On the dynamic slip boundary condition for Navier–Stokes-like problems

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

Strong admissibility for abstract dialectical frameworks

dialectical frameworks (ADFs) have been introduced as a formalism for modeling argumentation allowing general logical satisfaction conditions and the relevant argument evaluation. Different criteria used to settle the acceptance of arguments are called semantics. Semantics of ADFs have so far mainly been defined based on the concept of admissibility. However, the notion of strongly admissible semantics studied for abstract argumentation frameworks has not yet been introduced for ADFs. In the current work we present the concept of strong admissibility of interpretations for ADFs. Further, we show that strongly admissible interpretations of ADFs form a lattice with the grounded interpretation as the maximal element. We also present algorithms to answer the following decision problems: (1) whether a given interpretation is a strongly admissible interpretation of a given ADF, and (2) whether a given argument is strongly acceptable/deniable in a given interpretation of a given ADF. In addition, we show that the strongly admissible semantics of ADFs forms a proper generalization of the strongly admissible semantics of AFs.

Composition Vector Spaces as a New Type of Tri-Operational Algebras

The aim of this paper is to define and study the composition vector spaces as a type of tri-operational algebras. In this regard, by presenting nontrivial examples, it is emphasized that they are a proper generalization of vector spaces and their structure can be characterized by using linear operators. Additionally, some related properties about foundations, composition subspaces and residual elements are investigated. Moreover, it is shown how to endow a vector space with a composition structure by using bijective linear operators. Finally, more properties of the composition vector spaces are presented in connection with linear transformations.

Semi-Stable Semantics for Abstract Dialectical Frameworks

dialectical frameworks (ADFs) have been introduced as a formalism for modeling and evaluating argumentation allowing general logical satisfaction conditions. Different criteria that have been used to settle the acceptance of arguments are called semantics. However, the notion of semi-stable semantics as studied for abstract argumentation frameworks has received little attention for ADFs. In the current work, we present the concepts of semi-two-valued models and semi-stable models for ADFs. We show that these two notions satisfy a set of plausible properties required for semi-stable semantics of ADFs. Moreover, we show that semi-two-valued and semi-stable semantics of ADFs form a proper generalization of the semi-stable semantics of AFs, just like two-valued model and stable semantics for ADFs are generalizations of stable semantics for AFs.

Цветовая энергия некоторых кластерных графов

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

Hypersurfaces in a conformally flat space

The hypersurface of a space is one of the most important objects in a space. Many authors studied the various geometric aspects of hypersurfaces in a space form. The notion of conformal flatness is one of the most primitive concepts in differential geometry. Again, conformally flat space is a proper generalization of a space form. In this paper, we study the geometry of hypersurfaces in a conformally flat manifold. Then we have investigated some sufficient conditions imposed on the shape operator for which the hypersurface satisfies various pseudosymmetric-type conditions imposed on its conformal curvature tensor.

H-supplemented modules with respect to images of fully invariant submodules

Lifting modules plays important roles in module theory. H-supplemented modules are a nice generalization of lifting modules which have been studied extensively recently. In this article, we introduce a proper generalization of H-supplemented modules via images of fully invariant submodules. Let F be a fully invariant submodule of a right Rmodule M. We say that M is IF -H-supplemented in case for every endomorphism φ of M, there is a direct summand D of M such that φ(F) + X = M if and only if D + X = M, for every submodule X of M. It is proved that M is IF -H-supplemented if and only if F is a dual Rickart direct summand of M for a fully invariant noncosingular submodule F of M. It is shown that the direct sum of IF –H supplemented modules is not in general IF -H-supplemented. Some sufficient conditions such that the direct sum of IF -H-supplemented modules is IF -H-supplemented are given

A note on a generalization of injective modules

As a proper generalization of injective modules in term of supplements, we say that a module $M$ has the property (ME) if, whenever $M\subseteq N$, $M$ has a supplement $K$ in $N$, where $K$ has a mutual supplement in $N$. In this study, we obtain that $(1)$ a semisimple $R$-module $M$ has the property (E) if and only if $M$ has the property (ME); $(2)$ a semisimple left $R$-module $M$ over a commutative Noetherian ring $R$ has the property (ME) if and only if $M$ is algebraically compact if and only if almost all isotopic components of $M$ are zero; $(3)$ a module $M$ over a von Neumann regular ring has the property (ME) if and only if it is injective; $(4)$ a principal ideal domain $R$ is left perfect if every free left $R$-module has the property (ME)

On \({\mathcal{F}}\)-Contractions for Weak α-Admissible Mappings in Metric-Like Spaces

In the paper, we consider some fixed point results of F-contractions for triangular α-admissible and triangular weak α-admissible mappings in metric-like spaces. The results on F-contraction type mappings in the context of metric-like spaces are generalized, improved, unified, and enriched. We prove the main result but using only the property (F1) of the strictly increasing mapping F:0,+∞→−∞,+∞. Our approach gives a proper generalization of several results given in current literature.

Double Controlled Quasi-Metric Type Spaces and Some Results

Abdeljawad et al. (2018) introduced a new concept, named double controlled metric type spaces, as a generalization of the notion of extended b-metric spaces. In this paper, we extend their concept and introduce the concept of double controlled quasi-metric type spaces with two incomparable functions and prove some unique fixed point results involving new types of contraction conditions. Also, we introduce the concept of α−μ−k double controlled contraction and prove some related fixed point results. We give several examples to show that our results are the proper generalization of the existing works.