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The main work of this paper is to investigate two kinds of generalized focal surfaces and two kinds of evolutes generated by spacelike curve γ lying in lightlike surfaces in Minkowski three‐space. Applying the method of unfolding theory in singularity theory to our study, it is shown that there exist the cuspidal edge and the swallowtail types of singularities in each of two classes of generalized focal surfaces under certain conditions; the only cusps will appear in each of evolutes. Two new geometric invariants are presented to classify the singularities of generalized focal surfaces and evolutes. Much more importantly, we reveal the correspondence among the geometric invariants, the types of singularities on generalized focal surfaces and evolutes, the singularities of two kinds of evolutes, and the contact of γ with the osculating spheres. Finally, several examples are presented to demonstrate the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

Let tp (G) denote the number of paths in a graph G and let f: E → Z+ be an edge labeling of G. The weight of a path P is the sum of the labels assigned to the edges of P. If the set of weights of the paths in G is {1, 2, 3,..., tp (G)}, then f is called a Leech labeling of G and a graph which admits a Leech labeling is called a Leech graph. In this paper, we prove that the complete bipartite graphs K2, n and K3, n are not Leech graphs and determine the maximum possible value that can be given to an edge in the Leech labeling of a cycle. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple connected graph. A perfect matching (or Kekulé structure in chemical literature) of G is a set of disjoint edges which covers all vertices of G. The anti-forcing number of G is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by af(G). For every m ∈ N, the mth power of G, denoted by Gm, is a graph with the same vertex set as G such that two vertices are adjacent in Gm if and only if their distance is at most m in G. In this paper, we study the anti-forcing number of the powers of some graphs. [ABSTRACT FROM AUTHOR]

This paper introduces certain relations between 1-edge contraction and the total vertex stress and the confluence number of a graph. A main result states that if a graph G with ζ(G) = k ≥ 2 has an edge vivj and a ζ-set CG such that vi, vj ∈ CG then, ζ(G/vivj ) = k − 1. In general, either S(G/ei) ≤ S(G/ej ) or S(G/ej ) ≤ S(G/ei) is true. This observation leads to an investigation into the question: for which edge(s) ei will S(G/ei) = max{S(G/ej ) : ej ∈ E(G)} and for which edge(s) will S(G/ej ) = min{S(G/el) : el∈ E(G)}? [ABSTRACT FROM AUTHOR]

A signed graph (G, σ) is called a parity signed graph if there exists a bijective mapping f : V (G) → {1, . . ., |V (G)|} such that for each edge uv in G, f(u) and f(v) have same parity if σ(uv) = +1, and opposite parity if σ(uv) = −1. The rna number σ−(G) of G is the least number of negative edges among all possible parity signed graphs over G. Equivalently, σ−(G) is the least size of an edge-cut of G that has nearly equal sides. In this paper, we show that for the generalized Petersen graph Pn,k, σ−(Pn,k) lies between 3 and n. Moreover, we determine the exact value of σ−(Pn,k) for k ∈ {1, 2}. The rna numbers of some famous generalized Petersen graphs, namely, Petersen graph, Dürer graph, Möbius-Kantor graph, Dodecahedron, Desargues graph and Nauru graph are also computed. Recently, Acharya, Kureethara and Zaslavsky characterized the structure of those graphs whose rna number is 1. We use this characterization to show that the smallest order of a (4n + 1)-regular graph having rna number 1 is 8n + 6. We also prove the smallest order of (4n − 1)-regular graphs having rna number 1 is bounded above by 12n − 2. In particular, we show that the smallest order of a cubic graph having rna number 1 is 10. [ABSTRACT FROM AUTHOR]

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order 12p, where p is a prime. [ABSTRACT FROM AUTHOR]

A defective vertex coloring of a graph is a coloring in which some adjacent vertices may have the same color. An edge whose adjacent vertices have the same color is called a bad edge. A defective coloring of a graph G with minimum possible number of bad edges in G is known as a near proper coloring of G. In this paper, we introduce the notion of equitable near proper coloring of graphs and determine the minimum number of bad edges obtained from an equitable near proper coloring of some graph classes. [ABSTRACT FROM AUTHOR]

Most of the photonic valley-Hall edge states were constructed by changing structures to break the spatial inversion symmetry, restricting the practical application potential. In this paper, we construct a tunable topological valley-Hall edge state based on the large optical Kerr effect. It is demonstrated that topological phase transition happens by engineering the intensity of the injected pump and that a valley-Hall edge state can be generated at the interface between two regions with different topological properties. In addition, eigenfrequency and transmission characteristics of the edge state as a function of applied pump intensity are investigated. The topological protected valley-dependent transmission is studied under non-uniform distributed pump intensity. This work may open a new path toward designing reconfigurable all-optical metadevices. [ABSTRACT FROM AUTHOR]

Constantly sharp knives have applications in industrial cutting, clean shaving, and precision surgery. With the purpose of increasing blade sharpness, an ultrasonic knife was built, operating at 134-kHz continuous ultrasound with an average temperature increase of 0.4 ° C at the blade. The sharpness of the incisions made to a sheet of plain printer paper with a blunted blade under sonication were compared to those of the same blade without sonication and to industrially cut paper edges. The incisions from the sonicated blade were visibly sharper than those from the unsonicated blade, and at least as sharp as those from the industrially cut paper edge. [ABSTRACT FROM AUTHOR]

