The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively.

State-Burst Feedback Control for Fault Recovery of Input/State Asynchronous Sequential Machines

Static corrective controllers are more efficient than dynamic ones since they consist of only logic elements, whereas their existence conditions are more restrictive. In this paper, we present a static corrective control scheme for fault diagnosis and fault tolerant control of input/state asynchronous sequential machines (ASMs) vulnerable to transient faults. The design flexibility of static controllers is enlarged by virtue of using a diagnoser and state bursts. Necessary and sufficient conditions for the existence of a diagnoser and static fault tolerant controller are presented, and the process of controller synthesis is addressed based on the derived condition. Illustrative examples on practical ASMs are provided to show the applicability of the proposed scheme.

Bifurcation analysis of modified density gradient continuum traffic model

This paper proposes a new density gradient continuous traffic flow model, and analyzes the linear stability of the model, as well as the bifurcation type of the model. Numerical simulation of the new model verifies the usability of the model. From the perspective of system stability, the bifurcation analysis method is used to analyze the nonlinear traffic phenomena on the expressway. The equilibrium solution of the model is discussed. On this basis, Hopf bifurcation, saddle bifurcation and Bogdanov–Takens bifurcation are obtained, and the existence conditions and fractional types of Hopf bifurcation and saddle bifurcation are obtained. The traffic flow characteristics of Hopf bifurcation and saddle node bifurcation are analyzed.

EXACT PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

In the paper is given a method of finding periodical solutions of the differential equation of the form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes the greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions of $t$. This reduces $n$-periodic soluble problem to a system of $n+1$ linear equations, where $n=2,3$. Furthermore, by using the well known properties of linear system in the algebra, all existence conditions for $2$ and $3$-periodical solutions are described, and the explicit formula for these solutions are obtained.

National spatial data infrastructure (nsdi) of Ukraine: what are its actual, feasible and simultaneously "correct" models?

The actual, feasible and simultaneously "correct" models of digital NSDI of Ukraine are considered in the work. A model of the existed digital NSDI system of Ukraine is named “actual”. This model already differs from the model defined by the [1]. As the latter is unlikely to be implemented in the near future, the issue of the digital feasible NSDI model of Ukraine in the next five years, which would take into account the actual model, is especially acute. In addition to feasibility, such a model must also be "correct", what is proposed in the article. The correct is called a model, the truth of which can be established by inductive or deductive reasoning. To do this, the correct model must be formalized enough so that everyone can verify the authors’ reasoning independently. Understanding both actual and correct models of NSDI of Ukraine will help to properly organize and develop actual Spatial Infrastructure Activities (SpIA) in Ukraine, including the real[1] implementation of the [1]. Although the results of the article call into question its feasibility and substantiate an alternative viewpoint on the automation problem of NGDI/NSDI/SpIA. However, we are convinced that it is still possible to change the alternative viewpoint to a cooperative one, if by means of by-laws the models of NGDI (Law), NSDI (article) and, finally, SpIA are agreed upon To prove the "correctness" of the feasible NSDI model, the theory of Relational cartography and its two main methods are used: Conceptual Frameworks and Solution Frameworks. In addition, the correspondence between Relational cartography and Model-Based Engineering is used. Key words: NSDI; product model; process model; actual, feasible and «correct» model. [1] Real. 1. Which exists in reality, true. Is used with: reality, life, existence, conditions, circumstances, fact, danger, force, wages, income. 2. One that can be implemented, executed: a real plan, a real program, a real task, a real deadline. 3. Which is based on taking into account and assessing the real conditions of reality: a real approach, a real view, a real policy.- accessed 2021-feb-14, http://slovopedia.org.ua/32/53408/32016.html (Ukrainian).

A semilinear parabolic predator–prey system with time delay and habitat complexity effect

Based on the research on the predator–prey model with Holling type response function, a delayed predator–prey system with diffusion term and habitat complexity effect is established, and the effects of time delay and diffusion on dynamical behavior of the system are studied. First, taking habitat complexity as the parameter, the dynamical properties of the system without time delay are studied. By eigenvalue analysis, the sufficient conditions for locally asymptotic stability of the positive equilibrium and globally asymptotic stability of the boundary equilibrium are given, the existence conditions of Hopf bifurcation induced by diffusion term are discussed. In an appropriate range, diffusion makes a family of spatially homogeneous and inhomogeneous periodic solutions bifurcate from the positive equilibrium. Second, taking production delay as the bifurcation parameter, the existence conditions of Hopf bifurcation are given, the method to determine the bifurcation direction and the stability of bifurcating periodic solutions is given by using the center manifold theory and normal form method. Finally, the biological interpretations of the results are given, and some numerical simulations are given to verify the theoretical analysis results.

Riesz potentials and orthogonal radon transforms on affine Grassmannians

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡 j,k f on L p -functions, Fuglede type formulas connecting 𝓡 j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j,k f under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.