MP-CE Method for Space-Filling Design in Constrained Space with Multiple Types of Factors

Space-filling design selects points uniformly in the experimental space, bringing considerable flexibility to the complex-model-based and model-free data analysis. At present, space-filling designs mostly focus on regular spaces and continuous factors, with a lack of studies into the discrete factors and the constraints among factors. Most of the existing experimental design methods for qualitative factors are not applicable for discrete factors, since they ignore the potential order or spatial distance between discrete factors. This paper proposes a space-filling method, called maximum projection coordinate-exchange (MP-CE), taking into account both the diversity of factor types and the complexity of factor constraints. Specifically, the maximum projection criterion and distance criterion are introduced to capture the “bad” coordinates, and the coordinate-exchange and the optimization of experimental design are realized by solving one-dimensional constrained optimization problem. Meanwhile, by adding iterative perturbations to the traditional coordinate exchange process, the adjacent areas of the local optimal solution are explored and the optimum performances of the current optimal solution are retained, while the shortcomings of random restart are effectively avoided. Experiments in the regular space and constraint space, as well as experimental design for the terminal interception effectiveness of a missile defense system, show that the MP-CE method significantly outperforms existing popular space-filling design methods in terms of space-projection properties, while yielding comparable or superior space-filling properties.

A Different View on Dynamics of Space Curves Geometry

In this study, we define the X-torque curves, $$X-$$ X - equilibrium curves, X-moment conservative curves, $$X-$$ X - gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where $$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$ X ∈ T s , N s , B s and we examine these curves and we give their properties.

On expansive homeomorphism of uniform spaces

We study the notion of expansive homeomorphisms on uniform spaces. It is shown that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a Hausdor space and hence a regular space. Further, we characterize orbit expansive homeomorphisms in terms of topologically expansive homeomorphisms and conclude that if there exist a topologically expansive homeomorphism on a compact uniform space then the space is always metrizable.

STABILITYANALYSIS OF SCHWARZSCHILD-DAMOUR-SOLODUKHIN WORMHOLE

The equations of general relativity are nonlinear second-order partial differential equations, and, as a consequence, obtaining the exact solutions is a difficult problem. One of the solutions to this problem is to obtain models with a thin self-gravitating shell. This method is used to study most of the phenomena in the theory of gravity, where the reverse effect of matter on the geometry of space-time is a key factor. Another interesting problem that can be studied using the thin shell method is the «simulation» of a black hole. Consider a system consisting of a spherically symmetric Schwarzschild black hole and a thin shell surrounding it, located at a certain fixed distance from the black hole. From the viewpoint of gravitational physics, an observer at infinity is unable to distinguish a real black hole from a wormhole with a thin shell, in which the simulation condition is satisfied. Simulation of a black hole is possible only under sufficiently stringent conditions for the parameters of the model. In particular, the shell needs to be held at a fixed radius. In the general case, such a movement of the shell is non-geodesic, and external forces are required to hold it. The radius of the shell is also a parameter that determines the possibility / impossibility of simulation. In this paper, the radius is found for the case of a Schwarzschild black hole. In particular, the paper considers a model of a wormhole obtained as a result of gluing two space-times: a Schwarzschild black hole and a Damour-Solodukhin wormhole. The latter solution differs from the Schwarzschild black hole in the parameter of the dimensionless real deviation λ and is a twice asymptotically flat regular space-time. It is shown that they can be glued along a given radius. As a result, a thin shell is formed between two glued manifolds consisting of exotic matter. Cases are considered when the thin shell is stable. It turns out that zones corresponding to the «force» constraint are more restrictive than those corresponding to the «mass» constraint.

Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays

<p style='text-indent:20px;'>We study asymptotically autonomous dynamics for non-autonom-ous stochastic 3D Brinkman-Forchheimer equations with general delays (containing variable delay and distributed delay). We first prove the existence of a pullback random attractor not only in the initial space but also in the regular space. We then prove that, under the topology of the regular space, the time-fibre of the pullback random attractor semi-converges to the random attractor of the autonomous stochastic equation as the time-parameter goes to minus infinity. The general delay force is assumed to be pointwise Lipschitz continuous only, which relaxes the uniform Lipschitz condition in the literature and includes more examples.</p>

A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

New soft separation axioms and fixed soft points with respect to total belong and total non-belong relations

In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with respect to ordinary points, namely t t tt -soft pre T i ( i = 0 , 1 , 2 , 3 , 4 ) {T}_{i}\hspace{0.33em}\left(i=0,1,2,3,4) and t t tt -soft pre-regular spaces. The motivations to use these relations are, first, cancel the constant shape of soft pre-open and pre-closed subsets of soft pre-regular spaces, and second, generalization of existing comparable properties on classical topology. With the help of examples, we show the relationships between them as well as with soft pre T i ( i = 0 , 1 , 2 , 3 , 4 ) {T}_{i}\hspace{0.33em}\left(i=0,1,2,3,4) and soft pre-regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a t t tt -soft pre-regular space and demonstrate that it guarantees the equivalence of t t tt -soft pre T i ( i = 0 , 1 , 2 ) {T}_{i}\hspace{0.33em}\left(i=0,1,2) . Furthermore, we investigate the behaviors of these soft separation axioms with the concepts of product and sum of soft spaces. Finally, we introduce a concept of pre-fixed soft point and study its main properties.

Intuitionistic fuzzy based regular and normal spaces

In this paper, we define the notion of intuitionistic fuzzy based regular and normal spaces. We also study that classical regular and normal spaces are also intuitionistic fuzzy based regular and normal spaces but the converses are not true in general. This notion opens up a new conception of generalization of classical regular and normal spaces. The hereditary and topological properties of intuitionistic fuzzy based regular and normal spaces have been also investigated. Moreover, by setting some examples we show that every intuitionistic fuzzy based regular space as well as intuitionistic fuzzy based normal space need not be T1 spaces. Finally, it is shown that under some conditions the images and homeomorphic images are preserved in intuitionistic fuzzy based regular and normal spaces.