In order to solve the problem of low efficiency of image feature matching in traditional remote sensing image database, this paper proposes the feature matching optimization of multimedia remote sensing images based on multiscale edge extraction, expounds the basic theory of multiscale edge, and then registers multimedia remote sensing images based on the selection of optimal control points. In this paper, 100 remote sensing images with a size of 3619 ∗ 825 with a resolution of 30 m are selected as experimental data. The computer is configured with 2.9 ghz CPU, 16 g memory, and i7 processor. The research mainly includes two parts: image matching efficiency analysis of multiscale model; matching accuracy analysis of multiscale model and formulation of model parameters. The results show that when the amount of image data is large, feature matching takes more time. With the increase of sampling rate, the amount of image data decreases rapidly, and the feature matching time also shortens rapidly, which provides a theoretical basis for the multiscale model to improve the matching efficiency. The data size is the same, 3619 × 1825, which makes the matching time between images have little difference. Therefore, the matching time increases linearly with the increase of the number of images in the database. When the amount of image data in the database is large, a higher number of layers should be used; when the amount of image data in the database is small, the number of layers of the model should be reduced to ensure the accuracy of matching. The availability of the proposed method is proved. [ABSTRACT FROM AUTHOR]

In this paper, we use the edge distance random matrix method to analyze and study the dance movements of 3D animation characters and design a 3D animation character dance movement model. Firstly, each dance movement video in the dataset is divided into equal parts, while the segmented videos are subjected to the operation of accumulating edge features separately, and the edge features in all video images within each segment are accumulated into one image, and the directional gradient histogram features are extracted from them. Finally, a set of directional gradient histogram features are used to represent the local appearance and shape of the video dance movement. The reconstructed human movements are obtained mainly by fitting the 3D human coordinates in the image space using the human model and the estimated depth coordinates, which are currently more mature and are chosen from existing techniques. In other combinations, the performance of the method in this paper is better than the recognition results of the benchmark method, especially when the similarities of dance movements in the towel flower combination and the piece flower combination are too high. In response to the problem of the reconstructed action appearing to have ground penetration, slipping, and floating feet, a method is proposed to optimize the fitting human model foot problem according to the foot touching the ground. The experimental results show that the method can replace the traditional motion capture method to a certain extent, simplify the use of motion capture, and reduce the cost of motion capture. In the model deformation stage, to reduce the deformation quality problem during the model motion and improve the efficiency of weight calculation, a model deformation method combining double quaternions and bounded double tuning and weights are given. The static 3D model is tetrahedralization, and then the skeletal control points of the model are set and the weight of each skeletal segment to the model is calculated; next, the 3D model is mixed and bound with the skeletal data using the dual quaternion skinning algorithm; finally, the static 3D model motion is driven by the skeletal data. Experiments demonstrate that the method results in better deformation of the 3D model during rotation. [ABSTRACT FROM AUTHOR]

A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if G contains H as an immersion, then for every integer k , the number of vertices of degree at least k in G is at least the number of vertices of degree at least k in H. In this paper, we prove that this obvious necessary condition is "nearly" sufficient for graphs with no edge-cut of order 3: for every graph H , every H -immersion free graph with no edge-cut of order 3 can be obtained by an edge-sum of graphs, where each of the summands is obtained from a graph violating the obvious degree condition by adding a bounded number of edges. The condition for having no edge-cut of order 3 is necessary. A simple application of this theorem shows that for every graph H of maximum degree d ≥ 4 , there exists an integer c such that for every positive integer m , there are at most c m unlabeled d -edge-connected H -immersion free m -edge graphs with no isolated vertex, while there are superexponentially many unlabeled (d − 1) -edge-connected H -immersion free m -edge graphs with no isolated vertex. Our structure theorem will be applied in a forthcoming paper about determining the clustered chromatic number of the class of H -immersion free graphs. [ABSTRACT FROM AUTHOR]

The author introduced new concept of this paper, how to protect from the Corona virus. Here fuzzy related concept act a main role in complete bipartite coloring fuzzy graph networking. Because the minute type of Germs also find out and express. Many of the Mall, Cinema Theater, School, College area having a some of the Corona patient. From this kind of patient, easily spread with others. In this situation the author draw a complete bipartite coloring fuzzy graph to solve this problem. If the graph is complete all the vertices are adjacent. So each vertices have n-1 edges. Now the author convert all the Edges are frequency lines and Vertices are convert into some different like scanner, thermometer, sanitizer. This is one of the good idea to protect from corona virus to spread others. And also give an brief explanation of this paper. [ABSTRACT FROM AUTHOR]

For a hypergraph H$H$, define its intersection spectrum I(H)$I(H)$ as the set of all intersection sizes |E∩F|$|E\cap F|$ of distinct edges E,F∈E(H)$E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a k$k$‐uniform 3‐chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with k$k$. Despite the problem being reiterated several times over the years by Erdős and other researchers, the lower bound of three intersection sizes has remarkably withstood any improvement until now. In this paper, we prove the Erdős–Lovász conjecture in a strong form by showing that there are at least k1/2−o(1)$k^{1/2-o(1)}$ intersection sizes. Our proof consists of a delicate interplay between Ramsey‐type arguments and a density increment approach. [ABSTRACT FROM AUTHOR]

