Different Types of Neutrosophic Chromatic Number

New setting is introduced to study chromatic number. Different types of chromatic numbers and neutrosophic chromatic number are proposed in this way, some results are obtained. Classes of neutrosophic graphs are used to obtains these numbers and the representatives of the colors. Using colors to assign to the vertices of neutrosophic graphs is applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different types of edges from connectedness in same neutrosophic graphs and in modified neutrosophic graphs to define the relation amid vertices which implies having different colors amid them and as consequences, choosing one vertex as a representative of each color to use them in a set of representatives and finally, using neutrosophic cardinality of this set to compute types of chromatic numbers. This specific relation amid edges is necessary to compute both types of chromatic number concerning the number of representative in the set of representatives and types of neutrosophic chromatic number concerning neutrosophic cardinality of set of representatives. If two vertices have no intended edge, then they can be assigned to same color even they’ve common edge. Basic familiarities with neutrosophic graph theory and graph theory are proposed for this article.

Neutrosophic Chromatic Number Based on Connectedness

New setting is introduced to study chromatic number. vital chromatic number and n-vital chromatic number are proposed in this way, some results are obtained. Classes of neutrosophic graphs are used to obtains these numbers and the representatives of the colors. Using colors to assign to the vertices of neutrosophic graphs is applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using vital edge from connectedness to define the relation amid vertices which implies having different colors amid them and as consequences, choosing one vertex as a representative of each color to use them in a set of representatives and finally, using neutrosophic cardinality of this set to compute vital chromatic number. This specific relation amid edges is necessary to compute both vital chromatic number concerning the number of representative in the set of representatives and n-vital chromatic number concerning neutrosophic cardinality of set of representatives. If two vertices have no vital edge, then they can be assigned to same color even they’ve common edge. Basic familiarities with neutrosophic graph theory and graph theory are proposed for this article.

Chromatic Number and Neutrosophic Chromatic Number

New setting is introduced to study chromatic number. Neutrosophic chromatic number and chromatic number are proposed in this way, some results are obtained. Classes of neutrosophic graphs are used to obtains these numbers and the representatives of the colors. Using colors to assigns to the vertices of neutrosophic graphs is applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using strong edge to define the relation amid vertices which implies having different colors amid them and as consequences, choosing one vertex as a representative of each color to use them in a set of representatives and finally, using neutrosophic cardinality of this set to compute neutrosophic chromatic number. This specific relation amid edges is necessary to compute both chromatic number concerning the number of representative in the set of representatives and neutrosophic chromatic number concerning neutrosophic cardinality of set of representatives. If two vertices have no strong edge, then they can be assigned to same color even they’ve common edge. Basic familiarities with neutrosophic graph theory and graph theory are proposed for this article.

Synthesis, crystal structure and selected properties of K2[Ni(dien)2]{[Ni(dien)]2Ta6O19}·11 H2O

The new compound K2[Ni(dien)2]{[Ni(dien)]2Ta6O19}·11 H2O crystallized at room temperature applying a diffusion based reaction in a H2O/DMSO mixture using K8{Ta6O19}·16 H2O, Ni(NO3)2·6H2O and dien (diethylenetriamine). In the crystal structure, the Lindqvist-type anion [Ta6O19]8– is structurally expanded by two octahedrally Ni2+-centered complexes via three Ni–µ 2-O–Ta bonds thus generating the new {[Ni(dien)]2Ta6O19}4– anion. Two KO8 polyhedra share a common edge to form a K2O14 moiety, which connects the {[Ni(dien)]2Ta6O19}4– cluster shells into chains. The isolated [Ni(dien)2]2+ complexes are located in voids generated by the structural arrangement of the chains. An extended hydrogen bonding network between the different constituents generates a 3D network. The crystal water molecules can be thermally removed to form a highly crystalline dehydrated compound. Partial water uptake leads to the formation of a crystalline intermediate with a reduced unit cell volume compared to the fully hydrated sample. Water sorption experiments demonstrate that the fully dehydrated sample can be fully reconverted to the hydrated compound. The crystal field splitting parameters for the octahedrally coordinated Ni2+-centered complexes have been evaluated from an UV/Vis spectrum yielding D q = 1056 cm−1 and B = 887 cm−1.

Streaming Algorithms for Subspace Analysis: an Edge- and Hardware-oriented review

Subspace analysis is a basic tool for coping with high-dimensional data and is becoming a fundamental step in early processing of many signals elaboration tasks. Nowadays trend of collecting huge quantities of usually very redundant data by means of decentralized systems suggests these techniques be deployed as close as possible to the data sources. Regrettably, despite its conceptual simplicity, subspace analysis is ultimately equivalent to eigenspace computation and brings along non-negligible computational and memory requirements. To make this fit into typical systems operating at the edge, specialized streaming algorithms have been recently devised that we here classify and review giving them a coherent description, highlighting features and analogies, and easing comparisons. Implementation of these methods is also tested on a common edge digital hardware platform to estimate not only abstract functional and complexity features, but also more practical running times and memory footprints on which compliance with real-world applications hinges.

Streaming Algorithms for Subspace Analysis: an Edge- and Hardware-oriented review

Subspace analysis is a basic tool for coping with high-dimensional data and is becoming a fundamental step in early processing of many signals elaboration tasks. Nowadays trend of collecting huge quantities of usually very redundant data by means of decentralized systems suggests these techniques be deployed as close as possible to the data sources. Regrettably, despite its conceptual simplicity, subspace analysis is ultimately equivalent to eigenspace computation and brings along non-negligible computational and memory requirements. To make this fit into typical systems operating at the edge, specialized streaming algorithms have been recently devised that we here classify and review giving them a coherent description, highlighting features and analogies, and easing comparisons. Implementation of these methods is also tested on a common edge digital hardware platform to estimate not only abstract functional and complexity features, but also more practical running times and memory footprints on which compliance with real-world applications hinges.

Triangles of nearly equal area

Given any n points in the plane, not all on the same line, there exist two non-collinear triples such that the ratio of the areas of the triangles they determine, differs from 1 by at most $$O(\log n/n^2)$$ O ( log n / n 2 ) . If we furthermore insist that the two triangles have a common edge, then there are two with area ratios differing from 1 by at most O(1/n). This improves some results of Ophir and Pinchasi (Discrete Appl. Math. 174 (2014), 122–127). We also give some constructions for these and related problems.

Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.