, can be transformed into multi-criteria problems of stochastic optimization with mixed conditions. The new multi-step solutions search methodology for these problems is based on non-linear robust estimation methods. It can be achieved through hierarchical two-level decisions synthesis scheme development. At the first step, this scheme includes identification of surrogate models (in the form of regression equations). At the second step, the effective robust estimates are computed to determine unknown values; estimations of unknown quantities are carried out under a priori and parametric data uncertainties. Results of calculations of inverse and direct problems in engineering dimensional chains for two-stage axial compressors are presented. They were obtained using interactive computer systems for decision-making support “ROD&IDS”.

Method and Device Based on Multiscan for Measuring the Geometric Parameters of Objects

The article deals with the issues of improving the accuracy of measurements of the geometric parameters of objects by optoelectronic systems, based on a television multiscan. A mathematical model of a multiscan with scanistor activation is developed, expressions for its integral output current and video signal are obtained, and the mechanism of their formation is investigated. An expression for the video signal is obtained that reflects the dual nature of the discrete–continuous multiscan structure: the video signal can have a discrete (pulse) or analog (continuous) form, depending on the step voltage between the photodiode cells of the multiscan. A Vernier discrete–analog method for measuring the parameters of the light zone on a multiscan is proposed, in which in order to increase the accuracy of the measurements, the location of the video pulse is determined relative to the neighboring reference pulses of a rigid geometric raster due to the slope of the discrete structure of the multiscan. It is established that the Vernier method enables one to make precision measurements of the coordinates, dimensions, and movements of the light zones by an overlay on a video raster of reference pulses from cells—a uniform sequence of Vernier pulses with a recurrence interval, followed by determining the number of the Vernier pulse that coincides with the raster pulse. An optoelectronic device based on a discrete–continuous multiscan, implemented on the basis of the proposed Vernier method of measuring the coordinates of the light zones, which has a high sensitivity to movement, is characteristic of continuous structures, and has increased stability and linearity of the coordinate characteristics typical for discrete structures, is developed.