This paper aims to initiate a discourse that connects allomothers, endemic to African culture, with collective manymothering attachments from a psychoanalytic perspective. This paper illuminates the process by which, beginning with West Africa, Black mothers adapted and carried the process of mothering with them to provide consistent nurturing, responsiveness, and attunement to their infants' and children's needs. This process of extending caregiving responsibilities to the community at large, which I have labeled manymothering, has created generations of resistance and resilience that have supported Black people to the present. The psychoanalytic lens of othermothers serves as an adaptive familial structure that has been sustained through intergenerational resilient transmission. The linkage between culture and spirituality as a means of ameliorating trauma and promoting resilience was examined. [ABSTRACT FROM AUTHOR]

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let Γ be a chart, and we denote by Γ m the union of all the edges of label m. A chart Γ is of type (7) if there exists a label m such that w (Γ) = 7 , w (Γ m ∩ Γ m + 1) = 7 where w (G) is the number of white vertices in G. In this paper, we prove that there is no minimal chart of type (7). [ABSTRACT FROM AUTHOR]

This paper presents an automated high-speed die edge detection framework, based on improved K-Mean and landscape analysis, that is highly suitable to be implemented in precision engineering for optical non-destructive testing of die-level defects inspection in the semiconductor industry. This paper specifically aims to achieve high accuracy (or yield) of greater than 99.95% with stable performance within a short computation time (i.e. less than 20 ms). To demonstrate the applicability of the proposed framework, it is validated using three production units (i.e. Production unit A: 6000 units; Production unit B: 3500 units; Production unit C: 4000 units) and is benchmarked to two baseline edge detection methods, namely cross-correlation and normalised cross-correlation methods, as well as state-of-the-art vision libraries, recent works, and several conventional edge detection methods. The results obtained show that the proposed framework is capable of performing die edge detection with promising accuracy and stable performance by achieving 100.0% yield in all three production units, having outperformed the benchmarking methods. Also, the overall computation time (considering die edge detection and die rotation) of the proposed framework is short, at approximately 15 ms. [ABSTRACT FROM AUTHOR]

Edge detection consists of a set of mathematical methods which identifies the points in a digital image where image brightness changes sharply. In the traditional edge detection methods such as the first-order derivative filters, it is easy to lose image information details and the second-order derivative filters are more sensitive to noise. To overcome these problems, the methods based on the fractional differential-order filters have been proposed in the literature. This paper presents the construction and implementation of the Prewitt fractional differential filter in the Asumu definition sense for SARS-COV2 image edge detection. The experiments show that these filters can avoid noise and detect rich edge details. The experimental comparison show that the proposed method outperforms some edge detection methods. In the next paper, we are planning to improve and combine the proposed filters with artificial intelligence algorithm in order to program a training system for SARS-COV2 image classification with the aim of having a supplemental medical diagnostic. [ABSTRACT FROM AUTHOR]

This paper presents the application of the edge reversal method for a further accurate evaluation of the three-dimensional tool geometry of a single-point diamond tool with an atomic force microscope (AFM). In the edge reversal method, the tip radius of an AFM probe, which ranges from a few nm to several-ten nm, can be evaluated quantitatively by using measured AFM profile data of a tool cutting edge and an indentation mark generated by the tool cutting edge. The AFM tip radius obtained by the edge reversal method is utilized in this paper to reconstruct the three-dimensional profile of the cutting edge of a single-point diamond tool from the AFM data, which is obtained in measurement as the convolution of the tool edge profile and the shape of AFM tool tip. The validity of the obtained AFM tip radius in the edge reversal method is verified by employing it for the compensation of the AFM image of a two-dimensional (2D) grating artifact. Finally, the reconstruction of the three-dimensional profile of the cutting edge of a single-point diamond tool is carried out, and the cutting edge radius and the nose radius of the tool are evaluated quantitatively based on the reconstructed three-dimensional (3D) profile of the cutting tool. [ABSTRACT FROM AUTHOR]

This paper challenges legal geographers to go further in their claimed rejection of legal closure (the reduction of any situation to a purely legal dimension), by calling for greater exploration of how law's command is always acting alongside and potentially subordinated to other spatio‐cultural normative influences. The insights of this advocated approach are illustrated using examples drawn from the author's fieldwork conducted across a variety of outdoor attraction settings, which have explored how place‐managers frame, arrange, and present their sites by reference to individual goals and communal interpretive practices deriving from multiple normative drivers. It is argued that this plurality of influences requires place‐managers to act as edgeworkers, developing a finely tuned understanding of how to manage the potentially hazardous co‐presence of visitors' bodies and physical edges at their sites. This requires active balancing of their interpretation of (and compliance with) broad commandments for their sites to be "reasonably safe," with the requirements of other pressing normativities, including those set by landscape aesthetics that demand that visitors must be presented with an appropriately thrilling experience. Recent trends in the development of contemporary legal geographic analysis, each of which are suggestive of an increasing attentiveness to law's limits, are considered in order to develop this critique of conventional legal geographic practice and in support of the potentiality of the recommended new focus, specifically: legal geography's "contingency orientation" (Orzeck & Hae); emerging interest in the "edge of law" (Jeffrey); and in the purposeful creation of normative atmospheres and law's dissimulation there (Philippopoulos‐Mihalopoulos; Pavoni). Through these analytical lenses place‐manager's active and skilful edgework becomes both more visible and more understandable as a dynamic situational balancing of compliance with safety laws and with the satisfaction of thrill and desire. This short paper calls for legal geographers to give more attention to how place‐managers transpose abstract legal requirements onto their sites, and what interpretative and emotional resources they use to do so. It emphasises the importance of other normative drivers, alongside those of law, in the formation and maintenance of places. [ABSTRACT FROM AUTHOR]

