Narrow Results

The purpose of this paper is to introduce the notion of several kinds of products of rough transformation semigroups. We establish relationship through coverings and investigate some algebraic properties with regard to these products.

We prove an analog of the classical Hartogs extension theorem for certain (possibly unbounded) domains on coverings of Stein manifolds.

Subcube embeddability of the hypercube can be enhanced by introducing an additional dimension. A set of new dimensions, characterized by the Hamming distance between the pairs of nodes it connects, is introduced using a measure defined as the magnitude of a dimension. An enumeration of subcubes of various size is presented for a dimension parameterized by its magnitude. It is shown that the maximum number of subcubes for a Q_n can only be attained when the magnitude of dimension is n-1 or n. It is further shown that the latter two dimensions can optimally increase the number of subcubes among all possible choices.

In this paper we consider the location of stops along the edges of an already existing public transportation network. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network.

The positive effect of new stops is given by the better access of the potential customers to public transportation, while the travel time increases due to the additional stopping activities of the trains. The latter leads to a negative effect for the customers. The goal is to cover all given demand points with a minimal amount of additional traveling time, where covering may be defined with respect to an arbitrary norm or gauge. This problem is NP-hard, even in the special case of Euclidean distances. In this paper, we derive a finite candidate set leading to a discrete set covering problem. Moreover, we identify network structures in which the coefficient matrix of the resulting set covering problem is totally unimodular, such that the problem can be solved efficiently in this case. Extensions of the problem are also discussed.

Key parts of the main equipment in a gas turbine may likely be damaged due to operation under high temperature, high pressure, high-speed rotation, etc. Accordingly, the cost for maintenance increases and the damaged parts may cause generation to stop. The surface of a blade is thermal-sprayed, using powder with main compositions such as Ni, Cr, Al, etc, in order to inhibit hot oxidation. Conventional regular maintenance of the coating layer of a blade is made by FPI (Fluorescent Penetrant Inspection) and MTP (Magnetic Particle Testing). Such methods, however, are complicated and take a long time and also require high cost. In this study, defect diagnostics were tested on the coating layer of an industrial gas turbine blade, using an infrared thermography camera. Since the infrared thermography method can check a temperature distribution by means of non-contact on a wide range of areas, it can advantageously save expense and time as compared to conventional test methods. For the infrared thermography method, however, thermo-load must be applied onto a tested specimen and it is difficult to quantify the measured data. To solve the problems, this paper includes description about producing a specimen of a gas turbine blade (bucket), applying thermo-load onto the produced specimen, photographing thermography images by an infrared thermography camera, analyzing the thermography images, and pre-testing to analyze defects on the coating layer of the gas turbine blade.

Given a simply connected orthogonal polygon P, a polynomial time algorithm is presented to cover the polygon with the minimum number of rectangles, under the restriction that if A and B are two overlapping rectangles in the cover then either A - B or B - A is connected. The algorithm runs in O(n log n + nm) time, where n is the number of vertices of P and m is the number of edges in the visibility graph of P that are either horizontal, vertical or form the diagonal of an empty rectangle.

Given a commutative ring with identity $A$ and an $A$-module $M$, a subset $H$ of $M$ is a cyclic covering of $M$, if this module is the union of the cyclic submodules $[h]=\{ah:a\in A\}$, where $h\in H$. Such covering is said to be irredundant, if no proper subset of $H$ is a cyclic covering of $M$. In this work, an irredundant cyclic covering of $A^n$ is constructed for every Artinian commutative ring $A$. As a consequence, a cyclic covering of minimal cardinality of $A^n$ is obtained for every finite commutative ring $A$, extending previous results in the literature.

We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where M=P²×[0,1] is a cylinder over the real projective plane P², M=Σ×[0,1] is a cylinder over a surface Σ with boundary, and M=S¹×S². A finite covering p:N→M induces a map π₁(p)* between labeled chord diagrams that corresponds to taking the preimage p^-1(L)⊂N of a link L⊂M. The maps p^-1 and π₁(p)* intertwine the constructed universal Vassiliev invariants.

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J be a knot in the complement of K with lk(J, K) = 0. Suppose there is covering space , where V(J) is a regular neighborhood of J satisfying V(J) ∩ im(K) = ∅ and Σ is a connected compact orientable 2-manifold. Let K′ be a knot in Σ × (0, 1) such that π_J(K′) = K. Then K′ stabilizes to a virtual knot , called a virtual cover of K relative to J. We investigate what can be said about a classical knot from its virtual covers in the case that J is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.

An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p, q), that is the counterimage of L under the universal covering of L(p, q). If lens spaces are defined as a lens with suitable boundary identifications, then a link in L(p, q) can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from this diagram of L, we obtain a diagram of the lift in S³. Using this construction, we are able to find different knots and links in L(p, q) having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift.

A (left) quandle is connected if its left translations generate a group that acts transitively on the underlying set. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to constant quandle cocycles of Andruskiewitsch and Graña. A connected quandle is simply connected if it has no nontrivial coverings, or, equivalently, if all its second constant cohomology sets with coefficients in symmetric groups are trivial.

In this paper, we develop a combinatorial approach to constant cohomology. We prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from $4$ are simply connected. We also consider constant cohomology with coefficients in arbitrary groups.

For a virtual knot $K$ and an integer $r\ge 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots ,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r(r=0\mathrm{\text{ and }}2\le r\le m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.

In this paper we introduce the concept of a covering of a ffsm by another, admissible partitions and relations of a ffsm, μ-orthogonality of admissible partitions, irreducibile ffsm, and the quotient of a ffsm induced by an admissible partition of the state set.

The covering number of a finite group $G$, denoted $\sigma (G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $\sigma (G)$ for a family of primitive groups $G$ with a unique minimal normal subgroup $N$, isomorphic to $A_n^m$ with $n$ divisible by $6$ and $G/N$ cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.

An application of solvable structures to the reduction of ODEs with a lack of local symmetries is given. Solvable structures considered here are all defined in a nonlocal extension, or covering space, of a given ODE. Examples of the reduction procedure are provided.

In this paper, we study some covering properties in topological spaces defined by preopen sets. We introduce and investigate the properties of the pre-Menger property, the almost pre-Menger property and their star versions. It is shown that the pre-Menger and the semi-Menger [Covering properties defined by semi-open sets, J. Nonlinear Sci. Appl.9 (2016) 4388–4398] are independent properties.

A graph is called $H$-equicoverable if every minimal $H$-covering of it is also its minimum $H$-covering. In this paper, we consider how to characterize a graph to be $H$-equicoverable where $H$ is $P_4$. We obtained necessary and sufficient conditions for a graph contains a cycle with length greater than 3 but not contains any 3-cycles.

We give in §3 results concerning the preservation of locally uniformly convergence for sequences and of the normality and compactness for families of 1) continuous mappings and 2) homeomorphisms, by lifting or factorizing in the general case of regular coverings (=coverings, Definition 1). With this aim we present in §2.2. the lifting and factorizing properties by regular coverings pointing out when normal coverings (Definition 2) are needed.

The reduction theory is the most significant component of rough set theory. This paper for the first time employs the topological separability to analyze the reductions of covering rough sets. First, a definition is given to the covering separability to describe the classification ability of knowledge bases. Second, a connection is built between the separability and discernibility matrix. The knowledge bases which do not satisfy the separability are transformed to ones with separability via the application of the topological method, and then discernibility matrices with lower orders are reached. As a significant advantage, the method simplifies discernibility matrices to lower order, and in turn improves all reduction algorithm based on discernibility matrix.