The Sombor index of the graph G is a degree based topological index, deffned as SO =P uv2E(G) p dG(u)2 + dG(v)2, where dG(u) is the degree of the vertex u of G and E(G) is the edge set of G. In this paper we calculate SO of some graph transformations. [ABSTRACT FROM AUTHOR]

For a graph G with no isolated vertex, a covering total double Roman dominating function (CTDRD function) f of G is a total double Roman dominating function (TDRD function) of G for which the set fv 2 V (G)jf(v) 6= 0g is a vertex cover set. The covering total double Roman domination number ctdR(G) equals the minimum weight of an CTDRD function on G. An CTDRD function on G with weight ctdR(G) is called a ctdR(G)-function. In this paper, the graphs G with small ctdR(G) are characterised. We show that the decision problem associated with CTDRD is NP-complete even when restricted to planer graphs with maximum degree at most four. We then show that for every graph G without isolated vertices, oitR(G) < ctdR(G) < 2 oitR(G) and for every tree T, 2-(T) + 1 - ctdR(T) - 4-(T), where oitR(G) and -(T) are the outer independent total Roman domination number of G, and the minimum vertex cover number of T respectively. Moreover we investigate the ctdR of corona of two graphs. [ABSTRACT FROM AUTHOR]

Let G = (V;E) be a given graph of order n. A function f: V ! f0; 1; 2g is an independent Roman dominating function (IRDF) on G if for every vertex v 2 V with f(v) = 0 there is a vertex u adjacent to v with f(u) = 2 and fv 2 V: f(v) > 0g is an independent set. The weight of an IRDF f on G is the value f(V) = P v2V f(v). The minimum weight of an IRDF among all IRDFs on G is called the independent Roman domination number of G. In this paper, we give algorithms for computing the independent Roman domination number of G in O(jV j) time when G = (V;E) is a tree, unicyclic graph or proper interval graph. [ABSTRACT FROM AUTHOR]

Considerable scientific work was carried out along the years to understand the physics of the cutting process with the aim of a better control of these processes. This paper investigates the cutting force variation along the cutting edge in orthogonal cutting processes. This study can be helpful for a better prediction of tool wear in machining planning. Two sets of machining tests were carried out considering two different tool geometries. [ABSTRACT FROM AUTHOR]

The following very natural problem was raised by Chung and Erdős in the early 80's and has since been repeated a number of times. What is the minimum of the Turán number ex (n , H) among all r -graphs H with a fixed number of edges? Their actual focus was on an equivalent and perhaps even more natural question which asks what is the largest size of an r -graph that can not be avoided in any r -graph on n vertices and e edges? In the original paper they resolve this question asymptotically for graphs, for most of the range of e. In a follow-up work Chung and Erdős resolve the 3-uniform case and raise the 4-uniform case as the natural next step. In this paper we make first progress on this problem in over 40 years by asymptotically resolving the 4-uniform case which gives us some indication on how the answer should behave in general. [ABSTRACT FROM AUTHOR]

Given a positive integer n and an r-uniform hypergraph F, the Tur´an number ex(n, F) is the maximum number of edges in an F-free r-uniform hypergraph on n vertices. The Tur´an density of F is defined as π(F) = limn→∞ ex(n,F) (n r). The Lagrangian density of F is πλ(F) = sup{r!λ(G): G is F-free}, where λ(G) is the Lagrangian of G. Sidorenko observed that π(F) ≤ πλ(F), and Pikhurko observed that π(F) = πλ(F) if every pair of vertices in F is contained in an edge of F. Recently, Lagrangian densities of hypergraphs and Tur´an numbers of their extensions have been studied actively. For example, in the paper [A hypergraph Tur´an theorem via Lagrangians of intersecting families, J. Combin. Theory Ser. A 120 (2013) 2020–2038], Hefetz and Keevash studied the Lagrangian densitiy of the 3-uniform graph spanned by {123, 456} and the Tur´an number of its extension. In this paper, we show that the Lagrangian density of the 3-uniform graph spanned by {123, 234, 456} achieves only on K3 5. Applying it, we get the Tur´an number of its extension, and show that the unique extremal hypergraph is the balanced complete 5-partite 3-uniform hypergraph on n vertices. [ABSTRACT FROM AUTHOR]

Soda-Lime Glass (SLG) has been widely used in the creation of highly-functionalised optical devices. However, machining of SLG easily resulted in various defects. Among these defects, edge chipping (EC) can be treated as the most damageable in terms of both mechanical properties and optical performances. Although efforts have been paid to this topic, previous investigations (i) focused on edge chipping in low strain rate, which was 101-102 times lower than the ones in real machining, (ii) assumed edge chipping shape as circular, while the experiments in this work proved this assumption might be inaccurate in high strain rate circumstances, and (iii) were experimental efforts, resulting in limited knowledge of process principles. To fill this gap, this paper aims to propose an analytical model to calculate the size and the shape of edge chipping in scratch of soda-lime glass with the special consideration of strain-rate hardening effect. A series of experimental trials were performed to validate the model accuracy under different parameters. The details in scratching process were discussed by force analysis. A typical model application was given as well, which was designing the machining strategy based on the proposed model so that a required region with limited edge chipping can be obtained. Considering the research gaps mentioned above, the key findings in this paper are expected to be meaningful and helpful to provide references in the real fabrication of high-valued optical devices made of SLG. [ABSTRACT FROM AUTHOR]

In a connected graph G , the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) ∪ E (G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicyclic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles, where for each cycle C i of G we define a unicyclic subgraph G i of G in which C i is the only cycle. Applying the result for unicyclic graph to the subgraph G i of every cycle C i then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number. [ABSTRACT FROM AUTHOR]

This paper considers the leader-following consensus problem for delayed neural networks with multi-layer signed digraph topologies. Both structurally balanced and unbalanced topologies are considered. By using the structural balance theory and the pinning control strategy, some sufficient conditions are given to guarantee that the bipartite leader-following consensus can be reached in structurally balanced multi-layer signed neural networks, which depend on the node dynamics and the topology of the multi-layer signed network. In addition, a scheme is designed for selecting the pinning nodes and gains of each layer to achieve bipartite leader-following consensus. When the multi-layer signed network is structurally unbalanced, the asymptotic stability of the neural network is investigated by decomposing the structurally unbalanced network into a structurally balanced subgraph and some special edges. The conditions proposed in this paper are also suitable for addressing the consensus problem of traditional single layer signed neural networks. Numerical examples are given to illustrate the correctness and effectiveness of the proposed theorems and algorithms. [ABSTRACT FROM AUTHOR]

Recently, emerging edge computing technology has been proposed as a new paradigm that compensates for the disadvantages of the current cloud computing. In particular, edge computing is used for service applications with low latency while using local data. For this emerging technology, a neural network approach is required to run large-scale machine learning on edge servers. In this paper, we propose a pod allocation method by adding various graphics processing unit (GPU) resources to increase the efficiency of a Kubernetes-based edge server configuration using a GPU-based embedded board and a TensorFlow-based neural network service application. As a result of experiments performed on the proposed edge server, the following are inferred: 1) The bandwidth, according to the time and data size, changes in local (20.4–42.4 Mbps) and Internet environments (6.31–25.5 Mbps) for service applications. 2) When two neural network applications are run on an edge server consisted with Xavier, TX2 and Nano, the network times of the object detection application are from 112.2 ms (Xavier) to 515.8 ms (Nano); the network times of the driver profiling application are from 321.8 ms (Xavier) to 495.7 ms (Nano). 3) The proposed pod allocation method demonstrates better performance than the default pod allocation method. We observe that the number of allocatable pods on three worker nodes increases from five to seven, and compared to other papers, the proposed offloading shows similar or better response times in environments where multiple deep learning applications are implemented. [ABSTRACT FROM AUTHOR]

In practical applications of graph theory, indeterminacy factors always appear in graphs. Uncertain random graph was proposed via chance theory, in which some edges exist with degrees in probability measure and others exist with degrees in uncertain measure. This paper discusses the contributions of edges for connectivity of an uncertain random graph and proposes concepts about significance of edges, according to which edges are classified. In addition, this paper presents algorithms for calculating connectivity index and significance of edges of an uncertain random graph. Examples are given to illustrate algorithms and methods. [ABSTRACT FROM AUTHOR]

A d-dimensional framework is a pair (G, p), where G = (V , E) is a graph and p is a map from V to R d . The length of an edge u v ∈ E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in R d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in R d is globally rigid for some graph G on n ≥ d + 2 vertices (where d ≥ 2 ), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For d = 1 , 2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid. [ABSTRACT FROM AUTHOR]

A signed graph is a pair (G , Σ) , where G = (V , E) is a graph (in which parallel edges are permitted, but loops are not) with V = { 1 , ... , n } and Σ ⊆ E. The edges in Σ are called odd and the other edges of E even. By S (G , Σ) we denote the set of all symmetric n × n matrices A = [ a i , j ] with a i , j < 0 if i and j are adjacent and connected by only even edges, a i , j > 0 if i and j are adjacent and connected by only odd edges, a i , j ∈ R if i and j are connected by both even and odd edges, a i , j = 0 if i ≠ j and i and j are non-adjacent, and a i , i ∈ R for all vertices i. For a graph G = (V , E) with V = { 1 , ... , n } , we denote by N (G) the set of all symmetric matrices X = [ x i , j ] with x i , i = 0 for all i ∈ V , and x i , j = 0 if i and j are adjacent. A matrix A ∈ S (G , Σ) has the Strong Arnold Property if for all X ∈ N (G) , A X = 0 implies that X = 0. The parameters ξ of a signed graph (G , Σ) are the largest nullity of any matrix A ∈ S (G , Σ) that has the Strong Arnold Property. In a previous paper, we gave a characterization of 2-connected signed graphs (G , Σ) with ξ (G , Σ) ≤ 2. In this paper, we give a complete characterization of the signed graphs (G , Σ) with ξ (G , Σ) ≤ 2. [ABSTRACT FROM AUTHOR]

• In this paper, we consider the submodular edge cover problem with submodular penalties. • We give a 2Δ-approximation algorithm for the above problem, where Δ is the maximal degree of the graph G. • We transform the submodular edge cover problem into a submodular set cover problem and show that there exists a Δ + 1 -approximation algorithm. In this paper, we consider the submodular edge cover problem with submodular penalties. In this problem, we are given an undirected graph G = (V , E) with vertex set V and edge set E. Assume the covering cost function c : 2 E → R + and the penalty function p : 2 V → R + are both submodular with p non-decreasing, c (∅) = 0 and p (∅) = 0. The goal of the submodular edge cover problem with submodular penalties is to select an edge subset to cover some vertices and penalize the vertex subset containing uncovered vertices such that the total cost of covering and penalty is minimized. For this problem, we first give a 2Δ-approximation algorithm by using a primal-dual technique, where Δ is the maximal degree of the graph G. Then we transform this problem into a submodular set cover problem, and by applying a known result for the submodular set cover problem we conclude that there is an approximation algorithm with an approximation ratio Δ + 1. [ABSTRACT FROM AUTHOR]

This article traces the evolution of the concept of the leading edge in Kohut's work. The leading edge is defined as the growth-promoting dimension of the transference. The authors argue that although Kohut did not ever use the term explicitly in his writings—Marian Tolpin (2002), one of Kohut's gifted pupils, introduced the concept into the psychoanalytic literature in the form of the forward edge—the idea of the leading edge was already present in nascent form in Kohut's earliest papers and became ever more central as his psychology of the self evolved and the concept of the selfobject transference took center stage. Kohut, it is argued, could not fully develop the idea of working with the leading edge for fear of being accused of advocating for a corrective emotional experience in psychoanalytic treatment. However, in his posthumous empathy paper (1982) Kohut came as close as he could to endorsing the leading edge as pivotal in all psychoanalytic work. [ABSTRACT FROM AUTHOR]

With the integration of Artificial Intelligence (AI) and Internet of Things (IoT), the Federated Edge Learning (FEL), a promising computing framework is developing. However, there are still unsolved issues on communication efficiency and data security due to the huge models and unreliable transmission links. To address these issues, this paper proposes a novel federated edge learning system, called LightFed, where the edge nodes upload only vital partial local models, and successfully achieve lightweight communication and model aggregation. First, a novel model aggregation method Model Splitting and Splicing (MSS) and a Selective Parameter Transmission (SPT) scheme are proposed. By detecting the updating gradients of local parameters and filtering significant parameters, selective rotated transmission and efficient aggregation of local models are achieved. Second, a Training Filling Model (TFM) is proposed to infer the total data distribution of edge nodes, and train a filling model to mitigate the unbalanced training data without violating the data privacy of individual users. Moreover, a blockchain-powered confusion transmission mechanism is proposed for defending the attacks from external adversaries and protecting the model information. Finally, extensive experimental results demonstrate that our LightFed significantly outperforms the existing FEL systems in terms of communication efficiency and privacy security. [ABSTRACT FROM AUTHOR]

In this paper, we deal with very well-covered graphs. We first describe the structure of these kinds of graphs based on the structure of Cohen–Macaulay very well-covered graphs. As an application, we analyze the structure of local cohomology of the residue rings by the edge ideals of very well-covered graphs. Also, we give different formulas of regularity and depth of these rings from known ones and we finally treat the CM t property. [ABSTRACT FROM AUTHOR]

Let G be an edge-colored graph. A triangle of G is called rainbow if any two edges of the triangle have distinct colors. We use m (G) and c (G) to denote the number of edges of G and the number of colors appearing on the edges of G , respectively. Li et al. in 2014 proved that every edge-colored graph of order n with m (G) + c (G) ≥ n (n + 1) / 2 contains a rainbow triangle and this result is best possible. In 2019, Fujita et al. characterized all graphs G satisfying m (G) + c (G) ≥ n (n + 1) / 2 − 1 but containing no rainbow triangle. In this paper, we conjecture that every edge-colored graph of order n with m (G) + c (G) ≥ n (n + 1) / 2 + 3 (k − 1) contains k edge-disjoint rainbow triangles. We show that the conjecture holds for k = 2 and 3 and these results are best possible. Furthermore, we characterize all graphs G satisfying m (G) + c (G) ≥ n (n + 1) / 2 + 2 but not containing two edge-disjoint rainbow triangles. At the end, we propose a conjecture on the number of vertex-disjoint rainbow triangles in an edge-colored graph. [ABSTRACT FROM AUTHOR]

The Barát‐Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree T, there is a constant cT such that every cT‐edge‐connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T‐decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in Bensmail et al. (2019) that when T is a path with k edges, there is a constant dk such that every 24‐edge‐connected graph G with size divisible by k and minimum degree dk has a T‐decomposition. We show in this paper that when F is a coprime forest (the sizes of its components being a coprime set of integers), any graph G with sufficiently large minimum degree has an F‐decomposition provided that the size of F divides the size of G (no connectivity is required). A natural conjecture asked in Bensmail et al. (2019) asserts that for a fixed tree T, any graph G of size divisible by the size of T with sufficiently high minimum degree has a T‐decomposition, provided that G is sufficiently highly connected in terms of the maximal degree of T. The case of maximum degree 2 is answered by paths. We provide a counterexample to this conjecture in the case of maximum degree 3. [ABSTRACT FROM AUTHOR]

5G mobile network technology together with edge computing will create new opportunities for developing novel road safety services in order to better support connected and automated driving in challenging situations. This paper studies the feasibility and benefits of localized mobile network edge applications for supporting vehicles in diverse conditions. We study a particular scenario, where vehicle sensor data processing, required by road safety services, is installed into the mobile network edge in order to extend the electronic horizon of the sensors carried by other vehicles. Specifically, we focus on a LiDAR data-based obstacle warning case where vehicles receive obstacle warnings from the mobile network edge. The proposed solution is based on a generic system architecture. In this paper, we first evaluate different connectivity and computing options associated with such a system using ns-3 simulations. Then, we introduce a proof-of-concept implementation of the LiDAR-based obstacle warning scenario together with first results from an experimental evaluation, conducted both in a real vehicle testbed environment and in a laboratory setting. As a result, we obtain first insights on the feasibility of the overall solution and further enhancements needed. [ABSTRACT FROM AUTHOR]

This paper considers an edge minimization problem in saturated bipartite graphs. An n by n bipartite graph G is H-saturated if G does not contain a subgraph isomorphic to H but adding any missing edge to G creates a copy of H. More than half a century ago, Wessel and Bollob'as independently solved the problem of minimizing the number of edges in K(s,t)-saturated graphs, where K(s,t) is the "ordered"" complete bipartite graph with s vertices from the first color class and t from the second. However, the very natural "unordered"" analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When s = t, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Kor'andi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in K(s,t)-saturated n by n bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for s = t 1 with the classification of the extremal graphs. We also improve the estimates of Gan, Kor'andi, and Sudakov for general s and t, and for all sufficiently large n. [ABSTRACT FROM AUTHOR]

Let G be a simple graph, and let χ ′ (G) and Δ (G) denote the chromatic index and the maximum degree of G , respectively. A graph G is a critical class two graph if χ ′ (G) = Δ (G) + 1 and χ ′ (H) ≤ Δ (G) for every proper subgraph H of G. Let d ‾ (G) denote the average degree of G , i.e., d ‾ (G) = 2 | E (G) | / | V (G) |. Vizing in 1968 conjectured that d ‾ (G) ≥ Δ (G) − 1 + 3 / n if G is a critical class two graph of order n. In this paper, we prove that d ‾ (G) ≥ 3 4 Δ (G) − 8 if G is a critical class two graph. Let δ (G) denote the minimum degree of G. We show that there exist two functions D and D 0 such that for any ϵ ∈ (0 , 1) , if G is a critical class two graph with Δ (G) ≥ D (ϵ) and δ (G) ≥ D 0 (ϵ) , then d ‾ (G) ≥ (1 − ϵ) Δ (G). We will give two specific functions satisfying the statement above in the paper. Moreover, we show that if G is a critical class two graph and δ (G) ≥ (log ⁡ Δ (G)) 3 4 , then d ‾ (G) ≥ Δ (G) − o (Δ (G)). [ABSTRACT FROM AUTHOR]

The standardized communication protocol and the universal application of intelligent electronic devices in edge networks provide technical support to the realization of smart cities and also bring network security issues that cannot be ignored. How to analyze and quantitatively assess the vulnerability of the edge network has become an important issue that needs to be solved urgently in the edge network security risk control. Based on the analysis of the vulnerability mechanism, this paper proposes the selection criteria and generation methods of vulnerability indicators and of composite vulnerability indicators, respectively. In addition, this paper proposes an improved vulnerability severity rating method to analyze the vulnerability of the system and potential security threats. This paper describes the security risk status of vulnerable points based on the state transition of the state offensive and defensive graph and implements the description of the security risk status of attacks in terms of attack frequency, attack success rate and harm. Experimental results justify the effectiveness of the proposed solution. [ABSTRACT FROM AUTHOR]

New applications such as intelligent manufacturing, autonomous vehicles and smart cities drive large-scale deep learning models deployed in the Internet of Things (IoT) edge environments. However, deep learning models require substantial computations, storage and communication resources to run. It is generally difficult to deploy and execute a complete deep neural network (DNN) on a resource-constrained edge device. One possible solution is to slice the DNN into multiple tiles distributed to different edge devices, which can reduce the number of computations and quantity of data on each edge device. In this paper, we propose DISSEC, a distributed scheduling strategy for DNN inference on IoT edge clusters. DISSEC leverages spatial partitioning techniques through fusing the convolutional layers and dividing them into multiple partitions that can be executed independently, and proposes a method to express the dependencies between partitions. It further proposes a search algorithm based on heuristics to produce a distributed parallel strategy with the best overall inference execution latency. The evaluation shows that our strategy can fully utilize the edge device resources by cooperating with multiple edge devices to perform partitioning tasks in parallel. Furthermore, compared to the existing work scheduling strategy, our strategy reduces communication overhead by 20% and overall execution latency by 9% under different partitioning granularities and numbers of edge devices. [ABSTRACT FROM AUTHOR]

• Analyze the inconsistency collapse of existing cell-based differentiable neural architecture search (NAS) methods, and the search process is also time- and memory-consuming due to the optimization of all operations simultaneously. • Propose a fast differentiable neural architecture search with source-target consistency, named STC-NAS. Source-target consistency means the number of edges in the searched cell is the same as that in the target cell. As only the least number of edges is searched during the search process, the resource consumption is greatly reduced and search efficiency is improved. • Propose to use the ReLU function to independently update the architecture weights for avoiding passive competition among candidate operations, and use Jensen-Shannon divergence as an auxiliary loss for architecture parameters optimization. • Experimental results show that STC-NAS can directly search on the target super-network and task dataset without any proxy, thus improving the performance of the searched architecture. STC-NAS achieves 2.42% test error on CIFAR-10, 16.45% test error on CIFAR-100, and 24.2% test error on ImageNet, outperforming the state-of-the-art works. Neural architecture search (NAS) has shown very promising results for automatically designing network models. Most existing cell-based NAS approaches generate the target network model from a source super-network, which usually confront inconsistency issues. In this paper, we propose a new NAS method named STC-NAS, a fast neural architecture search with source-target consistency, so that not only the performance of the searched target model is improved but also the search process is boosted. Specifically, during the search phase, we sample the source super-network to let the samples be consistent with the target model. Moreover, we leverage the Jensen-Shannon divergence to ensure the samples are optimized in the direction of being more similar to the target model. Experimental results demonstrate that our method needs only 0.059 GPU-days to search on CIFAR-10. Benefited from its efficiency, STC-NAS can directly search the target super-network on the target task datasets, achieving 2.42% test error on CIFAR-10, 16.45% test error on CIFAR-100, and 24.2% test error on ImageNet datasets. [ABSTRACT FROM AUTHOR]

Quantisation is a commonly applied technique to improve the efficiency of a neural network on edge devices. Many applications also require that a machine learning model extend its learning cycle into the field. Training a quantised neural network on edge devices is a non-trivial task since the resource available on the edge is limited. Most of the quantisation-aware training methods maintain a quantised model, as well as an extra full-precision model, which is used to prevent large accuracy losses. The keep of the full precision model is based on the assumption that there are sufficient memory and energy supply to the machine for training. The assumption is contradictory to the situation of edge AI. In this paper, we propose the Adaptive Precision Training method (APT), which only keeps a quantised model. The challenge is that a quantised model has difficulty in learning, due to the quantisation underflow issue. APT employs a metric called G a v g to quantify the learning ability of each layer and dynamically adjusts per-layer bitwidth to ensure the model can learn effectively. Experiments on image classification and text classification tasks suggest that APT trains quantised models effectively with limited accuracy loss. Compared with the 8-bit traditional QAT method, APT saves 60-72.5% memory space for model parameters. We investigate the bitwidth necessary for effective training and gain preliminary insights into the relationship between architecture and its learning ability. • Adaptive Precision Training (APT) trains a quantised network without a master copy. • APT saves 60-72.5% memory for model parameter in training with limited accuracy loss. • Insights into the characteristics of building blocks in neural network architectures. [ABSTRACT FROM AUTHOR]

• A cooperative edge caching strategy based on energy-latency balance is proposed. • A dynamic service migration strategy based on deep Q-Network (DQN) is proposed. • The results show that our edge caching algorithm has lower latency and energy consumption. • The proposed service migration algorithm is superior in migration cost and success rate. Mobile edge computing sinks computing and storage capabilities to the edge of the network to provide reliable and low-latency services. However, the mobility of users and the limited coverage of edge servers can cause service interruptions and reduce service quality. A cooperative edge caching strategy based on energy-latency balance is proposed to solve high power consumption and latency caused by processing computationally intensive applications. In the cache selection phase, the request prediction method based on a deep neural network improves the cache hit rate. In the cache placement stage, the objective function is established by comprehensively considering power consumption and latency, and We use the branch-and-bound algorithm to get the optimal value. We propose an improved service migration method to solve the problem of service interruption caused by user movement. The service migration problem is modeled using a Markov decision process (MDP). The optimization goal is to reduce service latency and improve user experience under the premise of specified cost and computing resources. Finally, the optimal solution of the model is solved by the deep Q-Network (DQN) algorithm. Experiments show that our edge caching algorithm has lower latency and energy consumption than other algorithms in the same conditions. The service migration algorithm proposed in this paper is superior to different service migration algorithms in migration cost and success rate. [ABSTRACT FROM AUTHOR]